Australian Curriculum Mathematics - Measurement and Geometry: Year 6 - Ages 11-12 by Teacher Superstore

RIC–6099 6.7/951

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All material identified by is material subject to copyright under the Copyright Act 1968 (Cth) and is owned by the Australian Curriculum, Assessment and Reporting Authority 2013. For all Australian Curriculum material except elaborations: This is an extract from the Australian Curriculum. Elaborations: This may be a modified extract from the Australian Curriculum and may include the work of other authors. Disclaimer: ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. In particular, ACARA does not endorse or verify that: • The content descriptions are solely for a particular year and subject; • All the content descriptions for that year and subject have been used; and • The author’s material aligns with the Australian Curriculum content descriptions for the relevant year and subject. You can find the unaltered and most up to date version of this material at http://www.australiancurriculum.edu.au/ This material is reproduced with the permission of ACARA.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Foundation) Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 1) Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 2) Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 3) Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 4) Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5) Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

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School Order# (if applicable):

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Internet websites In some cases, websites or specific URLs may be recommended. While these are checked and rechecked at the time of publication, the publisher has no control over any subsequent changes which may be made to webpages. It is strongly recommended that the class teacher checks all URLs before allowing students to access them.

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AUSTRALIAN CURRICULUM MATHEMATICS RESOURCE BOOK: MEASUREMENT AND GEOMETRY (YEAR 6) Foreword Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6) is one in a series of seven teacher resource books that support teaching and learning activities in Australian Curriculum Mathematics. The books focus on the measurement and geometry content strands of the national maths curriculum. The resource books include theoretical background information, Resource sheets, hands-on activities and assessment activities, along with links to other curriculum areas.

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Using units of measurement .................................................... 2–61

• UUM – 1

Connect decimal representations to the metric system (ACMMG135) – Teacher information ................................................................................. 2 – Hands-on activities ................................................................................... 3 – Links to other curriculum areas ................................................................. 3 – Resource sheets .................................................................................... 4–6 – Assessment .......................................................................................... 7–8 – Checklist ................................................................................................... 9

• UUM – 2

Answers ............................................................................................ 58–61

Shape .................................................................................................. 62–79 • Shape – 1

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Format of this book ....................................................................... iv – v

Construct simple prisms and pyramids (ACMMG140) – Teacher information ......................................................................... 62–63 – Hands-on activities ........................................................................... 64–65 – Links to other curriculum areas ................................................................ 65 – Resource sheets ................................................................................ 66–75 – Assessment ...................................................................................... 76–77 – Checklist .................................................................................................. 78

Answers .................................................................................................... 79

• UUM – 3

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Solve problems involving the comparison of lengths and areas using appropriate units (ACMMG137) – Teacher information ................................................................................ 30 – Hands-on activities .................................................................................. 31 – Links to other curriculum areas ................................................................ 32 – Resource sheets ................................................................................ 33–36 – Assessment ...................................................................................... 37–38 – Checklist .................................................................................................. 39

• UUM – 4

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Location and transformation .............................................. 80–113 Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies (ACMMG142) – Teacher information ......................................................................... 80–81 – Hands-on activities .................................................................................. 82 – Links to other curriculum areas ................................................................ 83 – Resource sheets ................................................................................ 84–94 – Assessment ...................................................................................... 95–96 – Checklist .................................................................................................. 97

• L&T – 2

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Convert between common metric units of length, mass and capacity. (ACMMG136) – Teacher information ......................................................................... 10–13 – Hands-on activities ........................................................................... 14–16 – Links to other curriculum areas ................................................................ 16 – Resource sheets ................................................................................ 17–25 – Assessment ...................................................................................... 26–28 – Checklist .................................................................................................. 29

Introduce the Cartesian coordinate system using all four quadrants (ACMMG143) – Teacher information ......................................................................... 98–99 – Hands-on activities ................................................................................ 100 – Links to other curriculum areas .............................................................. 101 – Resource sheets ............................................................................ 102–108 – Assessment .................................................................................. 109–110 – Checklist ................................................................................................ 111

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Connect volume and capacity and their units of measurement (ACMMG138) – Teacher information ......................................................................... 40–41 – Hands-on activities .................................................................................. 42 – Links to other curriculum areas ................................................................ 43 – Resource sheets ................................................................................ 44–45 – Assessment ............................................................................................. 46 – Checklist .................................................................................................. 47

• UUM – 5 Interpret and use timetables (ACMMG139) – Teacher information ......................................................................... 48–49 – Hands-on activities .................................................................................. 50 – Links to other curriculum areas ................................................................ 51 – Resource sheets ................................................................................ 52–55 – Assessment ............................................................................................. 56 – Checklist .................................................................................................. 57

Answers ...................................................................... 112–113

Geometric reasoning ............................................................ 114–131 • GR – 1 Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles (ACMMG141) – Teacher information ..................................................................... 114–115 – Hands-on activities ................................................................................ 116 – Links to other curriculum areas .............................................................. 117 – Resource sheets ............................................................................ 118–127 – Assessment ...................................................................................128–129 – Checklist ................................................................................................ 130

Answers ................................................................................................. 131

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

iii

FORMAT OF THIS BOOK This teacher resource book includes supporting materials for teaching and learning in all sections of the Measurement and Geometry content strand of Australian Curriculum Mathematics. It includes activities relating to all sub-strands: Using units of measurement, Shape, Location and transformation, and Geometric reasoning. All content descriptions have been included, as well as teaching points based on the curriculum’s elaborations. Links to the proficiency strands have also been included. Each section supports a specific content description and follows a consistent format, containing the following information over several pages: • teacher information with related terms, student vocabulary, what the content description means, teaching points and problems to watch for • hands-on activities • links to other curriculum areas

• resource sheets • assessment sheets.

• a checklist

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Answers relating to the assessment pages are included on the final page of the section for each sub-strand (Using units of measurement, Shape, Location and transformation, and Geometric reasoning).

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The length of each content description section varies.

Teacher information includes background information relating to the content description, as well as related terms, desirable student vocabulary and other useful details which may assist the teacher.

Related terms includes vocabulary associated with the content description. Many of these relate to the glossary in the back of the official Australian Curriculum Mathematics document; additional related terms may also have been added.

What this means provides a general explanation of the content description.

the teacher would use—and expect the students to learn, understand and use—during mathematics lessons.

description.

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The proficiency strand(s) (Understanding, Fluency, Problem solving Solving or Reasoning) relevant to each content description are shown listed. in bold.

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What to look watchforforsuggests suggestsany any difficulties and misconceptions the students might encounter or develop.

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Hands-on activities includes descriptions or instructions for games or activities relating to the content descriptions or elaborations. Some of the hands-on activities are supported by resource Resource sheets. Where applicable, these will be stated for easy reference.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

FORMAT OF THIS BOOK Links to other curriculum areas includes activities in other curriculum areas which support the content description. These include Information and Communication Technology, English, Science, Languages other than English, History, Geography and the Arts. This section may list many links or only a few. It may also provide links to relevant interactive websites appropriate for the age group.

r o e t s Bo r e p ok u S Resource sheets are provided to support teaching and learning activities for each content description. The resource sheets could be cards for games, charts, additional worksheets for class use or other materials which the teacher might find useful to use or display in the classroom. For each resource sheet, the content description to which it relates is given.

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Cross-curricular links reinforce the knowledge that mathematics can be found within, and relate to, many other aspects of student learning and everyday life.

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o c . che e r o t r s super Each section has a checklist which teachers may find useful as a place to keep a record of the results of assessment activities, or their observations of hands-on activities.

Answers for resource sheets (where appropriate) and assessment pages are provided on the final page of each sub-strand section.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 1

Connect decimal representations to the metric system (ACMMG135)

RELATED TERMS

TEACHER INFORMATION

Metric system (SI)

What this means

• The system of measurement units using the decimal system, and from which the SI units are derived.

• Most of the units of measurement used in Australia are metric; the common exceptions are time and angle.

SI Units

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(from the French Système International d’unités; International System)

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Metre

• In the original French system, the metre was one ten-millionth of the distance between the North Pole and Paris.

Teaching points

For example, 92 means nine multiplied twice (or 9 x 9); 63 means 6 x 6 x 6 (six multiplied by itself three times). It is expressed as ‘nine to the power of two’ or ‘six to the power of three’.

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A power can also be called an ‘index notation’ or ‘exponent’. Powers of ten express how many times ten is multiplied by itself; for example, 101 (ten to the power of one) is 10. Other powers of ten are expressed as follows:

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• When using two units together, they are expressed as 2 cm 5 mm or 2.5 cm, 4 m 25 cm or 4.25 m, 3 L 375 mL or 3.375 L, 4 kg 200 g or 4.2 kg etc.

What to look for

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10 = 100, 10 = 1000, 10 = 10 000. Note: 10 = 1 Student vocabulary centimetres

• Students know the appropriate units of measurement for the attributes of length, area, volume, capacity and mass, and their size comparisons and conversions. For example, students know that there are 1000 millilitres in 1 litre, and therefore that a millilitre is a considerably smaller amount of liquid than a litre.

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• The power of a number tells how many times to use the number in a multiplication. It is usually expressed as a number above and to the right of the number which is multiplied.

• Students who are uncertain about the conversion rates between units; e.g. how many centimetres in a metre, millilitres in a litre etc. • Students unsure whether to multiply or divide by a power of ten.

metres

kilometres

square centimetres

square metres

square kilometres

hectares

millilitres

litres

cubic centimetres

cubic metres

grams

kilograms

• Students become aware that measures may be given in more than one unit; for example, the length of a table may be 1 metre and 20 centimetres, and that this can also be shown as 1.2 metres or 120 centimetres.

Powers of 10

• Students should know the common prefixes for the metric system: centi (hundredths), milli (thousandths) and kilo (thousand).

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• This is the modern metric form of measurement based around seven base units and the number ten. The base units that are appropriate at this year level are: metre (length) and kilogram (not gram; for mass). The second is also a base unit for time, but is not metric. The SI system was officially established in 1960 and is updated in terms of precision on a regular basis.

• Students need to be familiar with the metric units for length, area, volume, capacity and mass, and the conversion rates between them for each attribute.

powers of ten

decimal

metric

multiplication

division

decimal point

Proficiency strand(s): Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 1

HANDS–ON ACTIVITIES Estimate before measuring in all these activities • Students measure the length of various items in 2 units and convert between them. For example, a desk that is 1 metre and 350 centimetres long can be written as 1350 centimetres or 1.35 metres. Use the equivalent measures on page 5. • For further length examples, see pages 17–19 (UUM – 2). • Students calculate perimeters of shapes using different units for the lengths of the sides. Convert all the measures to the same unit before adding together. Find the perimeter and express in different units. • For further perimeter examples, see pages 33–38 (UUM – 3). • Discuss the units used for mass of various supermarket items. For example, fruit is usually sold by the kilogram, so if purchasing a bag of apples, it may be 2.4 kilograms, 2 kilograms and 400 grams or 2400 grams. Confectionery is usually sold by multiples of the gram; e.g. 200-gram blocks of chocolate. Some items use different units depending on the size of the containers. Look for ‘family size’ and other labels to indicate bigger packaging.

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• For further mass examples, see pages 20–21 (UUM – 2).

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• Supermarket packaging could also be examined for the units used in capacity. For example, cans of soft drinks are usually sold by millilitres (375 mL), milk can be by the millilitre (600 mL carton) or by the litre (2 L carton). Use supermarket catalogues to get an idea of the units used. • Students measure the capacity of different containers and compare. Some of the measures could be in litres and others in millilitres. Students convert to the same unit to compare. • For further capacity examples, see pages 22–23 (UUM – 2). • For examples of the metric system in volume and capacity, see pages 44–46 (UUM – 4).

• Name examples of computer files and give their size. Work out how many could be saved onto thumb drives of different capacity. Use the Powers of ten comparison chart on page 4.

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• This unit connects between units within each of the measures of length, area, volume, capacity and mass that offer the context to do these types of calculations. If students have a reason to do this within the context of measurement, they will see the benefits of being able to do so readily. Converting units to smaller units such as centimetres to millimetres requires students to be able to multiply, in this case by 10; metres to centimetres by 100 and kilometres to metres by 1000. Converting the other way, from the smaller units to larger ones; e.g. from metres to kilometres, will require division by the appropriate power of ten. This unit links with (Multiply and divide decimals by powers of 10 [ACMMG130] )

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Information and Communication Technology

• A YouTube™ link segment called Not the great big hen – Multiplying by 10, 100 is done in a game show format where students solve problems that involve multiplying by 10 or 100. It runs for just over 9 minutes, but would be an interesting introduction to a lesson on converting metric measures. It can be found at <http://www.youtube.com/watc h?v=v1G6PXayO3Q&feature=related> or type Not the great big hen into a search engine. • A very short game where students help a farmer cross the river by multiplying or dividing by powers of ten can be found at <http://kids.britannica.com/lm/games/GM_5_5/GM_5_5.htm>.

Links to other curriculum areas that are concerned with more specific units of measurement will be found in the entries: • Pages 17–28 (UUM – 2). • Pages 33–38 (UUM – 3). • Pages 44–46 (UUM – 4).

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 1

RESOURCE SHEET Powers of ten comparison chart Powers of ten

Number

Name

Prefix

1012

1 000 000 000 000

trillion

tera

109

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giga

1 000 000

million

mega

103

1000

thousand

102

100

hundred

one

0.1

one-tenth

0.01

one-hundredth

centi

10-3

0.001

one-thousandth

milli

10-6

0.000 001

one-millionth

micro

10-9

0.000 000 001

one-billionth

nano

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billion

106

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1 000 000 000

kilo

hecto

10-2

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deci

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Connect decimal representations to the metric system

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10-1

–

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100

Sub-strand: Using Units of Measurement—UUM – 1

RESOURCE SHEET Equivalent measures

kilogram (kg) 1

kilometre (km)

grams (g) is equivalent to

1000

metres (m)

litre (L)

millilitres (mL)

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⁄4 (or 0.75)

is equivalent to

750

⁄2 (or 0.5)

is equivalent to

500

⁄4 (or 0.25)

1 metre (m)

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CONTENT DESCRIPTION: Connect decimal representations to the metric system

⁄4 (or 0.75) m

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is equivalent to

250

is equivalent to

100 centimetres (cm)

is equivalent to

75 cm

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⁄2 (or 0.5) m

is equivalent to

⁄4 (or 0.25) m

is equivalent to

25 cm

1 centimetre (cm)

is equivalent to

10 millimetres (mm)

⁄2 (or 0.5) cm

is equivalent to

5 mm

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

50 cm

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Sub-strand: Using units of measurement—UUM – 1

RESOURCE SHEET A square metre 1. (a) Use 1 cm2 grid paper and cut out a 10 cm by 10 cm square. How many square centimetres does it have? (b) How many of 10 cm x 10 cm squares do you think would fit into a one-metre square? (c) Estimate how many square centimetres you think there are in a one-metre square.

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2. Place a metre ruler on the floor. Line up enough 10 cm by 10 cm squares to make one row along the whole length of the ruler.

Place a second metre rule at 90º (perpendicular) to the first. Line up 10 cm by 10 cm squares along its length.

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Use two more metre rules to complete the frame of the one-metre square. Continue adding 10 cm x 10 cm squares until the one-metre square is full. This is one square metre, because it is one metre long and one metre wide. 3. (a) How many of the 10 cm by 10 cm squares were needed to fill the square metre? (b) If you know how many square centimetres 10 cm x 10 cm square, how many are in a square metre? (c) Was your estimate correct? Yes 6

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Connect decimal representations to the metric system

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Assessment 1

Sub-strand: Using units of measurement—UUM – 1

NAME:

DATE: Show week for the dairy

The dairy in a country town sold its produce during show week. 1. Cheese: A 15 kg block of special show cheese was produced. Wedges of the cheese were sold each day, and at the end of the day the remaining cheese was weighed. Day

Mass of block at the end of the day 11.75 kg

Amount sold

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Sunday

3.25 kg

Monday

9.4 kg

Tuesday

7.1 kg

1 kg 550 g

Thursday

3500 g

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Wednesday

Saturday

1.5 kg

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Friday

1.25 kg

(a) Complete the rest of the table. (b) How much cheese was left at the end of the week? (c) On which day was the most cheese sold?

(d) On which day was the least cheese sold?

(e) On which day was the weight of cheese sold only 5 g less than on the previous day?

Chocolate

Strawberry

Vanilla

Sunday

3.8 L

4.5 L

2.3 L

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Monday Tuesday

Wednesday Thursday Friday Saturday

2200 mL

1250 mL

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Amount sold each day

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Day

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CONTENT DESCRIPTION: Connect decimal representations to the metric system

2. Flavoured milk: Every day, the dairy made a new batch of their flavoured milk. It came in chocolate, strawberry and vanilla flavours.

2.9 L

1500 mL

1300 mL

2 ⁄2L

3 ⁄4 L

3100 mL

31⁄2 L

4300 mL

Amount of each flavour sold

5.3 L 8L

5.4 L 9000 mL 12 L sold in total

(a) Complete the rest of the table. (b) At the end of the week, how much milk was sold in total? (c) Which flavour was sold the most overall? (d) Which day had the most sales? (e) Which day had the least? Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Assessment 2

Sub-strand: Using units of measurement—UUM – 1

NAME:

DATE: Getting ready for the school fete

Use your understanding of the metric system and your knowledge of decimals to solve the following problems. 1. Tricia bought 35 metres of ribbon to tie up cookie bags for the school fete. Each cookie bag needed 50 centimetres of ribbon. How many bags could Tricia decorate?

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2. A butcher used 15 kilograms of mince to make some Super Burgers. For each Super Burger, 250 grams of mince is required. How many Super Burgers could he make?

3. One class organised a run-athon as part of their fundraising. 100 metre tracks were set out on the oval. Each team had to run 1.5 kilometres altogether. How many trips along the track did each team have to make?

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5. Jason’s class organised the drinks. They made cordial from concentrate which came in 2-litre bottles. Each bottle needed 5 litres of water to be added. They used 4 bottles of cordial altogether. They sold the drinks in 250 millilitre cups. How many cups did they sell?

6. Parents ran a ‘soak the student’ stall. Using water from a 50 litre tank, students purchased buckets of water to throw over their friends; 50c for a 250 mL bucket or $1.50 for a litre bucket. If all the water was used, what is the maximum and the minimum amount of money that could have been raised?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Connect decimal representations to the metric system

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4. Bec’s class decided to make cakes for the fete. Each cake needed 300 grams of dough, and they made 40 cakes. How many kilograms of dough did they use?

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Checklist

Sub-strand: Using units of measurement—UUM – 1

Multiplies and divides by powers of ten

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STUDENT NAME

Knows that metric units for length, weight, volume and capacity are part of the metric system

Connect decimal representations to the metric system (ACMMG135)

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Note: Rubrics that highlight more specific criteria of measurement can be found in the three units of work that follow. Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 2

Convert between common metric units of length, mass and capacity (ACMMG136)

RELATED TERMS

TEACHER INFORMATION

Length

What this means

• The measure of a path or object in one dimension from end to end (i.e. 1-D).

• Students are aware of the different attributes of an object that can be measured, and that different metric units are used for each of them.

Mass

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Weight

• The force of gravity acting on an object, used to measure mass (actually measured in Newtons). Note 1: It is correct to use the verb ‘to weigh’. Note 2: At this year level, students may use the terms ‘weight’ and ‘mass’ interchangeably, although it is best if the teacher uses the correct terminology.

