r o e For 10 11 years t s B r e o p u S

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Maths Problem Solving Series

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• Strategies and techniques covering all strands

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By Susan Cull

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of the curriculum, with activities to reinforce each problem solving method.

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Illustrated by Terry Allen. © Ready-Ed Publications - 2002. Published by Ready-Ed Publications (2002) P.O. Box 276 Greenwood W.A. 6024 Email: info@readyed.com.au Website: www.readyed.com.au COPYRIGHT NOTICE Permission is granted for the purchaser to photocopy sufficient copies for non-commercial educational purposes. However, this permission is not transferable and applies only to the purchasing individual or institution. ISBN 1 86397 463 6

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Working Mathematically (WM) 3.3: Uses Guess and Check as a problem solving strategy. WM 3.3, Meas. 3.18: Measures area, perimeter and capacity by counting uniform units and reads whole number scales. WM 3.3, Space 3.7a & b: Visualises shapes and placement in order to solve spatial problems. WM 3.3, Chance & Data (C&D): Interprets given data and makes numerical statements and estimations based on the frequency of likely outcomes. WM 3.3: Uses diagrams to organise known data in order to proble-solve. WM 3.3, Meas. 3.19: Uses diagrams to visualise spatial problems. Explores and applies the concepts of perimeter, area and length. WM 3.3, Space 3.9: Represents problems diagrammatically. Arranges 2D shapes according to geometric criteria. WM 3.3, C&D 3.23: Uses diagrams to visualise problems and organise data; test predictions, justifying their choices. WM 3.3: Makes tables of data to help solve a problem. Decides what information needs to be represented in a table. Number 3.12: Identifies number patterns in word problems; follows a rule to generate number sequences. WM 3.3, Meas. 3.19: Tabulates numerical measurements of objects to solve problems. Explores the patterns and relationships between area, length and perimeters of 3D shapes. WM 3.3, Space 3.8: Visualises and organises spatial data. Interprets data in order to problem-solve. WM 3.3, C&D: 3.26: Displays and summarises data using frequencies and measurements. WM 3.3: Makes lists of data collected in order to solve a problem. Number 3.12: Identifies and continues number patterns; selects appropriate operation to solve a number problem. WM 3.3, Meas. 3.19: Directly compares measurements of a set of 2D objects (length, area and perimeter). WM 3.3, Space 3.7a: Identifies the features of 2D and 3D objects and looks for patterns. Explores networks and paths in order to find the shortest route. WM 3.3, C&D 3.26: Displays data in a list to summarise frequencies and many-to-one correspondence. WM 3.2, WM 3.3, Number 3.12: Makes and tests conjectures based on identified number patterns. WM 3.2, WM 3.3, Meas. 3.19: Explores and applies the concepts of perimeter, area and length of 2D shapes. WM 3.2, WM 3.3, Space 3.8: Visualises, arranges and explores patterns using 2D shapes. WM 3.2, WM 3.3, Space 3.8, C&D 3.24: Collects data about 2D shapes and looks for patterns in order to solve a spatial problem. WM 3.3, Number 3.11: Works backwards, using counting and order to solve number problems. WM 3.3, Meas 3.19: Uses area, length and time measurement concepts to solve a problem. WM 3.3, Space 3.9: Visualises and arranges 2D shapes so as to meet given geometric criteria; represents 3D objects in a meaningful way so as to solve surface area problems. WM 3.3, C&D 3.27: Organises known data into a table and interprets data by working backwards to solve a problem. WM 3.2, 3.3: Uses logical reasoning as a problem solving strategy, making conjectures based on given data. WM 3.2, 3.3, Meas. 3.19: Uses prior knowledge of the concepts of time, mass and area to logically solve problems. WM 3.2, 3.3, Space 3.10: Analyses and arranges 2D shapes, using logical reasoning to solve spatial problems concerning surface area and position. WM 3.2, 3.3, C&D 3.25: Organises data to solve a particular problem. Uses logical reasoning to formulate answers. WM 3.3, Number 3.14: Applies the concept of number to a range of simple problems; finds a pattern and then applies it to more complex situations. WM 3.3, Meas. 3.19: Applies measurement concepts to a range of simple problems; finds a pattern and then applies it to more complex situations. WM 3.3, Space 3.9: Analyses spatial arrangements; applies patterns to more complex spatial arrangements. WM 3.3, C&D 3.23: Understands and measures frequencies of events based on given data. Applies knowledge to a more complex situation.

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Related Outcomes

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The activities in this book refer to material from: Mathematics - a curriculum profile for Australian schools (1994) ISBN: 1 86366 213 8 This document is published by: Curriculum Corporation, St Nicholas Place, 141 Rathdowne St, Carlton VIC, 3053 www .cur riculum.edu.au/catalogue/ www.cur Page 2

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Rationale

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Problem Solving Strategies

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Create a Diagram Student Information Card Activities

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Use a Table Student Information Card Activities Make a List Student Information Card Activities

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Guess and Check Student Information Card Activities

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Work Backwards Student Information Card Activities Logical Reasoning Student Information Card Activities

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. tean Easier Version o Solve c . Student Information Card 41 c e r Activities h 42 er o t s super Answers 46

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Rationale Mathematical problem solving is the ability to use a variety of mathematical thinking processes and skills to interpret familiar and unfamiliar situations. Students completing mathematical problems will draw upon, and further consolidate a range of strategies, skills, known mathematical concepts and positive attitudes in order to solve the given problem. Strategies students can use include: y Guess and Check y Use a Table y Find a Pattern y Logical Reasoning

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Create a Diagram Make a List Work Backwards Solve an Easier Version

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Students indicate their understanding of the problem by choosing one or more of the above strategies and devising a plan. Appropriate skills and attitudes must be identified and used by the student to carry out the plan successfully. These skills and attitudes are important for students to develop, not only to solve the mathematical problem, but also to apply to other life situations. Appropriate skills and attitudes include:

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Making decisions Showing persistence Working collaboratively Being flexible

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Choosing technology (calculator, concrete materials) Working individually Showing initiative Developing clear written and verbal skills

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Students of age 10 and 11 enjoy working with their peers. When working collaboratively, students will need to verbally communicate clearly to explain their ideas and solutions to their peers. They will also develop their skills to express and listen to different ideas and views. Students will also know that working independently will give them the chance to firmly grasp the features of the problem.

