Striving to Improve: Mathematics - Angles, Shapes and Mensuration by Teacher Superstore

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Acknowledgements i. i-stock Photos. ii. Clip art images have been obtained from Microsoft Design Gallery Live and are used under the terms of the End User License Agreement for Microsoft Word 2000. Please refer to www.microsoft.com/permission.

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The Act allows a maximum of one chapter or 10% of the pages of this book, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that

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o c . che e r o t r s super Published by: Ready-Ed Publications PO Box 276 Greenwood WA 6024 www.readyed.net info@readyed.com.au

ISBN: 978 186 397 853 8 2

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Any copying of this book by an educational institution or its staff outside of this blackline master licence may fall within the educational statutory licence under the Act.

Reproduction and Communication by others

Contents Teachers’ Notes Curriculum Links

Angles

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Shapes And Mensuration

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Basic Measurement What Unit Is That? Length Conversions 1 Length Conversions 2 Units Of Capacity 1 Units Of Capacity 2 Units Of Mass 1 Units Of Mass 2 Converting Units Perimeter 1 Perimeter 2 Perimeter 3 Perimeter Of Polygons 1 Perimeter Of Polygons 2 Area 1 Area 2 Areas Of Rectangles 1 Areas Of Rectangles 2 Areas Of Triangles 1 Areas Of Triangles 2 Area And Cost Cubes And Volume Cubic Metres Volume Capacity

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

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7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

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Looking At Different Angles Naming Angles Measuring Angles 1 Which Angle Is Larger? Measuring Angles 2 Drawing Angles 1 Drawing Angles 2 Reflex Angles Intersecting Lines Parallel Lines Angles In A Triangle Scalene Triangles Isosceles Triangles Equilateral Triangles Snooker Angles Baseball Hits An Angle On Time

Answers

50-52

Teachers’ Notes This resource is focused on the Measurement and Geometry Strand of the Australian Mathematics Curriculum. It is intended for lower ability students and those who need further opportunity to consolidate these core areas in Mathematics. Each section provides students with the opportunity to consolidate written and mental methods of calculation, with an emphasis on process and understanding.

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The section entitled Angles enables students to review types of angles and naming angles. There is the opportunity to practise drawing angles and using angles within a context. Students then have the opportunity to investigate angles in a triangle and to also classify the different types of triangles. These activities are a useful way to scaffold a new unit of Mathematics and will help build confidence for lower ability students to attempt more challenging problems at their year level. The section entitled Shapes And Mensuration familiarises students with units of length, mass and capacity and provides activities to consolidate unit conversions using mental strategies. The activities then move on to exploring perimeter and area of rectangles and triangles and allow for a thorough consolidation of these foundational concepts. Students then engage with simple volume and capacity ideas.

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The activities can be used for individual students needing further consolidation in a mainstream classroom or as instructional worksheets for a whole class of lower ability students. The activities are tied to Curriculum Links in the Australian Curriculum ranging from grade levels of Year 4 through to Year 7 and are appropriate for students requiring extra support in Years 7, 8 and 9.

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It is hoped that Angles, Shapes And Mensuration will be used to help teachers provide appropriate resources and support to those students in greatest need. The book as a whole can be used as a programme of work for those students on a Modified Course or Independent Learning Programme. Activities are sufficiently guided so that students can work independently and at their own pace without constant supervision and guidance from the teacher.

Curriculum Links Calculate the perimeter and area of rectangles using familiar metric units (ACMMG109) Convert between common metric units of length, mass and capacity (ACMMG136)

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

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Calculate volumes of rectangular prisms (ACMMG160)

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Establish the formulas for areas of rectangles, triangles and parallelograms and use these in problem solving (ACMMG159)

Identify corresponding, alternate and co-interior angles when two straight lines are crossed by a transversal (ACMMG163) Investigate conditions for two lines to be parallel and solve simple numerical problems using reasoning (ACMMG164)

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• Classify triangles according to their side and angle properties and Demonstrate that the angle sum of a triangle is 180° and use this to find the angle sum of a quadrilateral (ACMMG166) describe quadrilaterals (ACMMG165)

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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)

Compare angles and classify them as equal to, greater than or less than a right angle (ACMMG089)

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Teachers’ Notes

Angles

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Types Of Angles

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Angle Properties

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Students explore the various types of angles and the conventions for naming and describing angles. Drawing angles, measuring angles and using a protractor is a central skill and allows for some hands-on work. These first activities allow for a thorough consolidation of angles before moving on to applying and analysing angles in a variety of polygons.

Students review some angle properties, including parallel lines, which serves as a foundation for future work in geometry. Students are encouraged to calculate unknown quantities mentally and without measuring the angles.

foundational work before moving onto other types of polygons.

Applying Angles

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A variety of different activities involving drawing and using angles is given as a fun means of working with angles. * With some angle measuring activities students may find it helpful to extend lines so that they can be matched with numbers shown on a protractor.

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* Please note that students may encounter slight variations with answers provided because of photocopying inconsistencies.

* Looking At Different Angles angle rays vertex

zz An acute angle is an angle less than 90°.

zz An angle is the amount of turn between two lines around a common point. The lines are known as rays and the point at which they meet is called a vertex.

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* Task a

zz An obtuse angle measures between 90° and 180°.

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zz A right angle is an angle that measures exactly 90°. They are often marked with a square at the angle.

Draw two examples of an acute angle and obtuse angle.

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Tick the angles below that are right angles. Draw a circle around the acute angles and put a cross inside the angles that are obtuse.

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* Task b

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8. 7.

* Naming Angles

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Example

Label the angles in each diagram below.

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* Task B: ANGLES in triangles

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* Task a

This angle has been labelled ÐQRS. The angle can also be known as angle R or ÐR.

Triangles have three angles. This triangle has been labelled DMNO. The angles can be known as ÐM, ÐN, ÐO. Note: The triangle is known as MNO rather than ONM, in keeping with alphabetical order.