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• The amount of matter an object contains, commonly measured in grams, kilograms and tonnes.

• Students are aware that within each of the attributes, there are different units that can be used, dependent of the size of the object and the level of accuracy required. For example, a bridge over a river may be measured in metres or even kilometres, but engineers designing and building it may need to have a level of accuracy of millimetres.

• Students decide the appropriate units for each attribute, rather than the teacher telling them what to use; e.g. students decide whether to use millimetres, centimetres, metres or kilometres to measure the height of a door.

• Students should know the common prefixes for the metric system: centi (hundredths), milli (thousandths) and kilo (thousand).

• This unit links to (Multiply and divide decimals by powers of 10 [ACMMG130] ).

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• The power of a number tells how many of that number are multiplied. It is usually expressed as a number above and to the right of the number which is multiplied. For example, 92 means two nines are multiplied twice (or 9 x 9); 63 means 6 x 6 x 6 (three sixs are multiplied). It is expressed as ‘nine to the power of two’ or ‘six to the power of three’.

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Teaching points

General • Estimate before measuring in all measurement activities.

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• It also links to the unit (Connect decimal representations to the metric system [ACMMG135] ).

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Powers of 10

• At this year level, students would be expected to be able to multiply and divide by powers of ten; thus enabling conversions from one unit of measure to another to be done quite easily.

• Be wary of using the flawed rule for multiplying by ten of ‘adding a zero’. When multiplying a decimal by ten, adding a zero would not necessarily give the correct answer; e.g. 0.5 x 10 is not 0.50. • When multiplying or dividing by powers of ten, it is the digits that ‘move’ to the left or right of the decimal point. The decimal point itself always remains in the same place. For example, to work out 34.58 ÷ 10, all of the digits move one place to the right, so the three tens now become three units, the four units now become four tenths, the five tenths become five hundredths and the eight hundredths now become eight thousandths. This gives a solution of 3.458.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 2

Convert between common metric units of length, mass and capacity (ACMMG136)

TEACHER INFORMATION (CONTINUED) Student vocabulary centimetre

• The metric units for mass are grams (g), kilograms (kg) and tonnes (t).

metre kilometre gram kilogram tonne

litre

• The metric units for capacity are millilitres (mL), litres (L) and kilolitres (kL).

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kilolitre

powers of ten decimal metric

multiplication division

• Note the use of the upper case ‘L’ for the abbreviation of litres; it is also used for mL and kL. This is to distinguish it from the number 1. • A gap is always left between the number of units and the abbreviation of the unit—e.g. 5 cm, 8 kg, 375 mL etc.—and no full stop is used at the end of the abbreviation (unless it is at the end of a sentence). • Abbreviations of metric units never use the ‘s’ at the end; e.g. ‘5 cm’ not ‘5 cms’; ‘8 kg’ not ‘8 kgs’; ‘375 mL’ not ‘375 mLs’. However, if the unit is written in full, the ‘s’ is needed; e.g. ‘5 centimetres’, ‘8 kilograms’ and ‘375 millilitres’.

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millilitre

• The metric units for length are millimetres (mm), centimetres (cm), metres (m) and kilometres (km).

• Students need to be fluent with conversions of common metric units of measure for length, mass and capacity (see page 12). Discussion could centre on needing to know about multiplying and dividing by ten or powers of ten. Discuss the meanings of the prefixes; e.g. milli, centi, kilo.

longer

longest short shorter

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shortest heavy

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heavier

heaviest light

lighter lightest holds more holds less holds the most holds the least

• Students need to develop ‘referents’ for length, mass and capacity. For example, students being aware that their little finger is approximately one centimetre in width; that a big stride is about a metre long; that a Base Ten small cube, if hollow, would hold one millilitre of water. Capacity is often problematic because of marketing, where a 2-litre cool drink bottle looks as if it holds more than a 2-litre tub of ice cream; although they may be familiar with the amount of liquid in a normal 375 mL can of soft drink. For mass, when holding an item to be estimated, most people mentally compare the item to something they know such as a tub of butter or a bag of sugar.

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decimal point

• When estimating the mass of an object, it usually needs to be handled. For example, you cannot make a reasonable estimate about the mass of a tin of ‘something’ without picking it up. It may be light if filled with polystyrene, or heavy if filled with nails.

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• Students need to know which units are appropriate for measuring different items. For example, we would not usually measure the length of a room in millimetres, the mass of a balloon in kilograms or the capacity of a bucket in millilitres. • When estimating the length, mass or capacity of a group of items, allow the students to adjust their remaining estimates after the first item has been measured. For example, a student may have estimated that a jar will hold 1.5 L of water, a cup 1.2 L and a jug 2.3 L. The jar is then measured and maybe found to only hold 300 mL. With this knowledge, the student may change the estimate of the cup to 220 mL and jug to 550 mL.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 1

Convert between common metric units of length, mass and capacity (ACMMG136)

TEACHER INFORMATION (CONTINUED)

Conversions Length

• National tests usually have questions where students are required to interpret scaled instruments. They need to have had many experiences actually doing this, not just watching.

10 mm = 1 cm 100 cm = 1 m

Length • There are 2 types of rulers: dead-end (where the ‘zero’ is level with the end of the ruler) and waste-end (where the ‘zero’ is situated a little way in from the end of the ruler).

1000 mm = 1 m 1000 m = I km Mass

r o e t s Bo r e p ok u S Dead-end

1000 g = 1 kg 1000 kg = tonne

1000 mL = 1 L 1000 L = 1 kL

Waste-end

0 1 cm

• Students may need specific teaching on how to use a ruler; we cannot assume they know. For example, they should have their non-writing hand spanned out so that the ruler is more stable and won’t move easily; they line up the item to be measured with the ‘zero’ on the ruler and read along to the end of item.

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Capacity

1 cm

• Ensure that all students’ rulers are in centimetres and millimetres, not inches.

• Drawing a line of a particular length is a different skill from measuring a line of a particular length. • Students need to be able to express distances in terms of metres and centimetres, or just in centimetres. When students are conversant with numbers to two decimal places, they can record distances using decimals—e.g. 1.3 metres—and know that this is the same as 1 metre and 30 centimetres.

• Students may need to be shown the correct way to use a trundle wheel; they need to make sure it starts at zero.

Mass

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• It is impractical for students to measure long distances (i.e. kilometres), but discussion could arise, for example, when going on an excursion. Students could also be expected to know how many metres are in a kilometre and be able to convert between the two units.

• Students should be given the opportunity to investigate the relationships between the units of measurement in mass, rather than be told the conversion rates. For example, using a pan balance, they could investigate how many 200 g or 250 g weights would be needed to balance 1 kg.

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• When estimating and measuring mass, the teacher needs to ensure that the masses of the items to be measured are not always obvious (i.e. that students cannot determine comparisons simply by looking). To do this, you need items that have similar masses but different volumes—e.g. a golf ball and a table tennis ball—and items that are the same size but have different masses; e.g. lidded tins filled with different materials such as sand and styrofoam.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 2

Convert between common metric units of length, mass and capacity (ACMMG136)

TEACHER INFORMATION (CONTINUED) • When using metric units for mass, students may use a pan balance and weights, kitchen scales or bathroom scales. Kitchen and bathroom scales may have a dial or a digital readout. It is a useful skill to be able read a dial, as there is no skill involved in reading a digital display.

r o e t s Bo r e p ok u S Capacity

• Students may need help with strategies for measuring capacity, particularly when not all graduations are marked on a measuring container. Many measuring containers mark only every 5 or 10 millilitres. What to look for

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• Students should be given every opportunity to handle the standard units of mass. Given such experiences, students will be better able to select appropriate units of measure as well as estimate mass. Parents can be encouraged to help at home by doing regular checks of the students’ mass, and by allowing them to help when weighing ingredients for cooking.

• Students who are uncertain about the conversion rates between units; e.g. how many centimetres in a metre, grams in a kilogram, millilitres in a litre etc.

• Students unsure of what attribute to measure; i.e. length, mass or capacity. • Students using inappropriate units of measurement.

• Students using inappropriate tools to measure; e.g. using a 30 cm ruler to measure the length of a basketball court, a medicine measure to calculate the capacity of a bucket or bathroom scales to weigh a pencil case.

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• Students having useful referents for length, mass and capacity. • Students unsure whether to multiply or divide when converting metric units.

o c . che e r o t r s super Proficiency strand(s): Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 2

HANDS-ON ACTIVITIES Estimate before measuring in all these activities Length • Students make their own simple measuring tools such as metre sticks, metre rulers and tapes. Students decide the graduations to be used; for example, drawing in every 5-millimetre mark, every centimetre mark, every 5-centimetre mark or every 10-centimetre mark. Discuss the need for accuracy. • In groups, students compile a list of items in the classroom that they estimate and measure between set distances. The students only get five minutes to write as many items as they can onto their list. Students then swap lists and check the other group’s list by actually measuring each of the items.

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Between 1 cm and 10 cm

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Less than 1 cm

Between 10 cm and 1 m

Between 1 m and 2 m

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• Encourage students to double-check their accuracy. Discuss the adage ‘measure twice, cut once’. The check may involve re-measuring using the same tool, or by using a different measuring instrument. For example, students could measure the length of a passageway using a trundle wheel, and then check their result by measuring with a metre ruler or retractable measuring tape. To work out the diameter of a cricket ball, they could place a Base Ten large cube at either side of the ball, and measure the gap between them.

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• Students explore how to measure difficult items; for example, working out how to determine the thickness of a sheet of paper. (Perhaps by making a wad of paper that is 1 centimetre high, and counting the number of sheets used. By dividing, this will give a decimal fraction of a centimetre.) • Measure a distance of ten metres in the playground. Discuss how many of these distances would be needed to make one kilometre. Compare how many of their strides would be needed to cover the ten metres, and how many would be needed for one kilometre. Students will need to be able to work out how many sets of ten metres are required to equal one kilometre; i.e. knowing the conversion rate of 1000 metres equals one kilometre (1000 m = 1 km).

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• If a student has a bike with an odometer, the distance around a basketball court or football oval could be measured. Students could suggest other ways this could be measured. Discuss how many times around the area they would have to go to cover 500 metres or one kilometre. Students could time the difference it takes to cycle around the area compared to walking or running around it. More able students may work out the time taken as a rate; e.g. it took 10 minutes and 25 seconds to walk 500 metres; 10 minutes and 25 seconds is the same as 625 seconds; 625 ÷ 5 is 125; so that is a rate of 125 seconds (or 2 minutes and 5 seconds) per 100 metres. • Students measure the perimeters of various shapes. See (UUM – 2).

• Use ‘The long and the tall’ Resource sheet, page 19, measuring different parts of the body with a tape measure. Compare relationships between different body parts; e.g. the length of the hand is equivalent to the length of the face, arm span is equivalent to height. Also, how different parts are used as non-standard measures; e.g. strides, hand spans and cubits for estimation.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 2

HANDS-ON ACTIVITIES (CONTINUED) Estimate before measuring in all these activities Mass: • Students should be given the opportunity to investigate the relationships between the units of measurement in mass, rather than be told the conversion rates. For example, using a pan balance, they could investigate how many 200 gram or 250 gram weights would be needed to balance 1 kilogram. • Use practical situations where possible; e.g. cooking activities. Look for recipes where some of the ingredient amounts are given in grams rather than cupfuls.

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• Find items that balance with specific weights; e.g. What items can you find that have the same mass as 50 grams?

• Once the items have been weighed, discuss how many of each item would be needed to make one kilogram; e.g. for the above items, you would need 120 two-centimetre cubes, 20 erasers or 1300 paper clips. • Students could record the mass of their school bag every day for a week. Discuss why there may be differences on different days. Weigh each of the bags when empty. Compare the masses of several students’ bags; both when they first get to school and when emptied. Discuss how many bags would be needed to have a total mass of five kilograms. • Use supermarket catalogues to investigate the different masses of food items; e.g. flour is sold in 500 gram packets or 1 or 2 kilogram packets; biscuits are usually sold in multiple grams such as 250 gram packets; fruit and vegetables are usually sold by the kilogram, but may be pre-packaged in either grams (e.g. mushrooms, berries) or kilograms (e.g. tomatoes, onions).

• Students graph the results of the prices of different-sized packets of the same product; e.g. buying cereal in 250 gram, 600 gram or 1 kilogram boxes. Students could investigate the unit cost of items; e.g. which is the best buy of the dog food packs below.

Fantastic dog food

$15.00

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$14.00 $12.00

Cost

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$10.00 $8.00

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$6.00

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$4.00 $2.00 $0.00

500 g

720 g

1.25 kg

2.5 kg

5 kg

Size of bags

• Students investigate how they could determine the mass of a small item such as a piece of pasta or a grain of rice. To do this, they would need to weigh a number of the items that can be easily read on a scale—e.g. 50 grams—and then divide that amount by the number of pieces they used. Discuss how many of the items would be needed to weigh one kilogram or 1.5 kilograms. • Look at nutrition labels on food products where relative contents are required to be stated per 100 grams of the product, so that the resulting figure can be easily read as a percentage. • Discuss tonnes and where their use might be appropriate; e.g. the contents of a truck, the mass of vehicles etc. Also discuss what they may look like in relation to a unit they know such as a kilogram. How many kilograms are in a tonne? (1000 kg) Look at a kilogram of sugar and the amount of space it takes up (its volume). How much room do you think we would need if you had a tonne of sugar? • Discuss milligrams and where their use might be appropriate; e.g. in the production of tablets, the mass of gemstones etc. How many milligrams are in a gram? (1000 mg) Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 2

HANDS-ON ACTIVITIES (CONTINUED) Estimate before measuring in all these activities Capacity • Use practical situations where possible; e.g. cooking activities. Look for recipes where some of the ingredient amounts are given in millilitres or litres rather than cupfuls. • Students need to be given experiences with containers that hold the same amount of liquid, but look quite different; e.g. a two-litre bottle of soft drink and a two-litre tub of ice-cream. Discuss the reason why the containers may have been designed this way. • Investigate whether containers do in fact hold what the labels state they do. For example, does a two-litre bottle of cordial actually hold two litres, or a little more or less? Discuss why this may be the case.

r o e t s Bo r e p ok u S

• Use supermarket catalogues to investigate the different capacities of items; e.g. milk is sold in 600 millilitre cartons or 1, 2 or 3 litre cartons; soft drinks in cans are usually sold in multiple millilitres such as 375 mL, or in bottles ranging from 375 mL to 1 L, 1.25 L and 2L bottles.

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• Using a large container, students estimate how far up the side one litre would come and mark it with an elastic band. Then using a 100 mL measuring cup, count out how many will be needed to make one litre, and pour that amount of water in. Discuss how many 100 mL cups were used and how close their estimates were to the actual amount. This could then be done with different containers and amounts; e.g. filling an eight-litre bucket using a 500-millilitre container.

• Discuss kilolitres and where their use might be appropriate; e.g. filling a swimming pool, amount of water in a rainwater tank etc. Also discuss what this may look like in relation to a unit they know, such as a litre. How many litres are in a kilolitre? (1000 L) Look at a litre of milk, and the amount of space it takes up (its volume). How much room do you think we would need if you had a kilolitre of milk? Students may be made aware of megalitres as a unit. There is an even larger unit of capacity, the megalitre, which is one million litres. It is used to record large amounts such as how much water in a dam.

English

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Information and Communication Technology

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• Read Who sank the boat by Pamela Allen. This story looks at a number of animals that decide to go for a row in a boat. As each animal jumps into the boat, it sits a little lower in the water. The ideas of balance and mass are mentioned, as well as the fact that it is the smallest animal that finally sinks the boat.

• An activity where students use a ruler to measure the length and mass of a parcel to determine the correct postage rate can be found at <http://www.kidsmathgamesonline.com/geometry/measurement.html>

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• A simple balance game can be found at <http://pbskids.org/cyberchase/math-games/poddle-weigh-in/>

• An activity where students read scales for masses up to 5 kg can be found at <http://www.ictgames.com/weight.html> • You can also get the scale to read masses up to 500 g at <http://www.ictgames.com/weight.html> • For scales up to 1 kg use <http://www.ictgames.com/weight.html>

• A game where students fill virtual pots with water using the least number of pours can be found at <http://pbskids.org/ cyberchase/math-games/can-you-fill-it/> Science • Every opportunity for science activities that involve measurement in the metric system should be encouraged. • Students investigate the masses of different dinosaurs, and compare them to a modern-day large animal such as an elephant. Languages • Students recognise the use of the prefixes associated with measures of length, mass and capacity. These include milli (as in millimetres, milligrams and millilitres), centi (as in centimetres), kilo (as in kilometres, kilograms and kilolitres) and mega (as in megalitres). • Students are aware of the term tonnes for large measures of mass.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 2

RESOURCE SHEET Units of length • Fill in your estimates for the length of each item below. Think about what equipment you will need to measure each of them and the units you will use. Item

Estimate

Actual length

Difference

Width of this piece of paper

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Width of the classroom door

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Teac he r

Height of the windows Length of your desk Height of the board Height of the door Height of your chair

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Length of the teacher's table

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CONTENT DESCRIPTION: Convert between common metric units of length, mass and capacity

Diameter of the bin

Height of the shelf

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• Measure the first item and write it in the table above. Work out the difference between your estimate and the actual measurement. Were you close? • If you think you need to, change the estimates for the other items. • Now measure the next item and calculate the difference between your estimate and the actual measurement. Alter remaining estimates if necessary. • Complete the table and add two more items.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 2

RESOURCE SHEET The king’s dilemma You will need:

Strips of paper tape A ruler

In a land far away, there lived a king whose realm was threatened by a group of seven fierce dragons. In desperation, he sought out his bravest knight and asked him to slay as many of the dragons as he could.

r o e t s Bo r e p ok u S

The king showed the knight a seven-metre bar of gold, and promised him one metre of gold for every dragon he slew. Each day for the next week, the knight went to battle with the dragons, and each night he returned with proof that he had killed one of them.

Cut a piece of paper tape that is 7 centimetres long (scale: 1 cm = 1 m).

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The king found that he could pay the knight his one metre of gold each night by making just two cuts in the gold bar. How did he do it?

Explore how the king was able to give the knight one metre of gold (one centimetre of your paper tape) per night, making only two cuts to the gold bar. Record your explanation and show the two cuts to demonstrate your solution.

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Convert between common metric units of length, mass and capacity

w ww

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m . u

Sub-strand: Using units of measurement—UUM – 2

RESOURCE SHEET The long and the tall 1. Use a tape measure to determine the lengths of your body. Record your results in centimetres and convert to metres. length (cm)

Body part height arm span inside leg

hand

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inside arm

length (m)

foot

face

cubit (elbow to tip of index finger)

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CONTENT DESCRIPTION: Convert between common metric units of length, mass and capacity

o c . che e r o t r s super

(b) Compare your results with others in your class.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 2

RESOURCE SHEET Congratulations – You win! You have a winning ticket from a fairground raffle and can choose one of the prizes below: • A one-metre high tower of 50c coins • A kilogram of $1 coins • A one-litre juice carton of 10-cent pieces

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• A three-metre line of 5c pieces

• 1 kilogram and 400 grams of 20c pieces

Teac he r

• A two-litre ice-cream carton of $2 coins.