Importantly, through the process of solving mathematical problems, students will need to apply and further develop their understanding of a range of mathematical concepts. The activities in this book will address the following concepts based on Student Outcome Statements: y Number y Space

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• y Measurement y Chance and Data

The purpose of this book is to provide 10 to 11 year old students with a range of mathematical problems, together with solutions.

The Problems

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Structure The mathematical problems in this book are divided into sections according to the strategy that is required to be used to help solve the problem. At the beginning of each section there is an explanation and example of the strategy. Within each section there is at least one problem to solve from each of the curriculum strands. Concrete Materials Students at 10 and 11 years of age should be encouraged to have access to, and use, concrete materials to aid their problem solving.

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Solutions Solutions are given for each problem. There may be other solutions that are correct.

Ideas with Activities The activities in this book have been produced so that they can be used in a variety of ways. The problems are presented as worksheets, for class use, use as homework exercises or even as questions on tests. Alternatively, the activities can be photocopied onto card and completed by individual students or by groups of students at convenient times. Problems can be tackled and discussed as a whole class, although students should be given opportunities to solve problems individually as well as part of a small group. Suggestions The following suggestions may assist with the teaching of problem solving skills: 1. Have a go at completing each problem solving activity in this book yourself. You will get to know what is involved in a problem and how you may help students to find a solution. 2. Ensure opportunities for success. Success in problem solving will ensure increased student enthusiasm. 3. Give students adequate time to think about, and if necessary, discuss each problem. 4. Encourage students to clearly set out their working and clearly write and/or verbalise their conclusions. 5. The main emphasis shouldn’t be placed on the final correct solution. Instead the student’s efforts at working out the problem should be recognized and praised even if the final answer is incorrect. Page 4

Problem Solving Strategies Students should be encouraged to follow a general problem solving procedure: 1. Read the problem carefully to understand what you are asked to find out and what information you have been given. 2. Choose a strategy and make a plan. 3. Carry out the plan and solve the problem. 4. Check the working out and make sure that your solution is actually answering the question.

Create a Diagram

Creating a diagram can help students to picture the problem and find the solution. To Create a Diagram, students must read the problem carefully and draw the information that has been given to them in the question. They can then work out the solution from the diagram they have drawn.

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Guess and Check

Use a Table

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The Guess and Check strategy can be helpful for many types of problems. When students use this strategy they will make a reasonable guess, based on the information that they have been given and then check to see if their guess is correct. Their guesses should get closer and closer to the answer, until the correct answer is found. Using a table is a good way to sort out and organize the information that has been given in the question. The information that has been set out in the table will hopefully lead students to the correct solution

Logical Reasoning

This strategy requires students to use the information they have been given in the question to eliminate possible solutions to finally discover the correct solution.

Make a List

Making a List is a strategy that will help students sort out the information that has been given in the problem. Once the students can see all of the possibilities for the solution, they can then attempt to solve the problem more easily.

© ReadyEdPubl i cat i ons Find a Pattern •f orr evi ew pur posesonl y•

When students use this strategy they look for a pattern from the information that has been given. Once the pattern has been identified the students can predict what will happen next and then continue the pattern to find the correct solution.

Work Backwards

Solve an Easier Version

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Working Backwards is an excellent strategy to use when the final outcome of the problem has already been given. Students just need to work out what the events were that occurred previously. Sometimes the problem is too difficult to solve in one step. When this happens the students will be able to make the problem more simple by dividing it into smaller and easier steps, re-wording the problem or using smaller numbers.

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Student Information Card

Guess and Check The Guess and Check strategy can be helpful for many types of problems. When you use this strategy you will need to make a reasonable guess, based on the information that you have been given, check your guess and check again if necessary. When you check, your guesses should get closer and closer to the answer, until you reach the correct answer itself.

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Example: Paul is 12 years old and his mum is three times as old. How many years must pass before his mum is twice as old?

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2. Make a guess. 4 years must pass.

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1. What do you know already? You know that Paul is 12 and his mum is 3 times as old as Paul, which must mean that Paul’s mum is 36 years old (3 x 12 = 36).

3. Check your guess. In 4 years time Paul will be 16 and Paul’s mum will be 40. Your guess is not correct because 40 is not twice or double 16. 4.

© ReadyEdPubl i cat i ons Make a guess. •f o rpass. r evi ew pur posesonl y• 10 years must

6. Make a guess. 12 years must pass.

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5. Check your guess. In 10 years time Paul will be 22 and Paul’s mum will be 46. Your guess is not correct because 46 is not twice 22 (but getting closer).

o c . che e r o t r s super Correct Solution.

7. Check your guess. In 12 years time Paul will be 24 years old and Paul’s mum will be 48. Your answer is correct since 48 is twice 24. 8.

Does your solution answer the question? Yes. 12 years must pass before Paul’s mum is twice as old as Paul.

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Guess and Check: Number

Strategy: Guess and Check

Strand: Number

1. Sandy has exactly \$6.00 in fifty and twenty cent coins. She has twice as many fifty cent coins as twenty cent coins. How many of each does she have? ................................................

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Strategy: Guess and Check 2. I am a two digit number. I am less than 40 and I am exactly divisible by 3 and 10. Guess and check to discover what number I am.

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o c . che e r o t r s super He washed a total of 17 cars and made

3. Mark washed cars on the weekend to raise money for a charity. For every two door car he washed he was paid \$3 and for every four door car he received \$5. \$73 for the charity.

How many of each type of car did Mark wash? ................................................ ................................................ Ready-Ed Publications

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Guess and Check: Measurement

Strategy: Guess and Check

Strand: Measurement

4. There are two rectangles whose perimeters are the same as their area. Find both rectangles. Show them on the grid below.

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Strategy: Guess and Check

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(Hint: Use centimetre grid paper to draw different rectangles until you find the answer.)

Strand: Measurement

5. How can you arrange five squares, each with sides of 1 centimetre, to make a perimeter of 12 centimetres? Draw your answer below. (Include all solutions.)

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Strand: Measurement

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6. You have two jugs marked 3 L and 7 L. They have no other markings. How would you use the two jugs to measure 4 L of water, exactly? Explain what you could do.

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7 Litre

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3 Litre

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Guess and Check: Space

Strategy: Guess and Check

Strand: Space

7. You have made four piles of cubes. The first pile has four more cubes than the second pile; the second pile has one cube less than the third pile; and the fourth pile has twice as many cubes as the second pile. If there are 20 blocks altogether, how many blocks are in each pile?