Label the triangles below and then write the name of the largest angle underneath. C

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e.g. This is DCDE

. te o Task c: ANGLES in diagrams c . * che e r In the diagram all the points have been labelled. In o t r s s rangles are not referred to by u a diagram such as this, pe G F

Largest angle is ÐC

just one letter such as ÐJ. Instead this angle is known as ÐKJM. The letter at the point of intersection (vertex) goes in the middle. Write down the names of all the angles in this diagram.

____________

* Measuring Angles 1 zz Angles are measured in degrees. This is usually expressed with this symbol °. A protractor is used to measure angles. Using a protractor follow the example below and then complete the activities.

How To measure an angle.

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Measure the angles below and write down the type of angle for each one, e.g. acute, obtuse or right.

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1. Place the centre of the protractor on the corner or sharpest point (vertex) of the angle. 2. Turn the protractor so that the base line runs along one of the lines that forms the angle. 3. You can then read the size of the angle from the position of the second line. For example this angle is approximately _______° 4. Most protractors number the angles both clockwise and anti-clockwise. Make sure that you start at 0 and follow the correct set of numbers.

3. size:. . . . . . . . . . . . . . . . .

type: . . . . . . . . . . . . . . .

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1. size:. . . . . . . . . . . . . . . . .

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4. size:. . . . . . . . . . . . . . . . .

5. size:. . . . . . . . . . . . . . . . .

6. size:. . . . . . . . . . . . . . . . .

type: . . . . . . . . . . . . . . .

7. size:. . . . . . . . . . . . . . . . .

8. size:. . . . . . . . . . . . . . . . .

9. size:. . . . . . . . . . . . . . . . .

type: . . . . . . . . . . . . . . .

type: . . . . . . . . . . . . . . . 9

* Task a

* Which Angle Is Larger?

Look at the angles below. Without using a protractor, circle the letter of the largest angle.

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a protractor find the actual angle size for each of the angles above. * Task B Using . t C = .......................... 2. O = ........................ K = . ........................... 1. A = . ......................

o c . e 3. W = ...................... N =c .......................... 4. G = . ....................... S= .............................. her r o t s s r u e p 5. R = ........................ Q = . ........................ 6. Z = ......................... U = ............................

7. E = ........................ V = ..........................

8. N = . ....................... P = .............................

9. T = ........................ K =............................ What word do the letters of the largest angles in each box spell out? _______________ Write a definition for this word_ _____________________________________________ 10

* Measuring Angles 2

Task a Measure angle ÐABC using a protractor. ÐABC = ___________

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Task B Look at the angles in the diagrams below. Find the size of these angles.

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M H

ÐGHI = ....................... ÐLMN = ........................ ÐQRS = ......................... ÐQSR = ......................

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Task C Measure the angles listed under each triangle.

. t e o ÐFBJ = . ................................ ÐRCA = . .................................. ÐYXZ = . ................................... c . c e he r ÐFJB = .................................. ÐARC = .................................... o ÐYZX = . ................................... t r s super

ÐJFB = . ................................ ÐCAR = .................................... ÐXYZ = ....................................

* Task D W

Measure the angles in the diagrams below.

ÐTDP = ...................... ÐWDV = .................... ÐGEK = ...................... ÐQOG = . ................. 11

* Drawing Angles 1

Task a * To draw an angle we first draw ourselves a straight line, usually a horizontal line. We then place the zero of our protractor on either the left or right side of the line and count around the number of degrees we need. Draw each of the following angles in the spaces below. b. 25°

c. 100°

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d. 135°

e. 75°

f. 165°

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a. 50°

a. 13°

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d. 61°

c. 142°

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* Task B

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* Task a

* Drawing Angles 2 Draw the following reflex angles. Be sure to indicate the correct angle. b. 270°

d. 190°

c. 330°

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f. 305°

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a. 210°

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d. 201°

c. 267°

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b. 324°

a. 216°

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* Reflex Angles

Reflex angles zz180° angles

A reflex angle is an angle between 180° and 360°. The reflex angle on the right measures 320°.

* Task a

Without using a protractor find the size of the reflex angles below.

* Task b

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25°

Find the size of the angles below by looking at the size of the reflex angle.

180° angles

360° angles

zz This is the full way around the circle. In a circle all angles drawn will always add up to 360°. 180° degree angles are in fact straight lines. zz What does a 180° degree angle look like?

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* Task C

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50°

x = ......°

o c . chfind Using a protractor the size of the angles in the circle below. e r e o t r s super a = ____________ ° b = _ ___________° c = _____________° d = _ ___________° e = _ ___________°

80°

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Now check if all the angles add up to 360°.

110°

* Intersecting Lines * Task a

Measure the angles in the diagram below.

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ÐACD = ...........................................

ÐBCE = ...........................................

ÐDCE = ............................................

ÐACB = ...........................................

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Complete this sentence to make up a new rule.

* Task b

Use this rule to find the size of angle x below.

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When two straight lines intersect_ _______________________________________ .

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1....................... 2....................... 3....................... 4....................... 5....................... 6....................... Task C: challenge - Find the values of all the other angles in each diagram above and write them in. Remember the rule: Angles along a straight line add up to 180°. 15

* Parallel Lines * Task a

In the diagrams below measure the size of angles x and y. Are they congruent? x y

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Find the size of these angles by using all the rules that you know. Remember: Angles made by intersecting lines are congruent. You do not need a protractor for this activity. 110°

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* Task b

x t ©R adyEdPubl i ca i ons 2 e 105° •f orr evi ew pur posesonl y•

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w = ............° z = ............° x = ............° y = ............°

the angles below. * Task c Measure . t

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a = ............° b = ............° c = ............° d = ............° What did you find? . .................................................................................................................... 16

* Task a

* Angles In A Triangle You will need two blank pieces of paper.

C A B Right Angle Isosceles

Scalene Equilateral

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Step 1: On the first piece of blank paper, draw and cut out four triangles as shown above. Step 2: Label the angles of each triangle e.g. ABC. Step 3: Carefully tear the corners off each paper triangle, and arrange and paste them on another piece of paper, like the example right.