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Which is the best prize to pick to give you the most amount of money? Show how you worked out the value of each prize. A one-metre high tower of 50c coins

A kilogram of $1 coins

A two-litre ice-cream carton of $2 coins

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Convert between common metric units of length, mass and capacity

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DOLLAR

A three-metre line of 5c pieces

o c . che e r o t r s super

1 kilogram and 400 grams of 20c pieces

DOLLAR

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DOLLAR

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A one-litre juice carton of 10-cent pieces

DOLLAR

DOLLAR DOLLAR DOLLAR

DOLLAR

DOLLAR DOLLAR DOLLAR

DOLLAR

Sub-strand: Using units of measurement—UUM – 2

RESOURCE SHEET What do they weigh? Find each of the items below and estimate their masses. Weigh the first item and write how many grams it weighs. Convert that amount to kilograms. Then take the next item and do the same thing. Carry on and fill in all the boxes below in the same way; that is, estimate, measure, write the mass in grams and convert the mass to kilograms. Items 10 pencils

Actual mass in grams

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Actual mass in kilograms

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4 lunchboxes

Estimated mass

A cricket ball 5 books

20 paperclips

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A box of Pattern Blocks

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1 litre of water

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CONTENT DESCRIPTION: Convert between common metric units of length, mass and capacity

3 Base Ten large cubes

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 2

RESOURCE SHEET Edible fruit Some fruit have parts that are inedible; for example, the rind an orange or the skin on a banana. We eat the skin and flesh of fruits like cherries and peaches, but we discard the stones. Grapes and tomatoes can be brought on the vine, but as this not edible, it too is thrown away. So how much of our fruit do we actually eat and which gives us the best value for money?

r o e t s Bo r e p ok u S

• Weigh each fruit whole and record the mass on the table below.

• Prepare the fruit for eating by removing, as economically as possible, all inedible parts. • Work out and record the ratio of edible to inedible fruit. • Record the mass of edible fruit as a percentage. Fruit

Total weight

Inedible weight

Ratio

Apple

inedible edible

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• Weigh each piece’s inedible mass and record on the table.

Percentage

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Watermelon Pear

Mandarin Peach

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Kiwifruit

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Strawberry Grapes Mango Which fruit has the least waste to fruit ratio? Which fruit has the most waste to fruit ratio? 22

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Convert between common metric units of length, mass and capacity

Orange

Sub-strand: Using units of measurement—UUM – 2

RESOURCE SHEET Jugs and more jugs 1. How much milk is in each of the jugs below? (a)

500 mL

(b)

(c)

(d)

500 mL

r o e t s Bo r e p ok u S

2. If a new two-litre carton of milk is opened each time, and the amount shown are poured into each jug, how much milk is left in the carton? 2L

500 mL

(d)

(a)

(b)

5L 3L

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(c)

500 mL

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3. If half a litre of juice is added to each of the jugs below, how much juice is now in each jug?

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CONTENT DESCRIPTION: Convert between common metric units of length, mass and capacity

(c)

(b)

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(a)

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(d)

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 2

RESOURCE SHEET Mixing punch At the school fete, a stall was set up to sell fruit punch drinks. The ingredients used to make the punch drinks were cordial concentrates, fruit juices and soda water to give them a bit of sparkle. The bottles came in different sizes.

250 mL

r o e t s Bo r e p ok u S

500 mL

300 mL

750 mL

What is the volume and how many bottles of each ingredient are required to make each punch flavour? The first one is done for you. Quantity of punch required

Ratio of ingredients

Orange zinger

50% orange juice 50% soda water

Fruit cocktail

Sharp and sweet

15 L

4 L soda water orange juice

2 soda water

orange juice

lime cordial berry juice

o c . che e r o t r s super orange juice

⁄4 orange juice 1 ⁄8 orange cordial 1 ⁄8 lime cordial 1 ⁄2 soda water 1

4 orange juice

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4 L orange juice

Bottles of each ingredient

1500 mL lime cordial 1.5 L orange cordial 1500 mL berry juice 1 L orange juice 1.5 L apple juice 8 litres soda water

orange juice

orange cordial

lime cordial

soda water

lime cordial

orange cordial

berry juice

orange juice

apple juice

soda water

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Convert between common metric units of length, mass and capacity

Tuttifruiti

⁄4 lime cordial 3 ⁄4 berry juice

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Citrus special

⁄5 orange juice 3 ⁄10 apple juice 1 ⁄10 orange cordial 3

Quantity of each ingredient

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Punch flavour

Sub-strand: Using units of measurement—UUM – 2

RESOURCE SHEET Measure hunt Find items in the classroom that would belong in each of the boxes below.

Between 1 mm and 5 mm

Between 11⁄2 metres and 2 metres

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Between 5 gram and 10 grams

Between 100 grams and 600 grams

Between 1 kilogram and 11⁄2 kilograms

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Between 10 cm and 50 cm

Between 11⁄2 litres and 2 litres

Write two items that are best measured using the following units:

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CONTENT DESCRIPTION: Convert between common metric units of length, mass and capacity

Between 50 mL and 1 L

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grams:

millimetres:

litres:

kilograms: kilometres:

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Between 5 mL and 10 mL

o c . che e r o t r s super

millilitres: tonnes: megalitres: metres: milligrams:

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Assessment 1

Sub-strand: Using units of measurement—UUM – 2

NAME:

DATE: Measuring lengths

1. Measure the three segments below and record your answers in millimetres. Then re-write your answers in centimetres. mm

(a)

(b)

mm cm mm

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(c)

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2. Re-write these lengths in centimetres. (b) 3 mm

(a) 62 mm

3. Re-write these lengths in millimetres.

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(a) 6 m

(b) 3.8 m

6. Order these lengths from shortest to longest: 124 cm

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2.06 m

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5. Re-write these lengths in centimetres.

12.1 m

121 cm

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7. What unit of measurement would you use to measure each of the following? (a) Your height

(b) The distance between Sydney and Melbourne (c) The width of the top of a screw (d) The length of a fingernail (e) The distance between Brisbane and London (f) The thickness of a piece of string 8. (a) Draw a line that is 15.5 centimetres long. (b) How many millimetres is that? 26

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Convert between common metric units of length, mass and capacity

(b) 12 cm (c) 0.6 cm © R . I . C . P u b l i c a t i ons 4. Re-write these lengths in metres. •f orr ev i ew pur pose sonl y• (a) 290 cm (b) 550 cm (c) 2400 cm (a) 7.4 cm

Assessment 2

Sub-strand: Using units of measurement—UUM – 2

NAME:

DATE: Measuring mass

1. Re-write these amounts in grams. (a) 1.5 kg

(b) 0.9 kg

2. Re-write these masses in kilograms. (a) 1000 g

(b) 50 g

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3. Circle the correct weight to balance the scales. (a) 375 kg

375g

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(b) 0.375 kg

(d) 3.75 kg

(e) None of these.

4. What is the mass of this box in grams? Re-write this in kilograms.

grams kilograms

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5. Order these masses from lightest to heaviest: 1200 g

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CONTENT DESCRIPTION: Convert between common metric units of length, mass and capacity

o c . che e r o t r s super 2.06 kg

0.1 kg

206 g

6. The price of bananas is $8.60 a kilogram. How much will these bananas cost altogether?

250 g

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Assessment 3

Sub-strand: Using units of measurement—UUM – 2

NAME:

DATE: Measuring capacity

1. Re-write these capacities in millilitres. (a) 2.3 L

(b) 0.9 L

2. Re-write these capacities in litres. (a) 47 mL

(b) 7900 mL

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3. Order these capacities from least to most:

0.1 kL

0.25 L

2.5 L

1750

1500

1500 1250

500 250

800 600

500

400

400 300

200

250 200 150 100

100

(a) How much liquid is in each container? C

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(b) Which container holds the most? (c) Which container holds the least?

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5. Container A can hold 3 L of liquid, container B 7000 mL and container C 20 L. Without partfilling any of the containers, explain how you could use the three containers to measure out exactly 5 L of water.

A 28

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

C R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Convert between common metric units of length, mass and capacity

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804 mL

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10 L

Checklist

Sub-strand: Using units of measurement—UUM – 2

Convert between common metric units of length, mass and capacity (ACMMG136) Converts between common metric units of … mass

capacity

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STUDENT NAME

length

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using Units units of measurement—UUM Measurement—UUM – 32

Solve problems involving the comparison of lengths and areas using appropriate units (ACMMG137)

RELATED TERMS

TEACHER INFORMATION

Perimeter

What this means

• A measure of the distance around the boundary of a twodimensional shape.

• Students know how to work out the perimeter and area of a rectangle.

Area

Appropriate units (metric)

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• For perimeter, use centimetres, metres and kilometres. For area, use square centimetres, square metres and square kilometres. Large areas of land may also be measured in hectares, though students would not be expected to work with these units at this year level.

• To work out the area of shapes other than rectangles or squares, students would overlay a 1cm2 grid and count the squares and part-squares or construct the shape to be measured on 1cm2 grid paper and count the squares or part-squares.

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• The amount of surface covered (i.e. 2-D; measured in square units).

• If using shapes other than rectangles or squares, students could investigate the differences when calculating the perimeters of regular and irregular polygons. They may also investigate the perimeters of curved shapes using string or similar to place around the boundary and then measure against a ruler.

• Students are familiar with the common metric units of measurement for length and area.

• There is no direct relationship between the area and perimeter of rectangles. Two rectangles with the same area may have different perimeters, and two rectangles with the same perimeter may have different areas.

• It is not necessarily the case that, as the perimeter of a rectangle is increased or decreased, the area of the rectangle is increased or decreased.

Teaching points

Student vocabulary

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• Students need to be fluent with conversions of common metric units of measure for length; see pages 10–28 (UUM – 2). Discussion could centre on needing to know about multiplying and dividing by ten, or powers of ten. Discuss the meanings of the prefixes; e.g. kilo, centi, milli. • Teachers need to stress that the formulas for working out the perimeters and areas of rectangles only pertain to rectangles, and are not necessarily appropriate for other shapes.

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• Use the language ‘square centimetres’, not ‘centimetres squared’; they mean different things. For example, 2 cm2 (2 square centimetres) means 2 lots of 1 cm x 1 cm squares, where 2 centimetres squared means a square with a length of 2 cm and a width of 2 cm (which is actually 4 square centimetres).

What to look for

perimeter area centimetres metres kilometres

• Students confused about the difference between perimeter and area. • Students believing that there is a direct relationship between the perimeter and area of a rectangles. • Students who use ‘square centimetres’ and ‘centimetres squared’ interchangeably.

square centimetres square metres square kilometres

Proficiency strand(s): Understanding Fluency

Problem solving Reasoning

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units Units of measurement—UUM Measurement—UUM – 32

HANDS-ON ACTIVITIES Estimate before measuring in all these activities • Estimate and then measure the perimeter of large areas within the classroom and in the playground. For example, What is the perimeter of the netball court? What is the perimeter of the wet area? Students could first use non-standard units such as paces or jumps, and then use metre rulers or trundle wheels to check. Discuss how you might want to record the lengths of each of the sides, and then add them all up at the end. This may help overcome the problem of losing count. • Try to take advantage of practical situations. For example, The grass on one section of the oval is to be replaced, so the gardener will rope it off until it is established. How much rope will be needed? By asking a question such as, The rope costs $6.25 a metre; how much would the total amount of rope cost?, this links to (Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies [ACMNA129] ).

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50 cm

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• Using a loop of string (approximately 50 centimetres long), have students make a variety of shapes with it on 1 cm2 grid paper, carefully trace around each one, then count the squares within each shape to work out the areas. Discuss the fact that the shapes all have the same perimeter but different areas, and look at what shapes have the largest or smallest areas. (A circle will give the largest area.)

P= A=

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• In a similar activity to the one above, students cut a number of strips of 1 cm2 grid paper that are 2 centimetres high and 24 centimetres long. Fold all the strips in half width-wise (to give extra strength). Fold the strips into a variety of regular and irregular polygons, using tape to join them. Put the polygons onto 1 cm2 grid paper and work out the area of each of them. Ask questions such as: Which shape has the largest area? Which has the smallest? What do you think will be the smallest area we can make that has a perimeter of 24 centimetres? What can we say about the perimeters of all the shapes?

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not to scale

• Make a variety of shapes with the same area and compare their perimeters.

• What is the difference between 1 metre square and 1 square metre? A metre square is a square whose sides each measure one metre in length. A square metre can be any shape as long as the total area of the shape is one square metre. For example, a rectangle with a length of 2 metres and a width of 0.5 metres will have an area of one square metre, even though it is not a metre square. In groups, students could investigate other shapes they can make that have an area of one square metre by using newspaper and tape and making a number of squares that are one metre by one metre. They could then cut these in half in different ways and tape them back together in a different format. As long as there are no gaps, additions or overlaps, the new shapes will still have an area of one square metre. This may help overcome a common misconception that a square metre (1 m2) must be in the shape of a square (i.e. a metre square).

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using Units units of measurement—UUM Measurement—UUM – 32

LINKS TO OTHER CURRICULUM AREAS Information and Communication Technology • Students can calculate the perimeters of shapes on the website <http://www.bgfl.org/custom/resources_ftp/client_ftp/ ks2/maths/perimeter_and_area/index.html> • A shape surveyor that looks at area and perimeter of rectangles can be found at <http://www.funbrain.com/cgi-bin/ poly.cgi> • The Shape Explorer can be found at <http://www.shodor.org/interactivate/activities/ShapeExplorer/> You can choose to have only rectangular shapes for this activity. It also has the option of looking at the areas and perimeters of the shapes students have worked with in table format.

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• A catchy YouTube™ song on perimeter and area can be found at <http://www.youtube.com/watch?v=D5jTP-q9TgI> It goes straight to multiplication for working out the area of a rectangle, so this could be used after students have ‘discovered’ how the formula works. • A YouTube™ Perimeter rap song can be found at <http://www.youtube.com/watch?v=wynwRcc5q_U&feature=related> It shows some of the measurements in inches and others in centimetres.

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• An activity where students use perimeter and area to launch a ship can be found at <http://pbskids.org/cyberchase/ math-games/airlines-builder> The instructions are a little unclear, but students will soon get the hang of it.

• A Design a Party planning activity where students look for particular rectangles with given areas and perimeters can be found at <http://www.mathplayground.com/PartyDesigner/PartyDesigner.html>

Science

• Collect a variety of leaves. Students try to find two that have the same perimeter or the same area. They then estimate the perimeter and area of each, check the perimeter by carefully following the edge with string and measuring it against a ruler, and check the area by overlaying each of them on a 1cm2 grid and counting the squares.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units Units of measurement—UUM Measurement—UUM – 32

RESOURCE SHEET Perimeter puzzles Estimate the perimeters of each of the shapes below. Then measure each one.

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Object

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Parallelogram

Rectangle

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CONTENT DESCRIPTION: Solve problems involving the comparison of lengths and areas using appropriate units

You may change your other estimates if you wish. Write all your results in the table. Then write how you calculated the perimeter of each shape. (Did you do it the same way for each of the shapes?)

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Estimated perimeter

Actual perimeter

Difference between actual and estimated perimeters

Trapezium

Oval

Hand

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using Units units of measurement—UUM Measurement—UUM – 32

RESOURCE SHEET Which is the biggest? The size of a 2-D shape can be measured by perimeter and by area. Does a large perimeter mean a large area?

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1. Record anwers in the table.

(a) Estimate the perimeter and area of each shape. (b) Measure the perimeter and area of each shape. Shape

Perimeter Estimate

Area Actual

Estimate

Actual

A B C D 34

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Solve problems involving the comparison of lengths and areas using appropriate units

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Sub-strand: Using units Units of measurement—UUM Measurement—UUM – 32

RESOURCE SHEET

This square is 1 unit long, and 1 unit wide. Its perimeter is 4 units.

1 unit

3 units

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This arrangement of 9 tiles has an area of 9 square units. Its perimeter is 12 units.

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1 unit

You will need:

1 unit

Constant area, changing perimeter 1 unit 9 square tiles

3 units

1. Using 9 square tiles for each, make 4 more arrangements with the given perimeters. Make a copy of each arrangement on the grid. Perimeter = 14 units

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(b) Area = 9 square units

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CONTENT DESCRIPTION: Solve problems involving the comparison of lengths and areas using appropriate units

(a) Area = 9 square units

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Perimeter = 18 units

(d) Area = 9 square units

Perimeter = 20 units

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using Units units of measurement—UUM Measurement—UUM – 32

RESOURCE SHEET

1 unit

3 units

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This arrangement of 9 tiles has an area of 9 square units. Its perimeter is 12 units.

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1 unit

Constant perimeter, changing area 1 unit This square is 1 unit long, and 1 unit wide. Its perimeter is 4 units.

3 units

(a) Perimeter = 12 units

Area = 8 square units

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(d) Perimeter = 12 units

Area = 7 square units

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(b) Perimeter = 12 units

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Area = 5 square units

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Solve problems involving the comparison of lengths and areas using appropriate units

1. Use square tiles to make four more arrangements, each with a perimeter is 12 units but using one less tile each time.

Assessment 1

Sub-strand: Using Units Measurement—UUM – 32 units of measurement—UUM

NAME:

DATE: What’s the size? 1. (a) Which one has the larger area? (b) How can you check? A

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(d) Which has the larger perimeter?

2. (a) What is the area of shape C? (b) What is the area of shape D?

D © R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

(e) What is the perimeter of shape D? (f) Which has the larger perimeter?

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3. (a) What is the area of shape E? (b) What is the area of shape F?

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CONTENT DESCRIPTION: Solve problems involving the comparison of lengths and areas using appropriate units

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(e) How can you check?

(d) What is the perimeter of shape E?

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4. (a) Draw two more shapes.

(e) What is the perimeter of shape F?

(f) Which has the larger perimeter?

(b) Record the perimeter and area of each shape.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Assessment 2

Sub-strand: Using Units Measurement—UUM – 23 units of measurement—UUM

NAME:

DATE: Complete the tables

1. (a) Use square tiles and the information in the table to make each rectangle. (b) Draw each rectangle on 1 cm2 grid paper. (c) For each rectangle, add the missing information to the table.

A B

Perimeter

Length of sides

12 cm2

14 cm

cm and

12 cm2

16 cm

cm and

16 cm2

16 cm

cm and

24 cm2

20 cm

cm and

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54 cm2

cm2

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Area

6 cm and 9 cm

28 cm

40 cm2

4 cm and 10 cm

44 cm

cm and

cm 26 cm cm and cm ©40R . I . C.Pu bl i cat i on s I cm 10 cm and •f orr ev i ew p24ucm r poses onl y•cm

20 cm2

75 cm2

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cm2

4 cm and

3 cm and 25 cm

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40 cm

10 cm and

2. (a) Use square tiles to make five different rectangles, each with an area of 36 cm2.

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(b) Draw these rectangles on 1 cm2 grid paper. (c) Calculate the perimeter of each rectangle.

(d) For each rectangle, add the missing information to the table.

Rectangle

Area

Length of sides

36 cm2

cm and

36 cm2

cm and

36 cm2

cm and

36 cm2

cm and

36 cm2

cm and

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

Perimeter

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Solve problems involving the comparison of lengths and areas using appropriate units

Rectangle

Sub-strand: Using units Units of measurement—UUM Measurement—UUM – 32

Knows that perimeter and area are not directly related

Calculates areas of non-rectangular shapes on 1 cm2 grid paper

Calculates areas of squares and rectangles

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STUDENT NAME

Calculates perimeters of shapes

Solve problems involving the comparison of lengths and areas using appropriate units (ACMMG137)

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 4

Connect volume and capacity and their units of measurement (ACMMG138)

RELATED TERMS

TEACHER INFORMATION

Volume

What this means

• The volume of an object is the total space occupied by the object (i.e. 3-D; measured in cubic units).

