Measure Up by Teacher Superstore

Forr Ages 10+ or e t s B

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Measure Up

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Using measurement © Ready EdPubl i cat i ons •f o rr evi ew pur pos esonl y• concepts to solve open-ended tasks. o c . che e r o t r s super

Written by Donelle Francesconi. Illustrated by Terry Allen. © Ready-Ed Publications - 2002

Published by Ready-Ed Publications P.O. Box 276 Greenwood WA 6024 Email: info@readyed.com.au Website: www.readyed.com.au COPYRIGHT NOTICE Permission is granted for the purchaser to photocopy sufficient copies for noncommercial educational purposes. However this permission is not transferable and applies only to the purchasing individual or institution. ISBN 1 87526 461 X

Links to Outcome Statements

Strand: Measurement Page

Related Outcome

National

6-11 Identifies some of the commonly used units and basic units; calculates some simple conversions; and uses prefixes appropriately. 12 Uses a scale to determine distances and compares distances.

M 3.1

Meas 4.18

M 3.4b

Meas 3.19

M 2.1

Meas 4.18

M 3.4

Meas 4.22

The student:

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Investigates and appreciates systems of measurement other than the metric system.

14-18 Uses straightforward arithmetic to calculate perimeters.

M 3.4

Meas 4.22

Appreciates that many different shapes may have the same perimeters.

M 4.4

Meas 4.22

Uses a fixed scale to construct a perimeter; differentiates between two and threedimensional shapes.

M 3.4b

Meas 4.19

M 3.4

Num 4.14

M 4.4

Meas 4.22

M 4.2

Meas 4.22 Num 4.14

M 4.2

Meas 4.22 Num 4.14

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Sketches mathematical shapes from a description; uses straightforward arithmetic to calculate perimeters; and calculates and compares costs based on price per quantity.

22-25 Uses straightforward arithmetic to calculate areas of simple and complex shapes. 26

Uses a variety of methods to determine the area of a shape.

Calculates the area of a complex shape; performs multi-level calculations; compares two situations.

Calculates the area of a complex shape; performs multi-level calculations; sources and compares two situations.

Compares the area of several shapes; uses the concepts of area and distance to solve a problem.

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Meas 4.22

30-32 Uses straightforward arithmetic to calculate surface areas of rectangular prisms.

M 3.4

Num 4.14

33-34 Uses formula or alternative means to solve a problem.

WM 3.4 M 3.4

WM 3.3 Meas 4.22 Num 4.14 WM 3.3 Meas 4.22

35-37 Uses straightforward arithmetic to calculate the capacity of rectangular prisms and cylinders.

M 4.2 M 4.3

Uses formula or alternative means to solve a problem; demonstrates an understanding WM 3.4 of capacity as a three-dimensional measurement. M 4.3 Appreciates and justifies that different shaped objects can have the same capacity; M 3.4b determines dimensions of a shape given specific criteria. M 4.4

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Uses formula to solve a problem; demonstrates an understanding of capacity as a three-dimensional measurement.

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41-42 Distinguishes between different measurement concepts; namely for one, two and three dimensional shapes.

M 4.2

WM 3.3 Meas 4.22 Meas 4.22

Meas 4.22

The activities in this book refer to material from: Outcomes and Standards Framework: Mathematics (1998) ISBN 0 7309 8671 3

Mathematics - a curriculum profile for Australian Schools (1994) ISBN: 1 86366 213 8

This document is published by: Education Department of Western Australia, Royal Street, East Perth, WA, 6000 www.eddept.wa.edu.au/centoff/outcomes/

This document is published by: Curriculum Corporation, St Nicholas Place, 141 Rathdowne St, Carlton VIC, 3053 www.curriculum.edu.au/catalogue/

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Contents 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

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Teachers’ Notes Basic Measurement: Exercises What Unit Is That? More About Prefixes Converting Between Units: Using Conversion Charts Using the Conversion Method by Following the Little Bumps! Measurement: Word Problems Peter’s School Run: Measurement Mini Task How Did They Measure That? Measurement Open-ended Task Perimeter: Exercises Perimeter of Circles - The Circumference: Investigation Circumference: Exercises Perimeter: More Exercises Perimeter and Circumference: Word Problems The Farmer’s Fence: Perimeter Mini Task My Maze: Perimeter Open-ended Task Area: Exercises Area: More Exercises Tricky Areas: Exercises Area: Word Problems How Big Is That? Area Mini Task The Paint Problem: Area Mini Task The Dirt In Carpet Cleaners: Area Open-ended Task Care For a Cup of Tea? Area Open-ended Task Surface Area: Introduction Surface Area: Exercises Surface Area: Word Problems Sticky Problem: Surface Area Mini task 4 Boxes 4 You: Surface Area Open-ended Task Capacity (Volume): Exercises Capacity: Word Problems Capacity: Tricky Word Problems The Price of Pencils: Capacity Mini Task The Case of The Music Man: Capacity Open-ended Task Chocolate Anyone? Capacity Open-ended Task Miscellaneous Exercises Miscellaneous Word Problems Answers

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Teachers’ Notes This book is directed towards developing process skills using a sound content base and so is directly in tune with outcomes-based courses. The aim of this book is to provide teachers with a plan for presenting outcomes-based, open-ended tasks to mathematics students of Years 5 - 7. The conceptual outcome chosen for this book is Measurement. Each measurement concept (basic units, conversions, perimeter, area, surface area and capacity) is presented as a series of four types of questions:

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1. Knowledge and understanding of mathematical concepts can be achieved by rigourbased exercises.