What did you find? .....................................................................................................................

* Task b

Circle all the triangles that match the given term.

Scalene

Isosceles

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Equilateral a b

Scalene Triangles * zz A scalene triangle is a triangle that has three sides of different lengths. It also has no equal angles.

Measure the angles in these three triangles. Write the angle sizes in the correct place in each triangle.

* Task b

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* Task a

Draw three scalene triangles using a ruler and then measure each of the angles using a protractor.

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* Task d

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o c . che e r o t r List some objects in your classroom or playground that include a scalene triangle. s s r u e p For example, the slide in the playground. G

Isosceles Triangles * zz An isosceles triangle has two equal sides.

* Task a

Measure the angles in each of the isosceles triangles below then write a rule about what you found.

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* Task b

Without using a protractor, find the missing angles in the these isosceles triangles.

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68°

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* Task c

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81°

75°

Calculate the missing angles in the symmetrical house below without using a protractor. Use the other rules you know of to work out the angles.

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o c . che e r o a = . t r s ............................. super s = . .............................

m = . .............................

r = . .............................

t = . .............................

i = . .............................

e = . .............................

Equilateral Triangles * zz Equilateral triangles have three sides of equal length and angles of the same size.

* Task a

Without measuring the equilaterals below, what size must each of the angles be? . Answer:_____

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* Task b

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Equilateral triangles are always the same shape, just different sizes. Measure the angles in the triangles below to see if they are equilaterals and tick the ones that are. You will have to be spot on with the protractor as some of them are pretty close!

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1. Draw a line in the lower part of the paper. 2. Using a compass place the point on the left end of the line and the pencil exactly on the right end of the line. 3. Carefully mark a small line by swinging the compass around. Make sure you do not change the width at which the compass is set. 4. Now repeat this action by placing the point of the compass on the other end of the line. 5. Join the ends of the line to the intersection of the two arcs (curves). 6. Measure the angle sizes using a protractor.

* Snooker Angles Pocket

* Task a

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Pocket

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zz Here is a diagram of a snooker table with only four pockets. The way in which it works is very simple. If you hit the ball it then bounces off the side cushion at a 90° angle until it finally lands in a pocket. In the example right, the track of the ball is mapped out for you.

Pocket

Using this method, work out which pocket each of the following balls will go into if they are hit in the direction of the arrow. Draw a circle around the pocket.

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* Baseball Hits Look at the baseball pitch below. The dots in the diamond indicate where the ball has landed. You may need to draw in some guide lines to help you measure.

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8°

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X batter

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* Task a

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Find the angle of the shot for each of the hits above. The first one has been done for you.

8° 1 = ................................ 2 = ................................ 3 = ................................ 4 = ........................... 5 = ................................ 6 = ................................ 7 = ................................ 8 = ........................... 22

* An Angle On Time

zz We see changing angles on a clock face every minute.

* Task a e.g.

What sized angle is shown for the times below? Hint: Use this clock face and pencil lines to help you.

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5.30 - 15° 11.00 ............ Noon ............ 3.00 ...............

* Task b

Draw the times in the clock faces below using your ruler. What angles are they showing?

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7.00 ............... 8.30 . ............. 9.00 ............... 3.45 ...............

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7.16 ................°

* Task c: clock quiz

. te are formed at exactly 6.00? _____ 2. How many degrees o c . 3. After 3.00, when isc the very next time that the hands forme a perfect right angle? h r _____ er o t s s r u e p 4. What is the reflex angle size when a clock’s hands say 4.30? ____ 1. How many degrees does an hour hand move between 11.00 and 12.00? _____

5. What kind of angle is formed by the clock hands at these times? Measure from the hour hand, e.g. 11.45 - reflex. a. 3.45 ..................................

b. 11.30 . ..............................

c. 9.33 ...................................

d. 7.45 . ................................

e. 12.45 ................................

f. 2.17 . ..................................

g. 1.55 ..................................

h. 5.58 ..................................

i. 12.37 ................................. 23

Teachers’ Notes

Shapes And Mensuration

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Units And Unit Conversion

Perimeter And Area

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Students explore the various types of “everyday” units involving length, mass and capacity. The emphasis is on developing mental strategies to fluidly move between different units and to understand the importance of uniform units when working with calculations.

An exploration of perimeter and area is important for a thorough understanding of these topics. Once grasped these activities focus on extending their understanding to work with rectangles and triangles. A calculator may be useful for some of these calculations and an emphasis on correct units is to be encouraged.

Volume And Capacity

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An introductory look at volume and capacity is provided here as an exploration of these concepts. These activities can serve as a foundation to further work on volume and 3D shapes.

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* Basic Measurement Exercises Try these exercises without any help. a) Name some units that we use to measure length._________________________

_ ________________________________________________________________

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b) Name some units that we use to measure mass.__________________________ _ ________________________________________________________________

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_ ________________________________________________________________

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c) Name some units that we use to measure volume.________________________

_ ________________________________________________________________

d) How many centimetres in 2.3 metres?_ _________________________________

e) Which is smaller, 2 litres or 200 millilitres? (circle correct answer) f)

g) Which is bigger, 45 millimetres or 45 centimetres? (circle correct answer)

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. te some or all of questions a) to c) correctly then o If you answered you c . know something about what type of units we can use to make various c e r measurements. h er o t s super If you answered some or all of questions d)

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h) How many milligrams in one gram?____________________________________

to h) correctly then you know something about the size of units.

* What Unit Is That?

zz For the remaining work on measurement, we will use the metric system. The basic unit for measuring length or distance is a metre.

* Task a

Link these measurements with their basic units by drawing a line.

length volume

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mass

gram

zz Sometimes the basic units are not the best to use because the numbers might be too large or too small.

For example, you wouldn’t measure the distance from Perth to Sydney in metres or the mass of an ant in kilograms! Instead, you’d use prefixes to the basic units.