• Students recognise that one millilitre (1 mL) of water is equivalent to one cubic centimetre (1 cm3) and has a mass of one gram (1 g).

Capacity

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Units of measurement

• For volume, the appropriate metric units are: cubic millimetres, mm3; cubic centimetres, cm3; cubic metres, m3.

• One cubic centimetre of material displaces one millilitre of water and one cubic centimetre of water weighs approximately one gram. • The mass of an object placed in a container of water may not affect the amount of water displaced. This means that putting two objects of the same volume but different masses will displace the same amount of water.

Teaching points

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• The amount a container can hold. This is different to volume, which is how much space it takes up. An example of this difference is if you consider an esky. The amount of room it takes up in a cupboard is its volume. The amount it can hold is its capacity.

• When an object is placed in a container of water, the amount of water displaced is its volume.

• Estimate before measure in all measurement activities.

• Students should be free to investigate the effects of placing objects in water and what happens to the water level. • Students need to mark off levels on a container to measure the water level.

• Students may need to be shown how to read the levels of water marked on a graduated container. • There are three ways to measure displacement:

– Partly fill a container, mark the water level, place the object into the container and measure the distance the water has risen.

millilitres litres kilolitres

cubic centimetres cubic metres

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Student vocabulary

– Place the object into an empty container, fill the container sufficiently to cover the object (or fill to the top of the container), remove the object and measure the difference. – Fill a container to the top with water, place in the object and catch and measure the overflow. • The common metric unit for volume are cubic millimetres (mm3), cubic centimetres (cm3) and cubic metres (m3).

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• The metric units for capacity are millilitres (mL), litres (L) and kilolitres (kL). • Note the use of the upper case ‘L’ for the abbreviation of litres; it is also used for mL and kL. This is to distinguish it from the number 1.

takes up more space takes up less space holds more holds less holds the most holds the least

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 4

Connect volume and capacity and their units of measurement (ACMMG138)

Conversions

TEACHER INFORMATION (CONTINUED)

Capacity 1000 mL = 1 L 1000 L = 1 kL Powers of ten decimal metric

decimal point

• Students need to be fluent with conversions of common metric units of measure for capacity. Discussion could centre on needing to know about multiplying and dividing by ten, or powers of ten. This links to (Connect decimal representations to the metric system [ACMMG135] ).

• Discuss the meanings of the prefixes; e.g. milli, centi, kilo.

What to look for

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• Abbreviations of metric units never use the ‘s’ at the end; e.g. ‘375 mL’ not ‘375 mLs’. However, if the unit is written in full, the ‘s’ is needed; e.g. ‘375 millilitres’.

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multiplication division

• A gap is always left between the number of units and the abbreviation of the unit—e.g. 5 cm, 8 kg, 375 mL etc.—and no full stop is used at the end of the abbreviation (unless it is at the end of a sentence).

• Students using inappropriate units of measurement.

• Students using inappropriate tools to measure; e.g. using a medicine measure to calculate the capacity of a bucket.

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o c . che e r o t r s super Proficiency strand(s): Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 4

HAND-ON ACTIVITIES Estimate before measuring in all these activities • Compare two objects by marking the level to which each one causes the water in a container to rise (its displacement). This can be done by placing a rubber band around the container to mark the water level once the first object has been immersed and another rubber band to mark the level once the first object has been removed and the second object immersed. The difference between the two rubber bands is the difference in volume of the two objects. This can later be done using a graduated container, so that the measures can be read without the use of rubber bands. See pages 44 and 45. • Pour water into a container and mark the level with a rubber band (or use a graduated container and take note of the level of water in it). Immerse an object into the water. Pour off and measure the water above the initial mark. This will be the object’s volume. Students can compare the volume of the objects by repeating this activity for each of them and comparing the results.

Pour off displaced volume and measure

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Level of water

rubber bands

Object added

• Use a container full of water standing in a tray or bucket. Students place the object to be measured into the container and measure the overflow of water in millilitres. Again, comparisons can be made by following this process for each of the objects.

Displaced volume

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Pour off and measure displaced volume

Cubic centimetre cubes in water

Millilitres displaced

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• Students use 1 cm3 cubes and measure the displacement when multiples are immersed in water. This can be done using any of the three methods above. Students could graph the results. Discussion could centre on the relationship between millilitres and cubic centimetres (1 mL occupies 1 cm3 of space).

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Number of cubes

5 cubes displaced 5 mL of water, 10 cubes displaced 10 mL, and so on. We think that 1 cube would displace 1 mL of water. This means that 1 cm3 is the same as 1 mL. • Discuss the displacement method for calculating the volume of irregular objects such as rocks. Encourage students to write about their findings. The rock displaced 65 mL of water, so its volume must be 65cm3. • Students investigate immersing objects of the same volume but different masses. For example, have 2 jars of the same dimensions, one filled with wet sand, the other empty. What do you think will happen when we put each of these into water one at a time? Do you think the water will rise more for one than the other, or the same for each? • Students investigate what happens when an object to be measured using displacement floats. This could be tested using a block of wood (rectangular prism) for which the volume could be readily calculated.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 4

LINKS TO OTHER CURRICULUM AREAS English • Read Who sank the boat by Pamela Allen. In this story, a group of animals decide to go for a row in a boat. As each animal jumps into the boat, it sits a little lower in the water. The ideas of balance as well mass are mentioned, as well as the fact that it is the smallest animal that finally sinks the boat. • Read Mr Archimedes’ bath, also by Pamela Allen. In this book, Mr Archimedes notices that when he and his animal friends get in and out of the bath, the water level changes.

Information and Communication Technology

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• An online quiz that tests students’ knowledge of volume and capacity can be found at <http://wps.pearsoned.com.au/ nsm56/29/7456/1908805.cw/content/index.html> • A website where students choose from three different objects to solve a capacity question can be found at <http:// www.bbc.co.uk/skillswise/game/ma23capa-game-taking-measures-capacity>

Science

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• Decanting puzzles can be found at <http://www.netrover.com/~kingskid/jugs/jugs.html>

• Students could investigate the effects of different solutions used in displacement experiments. The density of solutions will make a difference to how they displace objects. Links can be made to (Mixtures, including solutions, contain a ). combination of pure substances that can be separated using a range of techniques [ACSSU113]

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 4

RESOURCE SHEET Dunking Investigate the relationship between the volume of an object and its effect on the level of water in a container. Materials:

1 tall glass 4 different coloured rubber bands plasticine water

Procedure:

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2. Mark the original water level (O) with a coloured band.

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1. Fill the glass to approximately one-third of its capacity.

3. Discuss with a partner where you think the new water level will be when you add the plasticine. Add a different coloured band to show your prediction (P1).

4. Gently lower a ball of plasticine into the water. Mark the new level with a third rubber band (N1). How close was your prediction? 5. Take out the plasticine and make it into a different shape. What do you think will happen when you place the new shape into the water? Move the ‘prediction band’ to show where you think the new level will be (P2).

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On the diagram, mark the glass with approximate positions of the three water levels (O, N1, N2) and your two predictions (P1, P2).

o c . che e r o t r s super 1. the amount of water in the glass at each level?

2. the amount of water between the original and new levels?

rubber band 44

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Connect volume and capacity and their units of measurement

Results:

Sub-strand: Using units of measurement—UUM – 4

RESOURCE SHEET More dunking Investigate the relationship between the mass of an object and its effect on the level of water in a container. Materials:

2 identical, tall glasses rubber bands 2 small balls of the same size but different masses

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water Procedure:

2. Mark the water level on the glasses with rubber bands (O).

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1. Fill each glass to the same level, approximately one-third full.

3. Hold the two balls, one in each hand to determine which is the heavier.

4. Decide which glass each ball will be added to. Mark each glass with another rubber band to show the level you think the ball will bring the water to (P). 5. Carefully lower the two balls into the glasses, one at a time.

© R. I . C.Publ i cat i ons 1. On the each glass the positions of the two water levels (O, •diagram, f ormark r ev i e wwithp uapproximate r pose son l y • N) and your prediction (P). 6. Mark the new level of water in each glass with a third rubber band (N).

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CONTENT DESCRIPTION: Connect volume and capacity and their units of measurement

Results:

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What can you say about: 1. the amount of water in the two glasses at the three different levels?

2. the amount of water between the original and new levels?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Assessment 1

Sub-strand: Using units of measurement—UUM – 4

NAME:

DATE: Volume and capacity One millilitre (1 mL) of water has an equivalent volume of one cubic centimetre (1 cm3) and has a mass of one gram (1 g).

1. A piece of metal is lowered into a container full of water and the overflow is measured. If the overflow is 92 mL, what is the volume of the metal?

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2. A piece of stone with a volume of 125 cm3 is lowered into a measuring jug containing 1.5 litres of water. What is the new volume shown on the jug?

What is the volume of the water? weights equal 385g

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3. The two containers on the scale weigh exactly the same. The container on the right holds 385g of kitchen weights. The container on the left is filled with water.

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Metal 1

volume of metal

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4. Two pieces of precious metal are lowered into two identical containers each holding 500 mL of water. One piece of metal has a volume of 127 cm3, and the other displaces 450 mL of water. Use these facts to complete the table. Metal 2

127 cm3

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volume of water displaced

450 mL

new level in container

5. A bottle of milk contains 600 millilitres. How many cubic centimetres is that? Circle the answer. (a) 100 cm3

(b) 30 cm3

(d) 1000 cm3

(e) 3600 cm3

(c)

600 cm3

6. What is the mass of 375 millilitres? Circle the answer. (a) 375 mg (b) 375 g (c) 375 kg (d) impossible to determine without weighing 46

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Connect volume and capacity and their units of measurement

Checklist

Sub-strand: Using units of measurement—UUM – 4

Uses the correct units of measure for volume and capacity

Describes the results of displacement in terms of volume and capacity

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STUDENT NAME

Understands the connection between volume and capacity

Connect volume and capacity and their units of measurement (ACMMG138)

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 5

Interpret and use timetables (ACMMG139)

RELATED TERMS

TEACHER INFORMATION

am (ante meridiem)

What this means

• From the Latin words meaning before noon.

• It is important for students to become familiar with a variety of timetable formats. The 12-hour and 24-hour formats are most commonly used.

pm (post meridiem)

• From the Latin words meaning after noon.

60 seconds = 1 minute 60 minutes = 1 hour 24 hours = I day

Expressing two units

• For example, 1 day and 4 hours as 28 hours or 2 hours and 40 minutes as 160 minutes.

• Converting between 12- and 24-hour times can be difficult because of the non-metric nature of time. So 1630 is not 6:30 pm, but 4:30 pm. • Calculations of time difference can be quite difficult, again because of the non-decimal nature of time. For example, if using a timetable and calculating how long before the next bus, a calculator may actually hamper the process. If the bus arrives at 1625 and it is currently 1547, you cannot simply key 1625 into a calculator and subtract 1527; the result would be 78, which a child could incorrectly interpret as 78 minutes. In this case, the number of minutes until 1600 would be calculated first (13 minutes), and the extra 25 minutes until the desired time (1625) added to give a total waiting time of 38 minutes.

Teaching points

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Conversions of time

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• It is expected that students can tell the time to the nearest minute using both 12- and 24-hour clocks.

o’clock

half past

quarter past quarter to

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xx:25 (eg 3:25) xx:52 (eg 3:52) clockwise

• When writing 12-hour time, we use a colon between the hours and minutes; for example, two o’clock should be 2:00 not 2.00. When writing 24-hour time we generally do not use a colon, but use 4 digits; for example, 4:33 pm would be written as 1633. For times before 10 am, there is a zero at the start in 24-hour time; e.g. 0730 for 7:30 in the morning. Whether we say ‘oh’ or ‘zero’ depends on community practice, but it should always be in the written form. (Note: some sources do use a colon in 24-hour time.)

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Student vocabulary

• Standard abbreviations for units for time are seconds (s), minutes (min) and hours (h); the other units do not have standard abbreviations. Note: ‘sec’ and ‘hr’ are commonly used abbreviations for second and hour, but they are not the correct ones.

• Telling the time in both 12- and 24-hour formats should be practised daily, and treated incidentally whenever the opportunity arises.

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• Classrooms should have both an analogue clock and a digital clock, preferably side-by-side. Regularly seeing the two different displays for the same time of day helps students realise that there are two equally valid ways to read the time. There are some large clocks available commercially that clearly show the time in both formats.

am (ante meridiem) pm (post meridiem)

13:00

OFF

GIGA-BLASTER

second minute hour day

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 5

Interpret and use timetables (ACMMG139)

TEACHER INFORMATION (CONTINUED)

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• When using the 24-hour time format, times on the hour generally are said as ‘hundred’; e.g. 1100 would be eleven hundred and 0500 would be zero five hundred. Other times with a zero at the end would be spoken in tens; e.g. 1120 would be eleven twenty and 0530 would be zero five thirty.

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• The spoken time reflects the written digital time; e.g. in 12-hour time format, 11:28 would be said as eleven twenty-eight, not twenty-eight minutes after/past eleven. The time 7:31 would be said as seven thirtyone, not twenty-nine minutes to 8 or thirty-one minutes after/past seven. With times such as 11:05, whether we say oh instead of zero or whether we verbalise the zero at all, depends on community practice. However, the zero must be used in the written form. When ‘am’ and ‘pm’ are used, the individual letters are spoken.

• National tests usually have several questions on time, including calculating time differences.

What to look for

• Students unable to convert from 12- to 24-hour time.

• Students not using four digits when writing 24-hour time; e.g. writing 815 instead of 0815 for 8:15 am.

• Students using the wrong base (e.g. Base 10) for time calculations. • Students using a calculator inappropriately when subtracting one time from another to find the duration of an event.

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• Students unable to decide which operation is appropriate when calculating time problems.

o c . che e r o t r s super Proficiency strand(s): Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 5

HAND-ON ACTIVITIES • When converting from the 12-hour to the 24-hour clock, for any time after 12:59 pm (that is in the afternoon), we add 12; so 5:00 pm becomes (5 + 12) which is 1700, and 11:13 pm becomes (11 + 12, and the 13 minutes) which is 2313. • Where possible, base the use of timetables on real life situations such as class timetables, television guides, excursion plans, public transport timetables, students’ holiday plans etc. • Use a clock face format to display part of a timetable. This works well for a one-hour time frame, as the five-minute timeslots make it easy to calculate. Providing the information in this format is a way for students to be introduced to circle (pie) graphs. Students could then make their own circle graph for a different one-hour period, either during the school day or after school. (See page 54.) Ensure students understand that the timetable starts at 12 and continues clockwise but the 12 does not mean 12 o’clock.

r o e t s Bo r e p ok u S Discussion

Morning routine

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Between 9:00 and 10:00

Mental maths

Math activities

• Students plan a trip for a day out that involves using public transport. If possible, more than one form of public transport could be planned for, thus necessitating calculations about transit times and the viability of getting from one place to another in plenty of time.

• Offer students the opportunity to construct simple personal timetables. This may be for a period of one day, a weekend, a long weekend or the duration of a two-week school holiday.

Tues

Wed

0900

1200 1330 1500

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1030

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Thur

Fri

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Mon

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• Remove the class timetable from view on a Monday and ask students to fill it in as the days progress. At the end of the week, students could compare their timetables with the regular one, and discuss any differences and why they might have occurred; e.g. it was raining when we were due to have fitness on Wednesday morning, so we did it after lunch. • Look at timetables for public transport in the local area and discuss features of it. For example, What is the earliest train we could catch to the city? Is it earlier or later than the first train from the city? What time is the last train for the day? Why don’t they have any later trains? How far apart are the services? Does it vary at different times of the day? What is the longest time you would have to wait for a train? What is the shortest time between services?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 5

LINKS TO OTHER CURRICULUM AREAS English • Use the book Tick tock by James Dunbar as a stimulus book for discussion about different time periods. • Read Just a minute! by T Slater. • Read Clocks and more clocks by Pat Hutchins. Discuss what the times in the book would be if they were shown in 24-hour format. Discuss also the time differences between each of the rooms.

Information and Communication Technology • A stop-the-clock format for recording 24-hour time can be found at <http://www.bgfl.org/custom/resources_ftp/ client_ftp/ks2/maths/time/index.htm>

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• Stop the Clock, with a 24-hour option, can be found at <http://resources.oswego.org/games/StopTheClock/sthec5. html> Students have to match an analogue clock with its digital time display.

Languages

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• Students tell the time in another language, and in both 12- and 24-hour formats.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 5

RESOURCE SHEET 12- and 24-hour comparisons 12-hour clock

24-hour clock

12 midnight

0000

1 am

0100

2 am

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0200

3 am

0300 0400

5 am

0500

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4 am

6 am

0600

7 am

0700

8 am

0800

0900

10 am

12 noon

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1200

2 pm 3 pm 4 pm 5 pm

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1300 1400

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6 pm

1800

7 pm

1900

8 pm

2000

9 pm

2100

10 pm

2200

11 pm

2300

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Interpret and use timetables

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1 pm

Sub-strand: Using units of measurement—UUM – 5

RESOURCE SHEET The 24-hour clock 0001

1:00 am

0100

2:00 am

0200

3:00 am

0300

4:00 am

0400

5:00 am

0500

6:00 am

0600

7:00 am

0700

8:00 am

0800

9:00 am

0900

10:00 am

1000

11:00 am

1100

12:00 noon

1200

23 11

1:00 pm

1300

2:00 pm

1400

3:00 pm

1500

4:00 pm

1600

5:00 pm

1700

6:00 pm

1800

7:00 pm

1900

8:00 pm

2000

9:00 pm

2100

10:00 pm

2200

11:00 pm

2300

12:00 midnight

2400

r o e t s Bo r e p ok u S 21

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12:01 am

What is 9:25 am in 24-hour time? Look at the 24-hour clock. 9:00 am is shown as 0900, so 9:25 am would be shown as 0925.

(a) 6:00 am

(b) 4:40 pm

(d) 12:00 midnight

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(e) 12:53 pm

(f) 3:15 am

2. Write these 24-hour times as 12-hour times, using am or pm.

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(g) 2245

CONTENT DESCRIPTION: Interpret and use timetables

(i) 0455 (k) 0005

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© R. I . C.Publ i cat i ons What is 9:25 pm in 24-hour time? 9:00 pm is shown as 2100, so 9:25 pm would be shown as 2125. •f orr evi ew pur posesonl y• 1. Write the following 12-hour times as 24-hour times.

o c . che e r o t r s super (h) 1007 (j) 1836 (l)

2359

3. What is the time period between these times of the same day? (a) 10:05 am and 1325 (b) 1535 and 4:15 pm (c) 11:50 am and 2350 (d) 1455 and 9:20 pm (e) 7:25 pm and 1955 (f) 0430 and 4:10 pm

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Using units of measurement—UUM – 5

RESOURCE SHEET Clock face timetables A clock face timetable can be used to show how much time is spent on different activities. A one-hour timetable shows minutes spent. A 12-hour timetable shows hours spent. The starting point is always 12, but this does not mean 12 o’clock.

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1. This is a one-hour timetable for a doctor. It shows the time spent with each patient between 1:00 pm and 2:00 pm.

(b) How long was her longest appointment?

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(a) How many patients did she see in the hour?