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2. Adaptation of such concepts to more difficult situations, seemingly non-mathematical, can be learnt through tackling word problems.

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3. Mini tasks are long word problems that often require multiple steps. They usually have a definite answer though it may be achieved through a variety of methods.

4. The open-ended tasks in this book can be achieved on a variety of levels and cover a range of student outcomes. The final answer is generally not important. The purpose of such questions is to test not only mathematical skill, but also for students to achieve the outcomes related to problem solving, logic, lateral thinking, working in groups, creativity, testing options amongst others. More on open-ended tasks tasks:

This book is generic and so outcomes for specific curricula have not been specified. Teachers can attach their own outcomes to each open-ended task.

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recommended that initial tasks be non-assessed until students become more confident with them. If done as an assessment, an appropriate rubric should accompany the task sheet.

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Tasks are designed to be carried out in groups or individually. If the task is to be assessed as a group activity, it should be accompanied by a rubric that clearly states the role of each member of the group.

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The very nature of open-ended tasks implies they have no one correct answer. Some of the tasks presented may have a ‘best’ answer, but if students can give logical and valid details as to how they arrived at their solution, the aim has been achieved.

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The tasks have been chosen such that nearly all students should achieve, at some level. Teachers can expect to see a wide range of problem solving abilities revealed in their classroom. NB: The initial ‘measurement open-ended task’ is more traditional (a research assignment) although it is multi-levelled. As each measurement topic will be presented using the above progression, students will become familiar with the procedure. Thus, the teacher should be able to incorporate more student-directed lessons.

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Some of the specific outcomes that will be addressed in the book include: appreciating the role of mathematics in society; working mathematically using particular skills and processes; content-based outcomes (measurement). Some extra pointers: All answers involving the use of pi (π) were calculated using 3.14.

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There are miscellaneous exercises and word problems at the end of the book. Answers have been included for all the exercises and word problems. Some of the mini tasks have answers. There are not specific answers to the open-ended tasks.

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It is important that students set out their work in a clear manner. This not only helps them to follow a method logically, it makes it easier for teachers to follow students’ thought patterns. To this end, it may be necessary for some of the word problems to be done on lined paper. Most of the mini tasks and open-ended tasks should also be done on lined paper.

Teachers should encourage discussion before beginning preliminary open-ended tasks so students are given some direction and inspiration. If progression during the task is stilted, gentle guidance, brainstorming and group-work are useful tools to help re-ignite interest and confidence. Post-completion feedback is also vital to ensure continued improvement and success.

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Teachers and students sometimes find that the idea of tackling an open-ended task is somewhat daunting. Hopefully, this book, with its definite structure, will guide them to an achievable end.

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Basic Measurement Exercises R Try these exercises without any help. 1 . a) Name some units that we use to measure length. _________________

r o e t s Bo r e p ok b) Name some units that we use to measure mass. __________________ u S ____________________________________________________________ ____________________________________________________________

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____________________________________________________________

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

____________________________________________________________ ____________________________________________________________

© ReadyEdPubl i cat i ons e) Which is smaller, 2 litres or 200 millilitres? (circle correct answer) •f orr evi ew pur posesonl y• f ) How many kilolitres in 534 litres? _______________________________ g) Which is bigger, 45 millimetres or 45 centimetres? (circle correct answer)

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d) How many centimetres in 2.3 metres? ___________________________

h) How many milligrams in one gram? _____________________________

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

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What Unit Is That? R For the remaining work on measurement, we will use the metric system system. The basic unit for measuring length or distance is a metre. 1 . Link these measurements with their basic units

r o e t s Bo r metre e p o u mass litre k S length

gram

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volume

R 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.

2. What units would you use to measure the following objects?

grams, kilometres, kilolitres, kilograms

_________________________

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_________________________

Amount of cordial in a glass

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Length of a pencil

. _________________________ te o c Length of your classroom _________________________ . che e r o Amount of water in a pool r _________________________ t s super Mass of a flea

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

_________________________

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More About Prefixes R Circle the correct abbreviated (shortened) unit for the following: millimetres

g km

kL kg

kilolitres

g km

kL kg

litres centimetres

kilometres

g km

kilograms

g km

milligrams

g km

metres

g km

kg kg kg

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grams

cm m L g km kL r o e t s r emg cm m B mmp mL L o g o km kL u k mm mL mg cm m L g km kL S

kL kg kL kg

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mm mLy mg cmP mb Ll ga km kg ©R ead Ed u i c t i okLns R Name• three objects whose mass youu would measure ino kilograms. f o rr e vi e wp r po ses nl y• millilitres

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R Name two objects whose length you would measure in metres.

. t e of liquid measured in millilitres. co R Name two containers . che e r o t r s super

R Measure this line to the nearest centimetre and the nearest millimetre. mm cm

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Converting Between Units: Using Conversion Charts

R Sometimes we may know a measurement in one unit but need to change it to another unit. We call this converting units units. Conversion charts can be used.