What units would youy usetoE measure theu following objects? Choose from: Re ad dP bl i c at i on s * Task b © millimetres, millilitres, milligrams, centimetres, metres, •f o r r evi ew pur posesonl y• litres, grams, kilometres, kilolitres, kilograms

Amount of cordial in a glass

_______________________________________

Mass of a flea

_______________________________________

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Length of a pencil

. _______________________________________ te o c Length of your classroom _______________________________________ . che e r o t Amount of water in a pool __s _____________________________________ r s uper Distance from home to school

Mass of an exercise book

_______________________________________

Height of an ant

_______________________________________

Mass of a dog

_______________________________________

Amount of milk in a full carton

_______________________________________

* Length Conversions 1 * Task a

Converting millimetres to centimetres To convert mm to cm we divide by 10. Write this in symbols on the diagram right. To convert cm to mm we multiply by 10. Write this in symbols on the diagram right.

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Convert each of the following.

d) 20 cm = ______ mm

* Task b

b) 50 mm = _____ cm

c) 35 mm = _____cm

e) 7.2 cm = _______ mm

f ) 4 mm = _______cm

Converting centimetres to metres To convert cm to m we divide by 100. Write this in symbols on the diagram right. To convert m to cm we multiply by 100. Write this in symbols on the diagram right.

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a) 1 cm = ______mm

Convert each of the following.

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d) 351 m = _______cm

b) 200 cm = _______m

c) 42 m = ________cm

e) 527 cm = ________m

f ) 32 cm = _______m

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a) 5 m = ______cm

Converting metres to kilometres t * Task c. econvert m to km we divide by 1000. To

o c . Write this in symbols on the diagram right. c e hm wer r To convert km toe multiply by 1000. o t s r u pe Write this in symbols ons the diagram right. m

Convert each of the following. a) 3 km = _________m

b) 2000 m = _______km

c) 4.3 km = _______m

d) 355 m = _______km

e) 12 m = _______km

f ) 215 km = _______m 27

* Task a

* Length Conversions 2 Convert each of the following units, showing your working in the space provided.

a) 5 cm = ___________ mm

b) 40 mm = ___________ cm

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e) 4km = ___________ m

f) 8000m = ___________ km

g) 3.2m= ___________ cm

h) 420mm = ___________ cm

i) 7.2 km = ___________ m

j) 5cm = ___________ mm

m) 5m = ___________ mm

n) 4km = ___________ cm

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d) 7m = ___________ cm

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c) 120 cm = ___________ m

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o) 3.2 m = ___________ mm

p) 7.1 km = ___________ cm

s) 60000mm = ___________ m

t) 5000000cm=___________ km

u) 75000mm=___________ m

v) 750 000 cm = ___________ km

o c . q) 8km = ___________c mm r) 6.5 km =___________ mm e her r o st super

Units Of Capacity 1 * zz Here are some things that can measure 1 L (1 Litre).

Find some other things in your home that are 1 L. Draw them here and write what they are on the line.

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* Task a

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. tesome other things in your home that are less than 1 L. o Task b Find * c . che e r o t r s super

* Units Of Capacity 2

zz There are 1000 ml in 1 L. Here are some things that you can find around the home: 1 L: milk carton, soft drink bottle, dishwashing detergent. 500 ml: sauce bottle, shampoo. 375 ml: soft drink can. 250 ml: small juice bottle.

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Find some other things in your home that are 1 L. Draw them here and write what they are on the line.

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* Task a

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* Task c

ml =

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ml =

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Find some other things in your home that are less than 1 L and write the * Task b © Re a dunderneath. yEdPubl i cat i ons measurement for them •f orr evi ew pur posesonl y•

o c . che e r o t r s super

Circle your best estimate (guess) for each of these.

100 ml

375 ml

500ml

20 ml

20L

* Units Of Mass 1

zz Here are some things that can measure 1kg (1 kilogram).

There are 1000 g (grams) in 1kg.

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

* Task a

Circle whether you would measure these things in grams or kilograms.

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Teac he r

zz Here are some things that can measure less than 1kg.

grams

kilograms

grams

kilograms

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grams

kilograms

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. tesome things in the home (try the kitchen or laundry) that have o a label showing Task b Find c * . how much they weigh. Draw them here and circle whether they are in grams (g) or c e her r kilograms (kg). o t s super

g / kg =

g / kg

zz There are 1000 g in 1kg.

Find some things in the kitchen or laundry that have a label showing how much they weigh. Draw and label them below. Make sure you show if it is in grams (g) or kilograms (kg).

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

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Teac he r

* Task a

* Units Of Mass 2

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250 g

5 kg

20 kg

m . u

o c . che e r o t r s super 900 g

5 kg

35 kg

100 kg

500 g

1 kg

5 kg

40 kg

13 g

100 g

1 kg

6 kg

* Converting Units zz When converting between units, you move the decimal point the number and direction of the little ‘bumps’.

Example

How many km in 534.6 m?

r o e t s Bo r e p ok u S= .5 3 4. = .534 km. i.e. 534.6 m

 Start at the decimal point .  We are we going from ‘m’ to ‘km’.  Move the decimal three places to the left (we follow the ‘bumps’ from ‘m’ to ‘km’).

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* Task a Convert the following units. a) 235 m =_ __________________ km b) 0.087 g =_________________ mg © ReadyEdPubl i cat i ons c) 1.27 =__________________ m u d) 690s mL =_________________ •kmf o rr evi ew p r po es onl y• L e) 26 cm =____________________ mm f) 7568 mm =_______________ m

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g) 0.0037 kg =________________ mg

m . u

h) 2734 cm =________________ km

j) 35 km =_________________ m . te o kg c k) 250 mL =_________________ L l) 4 678 mg =__. ____________ che e r o t r s s r u e p m) 0.0016 km =______________ mm n) 1.26 L =_ _______________ mL i) 200 mg =_ _______________ g

o) 3.67 cm =________________ m

p) 0.127 g = _______________ mg

* Perimeter 1 zz Perimeter is the distance around an object. Look at the example below: 3 cm

Example

3 cm

To work out the perimeter, add up all the sides. 3 cm

3 + 3 + 3 + 3 = 12 cm

r o e t s Bo r e p ok u S 3 cm

Work out the perimeters of these shapes.