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Strategy: Guess and Check 8. Arrange 12 matches like this: (They make 5 squares - four small and 1 big.) How would you remove 4 matches to leave exactly two squares?

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Draw the piles in this space.

Strand: Space

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. tplaced on the dotted line A mirror e o c . che e r o t r would create this picture: s s uper

Strategy: Guess and Check 9.

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Draw a dotted line on this diagram to show where the mirror should be placed to create this picture. Ready-Ed Publications

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Guess and Check: Chance and Data

Strategy: Guess and Check

Strand: Chance and Data

10. Your friend has written you a number code to break. It looks like this = 11 = 63

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How would you write 97 using your friend’s code?

Strategy: Guess and Check

Strand: Chance and Data

11. Jenny has two dice. She rolled the two dice and added the two numbers showing together. How many different combinations of 2 numbers could Jenny have rolled to get an answer of 6? Write them below:

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Strategy: Guess and Check

Strand: Chance and Data

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12. Oranges cost \$4.00 per kilo and apples cost \$3.00 per kilo. How many kilograms of each fruit did Mrs Jackson buy if she spent a total of \$18.00 on fruit and had 5 kilos of fruit altogether? Kg 1

\$4.00

\$3.00

\$7.00

2

\$8.00

\$6.00

\$14.00

3

\$12.00

\$9.00

\$21.00

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Student Information Card

Create a Diagram Creating a diagram can help you to picture the problem and find the solution. To Create a Diagram, read the problem carefully and draw the information that has been given to you. Work out the answer from the diagram you have drawn.

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Example: Every morning you have to make a sandwich for your lunch. You have a choice of white or brown bread and can choose jam, honey, cheese, salad or chicken filling. How many different types of sandwiches could you choose to make?

2. Draw a diagram. jam honey cheese salad chicken

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1. What do you know already? You know from the question that you can make sandwiches with white or brown bread. You also know that you can use jam, honey, cheese, salad or chicken as a filling. You can only use one filling for each sandwich.

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brown bread © Re ad yEdPubl i cat i ons •f orr evi ew pur posesonl y•

3. Find the solution. From the diagram you can count how many different types of sandwiches you could make. You could choose from 10 different types of sandwiches. 4.

o c . che e Correct Solution. r o t r s Does your solution answer s theu question? Yes. You can make 10 types of r e p sandwiches.

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Create a Diagram: Number

Strategy: Create a Diagram

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Strategy: Create a Diagram

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1. A fireman climbed to the middle rung of his ladder to spray water into a burning building. The fireman spotted flames further up so he climbed up 5 more rungs. When the fire died down he climbed down 10 rungs. The fireman then saw smoke from a window above and moved back up 12 rungs. Finally the fire was put out and the fireman climbed 10 rungs to the top of the ladder and entered the building. How many rungs did the ladder have? Draw your diagram in the box.

Strand: Number

Strand: Number

2. During lunchtime every student in Jamie’s class had at least one drink. There were 14 students that had a drink of water, while 6 students, including Jamie, had a drink of water and a drink of orange juice. There are 25 students in Jamie’s class. How many students had a drink of orange juice only?

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o c . che e r o t r Draw the friendships bracelets below using the scorrect colours. s up er

3. Grace decided to make a friendship bracelet for her best friend. By threading two purple and two green beads onto a length of string, how many different patterns of purple and green friendship bracelets can she make?

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Create a Diagram: Measurement

Strategy: Create a Diagram

Strand: Measurement

4. The perimeter of the school’s rectangular playing field is 600 metres. If the length is twice the width, what is the length of the playing field?

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Strategy: Create a Diagram 5. The gardener is digging a hole and standing in it. He is 1.65 metres tall. At the moment the top of the hole is 25 centimetres below the top of his head. He needs to dig the hole to a total depth of 2.5 metres. How much further down must the gardener dig?

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Use this space to work in.

Strand: Measurement

Use this space to work in.

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6. If you have ten 2 centimetre rods, four 5 centimetre rods and two 10 centimetre rods, how many different pairs of lines can you make, using the rods, if the lines in each pair must be equal in length?

Draw them.

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Create a Diagram: Space

Strategy: Create a Diagram

Strand: Space

7. The shape below is made from 4 squares joined along their edges.

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How many more shapes can you make using 4 squares, joined along their edges? Draw them below.

© ReadyEdPubl i cat i ons Strand: Space Strategy: Create ae Diagram •f orr vi ew pur poseson l y• 8. You have five goldfish to be placed into three fishbowls numbered 1, 2 and 3.

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Draw diagrams in the space below to show how many fish you could put into the three bowls so that each bowl has at least one fish in it. Include all solutions.

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Create a Diagram: Chance and Data

Strategy: Create a Diagram

Strand: Chance and Data

9. A father and his two sons want to row to an island to go fishing. Their small row boat will only carry one adult or two children. What is the least number of boat trips needed to get everyone across?

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Strategy: Create a Diagram

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Strand: Chance and Data

10. Jeremy wants to design a striped flag using only blue, red and green. How many different combinations can Jeremy make if each colour can be used once, twice or three times in each flag? Draw the patterns below.

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11. At a school assembly there are 5 seats in a row. Alex must sit next to Sophie but not next to Luke. Luke will not sit next to Kylie. Kylie and Sophie always sit next to each other. Who is sitting next to John? Write the names on the backs of the seats.

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Student Information Card

Use a Table

Making and using a table is a good way to sort out and organise the information that you have been given in the question. You can then clearly see the information which will help lead you to the correct solution.

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Example: An ice cream shop has just opened and has a range of 5 ice cream flavours for customers to choose from. There is also a choice of 3 different toppings. How many different one-flavour, one-topping combinations can customers choose from?

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1. What do you know already? You know that there are 4 ice cream flavours and 3 choices of toppings. You are not allowed to have more than one flavour or topping. 2. Draw a table. Organise the information that you have been given into a table.

© Rea dyEdPFl.u1b l i cat i o ns Fl. 1 and To. 1 and To. 2 Fl. 1 and To. 3 Ice cream Fl. i 2e and 1 u Fl. and To.s 2 o Fl.l 2y and To. 3 •flavour f or2r ev wTo.p r p2o se n • Topping 2

Ice cream flavour 3

Fl. 3 and To. 1

Fl. 3 and To. 2

Fl. 3 and To. 3

Ice cream flavour 4

Fl. 4 and To. 1

Fl. 4 and To. 2

Fl. 4 and To. 3

Fl. = Flavour

To. = Topping

Ice cream flavour 1

Topping 3

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3. Find the solution. Use the information in the table to count the number of different combinations possible. There are 12 possible combinations available. 4.