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This chart may help you convert between units. The example has been done using the units for length.

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Length

In length, we often use centimetres (cm). Where do you think ‘cm’ would go in the chart above? Write it in pencil, then check with your teacher.

Mass

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R Complete the conversion charts for mass and volume below.

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R Use this page to help you with the following conversion exercises. Ready-Ed Publications

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Using the Conversion Method by Following the Little Bumps! Refer to the Conversion Charts on page 9… R When converting between units, you move the decimal point the number and direction of the little ‘bumps’.

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3 We are we going from ‘m’ to ‘km’. 3 Move the decimal three places to the left (we follow the ‘bumps’ from ‘m’ to ‘km’). i.e. 534.6 m = .5 3 4. = .534 km

1 . Try these…

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r o e t s Bo r For example…. e p ok How many km inu 534.6 m? S 3 Start at the decimal point .

b) u 0.087 gc =a _______________ © ReadyEdP bl i t i ons mg c ) 1.27 km = _______________ m d) 690 mL = _______________ L •f orr evi ew pu r posesonl y• a) 235 m =_________________ km

f ) 7568 mm = ______________ m

g) 0.0037 kg = _____________ mg

h) 2734 cm = _______________ km

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Check your answers!

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e) 26 cm = _________________ mm

How did you go? If you got them all right, you can go on with the next page. If you got some wrong, you should practise some more using the questions below.

o c . c e herg j) 35 s r i) 200 mg = _______________ kmt =_________________ m o s up er k) 250 mL = _______________ L l) 4 678 mg = ______________ kg m) 0.0016 km = _____________ mm

n) 1.26 L = ________________ mL

o) 3.67 cm = _______________ m

p) 0.127 g = _______________ mg

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Measurement Word Problems 1. A chemist needs to weigh 0.085 kg of a chemical but has a scale that only measures in grams. What would the reading on the scale say?

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r o e t s B r e oois this in kilometres? 2. Sally measured 400 m using a trundle wheel. How far p u k S 3. How many litres of milk would you have if you combined five, 600 mL containers?

4. What would be the total mass of six 120 g chocolate bars? Give your answer in kilograms.

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5 . Tarlie ran five times around a 400 m track. How many kilometres did she run?

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6. Jamie had five items in her pencil case. Her two biros weighed 80 g each, her calculator weighed 150 g, the eraser was 50 g and her ruler was 15 g. What was the total mass of the objects in her pencil case in grams and kilograms?

7. To dilute cordial, Stacie’s dad added 50 mL of cordial to 450 mL of water. How many litres of diluted cordial did he make?

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Peter’s School Run Measurement Mini TTask ask R Peter rides to school every day. The route he takes is shown below. The scale for this drawing is 1 cm = 0.5 km. So, if the distance on the map is 4 cm, this equals 2 km (4 x 0.5).

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School

Park

Peter’s house

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Jack’s house

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b) Sometimes, Peter rides across the park. If he rides this way, is his journey longer or shorter than along the footpath? By how much is his journey longer or shorter?

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c ) Every Tuesday, Peter goes to his friend Jack’s house after school to do his homework. How many kilometres does this add to his normal route?

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How Did They Measure That? Measurement Open-ended TTask ask Not all measurements use the units we have looked at. Grams, litres and metres are part of the metric system used in many countries around the world today to measure mass, volume and length. However, there are other systems currently being used by different countries. In the past, still more systems were used.

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r o e t s B r e ooof measurement as R Your task is to findp out about as many ancient systems you can. u k S For each unit of measurement you discover, you should include: The name of the unit. The quantity the unit measures (mass, length, volume etc.). The types of objects measured by the unit. The country of its origin (where it was first used; by which ancient people was it used?). 3 The period during which it was used. 3 Any conversions to modern metric units that you can find. 3 3 3 3

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Start your notes here:

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Perimeter Exercises R 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. What is the perimeter of these shapes? 3 cm

This means the sides are the same length.

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

1.5 cm

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3 cm

Answer: _______________________

8.5 cm

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0.8 m

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9.3 m

5.6 m

18 m Page 14

Answer: _______________________

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Perimeter of Circles

- The Circumference

Investigation R It’s not so easy to find the perimeter (called a circumference) of a circle. formula. You have to use a special number called pi (π) and a formula A formula is a short-hand way of writing a relationship between values. Mathematicians use them all the time because they simplify words. In this activity you will work out what π is!

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r o e t s Bo r e pstring, follow the outline of circleoAkuntil the end of 3 Using a piece of u your string joins up with the beginning. Try to be as accurate as possible. S 3 Now measure the length of string and write it in the table next to

‘circumference’. 3 Next, measure the length of the line stretching from one side of circle A to the other through the middle. Write this value in the table next to ‘diameter’. 3 Repeat for all the other circles.

D © ReadyEdPubl i cat i on s •f orr evi ewC pur posesonl y•

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Circumference (cm) Diameter (cm)

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R To calculate the value of π, divide the circumference of each circle by its diameter. π is a measure of how many times the diameter of a circle will fit around the circle’s edge. It is always the same number (approx. 3.14) for every single circle!