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Teac he r

* Task a

_____ + _____ + _____ + _____ = _____ cm

_____ + _____ + _____ + _____ = _____ cm © R e a d y E d P u b l i c a t i o n s 3 •2f orr evi ew pur posesonl y•

_____ + _____ + _____ + _____ = _____ cm

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* Task b

Measure all the sides and then work out the perimeters in cm.

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o c . = _____ cm + _____ che _____ + _____ + _____r e o r st super _____ + _____ + _____ + _____ = _____ cm

_____ + _____ + _____ + _____ = _____ cm 34

* Perimeter 2 zz Perimeter is the distance around an object. Look at the example below. 3 cm

Example

3 cm

To work out the perimeter, add up all the sides. 3 cm

3 + 3 + 3 + 3 = 12 cm

r o e t s Bo r e p ok u S 3 cm

Work out the perimeter of these shapes. These are in mm.

_______ + _______ + _______ + _______ = _______ mm

* Task b

Measure all the sides and then work out the perimeters in mm.

w ww

Shape 1

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Shape 1 Shape 2 Shape 3

* Task c

Shape 2

Shape 3

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* Task a

_______ + _______ + _______ + _______ = _______ mm

o c . che e r o t r s super

_______ + _______ + _______ + _______ = _______ mm _______ + _______ + _______ + _______ = _______ mm

Use cm to work out the perimeter of this page.

______ cm + ______ cm + ______ cm + ______ cm =

______ cm

Can you work it out in mm too? ______ mm + ______ mm + ______ mm + ______ mm =

______ mm 35

* Perimeter 3

zz The perimeter of an object is the length or distance around it. For most objects, you can just add up the lengths of the sides to find the perimeter.

* Task a

What is the perimeter of these shapes?

3 cm 1.5 cm

This means the sides are the same length.

r o e t s Bo r e p ok u S 1.5 cm

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3 cm

4 m

Answer:________________________

Answer:_ _______________________

8.5 cm

2 m

3 m

5 m

7 m

o c . che e r o t r s super 5.6 m

18 m

Answer:________________________

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0.8 m

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2 m

Answer:________________________

* Perimeter Of Polygons 1

Task a * Match the number of sides to the shape. Number of sides

Shape

triangle

decagon

.....................................................................

quadrilateral

pentagon

hexagon

nonagon

septagon

b) A regular octagon, side length 7 m.

.....................................................................

c) A quadrilateral of side lengths 4 cm,

5 cm, 6 cm, 8 cm.....................................

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Teac he r

a) A regular hexagon, side length 10 cm.

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

Task b * Find the perimeter of the following shapes.

d) A regular septagon of side length 8 km.

.....................................................................

e) A pentagon of side lengths 4 mm,

6 mm, 5 mm, 4 mm, 3 mm.................

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c d

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o c . che e r o t r f e s sup r

List the shapes in order of perimeter length. Start with the object with the shortest perimeter. __________________________________________________________________ __________________________________________________________________ 37

Perimeter Of Polygons 2 * Measure the perimeter of all the numbered polygons in the diagram below. Show answers to the nearest millimetre.

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

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

* Area 1 zz Here is one square cm. This means it measures 1 cm all around.

zz This grid shows 6 square cm across and 6 square cm down.

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

Another way to talk about this shape is to say that: The area is 36 cm².

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Teac he r

There are 36 squares in this shape.

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A: ______ cm²

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Re d yofEd i c at i onumber nsof squares outa the area the shapesP in u thisb gridl by counting the a Work * Task© for each shape. •f orr evi ew pur posesonl y•

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

B: ______ cm²

C: ______ cm²

D: ______ cm² 39

* Area 2 zz Here is one square cm. This means it measures 1 cm all around.

* Task a

Work out the area of the shapes in this grid by counting the number of squares for each shape. Some shapes have half squares. Add two half squares to make a whole one.

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Teac he r

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

A: ______ cm²

B: ______ cm²

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Which shape has the biggest area?

C: ______ cm² A

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A B C . tea rectangle that has an area of 12 cm². o On the grid, draw c . che e r o t r s su er p draw a triangle that has an area of about 9 cm². Task B Challenge: In the space below

Which shape has the smallest area?

* Areas Of Rectangles 1 * Task a

Calculate the area of the following rectangles using Area = Length x Width.

10 m

24 cm

.........................................

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

7 cm

.........................................

6 km .........................................

.........................................

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6 km

Subtraction

Method 1

Method 2

Addition 6m

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Look at the two methods below. Total area is calculated by working out the area of the rectangles that fit into the polygon.

Area of large rectangle

= l x w = 6 x 5 = 30 m2

= l x w = 6 x 3 = 18 m2

Area of small rectangle

=lxw

= l x w = 2 x 4 = 8 m2

w ww

m . u

Area of large rectangle

= 2 x 2 = 4 m2

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Total area = 30 - 4 = 26 m2

* Task b

15 m

Total area = 18 + 8 = 26 m2

o c . Use onec of the methods to find the area of these shapes.e her r o t s super 6 km

3m 5m

5 km 4 km

12 m

3 km

10 km

2 cm

a) ................................................. b).................................................. c).............................................. 41

* Areas Of Rectangles 2 * Task a

Find the area of the shaded regions:

a. 2 km 3 km

6 km

8 cm a. .............................

7 km

r o e t s Bo r e p ok u S b. ............................. c. .............................

2 cm

d. .............................

5 cm

4 cm 3 cm

3 cm

8 cm

2 cm

6 cm

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10 cm

2 cm

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An old tablecloth, measuring 0.9 m x 1.3 m, was used by Sarah to make a poncho style cape. She needed to cut a 20 cm x 20 cm hole for her head. How much material was left for the poncho?