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Use a T able: Number Table:

Strategy: Use a Table

Strand: Number

1. Mark is going on a school camp for three days. He packs an orange shirt, a green shirt, a black and a blue pair of shorts and a grey and a red jumper. Use the table to find how many different three-piece outfits he can wear while he is on camp. (Complete the table yourself.) Shorts

orange

blue

Jumpers grey

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Strategy: Use a Table

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e.g.

Shirts

Strand: Number

2. When two numbers are multiplied the answer (product) is 96 and when the two numbers are added together the answer (sum) is less than 30. Make a table to find all the possibilities for the two numbers. 1st number

2nd number

Product (x)

Sum (+)

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e.g. 48

Strand: Number

o c . che e r o t r Caleb s Total of 40 rocks su per

3. Jordan and Caleb are comparing their rock collections. Caleb has 8 more rocks in his collection than Jordan. There are 40 rocks altogether. How many rocks does each boy have? Jordan

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Use a T able: Measur ement Table: Measurement

Strategy: Use a Table

Strand: Measurement

4. Kimberley has some 5 cent and 10 cent coins in her purse. Use the table below to find how many of each coin Kimberley has if: Total coins

Total amount

3

20 c

9

5c

10c

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40 c

8

65 c 50 c

Strategy: Use a Table

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Strand: Measurement

5. Below you can see a square that is growing in size.

a) Work out the area and perimeter of each square and complete the table. 1cm

2cm

3cm

4cm

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Area (cm²)

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Side (cm) 2 . t 3e 4 5

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b) Continue the table to find the perimeter and area of squares whose sides are 6 cm, 7 cm and 10 cm. 6 7 10 Page 18

Use a T able: Space Table:

Strategy: Use a Table

Strand: Space

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Strategy: Use a Table

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6. If you make \$50 every time the hands of a clock are vertical and horizontal and form a 90 degree angle, how much would you make in 24 hours? Use a Table to organise your information.

Strand: Space

7. If a rectangle has an area of 120 cm² and its length and width are whole numbers, what are the possibilities for the two numbers? Length

Width

Area (120 cm²)

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Strand: Space

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8. If you had 24 lollies, in how many ways could you arrange those lollies so that there would be three piles with an even number of lollies in each pile? Pile 1 e.g.

2

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Use a T able: Chance and Data Table:

Strategy: Use a Table

Strand: Chance and Data

9. There are three teams in a soccer competition: Red, Green and Yellow. Each team plays both the other teams twice. A team gets 2 points for a win, 1 point for a draw and 0 points for a loss. Complete the table to show the results of the competition. Team

Games played

Red

Games won

Games drawn

Games Points Points lost for wins for draws

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0

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1

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Strategy: Use a Table

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Total points

Strand: Chance and Data

10. Use the table below to find the relationship between the numbers in the top line and the numbers in the bottom row. Once you have discovered the relationship then find the missing numbers. 1 3

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5

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What is the relationship?

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Strand: Chance and Data

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11. Jessica and Louise began to read the same book on Friday. Jessica reads 11 pages a day while Louise reads 7 pages a day. Draw a table using the information above to find: a) on which day Jessica will finish reading 62 pages; b) how many pages Louise has read by the end of the same day.

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Student Information Card

Make a List

This strategy can be used to help you sort out the information you have been given in the problem. Once the information has been sorted then you can look at all of the possibilities for the solution and then attempt to solve the problem more easily.

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Example: How many four digit numbers can be formed using the digits: 1, 1, 9 and 9?

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2. Make a List. List down all of the possibilities.

3.

1199 1919 1991 9191 9119 9911

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1. What do you know already? You know that you have to make four digit numbers and you can only use the numbers 1, 1, 9 and 9.

© ReadyEdPubl i cat i ons •the f osolution. rr evi ew pur posesonl y• Find Check your answers and count how many different numbers you can make. The answer is 6.

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4. Correct solution. Does your solution answer the question? Yes, you can make 6, four digit numbers using the digits 1, 1, 9 and 9.

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Make a List: Number

Strategy: Make a List 1.

Strand: Number

Katie and Donna began to deliver newspapers to earn some money. They worked a different number of days, but they still earned the same amount of money in the end. Make a List using the following information to find out how many days each girl worked: Katie earned \$15 a day; Donna earned \$10 a day; Donna worked five more days than Katie. ....................................................

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Strategy: Make a List

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Strand: Number

2. Write a list of all the possible sets of three different numbers that can be chosen from 1, 2, 3, 4, 5, 6, 7, 8, 9; that when you add them together the sum is equal to 15. How many sets of these contain the number 5? ....................................................

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3. Jason has a jar of 5 cent, 10 cent and 20 cent pieces. He wants to buy a 50 cent ice cream. In how many ways could he make change for 50 cents? Make a List.

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Make a List: Measurement

Strategy: Make a List

Strand: Measurement

4. These shapes are each made up of four squares. a)

b)

c)

d)

e)

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List the perimeter and area of each shape.

a) .................................................................................... b) ....................................................................................

c) .................................................................................... d) ....................................................................................

© ReadyEdPubl i cat i ons • f orr evi ew pur posesonl y• Which shape or shapes have the smallest perimeter?

e) ....................................................................................

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Strategy: Make a List

Strand: Measurement

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5. Melinda and Rachel are going to the movies. They are trying to decide which is the shortest way to get there. On another sheet list the different ways for Melinda and Rachel to get from their home to the movies. 2 km

Home

Supermarket 1 km

2k

4 km 1k m

km

2 km

Park 1

School

Ice Cream shop m

Movies

3 km 2

km

Fish & Chip shop

What is the shortest route? ...................................................................... Ready-Ed Publications

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Make a List: Space

Strategy: Make a List

Strand: Space

6. Gemma has two red balls, two blue balls, a green shoe box and a brown shoe box. She can place the balls in the boxes any way she likes. Make a list of all the different arrangements of balls in the boxes Gemma can make. ...................................................................... ......................................................................