2. This is a 12-hour timetable for an electrician.

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(c) He went to the Smith house at 3:00 pm. It was his 6th appointment. What time did he leave there? (d) What time did he start work in the morning?

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(b) How long was his shortest visit?

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Note: the 12 does not mean 12:00, just the beginning of the timetable.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Interpret and use timetables

3. Make your own 12-hour timetable for last Saturday.

Sub-strand: Using units of measurement—UUM – 5

RESOURCE SHEET Timetable problems This is the schedule for the Chip Off the Block bus company, which offers regular services between Block City and Block Beach and between Block City and Block Mine. All services take one hour. Block City to Block Beach

Block Beach to Block City

Block City to Block Mine

Block Mine to Block City

0400

0510

0415

0525

1500

1240

1515

0700

r o e t s Bo r e p ok u S 1830

1020

1740

0715

1845

0800

1900

1110

1810

0815

1915

1100

1930

1140

1910

1115

1945

1300

2000

1210

1940

1315

2015

0430 0500 0530

1600

0640

1310

0445

1615

0705

1325

1700

0700

1410

0515

1715

0725

1425

1730

0840

1510

0545

1745

0905

1523

1800

0920

1610

0615

1815

0935

1625

1035

1805

1125

1825

1205

1925

1225

2005

Example: Jess arrived at Block City bus station at 3:47 pm. How long did she have to wait to for the next bus to Block Mine?

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0600

1305

Convert the 12-hour time to 24-hour time: 3:47pm to 1547.

The next bus to leave Block City for Block Mine after 1547, is the 1615. From 1547 to the next hour, 1600, is 13 minutes.

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From 1600, there are another 15 minutes to wait until the bus leaves at 1615.

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13 minutes plus 15 minutes are 28 minutes. Jess had to wait 28 minutes for her bus. Explain your answers to the problems below.

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CONTENT DESCRIPTION: Interpret and use timetables

1. Kim wanted a bus from Block Beach to Block City. He checked his watch. It read 10:49 am. How long until the next bus?

2. Ella arrived at the Block Mine bus station at 11:33 am. She wanted to go to Block Beach. What is the earliest she could get there?

3. At the end of his day shift, Tim got to Block Mine bus station at 6:15 pm. How long before he reaches Block City? What time will he arrive there?

4. Victor wanted to get back to Block City from Block Beach by 8 pm. What is the latest time he should leave the beach?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Assessment 1

Sub-strand: Using units of measurement—UUM – 5

NAME:

DATE: Cra-zee TV guide

Below is the TV guide for Mondays on the Cra-zee television service (CZTV). Cra-zee 2

Cra-zee 3

6:00 News 6:35 Brekky business 7:15 Don’t be bored 8:15 Play me music (rpt) 9:00 Talented dog show 10:00 Talented cat show 10:50 Talented bird show 11:45 Pets and more pets 12:30 Lunch news 1:00 Snooze news 1:55 Cooking made silly 2:50 Soap opera magic 3:20 Cartoons for all 5:00 Early news 5:30 Monster movie 8:45 Late news 9:00 Late at night movie 10:30 Just joking

6:00 Wake up everyone 7:30 Morning news 8:30 Monkeys in trouble 8:50 Monkeys out of trouble 9:25 Monkey movie 12:30 Midday madness 1:15 Police car fashions 2:30 Football (rpt) 4:50 Golf made easy 6:00 News 6:30 Tomorrow today 7:00 Sing my song 8:15 Cowboy sing-along 9:35 Great houses of old 10:50 Late night cooking 11:30 Car racing (rpt) 12:50 Best buys

6:00 Cook a breakfast 7:20 Morning movie 8:20 Cartoons for critters 10:10 Kiddies sing-along 11:30 Cra-zee cartoons 12:00 Lunch with me 1:00 Silly soap show 2:10 Guess my secret 2:55 Win your weight 3:45 My time (rpt) 4:20 Just me and you 5:05 Cra-zee kids show 6:00 News for you 6:30 Game of the year! 7:30 The best animals show 8:40 Movie moments 10:00 Sporting highlights 11:00 Late night comedy

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Cra-zee 1

2. If you could record and watch them all, how long would it take?

6. Ed arrived home at 1345. What shows are on at that time?

3. Which channel has the longest movie?

7. What is the earliest cooking show on Monday? What is the latest?

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4. How many repeat shows are there, and how long do they go for altogether?

8. Marie tuned in at 1850. How long must she wait to watch a news show? What channel will be showing it?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Interpret and use timetables

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5. What is the longest sports show? How long does it run for?

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1. How many news shows are broadcast altogether?

Checklist

Sub-strand: Using units of measurement—UUM – 5

Calculates duration of time in 12- and 24-hour formats

Uses timetables presented in 12-hour and 24-hour format

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STUDENT NAME

Converts between 12-hour and 24-hour time

Interpret and use timetables (ACMMG139)

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o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Answers

Sub-strand: Using units of measurement

UUM – 1 Page 6 1. 2. 3.

Page 17 Resource sheet – Units of length

Teacher check

Assessment 1 – Show week for the dairy

Cheese (a)

Mass of block at the end of the day

Amount sold

Sunday

3.25 kg

11.75 kg

Monday

2.35 kg

9.4 kg

Tuesday

2.3 kg

7.1 kg

Wednesday

1 kg 550 g

5.55 kg

Thursday

3500 g

2.05 kg

Friday

0.55 kg (or 550 g)

1.5 kg

Saturday

1.25 kg

0.25 kg (or 250 g)

Teac he r

Day

Page 18 Resource sheet – The king’s dilemma Solution: Cut the bar into three pieces that are 1 metre, 2 metres and 4 metres long. Night 1

Give knight the 1-metre length of gold. (1 m)

Night 2

Take back the 1-metre length of gold and give the knight the 2-metre length of gold. (2 m)

r o e t s Bo r e p ok u S

(b) 250 grams or 0.25 kg (c) Thursday (d) Friday (e) Tuesday Flavoured milk

Night 3

Give knight the 1-metre length of gold. He now has 3 metres of gold. (2 m + 1 m)

Night 4

Take back all the gold and give the knight the 4-metre length of gold. (4 m)

Night 5

Give the knight the 1-metre length of gold. He now has 5 metres. (4 m + 1 m)

Night 6

Take back the 1-metre length of gold and give the knight the 2-metre length of gold. He now has 6 metres. (4 m + 2 m)

Night 7

Give the knight the final 1-metre length of gold. He now has 7 metres. (4 m + 2 m + 1 m)

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Resource sheet – A square metre

(a) 100 (b) 100 Teacher check (a) 100 (b) 10 000

Page 7 1.

UUM – 2

Page 19 Resource sheet – The long and the tall

Total Day Chocolate Strawberry Vanilla (a) sold Sunday 3.8 L 4.5 L 2.3 L 10.6 L Monday 2200 mL 1850 mL 1250 mL 5.3 L Tuesday 3.5 L 2.9 L 1.6 L 8L Wednesday 2600 mL 1500 mL 1300 mL 5.4 L Thursday 2½ L 3¼ L 2L 7.75 L Friday 4L 3100 mL 1.9 L 9000 mL Saturday 4300 mL 4.2 L 3½ L 12 L

Teacher check

(b) 58.05 L (c) chocolate (d) Saturday (e) Monday Page 8 1. 2. 3. 4. 5. 6.

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13.85 L 58.05 L sold in total

Teacher check. Answers will vary.

Page 21 Resource sheet – What do they weigh? Teacher check

Page 22 Resource sheet – Edible fruit Teacher check Page 23 Resource sheet – Jugs and more jugs 1.

(a) 250 mL (b) 1.5 L (c) 850 mL (d) 100 mL (a) 1.5 L or 1500 mL (b) 1.6 L or 1600 mL (c) 1.35 L or 1350 mL (d) 0.2 L or 200 mL (a) 4 L or 4000 mL (b) 1 L or 1000 mL (c) 1.1 mL or 1100 mL (d) 1.75 L or 1750 mL

o c . che e r o t r s super 2.

Assessment 2 – Getting ready for the school fete

70 bags 60 Super Burgers 15 trips 12 kilograms 112 cups maximum $100, minimum $75

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22.9 L

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Amount of each flavour sold

Page 20 Resource sheet – Congratulations – You win!

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Answers

Sub-strand: Using units of measurement

Page 24 Resource sheet – Mixing punch

Page 27 Assessment 2 – Measuring mass

Quantity Contents of punch required

Ratio of ingredients

Quantity of each ingredient

Bottles of each ingredient

Orange zinger

50% orange juice 50% soda water

4 L orange juice 4 orange juice 4 L soda water 2 soda water

3 L orange juice 3 orange juice ⁄5 orange juice 1.5 L apple 2 apple juice 3 juice ⁄10 apple juice 1 orange 1 ⁄10 orange cordial 0.5 L orange cordial cordial 3

Fruit cocktail

Sharp and sweet

⁄4 orange juice 1 ⁄8 orange cordial 1 ⁄8 lime cordial 1 ⁄2 soda water

2 L orange juice 2 orange juice 1 L orange 2 orange cordial cordial 1 L lime cordial 4 lime cordial 4 L soda water 2 soda water

15 L

1500 mL lime cordial 1.5 L orange cordial 1500 mL berry juice 1 L orange juice 1.5 L apple juice 8 litres soda water

1.5 L lime cordial 6 lime cordial 3 orange 1.5 L orange cordial cordial 1.5 L berry juice 5 berry juice 1 L orange juice 1 orange juice 1.5 L apple 2 apple juice juice 4 soda water 8 L soda water

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Citrus special

Tuttifruiti

3. 4. 5. 6.

1200 g

2.06 kg

2 lime cordial 5 berry juice

Page 28 Assessment 3 – Measuring capacity 1.

(a) 2300 mL (b) 900 mL (c) 14 250 mL (a) 0.047 L (b) 7.9 L (c) 1.005 L 0.25 L 804 mL 10 L 0.1 kL (a) A = 500 mL B=1L C = 350 mL D = 50 mL E = 150 mL (b) B (c) D Fill container B (7000 mL) and pour into container A (3L); that leaves 4 L in B. Pour that into container C and empty container A. Fill container B, pour into container A, empty container A and fill it again from container B. This leaves 1 L in container B. Pour this into container C, which now has 5 L (4 L + 1 L).

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0.5 L lime cordial 1.5 L berry juice

(a) 1500 g (b) 900 g (c) 14 250 g (a) 1 kg (b) 0.05 kg (c) 3.4 kg (b) 0.375 kg 800 g, 0.8 kg 0.1 kg 206 g $2.15

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⁄4 lime cordial 3 ⁄4 berry juice

3. 4.

Page 25 Resource sheet – Measure hunt Teacher check

Page 26 Assessment 1 – Measuring lengths

6. 7.

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(a) 100 mm (10 cm) (b) 98 mm (9.8 cm) (c) 7 mm (0.7 cm) (a) 6.2 cm (b) 0.3 cm (c) 9.8 cm (a) 74 mm (b) 120 mm (c) 6 mm (a) 2.9 m (b) 5.5 m (c) 24 m (a) 600 cm (b) 380 cm (c) 1210 cm 121 cm 124 cm 2.06 m (a) cm or m (b) km (c) mm (d) cm or mm (e) km (f ) mm (a) Teacher check (b) 155 mm

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Page 33 Resource sheet – Perimeter puzzles

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UUM – 3

Estimated perimeter

Object

Actual perimeter

o c . che e r o t r s super Parallelogram

Difference between actual and estimated perimeters

16.6 cm

Rectangle

18 cm

Trapezium

16.1 cm

Oval

17 cm

Hand

35.5 cm

Page 34 Resource sheet – Which is the biggest? 12.1 m

Shape A: Shape B: Shape C: Shape D:

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

P = 28 cm; A = 48 cm2 P = 23.4 cm; A = 24cm2 P = 20.5 cm; A = 20cm2 P = 21.7 cm; A = 30 cm2

R.I.C. Publications® www.ricpublications.com.au

Answers

Sub-strand: Using units of measurement

Pages 35 Resource sheet – Constant area, changing perimeter (a)

P = 14, A = 9

(b)

P = 16, A = 9

Page 38 Assessment 2 – Complete the tables Rectangle

P = 18, A = 9

(c)

P = 20, A = 9

(d)

Area

Perimeter

Length of sides

12 cm

14 cm

4 cm and 3 cm

12 cm2

16 cm

6 cm and 2 cm

16 cm

4 cm and 4 cm

24 cm

20 cm

4 cm and 6 cm

54 cm

30 cm

6 cm and 9 cm

40 cm2

28 cm

4 cm and 10 cm

44 cm

2 cm and 20 cm

40 cm

r o e t s Bo r e p ok u S 2

26 cm

5 cm and 8 cm

The areas are all the same, but the perimeters are different. Answers may vary.

H I

20 cm

24 cm

10 cm and 2 cm

Pages 36 Resource sheet – Constant perimeter, changing area

20 cm2

18 cm

4 cm and 5 cm

56 cm

3 cm and 25 cm

40 cm

10 cm and 10 cm

(c)

(d)

100 cm

Rectangle

Length of sides

Perimeter

36 cm

1 cm and 36 cm

2 (1 + 36) = 74 cm

36 cm

2 cm and 18 cm

2 (2 + 18) = 40 cm

36 cm2

3 cm and 12 cm

2 (3 + 12) = 30 cm

4 cm and 9 cm

2 (4 + 9) = 26 cm

6 cm and 6 cm

P = 12, A = 7

P = 12, A = 6

36 cm

P = 12, A = 5

Area

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(b)

75 cm

P = 12, A = 8

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(a)

40 cm

36 cm

2 (6 + 6) = 24 cm

Page 44 Resource sheet – Dunking 1.

Page 37 Assessment 1 – What’s the size?

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(a) B (b) Cut out and overlay (c) 44 cm2 difference (d) B (e) Use a ruler (a) 5.5 cm2 (b) 5 cm2 (c) C (d) 11.5 cm (e) 12 cm (f ) D (a) 4.5 cm2 (b) 4 cm2 (c) E (d) 8.5 cm (e) 8 cm (e) E Teacher check

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Page 45 Resource sheet – More dunking 1. 2.

The amount of water in each glass is exactly the same at each level. This amount of water is the volume of each of the balls. The volume of both balls is the same; the level the water rose is the same.

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Page 46 Assessment 1 – Volume and capacity 1. 2. 3.

92 cm3 1.625 L (or 1625 mL) 385 mL

5. 6.

The amount of water is exactly the same at each level. None has been added or taken away. The volume of water between the original and new levels is the same as the mass of the plasticine shape.

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The perimeters are all the same, but the areas are different. Answers may vary.

Metal 1

Metal 2

volume of metal

127 cm3

450 cm3

volume of water displaced

127 mL

450 mL

new level in container

627 mL

950 mL

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Answers

Sub-strand: Using units of measurement

UUM – 5 Page 53 Resource sheet – The 24-hour clock 1.

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Page 54 Resource sheet – Clock face timetables 1.

(a) 9 patients (b) 15 minutes (c) 1 (d) 5 minutes (a) 7 (b) 1 hour (c) 4:30 pm (d) 6 am Teacher check

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(a) 0600 (b) 1640 (c) 1200 (d) 2400 (e) 1253 (f ) 0315 (g) 10:45 pm (h) 10:07 am (i) 4:55 am (j) 6:36 pm (k) 12:05 am (l) 11:59 pm (a) 3 hours and 20 minutes (b) 40 minutes (c) 12 hours (d) 6 hours and 25 minutes (e) 30 minutes (f ) 11 hours and 40 minutes

1. 2. 3.

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21 minutes 1600 (4 pm) 1 hour and 10 minutes. He will arrive at 1925 (7:25 pm). 1810 (6:10 pm). If he catches the later bus at 1910, he will arrive at 2010 (8:10 pm).

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Page 56 Assessment 1 – Cra-zee TV guide 1. 2. 3. 4. 5. 6. 7.

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Page 55 Resource sheet – Timetable problems

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8 news shows 4 hours 45 minutes Cra-zee 1 (3 hours 15 minutes Monster movie) Four shows are repeats. Total is 5 hours (or 300 minutes). Football at 2:30 pm on Cra-zee 2. It runs for 2 hours and 20 minutes. Snooze news; Police car fashions or Silly soap show. The earliest is Cook a breakfast at 6:00 am on Cra-zee 3. The latest is Late night cooking at 10:50 pm on Cra-zee 2. It will be 1 hour and 55 minutes before she can watch the Late news at 8:45 pm on Cra-zee 1.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Shape—Shape – 1

Construct simple prisms and pyramids (ACMMG140)

RELATED TERMS

TEACHER INFORMATION

Net

What this means

• A flat two-dimensional pattern that can be folded to make a model of a three-dimensional object.

• The making of models can assist students’ understanding of the properties of objects.

Prism

• Students need to de-construct prisms and pyramids as well as construct them.

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Pyramid

• Students need experience making nets for themselves using equipment such as Geoshapes® and Polydrons™. This would come before any use of templates.

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• A three-dimensional object with parallel and congruent end faces, with the other faces rectangles. The shape of the pair of congruent end faces names the prism; e.g. rectangular prism, triangular prism (below) etc.

• The use of a variety of materials is important.

• Ready-made templates for nets of three-dimensional objects usually only show one way to produce that shape. Particular nets are often chosen because they use the least amount of card or because they can fit onto a photocopier. However, there are a variety of ways of producing nets for any one shape, and students need to be exposed to as many of them as possible. For example, the ‘standard’ net for a triangular prism (seen in the left column) is shown below left, along with a variation that students should be aware of through exploration with materials.

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• A three-dimensional object with a polygonal base and the other faces triangles with a common vertex called the apex. It is named after its base; e.g. triangular pyramid, square-based pyramid (below) etc.

• It is useful for students to produce their own nets and fold and tape them together to make their shapes. • Students should become familiar with a variety of nets for prisms and pyramids.

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• Students explore the different ways to make nets for the same shape. • There is more than one net for each geometric solid.

• Students look at nets with different designs on the faces and visualise what will be on each face when the solid is constructed. • National tests usually have a question that requires students to be able to associate three-dimensional objects with their nets. They also have questions where students visualise solids and their faces from different orientations.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Shape—Shape – 1

Construct simple prisms and pyramids (ACMMG140)

TEACHER INFORMATION (CONTINUED) Student vocabulary

• Students confused about the difference between a pyramid and a prism.

net cube prism

• Students limited to being able to find only one or two different nets for any shape.

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triangular prism

rectangular prism pentagonal prism hexagonal prism octagonal prism

• Students being given a net for a geometric solid and having difficulty visualising what the shape will look like when completed. • Students unable to visualise and draw models from different orientations. • Students having difficulty drawing three-dimensional objects on isometric dot paper.

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What to look for

pyramid

square-based pyramid tetrahedron

pentagonal pyramid hexagonal pyramid octagonal pyramid

vertices (one vertex)

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edges

o c . che e r o t r s super Proficiency strand(s): Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Shape—Shape – 1

HANDS-ON ACTIVITIES • Students identify and describe the differences between prisms and pyramids. • Students classify prisms according to the face at each end; and pyramids according to their base. • It is worth pointing out to students that most diagrams and texts show the ends of prisms as regular polygons or rectangles. However, if the faces are congruent and irregular, and connected by rectangles, the shape is still a prism. In the diagram (right), we would classify the object as an octagonal prism, even though it does not look like the usual octagonal prism we see, which has a regular octagon at each end.