................................................................................................................

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o c . c e her r Task c o * t s su r pofe Joe needs to cut four rectangles from one large sheet wood measuring 2400 mm x 1800 mm. ................................................................................................................

One rectangle is 140 mm x 900 mm. One is 800 mm x 180 mm, one is 540 mm x 300 mm and the other is 256 mm x 190 mm. How much wood was left after all this?

..................................................................................................................................................................... ..................................................................................................................................................................... ..................................................................................................................................................................... 42

Areas Of Triangles 1 * 4m Example 1 is a rectangle and we know

Example 1

that to find the area of a rectangle : Area = L x W =4mx3m

= 12 m2

Example 2

Teac he r

6 cm 12 cm

Area of

= ½ (L x W) = ½ (4 x 3) = ½ (12 m2) = 6 m2

Area of

= ½ (L x W) = ½ (6 x 12) = ½ (72 cm2) = 36 cm2

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

Example 3

The rectangle in example 2 has now been divided into two equal triangles, each one half of the rectangle. We use what we know about the area of rectangles to find the area of triangles. So the area of a right-angle triangle = ½ of the area of a rectangle.

Area of shape A =_ _______

w ww

7 mm

6 km

3 km

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9m Area of shape B =_ _______

3.4 m

o c . che e r o t r sF super Area of shape C =_ _______

6.4 mm

21.3 mm

m . u

4 mm

Area of shape D =________

5m 2m 7m

Area of shape E =_ _______

Area of shape F =_ _______ 3 cm

G 870 cm

H Area of shape G =_ _______

8 cm

4 cm Area of shape H =_ _______

* Areas Of Triangles 2

zz You know that a rectangle divided into two can make two right-angled triangles.

Do you remember? The area of a right-angled triangle is = ________________

Does this work for any triangle? Let’s find out.

Example

r e t s A r Bo Bo e p ok u S 3 cm

Area of

A = ½ (4 x 3) = ½ (12) =6

Area of

B = ½ (2 x 3) = ½ (6) =3

Total Area = 3 + 6 = 9 cm2

2 cm

Now work out the area of the whole triangle that includes triangle A and B. Area of whole

= ½ (l x h) = ½ (6 x 3) = ½ (18) = 9 cm2

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Teac he r

4 cm

w ww

3 mm

m . u

13 m . te o ............................................................................. ............................................................................. c . che e r o t r s sud) per c) 9.4 km 2 mm

2 km

4 km 5 km

............................................................................. .............................................................................

* Area And Cost

* Task A

It costs $36 per square metre to carpet a house. Find the cost of having carpet fitted to the following rooms.

3.8 m

2.4 m 1.6 m

2.7 m

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

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Teac he r

.................................................................................. .................................................................................. 5.9 m c) d) 2.3 m 3.2 m 5.6 m .................................................................................. ..................................................................................

* Task b

A farmer has a yield of 3.4 tonnes of wheat for every hectare. What is the yield in tonnes of the following fields? Round your answers to two decimal places.

384 m

560 m

296 mc © ReadyEdPub l i at i ons .................................................................................. .................................................................................. •f o r r e v i e w p u r p986 os esonl y• m 293 m 740 m

864 m

w ww

m . u

186 m

.................................................................................. ..................................................................................

It costs $2.26 per m to tile a floor. Find the cost of tiling the following areas. Task c. te *a) o b) 6.4 m 8m

c . che e r o t r s super

3.2 m

.................................................................................. .................................................................................. c) d) 4.2 m 2.1 m

4.7 m 3.6 m

.................................................................................. .................................................................................. 45

* Cubes And Volume

zz What is volume and surface area? I’m glad you asked. A simple explanation is that volume is the number of blocks used to make the shape, and the surface area is the number of sides that can actually be seen (including the bottom). For example, if you make this shape, it would have a volume of 3 cubes and a surface area of 14 squares.

Model A

* Task a

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

Model C

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Teac he r

Example

Model D

Use the shapes to complete the table. (Remember that the models are made with 1 cm cubes.)

w ww

Model

m . u

. te o c Task b How many different models can you make: . * c e herb. using 9 cubes? r a. using 5 cubes? _____________ _____________ o t s super Add the models made from 5 cubes to the table below.

Model

5 cubes

9 cubes

Volume

Surface Area

Which shapes have the greatest volume and least surface area? ________________________ 46

* Cubic Metres

1. A box has dimensions of 40 cm x 30 cm x 60 cm. a) What is the volume in cm3? b) Convert this to m3.

..........................................................................................

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

2. A shipping container is 17 m x 3.1 m x 3.5 m. Calculate its volume in m3. ....................................................................................................................................................................

Teac he r

....................................................................................................................................................................

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3. A refrigerator has dimensions of 160 cm x 45 cm x 35 cm. How many cubic metres . would it take up?

....................................................................................................................................................................

4. Fill in the missing numbers:

Width

Depth

41 cm

360 cm

240 cm

4.2 m

1.3 m

0.8 m

100 cm

60 cm

w ww

. te 75 cm

2.4 m

cm3

m . u

Length

0.24

o c 4.5 . che e r o 1.3 m r 84600 t s super 80 cm

40 cm

96000

3.2 m

5. An Olympic pool has a length of 50 m, a width of 10 m and a depth of 6 m. What is the volume in m3? 6. An ice box has dimensions of 0.9 m x 0.35 m x 0.4 m. What is its volume?

............................................................................................................................................................... 47

Volume * zz One litre of water weighs one kilogram. zz One millilitre of water weighs one gram. zz One cubic metre of water weighs one tonne.

* Task A

zz One cubic millimetre of water weighs one gram.

Give the weight of water in these containers. c.

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

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Teac he r

a. b.

a................................................ b................................................. c................................................... Give the weight of water in containers with these dimensions. 8 d. e. f. g. 8

Solve these problems:

w ww

* Task b

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d.................................... e..................................... f.................................. g...................................