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Circumference Exercises R If you know the diameter of a circle you can find out the circumference by multiplying the diameter by pi (π). This formula is:

π r o e t s Bodiameter. The radius r Sometimes, the radius of a e circle is given instead of the p ok is the line from the edge of the circle to the middle. So, the radius multiplied u by two will give the diameter. S If the radius is given, we can use the formula: circumference = 2 x

x D

x r

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circumference =

Some calculators have π written on them. Does yours? If not, use the approximation of 3.14.

R Find the circumference of these circles, semi-circles or quarter circles.

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6. 2 1.6 m

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Perimeter More Exercises R Find the perimeter of these shapes. 1.

0.6 cm

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3 cm

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0.2 cm

5 cm

Answer: _______________________

8 cm

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6 cm

Answer: _______________________

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8 cm

5 mm

6.5 cm

3 mm 7 mm

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1.5 cm

Answer: _______________________ Page 17

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Perimeter and Circumference Word Problems R Some of these are tricky! Take your time to think about them and draw a diagram if it will help. 1 . Sally’s circular above-ground swimming pool is in the shape of a circle with a diameter of 3 m. What is its circumference?

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r o e t s Bo r e p ok u S Irene has a rectangular pool. If its length is

8 m and its width is 7 m, what is its perimeter?

Extension: A path has been put around Irene’s pool. If the path is 1 m wide, what is its outer perimeter? Hint: Use the diagram.

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3. Mrs Hunter wanted to put a herb garden in her yard in the shape drawn below. If the square down the bottom had a length of 3 m and the top shape is a semi-circle, what was the total perimeter of the shape? Mrs Hunter decided to put a small fence around the herb garden to keep the rabbits out. If fencing costs $8.95 per metre, how much will it cost in total?

o c . che e r 4 . John decided to do some exercise. First he ran t 5o laps of the school playing r s sup field. The field was actually a circle withe ar diameter of 110 m. Then he ran three times around the school that was in the shape shown below. What was the total distance that John ran? 100 m 80 m 40 m Page 18

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The Farmer’s Fence Perimeter Mini TTask ask Farmer Frances needs to put a fence around a small paddock. The paddock has an unusual shape. The central shape was a rectangle with sides of 125 m and 85 m. Along one of the short sides, the paddock spread out into a semi-circle.

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r o e t s Bo r He had two ideas about fencing: e p o u k 1. One plan was to put a barbed-wire fence along the three straight sides, S and a limestone wall on the edge of the semi-circle. 2. His other idea was to put a wooden fence around the whole paddock.

R If the barbed-wire fencing cost $12 per metre, the limestone was $45 per metre and the wooden fence cost $26 per metre, which would be the cheaper alternative, Plan 1 or Plan 2? Use these lines to show your working out method:

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My Maze Perimeter Open-ended TTask ask The local council has decided to make a maze in one corner of the park (if you don’t know what a maze is, you should ask your teacher). They are having a competition - the best design for the maze wins $100. You decide to enter.

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r o e t s Bo r R The specifications for the maze are as follows: e p o u k 3 The scale should be 1 cm = 1 m. 3 The maze S must have at least two circular (or quarter-circular, semiR Your final report should include:

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circular) sections. 3 The total distance of the correct path in the maze from beginning to end, assuming no false paths are taken, must be no shorter than 200 m and no longer than 500 m. 3 The height of the walls/hedges do not have to be to scale.

3 A two-dimensional plan with all measurements for each section of the maze clearly indicated. 3 A three-dimensional model of the maze to the same scale as the plan.

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Diagram

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R Start your notes here:

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Area Exercises If you wanted to plant some grass in your front yard, you’d need to know how big the area is so you can buy the correct amount of grass. Painters need to know the area of the walls they are painting so they don’t buy too much or too little paint.

r o e t s Bo r e p ok u To find the area of various shapes, we use special formulae (plural of S formula). A formula is a short-hand way of writing a relationship between

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The space inside a two-dimensional shape is called its area area. The most common units for area are metres squared (m²) or centimetres squared (cm²). Large areas are measured in km² or hectares.

values. Mathematicians use them all the time because they simplify words.

length

height

R These are the formulae we will use will help us find the area of rectangles, triangles and circles.

radius

Area = length x width

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A = L x W

base Area = ½ x base x height

Area =

π (radius)²

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height

width

π r²

A = ½ x B x H . te o c R Use the formulae given above to find the area of these . shapes. che e r o3. 1. 2.r st super 4 cm

5 cm

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6 cm

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Area More Exercises R Find the area of these shapes. Don’t forget to use the correct formula and units. 1.

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6.3 cm

Answer: _______________________

4. 4m

7 cm

6. 1.3

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11 cm

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1.2 m

18 cm

Answer: _______________________ Page 22

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Tricky Areas Exercises R If a shape doesn’t look like a rectangle, triangle or circle, you may have to divide it up into sections to make it look like these shapes. Check this out! 5 cm

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5 cm

8 cm

9 cm

5 cm

Total Area = = = = =

A (rect.) LxW 8x5 40 56 cm²

+ + + +

A (tri.) ½xbxh ½x4x8 16

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AREA = ??