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• Students use open construction materials such as straws, toothpicks or skewers, and blu-tac or plasticine to make various prisms and pyramids. There are open construction sets are available commercially.

• Students use Polydrons™ or Geoshapes® to construct prisms and pyramids. The advantage of using this manipulative is that students can undo some of the joins and reconnect them in different ways to investigate the various nets that can be made for the shapes. • Students draw their prisms and pyramids from different orientations.

Model

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• Students use isometric dot paper as well as 1 cm2 grid paper as another form of recording the models they make. Students may need to be shown the best way to use isometric dot paper; for example, the paper should be used in landscape rather than portrait orientation.

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• Students investigate the number of different nets that can be made for a cube, drawing them onto 1 cm2 grid paper. There are 11 different possibilities. To assist in this process, students could use Geoshapes® or Polydrons™ and join six squares together, then ‘unjoin’ some of the edges and rejoin them in different ways. Once students have found all 11 nets, they could cut them out, fold them and check that each of them does indeed make a cube.

Note: There are other ways to join six squares together, but no others will fold to make a cube. One example is shown below.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Shape—Shape – 1

HANDS-ON ACTIVITIES (CONTINUED) • An extension of the investigation on the previous page is for students to use some (or all) of the nets made; and try to work out where the numbers 1–6 would go in order to turn them into dice. Students need to know that the sum of the numbers on opposite faces of a dice is always 7. They can then check if their predictions on where the numbers sit on their cubes actually work by cutting them up and folding them into cubes.

r o e t s Bo r e p ok u S Pentagonal prism

Pentagonal pyramid

V+F=E+2 10 + 7 = 15 + 2

V+F=E+2 6 + 6 = 10 + 2

LINKS TO OTHER CURRICULUM AREAS

Information and Communication Technology

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• Students investigate the number of edges, faces and vertices on a range of different prisms and pyramids. The teacher can ‘lead’ the students to Euler’s law: that for all polyhedra, the number of vertices plus the number of faces is equal to the number of edges plus two (V + F = E + 2).

• Printable nets for different solids can be found at <http://www.senteacher.org/wk/3dshape.php> Note that only one version of a net is shown for each solid, so students would still need to see other variations.

• An animated display of nets and their solids can be found at <http://www.learner.org/interactives/geometry/3d_ prisms.html> Again, there is only one version of each net shown. • A similar website to the one above, but for pyramids, can be found at <http://www.learner.org/interactives/ geometry/3d_pyramids.html> • The illuminations website has an interactive website where different solids can be selected and rotated, and their nets displayed. It still has the same problem as the sites above. It can be found at <http://illuminations.nctm.org/ ActivityDetail.aspx?ID=70>

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• Another website for looking at nets of solids can be found at <http://www.kidzone.ws/math/geometry/nets/index. htm> • A short quiz where students match a net with the solid can be found at <http://www.sadlier-oxford.com/math/ enrichment/gr4/EN0411b/EN0411b.htm>

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History and Geography

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• Discuss the historical significance of pyramids in different cultures around the world. Students could research their use, and the range of building materials and possible techniques used in their construction. They could also investigate the ways they were decorated inside.

The Arts

• Students make various prisms and pyramids using light cardboard, and decorate them in different ways.

Science • Students investigate crystals. • A website that offers information on crystals can be found at <http://chemistry.about.com/cs/sciencefairideas/a/ aa072903a.htm> • Another website from the same source, which explains how to make rock candy using crystals, can be found at <http:// chemistry.about.com/od/foodcookingchemistry/a/rockcandy.htm>

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Shape—Shape – 1

RESOURCE SHEET

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Net for a cube

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This is only one of many nets that can be made for a cube. How many others can you find?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Construct simple prisms and pyramids

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Sub-strand: Shape—Shape – 1

RESOURCE SHEET

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Net for a rectangular prism

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CONTENT DESCRIPTION: Construct simple prisms and pyramids

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This is only one of many nets that can be made for a rectangular prism. How many others can you find?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Shape—Shape – 1

RESOURCE SHEET

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Net for a square-based pyramid

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This is only one of many nets that can be made for a square-based pyramid. How many others can you find?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Construct simple prisms and pyramids

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Sub-strand: Shape—Shape – 1

RESOURCE SHEET

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Net for a tetrahedron

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CONTENT DESCRIPTION: Construct simple prisms and pyramids

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This is one of two nets that can be made for a tetrahedron. Can you find the other one?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Shape—Shape – 1

RESOURCE SHEET

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Teac he r

Net for a triangular prism

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This is only one of many nets that can be made for a triangular prism. How many others can you find?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Construct simple prisms and pyramids

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Sub-strand: Shape—Shape – 1

RESOURCE SHEET

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Net for a pentagonal prism

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CONTENT DESCRIPTION: Construct simple prisms and pyramids

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This is only one of many nets that can be made for a pentagonal prism. How many others can you find?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Shape—Shape – 1

RESOURCE SHEET

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Teac he r

Net for a pentagonal pyramid

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This is only one of many nets that can be made for a pentagonal pyramid. How many others can you find?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Construct simple prisms and pyramids

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Sub-strand: Shape—Shape – 1

RESOURCE SHEET

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Net for a hexagonal pyramid

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CONTENT DESCRIPTION: Construct simple prisms and pyramids

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This is only one of many nets that can be made for a hexagonal pyramid. How many others can you find?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Shape—Shape – 1

RESOURCE SHEET

r o e t s Bo r e p ok u S

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Teac he r

Net for a hexagonal prism

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This is only one of many nets that can be made for a hexagonal prism. How many others can you find?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Construct simple prisms and pyramids

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Sub-strand: Shape—Shape – 1

RESOURCE SHEET Dicey dice

1. Study the net of each dice and answer the question. If you are not sure of the answer, cut it out and construct the cube. (a) What is opposite the ♥ on this dice?

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♣

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(b) What symbol is on the bottom of this dice?

♦ ✢

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CONTENT DESCRIPTION: Construct simple prisms and pyramids

What face is opposite the ➢?

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(d) This net is folded to make the cube below. What shapes would be on the top and bottom of the cube?

(e) Draw a different net for a dice. Construct the cube to check that the net is correct. Draw the net again and add a design to each face. Challenge a partner to work out the position of each face on the constructed dice. Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Assessment 1

Sub-strand: Shape—Shape – 1

NAME:

DATE: Prisms and pyramids

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Draw three different prisms.

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Write everything you know about prisms.

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Draw three different pyramids.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Construct simple prisms and pyramids

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Write everything you know about pyramids.

Assessment 2

Sub-strand: Shape—Shape – 1

NAME:

DATE: Prism and pyramid nets

•

Match the nets in the left-hand column to the prisms and pyramids in the middle column. Write the matching letter in the box.

•

In the right-hand column, draw a different net for each 3-D shape in the middle column. Net 1 1.

Geometric solid

Net 2

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CONTENT DESCRIPTION: Construct simple prisms and pyramids

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G 8.

H Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Checklist

Sub-strand: Shape—Shape – 1

Constructs pyramids from nets

Constructs prisms from nets

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STUDENT NAME

Knows the features of prisms and pyramids

Construct simple prisms and pyramids (ACMMG140)

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Answers

Sub-strand: Shape—Shape – 1

Shape – 1 Page 75 Resource sheet – Dicey dice 1.

(b) ♥

(a) (c)

and

(d) (e) Teacher check

Page 76 Assessment 1 – Prisms and pyramids Teacher check Page 77 Assessment 2 – Prism and pyramid nets Net 1

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Geometric solid

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Teacher check

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Location and transformation— L&T – 1

Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies (ACMMG142)

RELATED TERMS

TEACHER INFORMATION

Transformations

What this means

• There are three main types of Euclidian transformations below. In these, the length, width, angle size and area do not change.

• The transformations of translations (slides), reflections (flips) and rotations (turns) describe movements of a shape or object.

Translation (slide)

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Reflection (flip)

• The mirror image of an object or shape, so that each point of the object or shape is the same distance from the mirror line (or plane of symmetry with a threedimensional object) as the same point on the image. The shape and size of the shape or object does not change.

• Students need to have experience transforming or moving real objects and shapes before doing so using digital technologies. • Rotations or turns can be of any size from less than 1° through to 360° (a full turn), where the object or shape ends back where it started.

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• The movement of an object or shape that changes position in a given direction. It remains oriented the same way. The shape and size of the shape or object does not change.

• A One-step move means that the object or shape only moves in one direction within the transformation. Combinations of moves (i.e. multistep moves) will also be investigated.

• Students need to be aware of the difference between reflectional (line) and rotational symmetry. A transparent mirror may be used when testing for reflectional symmetry, but is of no help when looking for rotational symmetry.

Teaching points

• When an object or shape undergoes a translation, reflection or rotation transformation, its size, shape and features do not change.

• Translation (slide) transformations can be the basis of tessellations, where the same shape is repeated without gaps or overlaps to create a pattern.

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• The process by which a shape or object changes position by rotating about a fixed point through a given angle. The shape and size of the shape or object does not change.

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• A clear plastic tool used with symmetry. It has the reflective quality of a mirror, but can also be seen through so that it reflects the front side of the shape onto the other side. (It is also known as a Mira or georeflector.)

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• A reflection (flip) transformation is performed around a ‘mirror’ line, which is generally drawn in for clarity.

Rotation (turn)

Transparent mirror

• A translation (slide) can be done in any direction, but without turning (rotating) the object or shape.

• A rotation (turn) transformation is performed about a point.

• When the object or shape has been transformed by a reflection, the mirror line indicates a line of symmetry, where any point on one side of the line is the same distance from the line as the equivalent point in the reflection.

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• The end result of a translation or reflection to a shape that has more than one line of symmetry can appear the same.

• The end result of a translation or reflection to a shape that has either one or no lines of symmetry will appear different.

• With two-dimensional shapes, folding and cutting are common ways to determine reflectional symmetry about a line. • Transparent mirrors and normal mirrors are useful tools for identifying reflectional (line) symmetry. They cannot be used for rotational symmetry. • Students may need to be shown the correct way to handle a transparent mirror. Directions for this can be found on page 86.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Location and transformation— L&T – 1

Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies (ACMMG142)

TEACHER INFORMATION (CONTINUED)

RELATED TERMS

Teaching points (continued)

Clockwise

• National tests usually include questions about translations. Students need many experiences cutting, folding and turning shapes to be able to visualise the results of these actions.

• A turn in the direction that the hands on a clock move.

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• There are an infinite number of degrees of turn that can be produced.

Anticlockwise

• A turn of 360° takes the shape or object back to where it started.

• A turn that is in the opposite direction to the way that hands on a clock move.

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• In most cases, a turn of 90° clockwise leaves an object in a very different position to a turn of 90° in an anticlockwise direction; in fact a half-turn (180°) different.

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Note: In Australia, we use the term ‘anticlockwise’ rather than the American equivalent ‘counterclockwise’.

• When a shape or object is rotated 180° (a half turn) it will be upside down.

• If a shape with at least two lines of symmetry is turned exactly 180°, it produces the same effect as when it is reflected (flipped). If the shape or object does not have two or more lines of symmetry, it will be upside down after a flip of 180°.

Tessellations

• Repeated patterns of shapes which completely cover a surface without gaps or overlaps.

Student vocabulary translate reflect rotate quarter-turn half-turn clockwise anticlockwise upside down line of reflection mirror line

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• Students who alter the shape or size of the object or shape as part of their transformation. • Students who move their shape or object beyond the transformation to be performed; e.g. rotating a shape while sliding it.

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• Students who are unaware that their shape has changed in some way after a reflection, and need to reflect it back to check. • Students incorrectly holding and using a transparent mirror. • Students who leave gaps when tessellating multiples of the same shape, or who overlap their shapes. • Students who confuse half and quarter turns.

• Students who confuse the directions of clockwise and anticlockwise. • Students who do not recognise the pattern when a shape is turned 90° or 180° several times, or who are unable to continue the pattern by drawing the next one or two elements.

Proficiency strand(s): Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Location and transformation— L&T – 1

HANDS-ON ACTIVITIES • Using manipulatives such as pattern blocks, students choose one piece and trace around it on paper. They then translate (slide) the shape in any direction and draw around it again. Discuss what has happened to the shape. What has changed? What has stayed the same? How can we record what has happened on the paper? Use the shapes again, but this time reflect and then rotate the shape by 90°. Again discuss what has happened to the shape each time. What has changed? What has stayed the same? How can we record what has happened on the paper? • In pairs, using pattern blocks, one student creates a pattern or design with about 8–10 pieces. Their partner now has to create the same pattern, but reflected about a line. A drinking straw could be used to designate the mirror (symmetry) line. Discuss the fact that the pattern or design has now been reflected. What has changed? What has stayed the same? How can we record action on the paper? Now the student makes another pattern or design, and the partner has to rotate the design 90° around a point and translate the design, as a two-step move. Again, discuss what has happened to the design with these transformations.

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• Starting with a simple shape, students explore multi-step transformations. For example, what happens to a rectangle if you reflect it and then rotate it 45°?

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• Students repeat patterns by translating, reflecting or rotating a design onto paper ruled into squares or rectangles.

• Students play guessing games where they trace around a shape and move it to another position by using a translation, reflection or rotation, then trace around it several more times.

2 is a translation of 1. 3 is a reflection of 2. 4 is a rotation of 3.

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• Tessellations: students investigate tessellations and the types of shapes that will tessellate. Students will be using the translation transformation. • Students create designs for wrapping paper, tiles, wallpaper etc. that make use of a combination of transformations. • Present students with patterns in which combinations of transformations have been utilised. Students identify the particular transformations used and discuss features such as the degrees of turn in a rotation.

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• Students make a template of half and quarter shaped design, and then reflect the shape. Draw around the template again, and this time rotate the shape. Discuss the difference and why it happens.

• For the tessellating activity on page 91, students will need 8 copies of a particular shape. • Fold a piece of coloured paper in half, then in half again. Place the shape on top and trace around it. Cut it out. This gives 4 copies. Now make another 4. Students investigate which regular polygons tessellate.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Location and transformation— L&T – 1

LINKS TO OTHER CURRICULUM AREAS Information and Communication Technology • An animated display that explains translations, reflections and rotations can be found at <http://www.learnalberta.ca/ content/me5l/html/math5.html> • A website that has multi-step transformation questions can be found at <http://au.ixl.com/math/year-6/reflectionrotation-and-translation> • An interactive reflection activity can be found at <http://www.primaryresources.co.uk/online/reflection.swf> • There is an interesting tessellating program at <http://www.pbs.org/parents/education/math/games/first-secondgrade/tessellation/>

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• Another website for tessellations is <http://nlvm.usu.edu/en/nav/frames_asid_163_g_2_t_3.html?open=activities> • The Illuminations site for tessellating shapes can be found at <http://illuminations.nctm.org/ActivityDetail.aspx?ID=27> • A similar web page to the one above, again from Illuminations, can be found at <http://illuminations.nctm.org/ ActivityDetail.aspx?ID=35>

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• An interactive kaleidoscope site can be found at <http://www.zefrank.com/dtoy_vs_byokal/>

History and Geography

• Integrate with Studies of Society and Environment. Students explore patterns used in present day or earlier cultures. Particular attention could be paid to pottery, jewellery, mosaics and murals.

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The Arts

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• Students design sets of tiles that make use of one or more of the transformations of translation, reflection and/or rotation.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Location and transformation— L&T – 1

RESOURCE SHEET Marmaduke the Magnificent Design artist Marmaduke has created a series of patterned squares to be used to decorate his 3D installation. Follow the directions to complete a row of each square. The first two are done for you.

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Reflect

Translate

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Translate

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Rotate 60° clockwise

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies

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Rotate 90°

Sub-strand: Location and transformation— L&T – 1

RESOURCE SHEET Double transformations A cartoonist makes his initial sketches on grid paper, and then transforms them. 1. Describe how each shape has been transformed. There will be combinations of translations (slides), rotations (flips) and/or rotations (turns). The first one is done for you. (a) The shape has been translated 3 squares to the right then rotated 90° clockwise about the point ● .

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(c)

2. Follow the instructions to draw each shape in the correct new position.

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CONTENT DESCRIPTION: Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies

(b)

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(a)

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Rotate the shape 180° about the point ● and then translate it 3 squares to the left.

(b)

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Translate the shape up 3 and right 7, then rotate 60° clockwise about the point. ●

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Location and transformation— L&T – 1

RESOURCE SHEET Using a transparent mirror 1. Hold the mirror on your desk, as shown in the picture. There is a ‘right’ and ‘wrong’ side for looking through the mirror. The ‘right’ side has a slanted edge, which is placed face down on the paper. In this position, the mirror's slanted edge will be touching the desk, and you can move it to the line of symmetry to show a reflection.

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Draw the lines of symmetry. (a)

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3. Use the mirror to check for lines of symmetry in the shapes below. They may have one line of symmetry, more than one line of symmetry or they maybe asymmetrical because they have no lines of symmetry.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies

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2. Place the mirror on the picture of the second butterfly below. Try to move the page rather than the mirror; move it until one half of the figure reflects onto the other half. Draw along the ridge with your pencil to mark in the line of symmetry. The first butterfly is done for you.

Sub-strand: Location and transformation— L&T – 1

RESOURCE SHEET Transparent mirror magic

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2. Make a mirror image of the cat, using the transparent mirror.

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CONTENT DESCRIPTION: Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies

1. Place the mirror facing the dog. Using a sharp pencil, make another picture of the dog reflected into the space provided.

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3. Draw the reflection of the house below. Then make three houses in a row. Place the transparent mirror at the end of the row. How many houses can you see now?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Location and transformation— L&T – 1

RESOURCE SHEET Tessellations: quadrilaterals Do all quadrilaterals tessellate? 1. To find out, cut out different quadrilaterals from cardboard. Make 5 or 6 copies of each and see if they will tessellate.

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3. Use 1 cm isometric dot paper to tessellate these quadrilaterals:

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies

r o e t s Bo r e p ok u S

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2. Use 1 cm2 dot paper to tessellate each of the following:

Sub-strand: Location and transformation— L&T – 1

RESOURCE SHEET

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CONTENT DESCRIPTION: Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies

1 cm2 dot paper

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Location and transformation— L&T – 1

RESOURCE SHEET

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies

r o e t s Bo r e p ok u S

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Teac he r

Isometric dot paper

Sub-strand: Location and transformation— L&T – 1

RESOURCE SHEET Regular polygons and tessellations A tessellation is the pattern made with one polygon used many times. A regular polygon is one with all sides the same length and all angles congruent. Do all regular polygons tessellate? 1. To find out, make 8 copies of each regular polygon below. Try to fit the copies of each shape together with no gaps or overlaps. 2. Glue the sets of polygons onto another sheet of paper showing the tessellation where possible. Record:

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(a) the name of the regular polygon used

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CONTENT DESCRIPTION: Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies

(b) whether the shape will or will not tessellate.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Location and transformation— L&T – 1

RESOURCE SHEET Tessellating patterns Tessellating patterns can be made using one or more shapes that fit together without gaps or overlaps. Tessellating patterns can also be made using irregular shapes. Follow the procedure to make shapes that can tessellate.

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1. Take a small rectangle of cardboard (about 10 cm x 7 cm).

3. Translate (slide) the top part down until edge A is next to edge B. Do not reflect (flip) the shape. Compare the area of the new shape to the area of the rectangle.

B A

4. Tape edge A to edge B (no overlapping).

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6. Translate (slide) the left part over the right part until edge C is next to edge D. Remember, don’t reflect the shape. Tape together. What do you notice about the area of this shape?