1. An aquarium has dimensions of 1.2 m, 80 cm and 50 cm. What weight of water will it hold?.............................................................................................

. te

o c . ch e 3. 1 m of a liquid chemical costs $4.38. r e o t r How much would it cost to fill a s container xs 2.2 m x 1.4 m?................................ up3.4emr

2. The local dam measures 31 km x 2 km x 74 m. How many tonnes of water could the dam hold?................................................................ 3

4. One litre of ice cream costs $3.23. How much would it cost to buy 256 litres for an ice cream parlour? . .........................

5. Ten ice cream cones can be made from 1 litre of ice cream. Each ice cream costs $1.20. How many ice creams can I make from the 256 litres? How much money will I get from the sale of the ice creams?......................................... 48

...............................................................................................................................................................

* Capacity

Word Problems

1. What is the capacity of a suitcase that has dimensions of 80 cm x 50 cm x 15 cm?

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Teac he r

r o e t s Boof 0.8 m. What is its r 2. A packing box is in the shape of a cube with a side e p ok capacity? u S

3. Ms Thomas wanted to fill a garden bed with good soil. If the garden bed had the dimensions as shown below and soil cost $20/m³, how much would it cost her?

Be careful of these units!

50 cm

soil 8m

Ms Thomas then wanted to spread mulch on top of the soil to a depth of 10 cm. What volume of mulch would she need?

w ww

1.5

. te

m . u

10 cm

o c . che e r o 4. Which has the larger capacity: a box with s a height of 4 cm, length of 13 _ t r s r u e p cm and width of 8 cm, or a cube with sides of 7.5 cm?

* Answers

Angles

Looking At Different Angles Page 7 Task b: Right angles - 4, 6, 9; obtuse angles 3, 5, 7; acute angles - 1, 2, 8. Naming Angles Page 8 Task A & b: Answers will vary.

Intersecting Lines Page 15 Task a: ∠ACD = 30°, ∠BCE = 30°, ∠DCE = 150° , ∠ACB = 150°. The opposite angles are equal in size. Task b: 1. x=125°, 2. x=37°, 3. x=117°, 4. x=45°, 5. x=94°, 6. x=90°

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

Task c: ∠FHI, ∠FHG, ∠GHJ, ∠HGJ, ∠GJM, ∠MJK.

Which Angle Is Larger? Page 10

N.B. Please note angles may vary slightly due to the printing process. These answers should be used as a guide.

Parallel Lines Page 16 Task a: x and y are congruent. Task b: w = 105°, z = 105°, x = 110°, y = 110° Task c: a = 45°, b = 135°, c = 150°, y = 30° Each pair of angles within the parallel lines adds up to 180°.

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Measuring Angles 1 Page 9 The angle in the example is 25°. Task a: 1 - 63°, acute, 2 - 133°, obtuse, 3 - 26°, acute 4 - 90°, right, 5 - 60°, acute, 6 - 165°, obtuse 7 - 28°, acute, 8 - 90°, right 9 - 120°, obtuse.

Teac he r

Task b: a = 60°, b = 70°, c = 55° Task c: a = 65°, b = 63°, c = 110°, d = 70°, e = 52°.

Angles In A Triangle Page 17 Checks diagrams on page.

Measuring Angles 2 Page 11 Task a: ∠ABC = 42° Task b: ∠GHI = 135° ∠LMN = 65° ∠QRS = 62° ∠QSR = 64°. Task c: ∠JFB = 60° ∠FBJ = 60° ∠FJB = 60° ∠CAR = 105° ∠RCA = 45° ∠ARC = 30° ∠XYZ = 65° ∠YXZ = 43° ∠YZX = 72° Task d: ∠TDP = 80° ∠WDV = 180° ∠GEK = 20° ∠QOG = 45°.

. te

Task b: a. g - 81°, m - 18 b.p - 75°, q - 30° c.g - 68°, m - 44

Task c: s - 130°, m - 25°, a - 58°, r - 58°, t - 32°, i - 32°, e - 58°

o c . che e r o t r s super

Drawing Angles 1 Page 12 Check Diagrams Drawing Angles 2 Page 13 Check Diagrams Reflex Angles Page 14 Task a: a = 335°, b = 240°, c = 340° 50

Isosceles Triangles Page 19 Task a: An isosceles triangle always has two congruent angles at the base of the two congruent sides.

m . u

w ww

Task b: 1. A = 47°, C = 56°, 2. O = 109°, K = 94° 3. W = 16°, N = 25°, 4. G = 133°, S = 120° 5. R = 110°, Q = 70°, 6. Z = 60°, U = 139° 7. E = 80°, V = 60°, 8. N = 71°, P = 45° 9. T = 90°, K = 87°. The letters of the largest angle spell out Congruent.

Scalene Triangles Page 18 Check diagrams of triangles. Task c: Scalene - b, f, g.

Equilateral Triangles Page 20 Task a: 60°. Task b: Equilateral triangles - a, e. Snooker Angles Page 21 Task a: a) lower left-hand pocket, b) top right-hand pocket, c) top right-hand pocket, d) lower left-hand pocket, e) top left-hand pocket, f ) top left-hand pocket. Baseball Hits Page 22 N.B. - Answers are approximate 1) 8°, 2) 31°, 3) 16°, 4) 43°, 5) 61°, 6) 66°, 7) 97°, 8) 46°

N.B. For measurement activities slight variations may occur because of photocopying inconsistencies.

An Angle On Time Page 23 Answers may vary, check diagrams. Task c - clock quiz: 1) 30°, 2) 180°, 3) 3.33, 4) 315° 5) a) obtuse b) obtuse c) reflex d) acute e) reflex f ) acute g) reflex h) obtuse i) reflex.

Shapes And Mensuration

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What Unit Is That? Page 26 Task a: length-metre volume-litre mass-gram Task b: centimetres millilitres milligrams kilometres metres kilolitres grams millimetres kilograms litres

Units Of Capacity 1 Page 29 Task a: Examples of things that are 1 L are: milk, juice, soft drink, cordial, sauce, washing up liquid, etc. Task b: Examples of things that are less than 1 L are sauces, small drink bottles, shampoo, small milk cartons, etc.