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Strategy: Make a List

Strand: Space

7. Cameron can choose several different routes to walk to school.

Exactly how many different routes can he choose from? ............................ Playground

Shops

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Strand: Space

o c . che e r o t r pentagonal rectangular pentagonal s s r u e p pyramid prism prism

8. Observe the 3D shapes below and make a list of the number of corners each shape has. Circle the name of the shape which has the most corners? cube

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Make a List: Chance and Data

Strategy: Make a List

Strand: Chance and Data

9. There are 3 girls and 3 boys in the school student council. The principal wants a group of 3 student councillors to represent the school at the community’s ANZAC ceremony. Of the possible groups that could be chosen, how many could contain at least one boy? Make a List below. ....................................................

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Strategy: Make a List

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Strand: Chance and Data

10. How many three-digit numbers can be made using the digits 0, 1, 2 and 3 if no repetitions are allowed? Write them below. ....................................................

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11. Using a deck of 52 playing cards, how many ‘four of a kind’ combinations can you make?

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Student Information Card

Find a Pattern

When you use this strategy you will have to look for a pattern in the information that you have been given. Once you have found the pattern you can predict what will come next and what will happen again and again. You can then continue the pattern to find the correct solution.

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Example: Mandy had finally saved enough money to buy a worm farm. The worm farm could hold a total of 1000 worms. When Mandy bought the worm farm, there were 15 worms in the farm. If the number of worms in the farm was 30 in the second week Mandy had the farm and 60 the week after that, in how many weeks time will the farm be full? 1. What do you already know? You know that Mandy started with 15 worms. In the second week she had 30 worms in her worm farm, and then she had 60 worms in the third week. 2. Look for a pattern. Week: 1 2 No. Worms: 15 30

3E 4P 5b 6a 7o 8 © Readyd u l i c t i n s 60 120 240 480 960 1920 •you f o rr evthe i e w pur phave os es on l y• (When write down information you been given you

can notice the number of worms doubling each week. Once you’ve found that pattern, you can continue the pattern to find the solution.)

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3. Find the solution. Use the numbers in the pattern you have continued to find the solution. The worm farm will be full during the 8th week.

4. Correct solution. Does your solution answer the question? Yes. Mandy’s worm farm will be full in 8 weeks.

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Find a P atter n: Number Patter attern:

Strategy: Find a Pattern

Strand: Number

1. Use a calculator to find the product of the following numbers: 7 x 9, 77 x 99, 777 x 999 Observe the pattern in the answers and, without a calculator, try to predict what two numbers give the product of 77 762 223.

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Observe the pattern again and, without a calculator, predict the product for 77 777 x 99 999.

Strategy: Find a Pattern 2. Find the pattern.

a) 3, 4, 7, 11, 18, 29, ____, ____, ____

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© ReadyEdPubl i cat i ons c)• 102, 105, 111, 114, 120, p 123, ____, ____, ____ f o rr ev i ew u129, r po se s____, on____, l y• b) 3, 7, 16, 32, 57, 93, ____

d) Make a number pattern of your own. Ask a classmate to work out the pattern.

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o c . ch e 8, 4, 9 10, 5, 10 12, 6, 11 6, ____, ____ r er o st super Explain the rule:

3. In each set of numbers, every set follows the same rule. Find the rule and complete the final set.

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Find a P atter n: Measur ement Patter attern: Measurement

Strategy: Find a Pattern

Strand: Measurement

4. Ben made up a rule to use whenever someone in his class borrowed one of his pens. If his classmate borrowed his pen for 1 day, it would cost 1 cent; for 2 days it is 2 cents; for 3 days it is 4 cents; for 4 days it is 8 cents, and so on. If Ben’s friend Peter owed him \$1.28, for how many days must Peter have borrowed Ben’s pen? Complete the table to find the pattern. Day

1

2

Amount 1

2

1

3

4

4

8

7

15

5

6

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Strategy: Find a Pattern 5.

Strand: Measurement

This is a triangular grid. It is made up of 1 triangle and has an area of 1 triangle. Look at this diagram. Since 4 grids fit into this larger triangle, its area is 4 triangular grids.

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Fill in the table and discover the patterns to answer the following question: What is the perimeter and area of a triangle whose sides measure 9 units?

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6

4

3 4 5

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Find a P atter n: Space Patter attern:

Strategy: Find a Pattern

Strand: Space

6. Boxes of various shapes and with different patterns on them have been moved to a different position either by turning them end-over-end or pushing them along. Observe the boxes below, and the patterns, to work out how the boxes have been moved (either turned or pushed). Describe the pattern. a)

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Turn

c)

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Strategy: Find a Pattern 7.

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many small triangles would you have altogether?

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Strand: Space

8. a) How many squares are contained in a 2 x 2 square grid? .................. b) How many squares are contained in a 3 x 3 square grid? .................. c) How many squares are contained in a 4 x 4 square grid? .................. d) How many squares are contained in a 7 x 7 square grid? .................. Ready-Ed Publications

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Find a P atter n: Chance and Data Patter attern:

Strategy: Find a Pattern

Strand: Chance and Data

9. Without actually drawing or making the pattern, but instead finding the patterns in the table, work out the number of steps in the largest stair pattern that can be made if you have 100 matches. No. Steps 1

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10 18

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28

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40

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10. Joining the midpoints of the sides of a square forms a smaller square (see the diagram). The second square has half the area of the first square.

Repeating this gives this pattern. If the third square in the pattern has a side with a length of 8 cm, find the area of the sixth square.

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o c . e Use this space c toh show your r er o working: t s super 8 cm

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Student Information Card

Work Backwards Working backwards is an excellent strategy to use when the final outcome of the problem has already been given. All you need to do is work out what occurred previously to produce the given result.

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Example: Your mum went into the city to go shopping. When she got home she looked in her purse and found that all she had left were two one dollar coins, a twenty dollar note, a five dollar note, one fifty cent coin, two twenty cent coins and a ten cent coin. She remembered that she had only spent money twice that day. She spent \$15.05 on petrol and some money on food. When she counted the money left in her purse, your mum realised that it was exactly half of the amount of money she left home with in the morning. How much money did she spend on food during her shopping trip in the city?

1. What do you already know? You know that your mum was left with two one dollar coins, a twenty dollar and five dollar note, one fifty cent, two twenty cent and one ten cent coins. She had spent \$15.05 on petrol and some money on food. The amount she had left was half of the amount she had at the beginning of the day.