8 cm

This shape can be divided into a rectangle and a triangle.

4 cm

5 + 4 = 9

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. te o Answer: ________________________ Answer: ________________________ c . che e r o 1. 2.r 3. t s super 10 m

3.2 cm

1.8 cm

Answer: _____________ Ready-Ed Publications

1.5 cm

3 cm

1 cm

Answer: _____________

Answer: _____________ Page 23

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Area Word Problems 1. Colleen needed to carpet her rectangular living room. If the room is 8 m long and 4 m wide, how much carpet will she need? If the carpet Colleen wants is $29.50 per square metre, how much will it cost her?

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r o e t s Bo r e p ok u S 2. Nigel and his dad were paving their back yard according to the plan below. How many square metres of paving will they have to buy? 8m

4m 2m

2m 2m

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3 . A kite is in the shape of a diamond, as shown below. What is the area of the kite?

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4. A garden feature is to be in the shape of a circle with a 1.5 m radius. What will be the total area of the feature?

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How Big Is That? Area Mini TTask ask

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R Use at least two different techniques to find the area of this circle in cm².

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R Describe the techniques you used here.

Technique 2

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The Paint Problem Area Mini TTask ask Mr Harvey wants to paint the side of his house, shown below.

r o e t s Bo r e p ok u S 2.5 m

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He has decided to paint one layer in white undercoat and two layers of the colour Farmhouse Green.

3 Harry’s Hardware sells Farmhouse Green for $43 for a 4 L tin and 1 L tins of undercoat for $10 each. 3 Paulie’s Paints has 2 L of Farmhouse Green for $25. They sell white undercoat for $18 for 2 L.

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One litre of paint or undercoat will cover 4 m² of wall. Mrs Harvey wants her husband to work out if it’s cheaper to get the paint and undercoat from Harry’s Hardware or Paulie’s Paint (it’s best to get everything from the same shop). Unfortunately, Mr Harvey is not very good at maths! Can you help him decide which is the best deal?

o c What is the total areac of the front of the house? _________________________ . e hinr r o What is the area that will bee white undercoat? ________________________ t s super

R You can use these questions to help …

What is the area that will be in Farmhouse Green? ______________________ How many tins, of each type of paint, will he have to buy from each shop? Remember that he cannot buy half a tin, so he may have some paint left over. _________________________________________________________________ What is the cost from each shop? _____________________________________ _________________________________________________________________ Page 26

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The Dirt In Carpet Cleaners Area Open-ended TTask ask Carl’s Carpet Cleaning Service is having a sale. They will clean your carpets for $1 per square metre. Sam’s Spotless Carpets are also having a sale. They will clean your carpets at a flat rate of $25 per room, regardless of the size of the rooms.

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r o e t s Boor magazines to r R You are to use plans of houses from the newspaper e p to determine which is the best ok conduct an investigation deal for a variety u of different house/unit S sizes.

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Your report on the investigation should be clearly presented so that the following information is easy to find.

3 The area of individual rooms in each house/unit. 3 The total area of carpeting in each house/unit. 3 The cost of cleaning each house/unit using both Carl’s Carpet Cleaning Service and Sam’s Spotless Carpets.

Note:

For each plan, assume that every room is carpeted except the kitchen, bathroom, toilet and laundry. Hallways are cleaned for free by both services, so you don’t need to include them in your calculations. R Your investigation should try to answer the following questions.

R Start your notes here:

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3 Is there any relationship between the size of the rooms and the best deal? 3 Is there any relationship between the number of rooms and the best deal?

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Care For a Cup of Tea? Area Open-ended TTask ask R You are serving afternoon tea to some of your friends and need to carry the cups of tea and cake from the kitchen to your guests outside. You would prefer to make one trip only. You have a choice of three trays to use - these are shown below. The plates you must place on the trays are also shown below. The diagrams are NOT drawn to scale.

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r o e t s B r e oo Which tray would you use? p You can use any method you like, but, you must have proof k that the tray you u Stea things on it in a safe manner. choose will fit all the

R You should consider this information also.

3 The cups all fit in the saucers and the saucers fit on the cake plates. They can be carried safely on the tray in this way. 3 It is not safe to stack more than two cups together. 3 No plates or saucers can go on the cake tray as it has cake on it! 3 The saucers can be stacked on the plates.

20 cm

Cake Tray

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20 cm

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o c . base of cup che 4 x e r o r st super 2.5 cm

7.5 cm

12 cm

4 x

saucer

9.5 cm

4 x

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cake plate

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Surface Area Introduction When you wrap up a present that is in a box, you need to have enough paper to cover the entire outside area of the box (and a bit more to secure it properly). The outside area of a three-dimensional shape is called its surface area area. Like area, surface area is measured in cm² or m².

r o e t s Bo r e p ok u S below. How many different sides does this shape have? R Look at the shape 7 cm

10 cm

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For these exercises, we will just find the surface areas of rectangular prisms (like a novel or a box).