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Basic Measurement Page 25 a) metres, kilometres, inches, miles, etc. b) grams, kilograms, pounds, ounces, etc. c) litres, millilitres, cups, pints, etc. d) 230 cm e) 200 millilitres f ) 0.534 kilolitres g) 45 cm h) 1000 mg

Teac he r

Length Conversions 2 Page 28 a) 50 mm, b) 4 cm, c) 1.2 m, d) 700 cm, e) 4000 m, f ) 8 km, g) 320 cm, h) 42 cm, i) 7200m, j) 50mm, k) 2.45 m, l) 32000m, m) 5000mm, n) 400000cm, o) 3200 mm, p) 710000 cm, q) 8000000mm, r) 6500000mm s) 60m, t) 50km, u) 75m, v) 7.5 km

Units Of Capacity 2 Page 29 Task a: Examples of things that are 1 L are: milk, juice, soft drink, cordial, sauce, washing up liquid, etc. Task b: Examples of things that are less than 1 L are sauces, small drink bottles, shampoo, small milk cartons, etc. Task c: Best estimate: soft drink = 375 ml; laundry bucket = 7 L

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Task a: Comb = g; computer = kg; CD = g; chair = kg; person = kg Task b: Answers will vary for the second part of this question. Units Of Mass 2 Page 32 Task a: Answers will vary for the first part of this question. Task b: Best estimate: cereal = 250 g; child = 35 kg; stereo = 5 kg; pencil = 13 g

o c . che e r o t r s super

Length Conversions 1 Page 27 Task a: a)10 mm, b) 5 cm, c) 3.5 cm, d) 200 mm e) 72 mm, f ) 0.4 cm Task b: a) 500 cm, b) 2 m, c) 4200cm, d) 35100 cm e) 5.27 m , f ) 0.32 m Task c: a) 3000m, b) 2 km, c) 4300 m, d) 0.355km e) 0.012 km, f ) 215 000m

Converting Units Page 33 Task a: a) 0.235 km b) c) 1270 m d) e) 260 mm f ) g) 3700 mg h) i) 0.200 g j) k) 0.250 L l) m) 1600 mm n) o) 0.0367 m p)

87 mg 0.690 L 7.568 m 0.02734 km 35 000 m 0.004678 mg 1 260 000 127 mg

Note: the printing process may distort shapes.

Perimeter 1 Page 34 Task a: 8 cm, 14 cm, 12 cm; Task b: 9 cm, 16 cm, 14 cm. Perimeter 2 Page 35 Task a: 80 mm, 115 mm; Task b: 160 mm, 90 mm, 60 mm. Task c: Page is approximately 102 cm or 1005 mm.

Areas Of Triangles 2 Page 44 Task a: a) 3 mm2, b) 39 m2, c) 9.4 km2, d) 10 km2.

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Perimeter Of Polygons 1 Task b: a) 60 cm, b) 56 m, c) 23 cm, d) 56 km, e) 22 mm. Task c: a) 105 mm, b) 40 mm, c) 95 mm, d) 120 mm, e) 180 mm, f ) 134 mm, g) 213 mm. Order: b, c, a, d, f, e, g.

Area And Cost Page 45 Task a: a)$138.24, b) $369.36, c) $679.68, d) $231.84 Task b: a) 140.90 tonnes, b) 38.65 tonnes c) 9.26 tonnes, d) 289.65 tonnes Task c: a) $81.36, b) $46.28, c) $9.97, d) $38.24.

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Perimeter 3 Page 36 1. 9 cm 2. 16 m 3. 23 cm 4. 16 m 5. 19.6 m 6. 32.9 m

Teac he r

Areas Of Triangles 1 Page 43 Task a: a)14 mm2, b) 121.5 m2, c) 9 km2, d) 10.2 m2, e) 68.16 mm2, f ) 24.5 m2, g) 3.915 m2, h)44 cm2.

Cubic Metres Page 47 1.72 000 cm3, 0.072 m3, 2. 184.45 m3, 3. 0.25 m3, 4a) 3542400 cm3, 3.54 m3, b) 4368000 cm3, 4.37 m3, c) 40 cm, 240000cm3, d) 30 cm, 0.096 m3, e)1.875 m, 4500000 cm3, f ) 0.027 m, 0.086 m3. 5. 3000m3 6 ..126m3

Area 1 Page 39 Task a: A = 6cm²; B = 14cm², C = 10cm², D = 8cm².

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Area 2 Page 40 Task b: A = 3cm², B = 4cm², C = 7.5cm². Biggest is C; Smallest is A.

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Area Of Rectangles 1 Page 41 Task a: a)20 m2, b) 30 m2, c) 168 cm2, d) 36 km2, Task b: a) 81 m2, b) 24 cm2, c) 128 km2.

Area Of Rectangles 2 Page 42 Task a: a)36 km2, b) 40 cm2, c) 52 cm2, d) 83mm2 Task b: 1.13 m2 Task c: 3839360 mm2.

Volume Page 48 Task a: a.500 g, b. 800 g, c. 300 g, d. 512 g, e. 5376 kg, 5.376 tonnes, f. 5400 g, g. 2.3 tonnes. Task b: 1.0.48 tonnes or 480 kg, 2. 4 588 000 tonnes, 3. $45.87, 4. $826.88, 5. 2560 ice creams, $3072.00.

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Perimeter Of Polygons 2 Page 38 a)345 mm, b) 62 mm, c) 125 mm, d) 123 mm, e) 80 mm, f ) 80 mm, g) 112 mm, h) 94 mm, i) 120 mm, j) 55 mm, k) 55 mm, l) 48 mm.

Capacity Page 49 1. 60 000 cm³ 2. 0.512 m³ 3. 120; 20m3 4. The cube (»422 compared with 416).