D C

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7. Trace around your shape onto a sheet of paper. 8. Now fit the shape into (and next to) the shape you have drawn. Draw around it again. Repeat several times to make an interesting tessellating pattern.

9. Look at your pattern. Can you see an animal in it? Or a face? Draw extra lines to make it more like an animal etc. and repeat for each shape.

D C B A

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies

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2. Draw a wavy line across it. Cut along the line.

Sub-strand: Location and transformation— L&T – 1

RESOURCE SHEET Reflecting Ninja robot

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CONTENT DESCRIPTION: Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies

Place pattern blocks over the shapes below. Complete the pattern by reflecting the shapes around the mirror line. Draw around the reflected shapes to record the result.

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mirror line Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Location and transformation— L&T – 1

RESOURCE SHEET Rotating pinwheel Place pattern blocks over the shapes below. Rotate the shapes 90° clockwise to create the pattern for the next quarter.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies

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Draw around the shapes. Repeat for the remaining quarters.

Assessment 1

Sub-strand: Location and transformation— L&T – 1

NAME:

DATE: Transformation check

1. Describe the movements (translation, reflection, rotation, up, down) to explain the transformations to the new positions. (a)

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(c)

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CONTENT DESCRIPTION: Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies

(b)

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3. In the empty tank, draw the fish in its new position according to the transformation instructions.

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(a) Rotated 90°, then translated right

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Assessment 2

Sub-strand: Location and transformation— L&T – 1

NAME:

DATE: Pattern block symmetry

2. Use the same design below. Now make a rotation of the design through 180°. Draw in the resulting shape.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies

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1. Use pattern blocks to make the design below. Then make a reflection of the design. Draw in the resulting shape.

Checklist

Sub-strand: Location and transformation— L&T – 1

Understands tessellations

Recognises completed combined transformations

Combines transformations

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STUDENT NAME

Translates, reflects and rotates shapes

Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies (ACMMG142)

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Location and transformation— L&T – 2

Introduce the Cartesian coordinate system using all four quadrants (ACMMG143)

RELATED TERMS

TEACHER INFORMATION What this means

Coordinates (Cartesian coordinates

Quadrants

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• Sectors of a coordinate plane determined by the x-y axes. The four quadrants, by convention, are numbered in the order below. 1st quadrant

• The set of counting numbers, their opposites and zero; ie –2, –1, 0, 1, 2 …

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Origin

• When looking at Cartesian coordinates, we usually consider this to mean the identification of a point at an intersection of an imaginary line perpendicular to the x-axis and another line perpendicular to the y-axis at designated distances from the origin (where the two axes intersect). It is written as two numbers or letters or similar, separated by a comma and written in a bracket; e.g. (3, –2). If either of the points contains a 0, the point will fall on either the x-axis or y-axis, if the point is (0, 0), the point will be located at the origin. • When reading most maps (e.g. a city map or atlas), it is the gaps that are labelled in the grid, so finding a location such as H3 means that there is a cell that is bounded by lines, where the vertical reference is along as far as ‘H’ and the horizontal reference is up ‘3’ on the vertical axis.

3rd 4th quadrant quadrant

Integers

• Students being able to give and receive directions to determine location.

• The point where the x and y axes meet; (0, 0).

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• When using coordinates for grid references, name the horizontal axis (x-axis) first followed by the vertical axis (y-axis). • Students understand that there is are points on the x- and y-axis that have 0 as one of their two coordinates, which indicates that they actually sit on one of the axes (or their intersection, if the coordinate is [0, 0])

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2nd quadrant

• Students need to have a sound understanding of negative numbers.

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• A pair of numbers or symbols that represent a position on a grid. Understanding of this concept of naming coordinates is essential in later years when graphing functions in algebra and trigonometry.

• It is expected that students have already had experience using simple coordinates for determining the position of an item on a map, but only using one of the four quadrants. They will not have met a grid with negative numbers before.

• Before using maps and coordinates, students need to understand: – the need for a horizontal and vertical axis

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– that when reading points on a map, the horizontal axis is always read before the vertical axis – that any specific location on a map can be found using both the horizontal and vertical coordinates on the grid.

• The relationship between degrees and compass directions needs to be highlighted; e.g. from West to North is a turn of 90° clockwise. • Orienteering activities offer a useful practical context for using coordinates. • Note: In the Year 7 national tests, there is usually a question that involves coordinates. One of the possible answers is always the reverse order to the correct answer. So, for example, if the correct answer was (2, 4), one of the choices would appear as (4, 2).

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Location and transformation— L&T – 2

Introduce the Cartesian coordinate system using all four quadrants (ACMMG143)

TEACHER INFORMATION (CONTINUED) Student vocabulary What to look for

coordinate horizontal axis vertical axis

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one axis; two axes scale legend

degrees

• Students confused about the quadrant in which a particular coordinate would be located. • Students unsure of the use of negative numbers needed in the 2nd, 3rd and 4th quadrants. • Students unable to locate coordinates on either, or both, of the axes; e.g. (–3, 0), (0, 5), (0, 0).

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• Students who, when using coordinates, use the vertical axis (y-axis) first followed by the horizontal axis (x-axis), instead of the other way around.

compass north

south east

north-east

north-west south-east

south-west

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clockwise

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anti-clockwise left

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right

millimetres

centimetres metres kilometres

o c . che e r o t r s super Proficiency strand(s): Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Location and transformation— L&T – 2

HANDS-ON ACTIVITIES • Display chart Teacher demonstrates the two axes expanded to allow negative numbers as coordinates, giving the four quadrants. Discuss the order of naming the four quadrants, and what is different about this layout to the grids they have used in previous year levels. Students need to be reminded of the importance of having the x- and y-coordinates in the correct order when describing a location. Demonstrate the placement of coordinates into the four quadrants, and discuss the situation when a zero is involved. For example, the coordinate (–2, 0) lies is on the x-axis, the coordinate (0, –4) lies on the y-axis, and the coordinate (0, 0) lies at the origin (where the two axes intersect). On page 102 there is a chart that shows this information. It could be enlarged and laminated for classroom display. • Battleships

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This game is generally played in pairs, with each player having two grids; one for their ships’ deployment and to record any ‘hits’ on them, the other to record the shots they make onto their partner’s grid. (It may help to laminate copies for multiple uses.) The size and layout of the grid may be determined by the teacher, but is usually 10 x 10 squares, with the individual squares or cells identified by coordinates. Before play starts, each player arranges their ships onto their grid, without their partner seeing. There are five ships that are to be deployed: an aircraft carrier (5 points), a battleship (4 points), a submarine (3 points), a destroyer (also 3 points) and a patrol boat (2 points). The boats are placed over consecutive gridlines horizontally or vertically (not diagonally) and may not cross or overlap each other. Students take turns nominating a coordinate; for example (3, 4). The partner must stay whether this corresponds with part of one of their ships on their grid, saying ‘hit’ or ‘miss’. The boat is not considered ‘sunk’ until all the coordinates for it have been ‘hit’. Players take turns taking shots at each other’s grids. The winner is the first player to ‘sink’ all of the other player’s ships. See page 104 for Battleship grids. Note: there are a number of variations to this game, including the layout of the grid, the types of ships and the number of coordinates designated to each of them. 5 4 3 2

–5 –4 –3 –2 –1

1 –1

–2 –3 –4 –5

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• Have students formulate directions for others to follow—for example, from the classroom to the oval—and then draw diagrams showing the route they take. Encourage the use of directional language, particularly the use of degrees of turn and compass directions. • Introduce orienteering activities. Students use compass directions, find distances and face a bearing. The website for Orienteering Australia is <http://www.orienteering.asn.au/newcomers/>

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• Students find features on a map using positive and negative coordinates. Then students construct their own maps, using all four quadrants, and pose questions about some of the features to other students. The teacher may give the students grid paper already marked with the x- axis and y-axis labelled, or students may construct their own (see page 109).

Shipwreck Rocky inlet

6 5

Pretty beach

Reef

3 2

General store

1 –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1

Village

–1

–2

3 4 5

Scary cove 8 9 10 11 12

Harbour

–3 –4

Cliffs

–5 –6

• Play games and make codes on a grid that uses all four quadrants (see page 106).

100 Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Sub-strand: Location and transformation— L&T – 2

LINKS TO OTHER CURRICULUM AREAS Mathematics • Locate and represent integers on a number line. Introduce negative numbers – integers are the set of counting numbers, their opposites, and zero; i.e. –2, –1, 0, 1, 2, … This links to (Investigate everyday situations that use integers ). [ACMNA124]

Information and Communication Technology • A game where students have to park a car in a particular spot identified by a coordinate, and that uses all four quadrants, can be found at <http://www.sciencekids.co.nz/gamesactivities/math/grids.html> This site also helps students understand what happens to the coordinates as you head in different compass directions.

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• A game where students move Billy Bug to specific locations in order to find food can be found at <http://resources. oswego.org/games/BillyBug2/bug2.html> This game uses all four quadrants. • ‘Catch the fly’ is a game, similar to the one above, where students name a particular coordinate so that the frog can catch a fly. It can be found at <http://hotmath.com/hotmath_help/games/ctf/ctf_hotmath.swf>

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• A site where students translate shapes on a four-quadrant grid can be found at <http://au.ixl.com/math/year-6/ translations-graph-the-image> This links with (L&T – 1).

• A similar website to the one above, from the same group, looks at reflecting shapes on a four-quadrant grid. It can be found at <http://au.ixl.com/math/year-6/reflections-graph-the-image>

• A third website in the series, this time rotating the shapes, can be found at <http://au.ixl.com/math/year-6/rotationsgraph-the-image> • A website that gives a 5-step introduction to coordinates can be found at <http://www.bbc.co.uk/schools/ks3bitesize/ maths/algebra/coordinates/revise1.shtml> • There is a Battleships game, but no use of coordinates at <http://www.primarygames.com/puzzles/strategy/ battleship/>

• A similar Battleship game, again without coordinates, can be found at <http://www.hubworld.com/battleship/games/ board/battleship> It may be useful as the introduction to a lesson on coordinates.

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The Arts

• Students create coordinate pictures using, for example, 12 points, 18 points, 20 points etc. Students give the coordinates used.

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• A similar game to Battleships, but finding dinosaur bones at an archaeological site, can be found at <http://www.counton.org/games/virtualmathfest/ dinosaur.html> This game requires students to select coordinates, but not in the four quadrants.

F G H I J K L M N O P Q R S

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A B C D E F G H I J K L M N O P Q R S

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Giraffe Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

101

Sub-strand: Location and transformation— L&T – 2

RESOURCE SHEET Cartesian coordinates and the four quadrants Cartesian coordinates: A pair of numbers or symbols that represent a position on a grid. Quadrants: Sectors of a coordinate plane determined by the x-y axis. The four quadrants: By convention, are numbered in the order below.

r o t s 1steB r e quadrant oo p u k S 3rd quadrant

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2nd quadrant

4th quadrant

Naming the coordinates: Coordinates are named with the x-axis first, then the y-axis. Below are some examples:

4 (4, 3)

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(0, 2)

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3 2 1

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–2 –3

(–5, –4)

(2, –3)

–4 –5

102 Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Introduce the Cartesian coordinate system using all four quadrants

Sub-strand: Location and transformation— L&T – 2

RESOURCE SHEET Blank four quadrant grid 10 9 8

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–8

5 4 3 2 1

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–7

–6

–5

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–3

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CONTENT DESCRIPTION: Introduce the Cartesian coordinate system using all four quadrants

–2

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–8 –9

–10

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

103

Sub-strand: Location and transformation— L&T – 2

RESOURCE SHEET Battleship grids

5 Your ships

4 3

–3

–2

Battleship

4 long

–1

Submarine

3 long

Destroyer

3 long

Patrol boat

2 long

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–4

5 long

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–5

Aircraft carrier

–2 –3

4 5 long

Battleship

4 long

Submarine

3 long

Destroyer

3 long

Patrol boat

2 long

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Aircraft carrier

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–2 –3 –4 –5 104 Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Introduce the Cartesian coordinate system using all four quadrants

Enemy ships

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Sub-strand: Location and transformation— L&T– 2

RESOURCE SHEET Cover the square A game for 2–4 players. You will need:

A game board (below) on which the coordinates describe a square not a point. A different coloured pencil for each player 2 different coloured 6-sided dice 1 dice labelled +, +, +, -, -, -.

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Rules: Decide which of your two coloured dice is the vertical coordinate and which is the horizontal one. Players take it in turns to throw one coloured dice and the positive/ negative dice; then throw the other coloured dice and the positive/negative one. This gives a coordinate such as (2, –3).

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Shade the correct cell on the board in your colour.

If the cell is already taken, the player misses that turn. When all the cells in the shaded square are filled or when the time is up, the player with the most coloured cells in the shaded square is the winner.

–1

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CONTENT DESCRIPTION: Introduce the Cartesian coordinate system using all four quadrants

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–3 –4 –5 –5

–4

–3

–2

–1

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

105

Sub-strand: Location and transformation— L&T – 2

RESOURCE SHEET Secret agent code To decipher the two messages at the foot of the page, follow the coordinates in order to locate the letters on the grid. The x-axis coordinate is named first. Check: To find (–4, –2) run your finger along the x-axis of the code grid to –4, then straight down to –2 on the y-axis and you will find the letter O.

H r o e t s Bo r e okX Pp u S

C J

S2 1

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–2

M –3 –4

Y o

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Message 1: (–4, 0) (3, 5) (–1, –2) (0, 2) ❖ (–4, 0) (–1, –2) (0, –3) (2, –2) ❖ (0, –3) (4, –4) ❖ (–3, 5) (–2, –4) (–4, 0) ❖ (–2, –4) (–4, 0) (2, –2) ❖ (–4, 0) (3, 5) (2, –2) ❖ (3, 5) (–4, –2) (0, –3) (2, –2) (1, 1) (–4, –2) (0, –1) (2, 3), ❖ (4, 0) (–4, –2) (–4, 0) ❖ (0, –3) (4, –4) ❖ (–1, 0) (–4, –2) (2, 0). Message 2: (–1, –2) ❖ (–4, 0) (3, 5) (–1, –2) (4, 0) (2, 3) ❖ (1, 1) (2, –2) ❖ (3, 5) (–2, –4) (5, –1) (2, –2) ❖ (0, 2) (–4, –2) (–3, 2) (5, –1) (2, –2) (–1, 0) ❖ (–4, 0) (3, 5) (2, –2) ❖ (–3, 5) (–4, –2) (–1, 0) (2, –2). Write a short, coded message to a friend.

106 Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

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CONTENT DESCRIPTION: Introduce the Cartesian coordinate system using all four quadrants

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–5

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Sub-strand: Location and transformation— L&T – 2

RESOURCE SHEET The muddy swamp monster

r o e t s Bo r e p Monster Beach ok u S

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To play The muddy swamp monster, each player requires a different coloured token and the two dice on page 108 must be constructed. The dice with numerals indicates how many squares to move across. Positive numbers for moves to the right. Negatives numbers for moves to the left. The dice with words indicates vertical moves. Positive numbers for moves up. Negatives numbers for moves down. If a player lands in a ‘Muddy monster hole’ they return to ‘Start’. The winner is the first to reach one of the beaches.

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Smelly Beach

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Start

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Scary Beach

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CONTENT DESCRIPTION: Introduce the Cartesian coordinate system using all four quadrants

Swampy Beach Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

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Sub-strand: Location and transformation— L&T – 2

RESOURCE SHEET The muddy swamp monster dice

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r o e t s Bo r e p ok u 2 –3 –2 S 3

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o c . c e her two st r –two three –three o super –one

108 Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

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CONTENT DESCRIPTION: Introduce the Cartesian coordinate system using all four quadrants

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one

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•f orr evi ew pur posesonl y•

Assessment 1

Sub-strand: Location and transformation— L&T – 2

NAME:

DATE:

Knowing about coordinates 1. A rectangle has been constructed on the grid below. Three of the points are (–2, 4), (2, 4) and (2, –1). Mark in these points on the grid. . Mark the vertex on the grid. Join the

The coordinates for the 4th vertex is points to make the rectangle.

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–5

–4

–3

–2

–1

–2 –3

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2. On the grid above, draw a square in a different coloured pencil. One diagonal of the square is from the vertices (3, –3) to (–3, 3). Mark these two points on the grid. and

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The second diagonal of the square is between the vertices Mark these two points on the grid. Join the points to make the square.

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CONTENT DESCRIPTION: Introduce the Cartesian coordinate system using all four quadrants

3. In the map below, describe the features that can be found at (–8, 3), (–6, –3), (5, 3) and (4, –2).

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Shipwreck Island

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cky

Harbour

Village

Hills

–12–11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 Red River Dark Marsh

Cl iff s

ach

Blue River

2 1

1 –1

Big Mountain

–2 –3

Campsite

Little River

ng Lo

h ac

9 10 11 12

–4 –5

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109

Assessment 2

Sub-strand: Location and transformation— L&T – 2

NAME:

DATE: Spy secret message

To decipher the two messages at the foot of the page, follow the coordinates in order to locate the letters on the grid. The x-axis coordinate is named first. Check: To find (–4, –2) run your finger along the x-axis of the code grid to –4, then straight down to –2 on the y-axis and you will find the letter O.

C J

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S2 1

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W 1

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–2

M–3

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–4

Message 1: (–4, 0) (3, 5) (2, –2) ❖ (–2, 4) (–2, –4) (0, 2) (0, 2) (1, 1) (–4, –2) (0, –1) (–1, 0) ❖ (2, –4) (–4, –2) (0, –1) ❖ (0, –3) (4, –4) ❖ (2, –4) (–1, –2) (–3, 2) (2, –2) ❖ (–1, –2) (0, 2) ❖ (0, –3) (–2, –4) (–4, 0) (3, 5) (0, 2). Message 2: Put this message into code—I do not think I can eat this piece of paper.—and write your name in code at the end.

110 Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

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CONTENT DESCRIPTION: Introduce the Cartesian coordinate system using all four quadrants

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Checklist

Sub-strand: Location and transformation— L&T – 2

Can locate coordinates on a four-quadrant grid

Understands the use of a four-quadrant grid

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STUDENT NAME

Knows to use the x-axis before the y-axis when locating a point

Introduce the Cartesian coordinate system using all four quadrants (ACMMG143)

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Answers

Sub-strand: Location and transformation

Page 93 Resource sheet – Reflecting Ninja robot

L&T – 1 Page 84 Resource sheet – Marmaduke the Magnificent Rotate 90°

Reflect

Translate

Rotate 90°

Translate

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Page 94 Resource sheet – Rotating pinwheel

Rotate 60° clockwise

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Reflect

Page 85 Resource sheet – Double transformations 1.

(b)

Translated 5 squares right; then reflected.

(c)

Translated 6 squares right and 2 squares down, then reflected.

(a)

Page 95 Assessment 1 – Transformation check 1.

(a)

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(b)

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(b)

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Page 87 Resource sheet – Transparent mirror magic Teacher check Page 91 Resource sheet – Regular polygons and tessellations 2.

Translated 3 squares right, 1 square down, then rotated 180°; or translated 3 squares right, then reflected.

(c)

Translated 1 square right, 2 squares down, then reflected.

Page 86 Resource sheet – Using a transparent mirror Teacher check

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Translated 7 squares to the right and 2 squares down, then rotated 90° clockwise.