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2. Work backwards and find the solution. The first step would be to work out the amount of money your mum had at the end of the day and work backwards from there. y

\$20.00 + \$5.00 + \$1 + \$1 + 50 c + 20 c + 20 c + 10 c = \$28.00

y

If \$28.00 is half of what your mum started with then your mum must have started with 2 x \$28.00 = \$56.00

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o c You know that your mum spent \$15.05 on petrol so;. c e he r \$56.00 - \$15.05 = \$40.95 o t r s s r u e p You know that your mum had \$28.00 left so; \$40.95 - \$28.00 = \$12.95 your mum must have spent \$12.95 on food that day.

3. Correct solution Check your calculations. Does your solution answer the question? Yes. Your mum spent \$12.95 on food during her shopping trip in the city.

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Work Backwar ds: Number Backwards:

Strategy: Work Backwards

Strand: Number

1. Joe emptied his wallet at the end of his busy day. He had gone out with his friends and had spent some of the money he had been given for his birthday. When he emptied his wallet he found that all he had left were 2 five cent coins, 1 fifty cent coin, 1 twenty cent coin, 2 ten cent coins, 2 two dollar coins and 2 ten dollar notes. When he added up the money he realised that he had spent half of the amount of money he had started with that morning. Joe had only spent money twice that day - \$14.35 to get into the movies and some on lunch.

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How much money did he spend on lunch that day?

Strategy: Work Backwards

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2. My dad is now five times older than me. How many years ago was it when I was 2 years old and my dad was 34? Use a table to help you solve the problem.

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3. a) What number am I? If you add 6 to the number and then halve the result the number you get is 7.

b) What number am I? If you add 9 to the number and then multiply it by 3 the number you get is 36. .............................................

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Work Backwar ds: Measur ement Backwards: Measurement

Strategy: Work Backwards

Strand: Measurement

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Strategy: Work Backwards

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4. When Jim was walking across a bridge he noticed a wooden marker in the water. The water level was up to the –2.4 metre mark. Jim knew that the level at which the river flooded was marked on the wooden marker as being zero. A week later, after some heavy rainfall, Jim noticed the water level of the river had changed to 2.2 metres on the marker. By how many metres had the water level risen due to the heavy rain?

Strand: Measurement

5. Sarah and Catherine arrived home after school at 4.45 pm. They didn’t walk straight home from school, but stopped at a few places on the way. It took Sarah and Catherine 5 minutes to walk from school to the ice cream shop. They took 10 minutes to choose and eat an ice cream at the shop. They then walked 7 minutes to their friend Emma’s house. Together the 3 girls walked 3 minutes to the park and played together there for 30 minutes. They walked back to Emma’s house and listened to some music for 20 minutes. It then took Sarah and Catherine 12 minutes to walk home. At what time did the two girls leave school?

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Strand: Measurement

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6. Lara’s dad cut up a length of rope. He cut the length of rope in half and used one half. He then cut off one third of the other half and used it. If this piece of rope was 10 metres long, how long was the original rope?

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Work Backwar ds: Space Backwards:

Strategy: Work Backwards

Strand: Space

7. a) How many matches have been removed from the first arrangement to form the second arrangement? .....................................................

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b) How many triangles are in the original arrangement? .........................

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Work Backwar ds: Chance and Data Backwards:

Strategy: Work Backwards

Strand: Chance and Data

9. Travis has been able to save \$60 to buy Christmas presents for his friends and family. To save this money Travis put \$4 into his money box each week. Work backwards and complete the data in the table to find out how much Travis had saved 6 weeks ago. Weeks

1 week 2 weeks 3 weeks 4 weeks 5 weeks 6 weeks 7 weeks ago ago ago ago ago ago ago

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\$60

Strategy: Work Backwards

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Amount Saved

This week

Strand: Chance and Data

10. Mrs Samson conducted a survey in her class to find out what sort of pets were owned by the students in her class. There were 28 students in her Year 5 class. 6 students owned a cat only; 10 students owned a dog only; Half the amount of students who owned a dog owned a cat and a dog; Half the amount of students who owned a cat owned a dog, a cat and a bird; No students owned a cat and a bird; 2 students owned a bird only; 1 student had no pets at all.

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Student Information Card

Logical Reasoning This strategy will help you to use the information you have been given in the question to eliminate possible solutions to finally discover the correct solution. Example: What is the largest two digit number that 3 will divide into (divisible by 3) whose digits differ by 2?

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1. What do you know already? You know that the number must be less than 100 as it is a two digit number. It is divisible by 3. The digits of the number differ by two.

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2. Eliminate possibilities. Firstly, you could write down the numbers from 10 to 99 and then eliminate possibilities. a) Eliminate the numbers that are not divisible by 3 with: — b) Eliminate the numbers whose digits don’t differ by 2 with: 10 11 12 13 14 15 16 17 18 19 20

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93 94 95 96 97 98 99 3. Find the solution. Look at the remaining numbers. Which number is the highest? 75 Is your answer a two digit number? Yes Is your answer divisible by 3? Yes Do the digits in your number differ by two? Yes. Is 75 the highest remaining number? Yes Then 75 is the correct solution.

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Logical Reasoning: Number

Strategy: Logical Reasoning

Strand: Number

1. In the middle of a round pond lies a water-lily. The water-lily doubles in size every day. After exactly 20 days the pond will be completely covered by the water-lily. After how many days will half of the pond be covered by the water-lily?

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Strategy: Logical Reasoning

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2. Sandra, Amanda, Melanie and Cassandra are all best friends so they buy presents for each other for their birthdays. How many presents are bought altogether and how many presents must each person buy?

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Strand: Number

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3. What number am I? I am a two digit number. The sum of my digits is 3. I am exactly divisible by 5 and 2.

What number am I? I am a square number. I am a two digit number. The sum of my digits is 7. I am divisible by 5.

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Logical Reasoning: Measurement

Strategy: Logical Reasoning

Strand: Measurement

4. Find the area of this shape and explain how you worked out your answer. The area is .................... . I worked this out by: .....................................

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Strategy: Logical Reasoning

Strand: Measurement

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5. Sally’s mum wanted to bake some chocolate chip cookies. The problem was that she had to bake the cookies for 10 minutes but she only had two egg timers - a 4 minute timer and a 7 minute timer. Sally’s mum decided that she wouldn’t be able to make the cookies after all since she couldn’t bake them for the correct time. Sally explained to her mum that she could still bake the cookies for the correct time using the two egg timers. Explain below how Sally’s mum could use the timers to measure exactly 10 minutes?

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Strand: Measurement

more than Jenny. Jenny weighed 6 kg less than Shane.