5 cm

That’s right, it has three different sides:

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1 . The top, which is the same as the_______________________________

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2 . The front, which is the same as the _____________________________

3 . The right side, which is the same as the__________________________

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o c . Side Length (L) Width (W) Area (LxW) Number of Total area c e r (cm) h (cm) (cm²) o sides (cm²) er t s 2 s uper Top/bottom 10 7 70 140

R Complete this table to help you find the surface area of the whole shape.

Front/back

Right/left

R So, the surface area for the whole shape is the sum of the total areas. _____________ + ______________ + ______________ = ___________ cm² Ready-Ed Publications

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Surface Area Exercises R Find the surface area of the following rectangular prisms. You may wish to use a table like the one in the previous exercise, or you may have your own system. Remember to set your work out clearly so it is easy to understand. 2. r o e t s Bo r e p ok u S 1.2 m

8 cm

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m 12 c

1 cm

m 1.2

1.2 m

Answer: _______________________

68 cm

cm 85

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125 cm

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4 cm

m 6c

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Answer: _______________________ 5 . Extension

5 cm

Find the surface area if this box has no top.

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2 cm

7 cm

6 cm Page 30

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Answer: _______________________ Ready-Ed Publications

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Surface Area Word Problems R Rectangular prisms are quite difficult to draw. At the bottom of this page is a general box. You may want to write the dimensions for each question on the shape to help you. If you use pencil, you can use this same shape_ for each question. Obviously, the shape will not be to scale.

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1 . Irene wanted to paint a jewellery box for her sister. If the dimensions of the box were 12 cm by 8 cm by 10 cm, what is the total surface area she will have to paint?

2. A box is 20 cm by 40 cm by 35 cm. What is the minimum amount of paper needed to wrap the box (with no overlap).

3. A kite is to be made into the shape of a box.

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a) If the dimensions of the kite are to be 85 cm by 1.2 m by 55 cm, how much material will be needed (in m²)? b) If the kite material costs $15.20 per square metre, how much will it cost?

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Sticky Problem Sur face Area Mini TTask ask Surface Mrs Brooks in the library has a problem. She has to cover some textbooks with Contact® book covering and she wants to make sure none of the Contact is wasted.

r o e t s Bo r e p of the rectangular piece of Contact ok will she need R What are the dimensions u for each book?S Teac he r

3 The books are 20 cm long, 15 cm wide and 4 cm high. 3 She wants to use a 3 cm overlap of Contact.

R Start your notes here:

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Hint: Try to visualise this problem before your begin. Maybe you can look at some textbooks that have been covered with Contact.

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4 Boxes 4 You Sur face Area Open-ended TTask ask Surface Imagine you have bought your mum four presents for her birthday. Each gift has its own box and you decide to wrap all four boxes in the one piece of paper to surprise your mum.

r o e t s Bo r e p ok u S 15

3 cm

12 cm

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10 cm cm

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5 cm

10 cm

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R Below are the four boxes with their dimensions written on them.

o c . chhow R Your task is to decide you will stack the boxes r soe the least amount of e o t wrapping paper is used. You want the wrapping to be neat, so you cover r s s r u e p each exposed side perfectly, i.e. all edges are at right-angles, not sloped. 15

10 cm

You must clearly show the calculations and reasoning that led you to decide on this method of stacking. You will only get full marks if you do this. If your teacher cannot understand your reasoning, you may be asked to explain verbally. You may use any method you wish to help you solve this problem. Ready-Ed Publications

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Capacity (Volume) Exercises If you wanted to know how much stuffing to put in a cushion, you would need to know the capacity of the cushion. The amount of space that a three-dimensional solid fills is called its capacity. Capacity is usually measured in cubic metres (m³) or cubic centimetres (cm³).

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r o e t s Bo r volume Sometimes capacity is called volume. e pare quite difficult to visualise. ok Three-dimensional shapes u Swe will only study two types of three-dimensional shapes. R In these activities,

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Once again, we will use mathematical formulae to find the capacity of shapes.

1. The rectangular prism - like a book or a box.

Capacity = width x length x height

height

Cap. = W x L x H © Read y E d P u b l i cat i ons h t g n e l width •f orr evi ew pur posesonl y• A cube is a special rectangular prism that has all sides of the same length. radius

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2 . A cylinder - like a toilet roll or a drink can. Capacity = π x radius² x height height

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Cap. =

x r² x H

o c . R Find the capacity of shapes. cthese e her c) st r o a) b) d) super

1.2 m

2 7.

8 cm 4

2m 3m

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1 cm

3.6 m cm 10

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

3. A cylindrical water tank has a height of 12 m and a radius of 4 m. What is its capacity?

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4 . A wine bottle box is in the shape of a cylinder. If it has a diameter of 9 cm and a height of 30 cm, what is its capacity?

o c . c e 5 . Which has the largerh capacity: a box with a heighto ofr 4 cm, length of 13 cm e t r s s up and width of 8 cm, or a cube with sides of 7.5 cm? er

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Capacity Tricky W ord Problems Word 1. 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?

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Be careful of these units! 10 cm soil

1.5

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50 cm

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?