(a) square, regular pentagon, regular decagon, regular octagon, regular hexagon, triangle (b) The equilateral triangle, square and regular hexagon will tessellate. The regular pentagon, regular octagon and the regular decagon will not tessellate.

(a)

(b)

112 Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

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Answers

Sub-strand: Location and transformation

Page 96 Assessment 2 – Pattern block symmetry

Page 110 Assessment 2 – Spy secret message

Message 1: The password for my file is maths. Message 2: (–1, –2) ❖ (–1, 0) (–4, –2) ❖ (4, 0) (–4, –2) (–4, 0) ❖ (–4, 0) (3, 5) (–1, –2) (4, 0) (2, 3) ❖ (–1, –2) ❖ (–3, 5) (–2, –4) (4, 0) ❖ (2, –2) (–2, –4) (–4, 0) ❖ (–4, 0) (3, 5) (–1, –2) (0, 2) ❖ (–2, 4) (–1, –2) (2, –2) (–3, 5) (2, –2) ❖ (–4, –2) (2, –4) ❖ (–2, 4) (–2, –4) (–2, 4) (2, –2) (0, –1).

r o e t s Bo r e p ok u S

L&T – 2

Page 106 Resource sheet – Secret agent code Message 1: This time my cat ate the homework, not my dog. Message 2: I think we have solved the code. Message to a friend: Teacher check

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Page 109 Assessment 1 – Knowing about coordinates 5

(–2, 4)

(2, 4)

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–4

–3

–2

–1

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(–2, –1)

–1

–3

–5 5

(–2, 4)

(2, 4)

(–3, 3)

(2, –1)

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–2

–4

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(3, 3)

3 2 1

–5

–4

–3

–2

(–2, –1)

–1

(2, –1)

–2

(–3, 3)

–3

(3, –3)

–4 –5

Shipwreck Island, Dark Marsh, Cliffs, Little River

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113

Sub-strand: Geometric reasoning—GR – 1

Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles (ACMMG141)

RELATED TERMS

TEACHER INFORMATION What this means

Angle

• Two lines with a common end point called a vertex, or the extent of rotation about a point.

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Acute angle

• Students will have been introduced to the protractor in Year 5. Some students may still need to be shown how to use a protractor correctly. It is usually better to use a 180° protractor, as it is the most commonly used one. Students may be introduced to a 360° protractor (preferably with a moveable line) if desired. • Angles are generally named in one of three different ways. In the example below, we can describe the acute angle as ABC, CBA or B. A B

• Less than 90°.

Obtuse angle

• Greater than 90°, but less than 180°.

Teaching points

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Right angle 90°

• Students should have a sound understanding of the relative size of angles compared to the right angle.

• Angles are classified by their size in their relationship to the right angle (90°). (See definitions left.) • Show all angles in different orientations, especially right angles. This is to help avoid a common misconception that we can have right angles and ‘left angles’.

Straight angle

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Reflex angle

• Greater than 180°, but less than 360°.

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Perigon (one rotation)

• A full turn to end up at the start; 360°.

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• Exactly 180°.

• It is important to make angles with different arm lengths so that students realise that the length of the arms does not affect the size of an angle. • Students learn about different types of pairs of angles; e.g. vertically opposite angles, complementary angles and supplementary angles.

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• In order to be able to determine unknown angles, students may need to know about the relationships between some angles; e.g. what is meant by vertically opposite angles, supplementary angles and complementary angles. (See definitions on the next page.) It is also expected that students know that the three interior angles in any triangle add to 180°, and that the four interior angles in any quadrilateral add up to 360°.

114 Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

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Sub-strand: Geometric reasoning—GR – 1

Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles (ACMMG141)

RELATED TERMS (CONTINUED) Vertically opposite angles

• When two line segments intersect, the two sets of opposite angles formed are called vertically opposite angles. Opposite angles A and B below are congruent angles (they measure the same number of degrees), and angles C and D are congruent.

What to look for

• Two angles that together make 180°; e.g. two right angles are supplementary; angles ABC and CBD below are supplementary. C

• Students confused by the length of the arms of an angle, thinking that an angle with short arms is less than an angle of lesser degrees but with longer arms. Students with this misconception would judge that the first angle below is larger than the second, because of the length of the arms. In fact, the second angle is larger.

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• National tests at Year 7 level always include questions on angles. Some may ask students to identify types of angles, some ask students to decide on the size of an angle where a protractor is shown, and others ask students to identify an angle within a diagram where some of the angle measures are given.

• Students know the different types of angles and their properties.

Supplementary angles

Teaching points (continued)

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TEACHER INFORMATION (CONTINUED)

Complementary angles

G F

Student vocabulary

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right angle (90°)

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acute angle

• Students know how to read a protractor (e.g. using the inner or outer scale) and how to use the protractor to construct specific angles.

• Students compare their measure to whether the angle they are measuring is greater than or less than 90° (i.e. whether it is acute or obtuse), or whether it is greater than 180° (i.e. is a reflex angle).

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• Two angles that together make a right angle; e.g. angles EFG and GFH below are complementary.

• Students aware that an angle on a straight line is 180° (two right angles) and a full rotation or full turn is 360° (which is four right angles).

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obtuse angle reflex angle

straight angle (180°) full rotation (360°) degrees protractor base line centre mark inner scale outer scale

Proficiency strand(s):

intersecting

Understanding

vertically opposite

Fluency

complementary angles

Problem solving

supplementary angles

Reasoning

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Sub-strand: Geometric reasoning—GR – 1

HANDS-ON ACTIVITIES • Students practise measuring angles using a protractor (see pages 120 and 121), and then practise constructing angles (see page 122). Check that students are competent at measuring acute and right angles, move on to obtuse angles and then reflex angles. (See page 123 and practice sheet on page 124.) • Students use pairs of Geo-strips™ to construct angles. These can be made so that there is a scissor action at the join that allows the angles to be adjusted. Once students have made a number of different-sized angles with Geo-strips™ of different lengths (so that the arms of the angles are different sizes), students can estimate their size and place them in order from the one that has the smallest angle to the one that has the largest. Students trace around the angles that they make with the Geo-strips™, and then measure them using a protractor. From there, students can place the angles in order from smallest to largest. Discuss whether the order was what they expected, and why/why not.

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• Students explore the idea of vertically opposite angles using equipment such as Geo-strips™ or straws and drawing pins. Discuss what happens to the pairs of opposite angles when the intersecting strips or straws are moved. Students trace the angles produced, and superimpose one over the other to compare their sizes. Discuss whether the opposite angles are always congruent.

• Students could look at a simple version of proof of the fact that vertically opposite angles are always congruent, if they know about supplementary angles. In the diagram below, angles A and D are supplementary (i.e. the combined total of their angles is 180°). Also angles A and C are supplementary, as their combined total also equals 180°. As angle A is common to both pairs of supplementary angles, the other two angles (angles C and D) must be congruent.

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• Students investigate the exterior angles of polygons by extending the edges and using a protractor to measure. Discuss their findings. What can we say about each of the pairs of angles?

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• Students construct a variety of regular and irregular polygons and then investigate angles that are formed by diagonals. They could check for congruency either by using a protractor or by cutting the polygons and superimposing the angles. Start with rectangles, then regular pentagons, hexagons etc. and finally use irregular polygons. 60°

60°

30°

20°

120°

60° 60° 60°

• Students use their knowledge of the sum of angles in a triangle (which is 180°) to find a missing angle, if given the other two angle measures. They could also find the missing angle on any quadrilateral if given the other three, knowing that the sum of the angles on any quadrilateral is 360°. (See pages 125 and 126 for activities that lead students towards ‘discovering’ these relationships.) • Students use their knowledge of the complementary angles (two angles that total 90°), supplementary angles (two angles that total 180°) and vertically opposite angles to solve missing angle problems.

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Sub-strand: Geometric reasoning—GR – 1

LINKS TO OTHER CURRICULUM AREAS English • Read What’s your angle Pythagoras? by Julie Ellis. This book is a fictional look at how Pythagoras may have found out about right-angled triangles. Although it looks at Pythagoras’ theorem, which is beyond Year 6, it would provoke an interesting discussion on the usefulness of knowing about angles. Information and Communication Technology • One of the best websites for students to explore properties of angles, and a multitude of other geometric concepts, is GeoGebraPrim, which can be found at <http://www.mrlsmath.com/geogebra-prim-a-simpler-version-of-geogebra-foryounger-students/> This site explains some of the benefits of using this dynamic program, and you can download the free software directly from the site.

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• There is also a website called GeoGebra Help for Beginners at <http://www.freetech4teachers.com/2010/11/geogebratutorials-for-beginners-and.html#.UJTK1hwYeB4>

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• A website with a movable protractor in which students estimate then measure angles can be found at <http://www. iboard.co.uk/iwb/Angle-Inside-Shape-Estimate-and-Measure-533>

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• An Illuminations website that allows students to investigate angles in different polygons and then alter the proportions, and therefore the sizes of the angles in them, can be found at <http://illuminations.nctm.org/ActivityDetail.aspx?ID=9> • Complementary and supplementary angles can be explored at <http://au.ixl.com/math/year-6/complementary-andsupplementary-angles>

• Another website that looks at complementary angles can be found at <http://www.aaamath.com/geo-comp-ang. htm?> • A similar website to the one above, but for supplementary angles is <http://www.aaamath.com/geo-supp-ang.htm> • A nice, though slightly gimmicky way to remember the difference between complementary and supplementary angles can be found on the YouTube™ video <http://www.youtube.com/watch?v=GO20ZgUzlc0>

• There is an animated demonstration on how to draw a 50° angle using a 180° protractor, which students can then replicate using a virtual protractor. It can be found at <http://www.mathsisfun.com/geometry/protractor-using.html> • Another website that allows students to measure angles using a 180° protractor can be found at <http://www. mathplayground.com/measuringangles.html> It has a tolerance of only 1°. • Yet another website that gives a short tutorial and them some practice measuring angles, with a 180° protractor can be found at <http://lrrpublic.cli.det.nsw.edu.au/lrrSecure/Sites/LRRView/13231/applets/protractor.htm>

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• There is an interesting ‘Alien angles’ game at <http://www.mathplayground.com/alienangles.html>

• Mission 2110 Roboidz game: This game is somewhat corny, but uses angles including that the three angles of a triangle equal 180°. It can be found at <http://www.bbc.co.uk/bitesize/ks2/maths/shape_space/angles/play/>

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• Wally the penguin demonstrates turning 90°, 180°, 270° and 360° at <http://www.kerpoof.com/#/view?s=iai1000001>

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Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET Mix and match angles (1)

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Straight angle

90°

180°

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Right angle

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Acute angle

360°

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Full turn

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118 Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

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CONTENT DESCRIPTION: Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles

Cut out the 27 cards on this page and page 119. Mix them up and try to put them back in the correct groups.

Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET

135°

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Reflex angle

200°

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Obtuse angle

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opposite angles

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CONTENT DESCRIPTION: Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles

Mix and match angles (2)

. te Complementary o 15°, 75° c . angles c e her r o t s super Supplementary angles

120°, 60°

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Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET Using a 180° protractor A standard 180° protractor is used to measure angles between 0° and 180°.

• base line

• outer scale marked from left to right

• inner scale marked from right to left

• centre mark.

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inner scale

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outer scale

base line

centre mark © R. I . C .Publ i cat i ons To measure a particular angle: •f orr evi ew pur posesonl y• 1. Place the protractor with the base line along one arm of the angle to be measured and the centre mark on the vertex (the corner itself).

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3. Use the scale that will give the correct reading for the type of angle. 4. Read the number on the scale where the second arm of the angle lies.

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2. Decide whether the angle is acute (less than 90°) or obtuse (between 90° and 180°).

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This angle is an acute angle. It is 30°.

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CONTENT DESCRIPTION: Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles

Its features are the:

Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET Measuring acute and obtuse angles

Remember to work out beforehand whether you would expect each angle to be an acute angle, a right angle or an obtuse angle. Circle your answer. (b) acute angle

right angle

obtuse angle

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(a) acute angle

(d) acute angle

right angle © R . I . C . P u b l i c a t i o ns obtuse angle obtuse angle x x • •f orr evxi e w pur posesonl y = right angle

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(e) acute angle right angle obtuse angle

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CONTENT DESCRIPTION: Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles

1. Use your knowledge about measuring angles with a protractor to measure the following six angles.

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acute angle

right angle

obtuse angle

x x= x x=

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Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET Constructing angles

80 70 110

100

110 70

120 13 0

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150

160

100

110

120

0 12

0 14

160

150

30 170

4. Remove the protractor. Use a ruler to join the dot to the end of the line where the centre mark had been.

180

170

160

14 0

0 15

0 13

13 0

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110

180

170

160

180

14 0

0 15

0 13

3. Read around the scales of the protractor until you reach the size of angle you want to construct. Mark the paper with a dot at the point.

0 14

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2. Place the base line of the protractor on the line with the centre mark in the middle. Mark it with a dash.

70 100

110

100

110 120

0 12

50 40 14 0 150

160

170

180

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180

160

170

5. Use the protractor to check that the angle is correct.

6. In the space below, make these angles using the method above. Remember to look at whether the angle is acute (less than 90°) or obtuse (more than 90°). (a) 65°

(d) 12°

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(e) 165°

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CONTENT DESCRIPTION: Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles

1. Using a ruler and pencil, draw a straight line about 15 cm long.

Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET Measuring reflex angles (1)

This is a reflex angle (ABC). It is equivalent to 360° minus the acute angle.

This is a reflex angle (JKL). It is equivalent to 360° minus the obtuse angle.

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To measure a reflex angle:

D B

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1. Extend one arm of the angle beyond the vertex.

J D

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M © R. I . C.Publ i c at i ons 2. Use a protractor to measure the angle from the extended arm to the other arm of the angle. •f orr evi ew pur posesonl y• C

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CONTENT DESCRIPTION: Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles

Reflex angles are angles that are greater than 180° but less than a full turn (360°).

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M ABD = 135° JKM = 45° 3. Finally, add 180° to the measured angle. This is because we need to add the straight angle that forms part of the reflex angle. A

D B

K M

C 135° + 180° = 315°

45° + 180° = 225°

So angle ABC is 315° and angle JKL is 225°.

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Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET Measuring reflex angles (2)

Remember to work out beforehand exactly which angle is to be measured. (a)

(b)

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angle =

(c)

angle =

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(e)

angle =

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angle =

o c . che e r o t r s super (f) angle =

124 Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles

1. Use your knowledge about measuring angles using a protractor to measure the following six angles.

Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET What’s your angle? (1)

180°

Remember – a straight line is an angle of 180° 360°

1. Copy the triangle (right) onto coloured paper and cut them out. Label the angles A, B and C.

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2. Carefully tear off the three angles.

4. Draw a different type of triangle on coloured paper.

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Label the angles.

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3. Fit the angles together to make a straight line.

w ww

CONTENT DESCRIPTION: Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles

– the angle around a point is 360°.

angle A + angle B + angle C = 180°

o c . che e r o t r s super

Cut the triangles out, carefully tear off the angles and glue them in place in the box to show a straight line. L

angle L + angle M + angle N = N

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

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125

Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET What’s your angle? (2)

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2. Carefully tear off the four angles and place them together so you have one of each angle joining the next. They form a full turn angle, which is 360°.

3. Glue them in place in the box below.

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angle A + angle B + angle C + angle D = 360° 4. (a) Draw your own different quadrilateral (a shape with four straight sides). (b) Cut it out and label all the angles. Carefully tear them off.

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o c . che e r o t r s super

angle K + angle L + angle M + angle N =

126 Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles

1. Copy the square onto coloured paper. Label the angles.

Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET

A B

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Angle A

Angle C

Angle B

Angle D

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1. Angles A and B and angles C and D are called vertically opposite angles because their vertices are opposite each other. Vertically opposite angles are formed when a pair of lines intersect. Use a protractor to measure the angles.

2. (a) Use a protractor to measure the angles.

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Angle F

(b) What have you discovered about angles E and F and angles G and H?

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CONTENT DESCRIPTION: Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles

Vertically opposite angles

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3. (a) Draw a pair of intersecting lines and measure the angles.

(b) Do the size of the angles confirm what you have discovered? Yes

Angle Angle Angle Angle

4. Write what you have found out about vertically opposite angles on a seperate sheet of paper. Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

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127

Assessment 1

Sub-strand: Geometric reasoning—GR – 1

NAME:

DATE: Finding angles

You will need:

2 strips of cardboard (about 15 cm by 4 cm) a split pin

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1. Use the angle guide to find examples of the angles given below.

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• Use a sharp pencil to make a hole at one end of the 2 strips of card. Push the split pin through the holes to join them. Fix the back of the pin so that the 2 strips can be opened in and out, like a pair of scissors, to make different sized angles.

2. Trace the angles from the angle guide in the table. Measure the angle using a protractor. Description

Acute angle

Where you found it

Sketch of tracing

Degrees

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Obtuse angle

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Right angle

o c . che e r o t r s super

Straight angle

Reflex angle

128 Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles

When you start looking, you will see different angles all around you. Make an angle guide to find different types of angles in your room.

Assessment 2

Sub-strand: Geometric reasoning—GR – 1

NAME:

DATE: Knowing about angles: no protractors

1. Use your knowledge of angles to calculate the missing angles. Remember not to use a protractor. (b)

30°

20°

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(c)

(d)

135°

85°

25°

60°

(e)

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(g)

. te x

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w ww

CONTENT DESCRIPTION: Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles

(a)

(h)

80°

o c . che e r o t r s super 35°

145°

(i)

95°

(j)

70°

50° x

x= Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

x= R.I.C. Publications® www.ricpublications.com.au

129

Checklist

Sub-strand: Geometric reasoning—GR – 1

Calculates missing angles problems

Understands vertically opposite angles

Understands complementary and supplementary angles

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STUDENT NAME

Knows the sum of angles in a triangle and quadrilateral

Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles (ACMMG141)

w ww

. te

m . u

o c . che e r o t r s super

130 Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

R.I.C. Publications® www.ricpublications.com.au

Answers

Sub-strand: Geometric reasoning

GR – 1

Page 127 Resource sheet – Vertically opposite angles

Page 121 Resource sheet – Measuring acute and obtuse angles

(a) 45° (acute angle) (b) 75° (acute angle) (c) 15° (acute angle) (d) 135° (obtuse angle) (e) 90° (right angle) (f ) 160° (obtuse angle)

3. 4.

Page 128 Assessment 1 – Finding angles

Page 122 Resource sheet – Constructing angles 6.

(a) 65°

Angle A = 120° Angle C = 60° Angle B = 120° Angle D = 60° (a) Angle E = 150° Angle G = 30° Angle F = 150° Angle H = 30° (b) The pairs of vertically opposite angles are congruent (equal). Teacher check Teacher check Teacher check

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Page 129 Assessment 2 – Knowing about angles: no protractors

(b) 90°

(a) 90° (c) 80° (e) 130° (g) 115° (i) 70°

(b) 130° (d) 65° (f ) 40° (h) 60° (j) 130°

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(e) 165°

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Page 124 Resource sheet – Measuring reflex angles (2) 1.

(a) 350° (d) 310°

(b) 280° (e) 270°

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(d) 12°

o c . che e r o t r s super (c) 255° (f ) 195°

Page 125 Resource sheet – What’s your angle? (1) 1–3. Teacher check 4. angle L + angle M + angle N = 180°

Page 126 Resource sheet – What’s your angle? (2) 1–3. Teacher check 4. angle K + angle L + angle M + angle N = 360°

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

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