How much more does Dave weigh than Shane? ..................................................................

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Logical Reasoning: Space

Strategy: Logical Reasoning

Strand: Space

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7. In your classroom the desks are arranged into rows, with an equal amount of desks in each row. Your desk has three desks to its left and four to its right. It is third from the front and second from the back. How many desks are there altogether in your classroom? Draw the classroom layout below.

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8. A cube is painted and then cut into 64 smaller equal sized cubes.

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How many smaller cubes have:

a) no faces painted?................................................................. b) exactly one face painted? ..................................................... c) three or more faces painted? ................................................

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Logical Reasoning: Chance and Data

Strategy: Logical Reasoning

Strand: Chance and Data

9. You and your sister have to catch the 8 o’clock bus to get to school in time. Your watch is 10 minutes slow, but you think it is 10 minutes fast. Your sister’s watch is 10 minutes slow, but she actually thinks it is 5 minutes slow. If you and your sister believe your watches and leave to catch the bus, who misses the bus and by how many minutes? Explain your answer.

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Strand: Chance and Data

10. Harry was given a bag of jellybeans. He grabbed a handful and counted how many he took. He found that he had 20 less jellybeans in his handful than were left in the bag. Harry then decided to count how many of each colour of jellybean he had grabbed out of the bag. He counted 3 pink, 5 green, 4 red, 2 yellow and 1 orange. How many jellybeans were there originally in the whole bag of jellybeans?

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o c . che e List the different exact weights you can measure using the three r o t r weights. You can use the weights on their own or put them together to s s r u e p combine the weights.

11. You have been given three weights. The first one weighs 1 gram, the second weighs 5 grams and the last one weighs 8 grams.

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Student Information Card

Solve an Easier Version Sometimes a problem is too difficult to solve in one step. When this happens you may be able to make the problem simpler by dividing the problem into smaller, easier steps, by rewording the problem or by using smaller numbers.

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Example: A crowd of people watching the Christmas Pageant fills the footpaths on both sides of the street for a distance of 2 kilometres. The footpaths are 3 metres wide and an average person needs 1 square metre to stand on. What would be a good estimate of the crowd?

2. Solve an Easier Version. You can break this problem into smaller steps and use easier numbers; you can work out that 3 people would take up 1 metre of the footpath.

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1. What do you already know? You know that the crowd stretches for 2 kilometres, the footpath is 3 metres wide and an average person needs 1 square metre to stand on.

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If 3 people take up 1 metre of the footpath, how many would take up 10 metres? 10 x 3 = 30 If 30 people take up 10 metres, how many take up 100 metres? 10 x 30 = 300 or 100 x 3 = 300 How many people would take up 1000 metres or 1 km? 1000 x 3 = 3000

3. Solve the problem. Use what you have found out to now solve the original, more difficult problem. 2000 x 3 = 6000. Now the crowd is on both sides of the street so 6000 x 2 = 12 000.

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Check your calculations. Does your solution answer the question? Yes. A good estimate of the crowd would be 12 000 people stretching 2 kilometres on both sides of the street.

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Solve an Easier V ersion: Number Version:

Strategy: Solve An Easier Version

Strand: Number

1. A Town Council sent 10 workers to empty rubbish bins in their area. If those 10 workers can empty 1000 bins in 4 days, how many days would it take for them to empty 4950 bins? ...................................... ......................................

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Strategy: Solve An Easier Version

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2. A passenger train had 9 carriages, each carrying the same number of passengers. Three carriages broke down so each of the other carriages had to carry 6 more passengers. How many people were in each carriage originally? You might like to draw a diagram to help you.

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3. In a high school there are 1000 students. Each class has 25 students and 1 teacher. If each student must attend 6 classes a day and each teacher must teach 5 classes a day, how many teachers are there in the school? Hint: Start by working out how many classes of 25 students there are if there are 1000 students altogether. Then work out how many classes are given in a day if each student must attend 6 classes a day.

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Solve an Easier V ersion: Measur ement Version: Measurement

Strategy: Solve An Easier Version

Strand: Measurement

4. James ran in the 200 metre race at his school sports carnival. If James ran the 200 metre race in 63 seconds, how long would it take James to run 2 kilometres if he was able to run at the same speed?

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Strategy: Solve An Easier Version 5. If 500 grams of flour is used to make a 25 centimetre cake, how many grams are used for a 40 centimetre cake?

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6. Janet’s mum has made and decorated 25 circular jewellery boxes for the school’s Mother’s Day stall. Janet’s mum wants to complete the boxes’ decorations by gluing a length of ribbon around the circumference of each box. If the circumference of each box is 30 centimetres and the ribbon costs 60 cents per metre, how much will Janet’s mum have to spend to buy enough ribbon to decorate the 25 jewellery boxes? ......................................

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Solve an Easier V ersion: Space Version:

Strategy: Solve An Easier Version

Strand: Space

7. How many squares in this picture?

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8. How many different paths can be traced from point A to point B if the movement can only be vertical or horizontal? ......................................

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9. The following diagram is made up of many triangles. How many triangles are there in this figure?

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Solve an Easier V ersion: Chance and Data Version:

Strategy: Solve An Easier Version

Strand: Chance and Data

10. There are five teams in a hockey competition. Each team plays each other team at their home ground. How many hockey games are played in a season? ...................................... Write the fixtures for the season.

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11. Ten children are sitting in a circle. Each child shakes hands with everybody else except the children sitting on either side of them. How many handshakes are made? Draw a diagram to help you.

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12. There are 4 students learning to play the trumpet. The trumpet teacher told the students that they will each have the chance to perform a duet with each of the other students. How many different duet performances will there be?

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List them below.

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Answers Guess and Check 1. Sandy has ten 50 cent coins and five 20 cent coins. 2. I am the number 30. 3. Two door cars: 6; Four door cars: 11. 4. 2 possibilities: 4 x 4 unit square Area and perimeter = 16

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There are 8 arrangements.

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6 x 3 unit rectangle Area and perimeter = 18

Fill the 7 L bucket. Pour water to fill the 3 L bucket. There will now be 4 L left in the 7 L bucket. There will be 7 cubes in pile one, 3 cubes in pile two, 4 cubes in pile three and 6 cubes in pile four. Arrangement is as below:

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The mirror would be placed on the dotted line.