© ReadyEdPubl i cat i ons Nat bought two gift boxes that she wanted to fill with Smarties to give to friends. The rectangular prism for Jane 8s cm • f o rr evi ewboxp u r piso ewide, so11ncm l ylong •and

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5 cm high. The cylindrical box for Fiona has a radius of 3 cm and a height of 12 cm. If she filled each box completely with Smarties, who would get the most chocolate?

o c . c e he 3. A cylinder was fitted into r a cube-shaped box of side length 15 cm. There o t r was no room either on top of s or u to the sider ofs the cylinder once it was in e p the box. How much space was left unfilled?

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The Price of Pencils Capacity Mini TTask ask Two competing pencil manufacturers have different shapes for their pencils. 3 Pencil’s A’Plenty pencils are cylindrical with a length of 18 cm and a radius of 0.4 cm. 3 Not Just Pencils pencils are also cylindrical with a length of 11 cm and a radius of 0.5 cm.

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r o e t s Bo r e pof pencil has a radius of 0.15 cm. ok The ‘lead’ in both types u Seach pencil costs 0.2 cents/cm³ and the lead filling costs R If the wood for

HINT HINT:: Draw diagrams to help you. R Start your notes here:

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0.3 cents/cm³, which type of pencil is the cheapest to make. Remember that the lead sits inside the wood!

Diagrams

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The Case of The Music Man Capacity Open-ended TTask ask DJ Smoosch takes 25 CDs and 15 cassette tapes when he plays at school dances. He has lots of other equipment to carry, so wants to fit the CDs and cassette tapes in the smallest case possible.

r o e t s Bo r e othink Your final answeru willp be the dimensions of the case you is the k smallest he could Suse.

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R Use the actual dimensions of CDs and cassette tapes to investigate the various ways DJ Smoosch could pack his gear.

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R You will need to present a report of your findings. In your report include the following:

3 Your method for finding the case size. How did you work it out? 3 All the necessary calculations to prove that the dimensions of the case you chose will give the smallest volume. So, you will need to include the calculations for the case sizes that you rejected also. 3 A calculation showing the amount of ‘wasted space’, if any, of your chosen case size. This is the amount of space inside the case that is empty.

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Chocolate Anyone? Capacity Open-ended TTask ask A chocolate company, Tempting Treats, decides to do a promotion at Christmas time. They want to fill boxes and cylinders with their chocolate and they want lots of different-sized containers.

r o e t s Bo r e pspecifications so they will fit o The containers have some on the supermarket u k shelves: S

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The chocolate containers will all cost the same amount of money, so they all have to be the same capacity.

The capacity will be 1 728 cm³. The maximum allowable height for each container is 36 cm. The maximum allowable length/width for each container is 16 cm. The containers must LOOK different, so it is no good just varying the length by only 1 cm from one box to the next.

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3 3 3 3

R The company hired you to investigate the different dimensions of boxes and cylinders they can use.

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3 The company will need at least six different-sized containers. That is, three boxes and three cylinders. 3 They will need to have a list or table of the final dimensions you come up with. 3 They will also need to see all your calculations.

Diagrams

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Miscellaneous Exercises 1 . Find the surface area and capacity of this shape. Don’t forget to use the correct units.

r o e t s Capacity Bo= _______________ r e p ok u S Surface area = _______________

cm 5 . 2

4.8 cm

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3.2 cm

2 . What is the perimeter of this shape? 2m

Perimeter = _______________

3. Find the perimeter and area of this shape.

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7.1 m

Perimeter = _______________

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Area = _______________

4 . How many grams in: 23 kg? _____________

134.5 mg? __________

How many litres in:

6578 mL? __________

0.089 kL? ___________

How many mm in:

0.5 cm? ____________

1.46 m? ____________

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Miscellaneous Word Problems R Draw diagrams in the spaces to help you solve these: 1 . A cat ran twice around a rectangular house that was 30 m long and 20 m wide. How far did the cat run?

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total surface area of tiling required?

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r o e t s Bo r e p ok 2 . A swimming pool is 5 m long, 4 m wide and 6 m deep. u S a) What is the volume of the pool? b) If the pool sides and base is to be tiled, what is the

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3 . A circular area 8 m in diameter is to be brick-paved. If brick paving costs $15 per square metre, how much will it cost in total?

4 . A feature is to be painted on Mrs Flurrey’s living room wall. The shape of the feature is a square with a semi circle on the top of it. If the length of the square is 0.8 m, what is the total area of the shape?

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Answers Page 6 - Basic Measuremen Measurementt 1a) 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

Page 14 - P erimeter: Perimeter: 1. 9 cm 2. 3. 23 cm 4. 5. 19.6 m 6.

Exercises 16 m 16 m 32.9 m

r o e t s Bo r e p ok π u S

Page 16 - Circumference: Exercises (using = 3.14) 1. 47.1 cm 2. 4.33 cm 3. 12.56 cm 4. 5.97 cm 5. 5.09 m 6. 11.78 m

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Page 7 - What Unit Is That? 1. length gram volume metre mass litre 2. centimetres millilitres milligrams kilometres metres kilolitres grams millimetres kilograms litres

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Page 12 - P eter ’s School R un Peter eter’s Run a) 6.25 km b) 5.5 km; 0.75 km shorter. c) 8.75 km; 2.5 km longer.