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Create a Diagram 1. The ladder had 35 rungs altogether. 2. 5 students had orange juice only. 3. Grace can make 6 different patterns: G = Green P = Purple GGPP GPPG GPGP PGPG PGGP PPGG 4. The length of the playing field is 200 m. 5. The gardener must dig a further 1 m and 10 cm. 6. There would be 6 different pairs of rods: 1 x 10cm and 2 x 5cm 1 x 10cm and 5 x 2cm 10 x 2cm and 2 x 10cm 5 x 2cm and 2 x 5cm 2 x 10cm and 4 x 5cm 7. There are 4 more shapes you can make:

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11. Jenny could have rolled the following combinations of dice: 2 and 4; 5 and 1; or 3 and 3. 12. Mrs Jackson bought 3 kg of oranges and 2 kg of apples.

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You can arrange the fish in 6 different ways:

9. The least number of boat trips would be 5. 10. Jeremy could make 27 different combinations for his flag. 11. John is sitting in between Alex and Luke. Page 46

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Use a T able Table 1. Mark could wear 8 different three-piece outfits. 2. All possibilities include: 4 and 24; 6 and 16; 8 and 12. 3. Caleb has 24 rocks and Jordan has 16 rocks. 4. 5 c 10 c Total coins Total amount 2 1 3 20 c 3 6 9 75 c 6 1 7 40 c 3 5 8 65 c 8 1 9 50 c 5.a) Side (cm) Perimeter (cm) Area (sq. cm) 1 4 1 2 8 4 3 12 9 4 16 16 5 20 25 b) 6cm: P = 24 A = 36 7cm: P = 28 A = 79 10cm: P = 40 A = 100 6. 90 deg. Angle \$50 3.00am \$50 9.00am \$50 3.00pm \$50 9.00pm \$50 Total = \$200 7. Length Width Area 1 cm 120 cm 120 cm2 2 cm 60 cm 120 cm2 3 cm 40 cm 120 cm 2 4 cm 30 cm 120 cm2 5 cm 24 cm 120 cm2 6 cm 20 cm 120 cm2 8 cm 15 cm 120 cm2 10 cm 12 cm 120 cm2 Pile 2 Pile 3 8. Pile 1 2 4 18 2 6 16 2 8 14 2 10 12 4 4 16 4 6 14 4 8 12 4 10 10 6 6 12 6 8 10 8 8 8 There are 12 different arrangements of lollies. 9. Team Games Games Games Games Points for Points for Total played won drawn lost wins draws points Red 4 3 1 0 6 1 7 Green 4 2 1 1 4 1 5 Yellow 4 0 3 1 0 3 3 10. 1 2 3 4 5 9 12 18 3 5 7 9 11 19 25 37 Relationship: In the first column the number has been increased by 2, in the third column the number is increased by 3, the fourth column by 4 and so on. 11. Jessica will finish reading 62 pages on Thursday. Louise would have read 42 pages by the end of the same Thursday. Make a List 1. Katie worked for 10 days and Donna worked for 15 days. They each earned \$150 each. 2. There are 7 sets of numbers that when added together make 15. 3 of those sets contain the number 5: 9,1,5 8,1, 6 7,2,6 9,2,4 8,2,5 7,3,5 8,3,4 3. There are 12 ways to make 50 cents. 4. e) has the smallest perimeter. (Perimeter = 8cm.) 5. The shortest way to get to the movies is: home - supermarket - ice cream shop - movies. Total distance = 5 km. 6. Gemma can make 8 arrangements. 7. Cameron can choose 7 different ways to get to school. 8. The pentagonal prism has the most corners.

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Answers cont. Make a List cont. 9. 19 possible groups of student councillors could be chosen. 10. 24 three-digit numbers can be made. 11. 13 different combinations of ‘four of a kind’ can be made. Find a P attern Pattern 1. a) 63, 7623, 776223 b) 7 777 x 9 999 c) 7 777 622 223 2. a) 47, 76, 123 b) 142 c) 132, 138, 141, 147, 150 3. 3, 8. The rule: Halve the first number and then add 5. 4. Peter had borrowed Ben’s pen for 8 days. 5. Perimeter = 27 units. Area = 81 triangular units. 6. a) Turn, turn, turn. b) Turn, push, turn. c) Push, turn, turn. 7. There would be 100 small triangles altogether. 8. a) 4 b) 9 c) 16 d) 49 9. The largest stair pattern you could make if you had 100 matches would be an 8 step pattern (a 9 step pattern uses 108 matches). 10. The area of the sixth square = 1cm2.

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Work Backwards 1. He spent \$10.65 on lunch. 2. It was 6 years ago. 3. a) The original number is 8. b). The original number is 3. 4. The water level had risen by 4.6 m. 5. The girls left school at 3:15pm. 6. The rope was originally 30 metres in length. 7. a) 6 matches have been removed. b) There are 10 triangles in the original arrangement. 8. There would be 12 smaller cubes with 2 sides painted. 9. Travis had saved \$36 six weeks ago. 10. 2 students owned a dog and a bird.

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Solve An Easier V ersion Version 1. It would take 20 days for 10 men to clear 4950 bins. 2. There were 12 people in each carriage in the beginning. 3. There are 48 teachers in the school. 4. It would take James 10 minutes and 30 seconds. 5. 800g would be used. 6. Janet’s mum will need 750 cm of ribbon which will cost a total of \$4.50. 7. There are 30 squares in the picture. 8. 16 different paths can be traced. 9. There are 27 triangles altogether in this figure. 10. 40 hockey games are played. 11. 70 handshakes are made. 12. There will be 6 duet performances.

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Logical Reasoning 1. After 19 days, half the pond will be covered by the water-lily. 2. 12 presents are bought altogether. Each person must buy 3 presents each. 3. 30, 25. 4. The area of the shape is 20 cm². The regions outside the 5cm x 4cm rectangle match the regions inside the rectangle making it a complete rectangle. 5. Start both timers together. When the 4 min. timer runs out put the cookies into the oven and let the 7 minute timer continue. This will run for a further 3 mins. When the 7 min. timer runs out, turn it over and let it start again to run for 7 mins. This will make a total of 10 mins. 6. Dave weighs 8 kg more than Shane. 7. There are 32 desks altogether in the classroom. 8. a) 8 b) 24 c) 8. 9. You will miss the train by 20 minutes. 10. There were 50 jellybeans originally in the bag. 11. You could measure 6 exact weights: 1, 5, 6 (1g + 5g), 8, 9 (1g + 8g) and 13 (5g + 8g).