Page 17 - P erimeter: More Exercises Perimeter: 1. 2.4 cm 2. 17.71 cm 3. 38 cm 4. 10.05 cm 5. 23.14 mm 6. 23.36 cm

Page 18 - P erimeter & Circumference Perimeter 1. 9.42 m 2. 30 m; ext. 34 m 3. 13.71 m; $122.70 4. 2087 m or 2.087 km

Page 9 - Converting Between Units

kg kL

cm mm

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125 m

Barbed-wire: 335 m; $12/m; cost = $4020 Limestone: 133.45 m; $45/m; cost =$6005.25 Wooden: 468.45 m; $26/m; cost = $12179.70 Total of barbed wire + limestone = $10025.25 Plan 1 (barb wire + limestone) is cheapest.

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Page 10 - Using the Conversion Method 1a) 0.235 km b) 87 mg c) 1270 m d) 0.690 L e) 260 mm f) 7.568 m g) 3700 mg h) 0.02734 km i) 0.200 g j) 35 000 m k) 0.250 L l) 0.004678 mg m) 1600 mm n) 1 260 000 o) 0.0367 m p) 127 mg Page 11 - Measurement: W ord P roblems Word Problems 1) 85 g 2) 0.400 km 3) 3 L 4) 0.72 kg 5) 2 km 6) 375 g, 0.375 kg 7) 0.5 L Page 42

133.45 m

85 m

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mm kL L cm g km kg mg m mL

Page 21 - Area: Exercises 1. 20 cm² 2. 3 m² 3. 113 cm²

Page 22 - Area: More Exercises 1. 39.7 cm² 2. 706.5 m² 3. 42 cm² 4. 12.56 m² 5. 27.5 cm² 6. 0.78 m² 7. 76.9 cm² 8. 254 cm² Page 23 - T ricky Areas: Exercises Tricky 1. 36 m² 2. 26.28 cm² 3. 4.15 cm² 4. 6.75 cm² 5. 16.86 m² Ready-Ed Publications

Page 24 - Area: W ord P roblems Word Problems 1. 32 m²; $944 2. 40 m² 3. 1700 cm² 4. 7.065 m² 5. 32 m²

Page 35 - Capacity: W ord P roblems Word Problems 1. 60 000 cm³ 2. 0.512 m³ 3. 602.9 m³ 4. 1907.55 cm³ 5. The cube (≈422 compared with 416). Page 36 - Capacity: T ricky W ord P roblems Tricky Word Problems 1. 6 m³ @ $20/m³ = $120; 1.2 m³ of mulch 2. Jane: 440cm³ / Fiona: 339cm³ ∴ Jane would get the most chocolate. 3. box: 3375 cm³ / cylinder: 2649 cm³ ∴ Extra space = 726 cm³

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Page 29 - Surface Area: Introduction 1. bottom 2. back 3. left side L W Area Sides Total front/back 7 5 35 2 70 right/left 10 5 50 2 100 140 + 70 + 100 = 310 cm²

Page 37 - The P rice of P encils Price Pencils Pencils A’P Not Just P. capacity (cm³) - pencil 9.0432 8.635 - lead 1.2717 0.77715 - wood 7.7715 7.8578 cost (cents) - lead (x 0.3) 0.3815 0.2331 - wood (x 0.2) 1.5543 1.5716 Total cost 1.94 c 1.80 c ∴ Not Just Pencils is cheapest to make.

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Page 26 - The P aint P roblem Problem Paint Area of house = 10.5 m² ∴ white u/coat = 1 x 10.5 = 10.5 m² ∴ green u/coat = 2 x 10.5 = 21 m² If 1L covers 4 m² then: white u/coat = 10.5 ÷ 4 = 2.625 ∴ 3 L green u/coat = 21 ÷ 4 = 5.25 ∴ 6 L Harry’s: 3L of white = 3 x 1 L = 3 x $10 = $30 6L of green = 2 x 4 L = 2 x $43 = $86 Total cost = $116 Paulie’s: 3L of white = 2 x 2 L = 2 x $18 = $36 6L of green = 3 x 2 L = 3 x $25 = $75 Total cost = $111 ∴ They should buy the paint from Paulie’s paints.

Page 34 - Capacity: Exercises a) 24 m³ b) 80 cm³ c) 16.28 m² d) 203.5 cm²

Page 31 - Surface Area: W ord P roblems Word Problems 1. 592 cm² 2. 5800 cm² 3. 4.295 m²; $65.28

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Page 41 - Miscellaneous W Word ord Problems 1. 200 m 2a)120 m³ b) 128 m² 3. $753.60 4. 0.8912 m² 5. 1.982 kg (1982 g)

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Page 32 - A Sticky P roblem Problem Note: Only the front, back and spine need to be covered with the 3 cm overlap. The dimensions are as shown. 4

Area = 56 m² 0.1345 g 89 L 1460 mm

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Page 30 - Surface Area: Exercises 1. 232 cm² 2. 8.64 m² 3. 49810 cm² 4. 124 cm² 5. 332 cm²

2. 21.81 m 3. P = 36 m 4. 23 000 g 6.578 L 5 mm

20 15 21O

50 cm

∴ Dimensions of Contact = 50 x 21 cm Ready-Ed Publications

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