Maths Problem Solving Series: Book 1

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Maths Problem Solving Series © ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

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By Val Morey

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Strategies and techniques covering all strands 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 462 8

Rationale This set of problem solving activities supports the aims of the Curriculum Corporation’s document, Mathematics - a curriculum profile for Australian schools - through which students learn to use the various strands of Mathematics to “describe, interpret and reason about their social and physical world”.

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An outcomes approach to mathematics education means students and teachers need many and varied opportunities for students to demonstrate their learning and their understanding. To be as accurate as possible in determining students’ current level of numeracy, those students must be given opportunities to demonstrate their understanding through activities which really show what they are able to do, as well as what they are not. Scoring 100% in a page of exercises tells teachers that a student has learnt what was taught, and can be useful information for checking that criteria have been met, but may not be a demonstration of their true or full mathematical understanding.

In order to make more accurate assessments of students’ understanding, and therefore place them carefully within the levels described in the curriculum document, activities are needed which address two important aspects: y Students must be able to show how well they can apply concepts or processes learnt to different contexts in order to solve a problem; and

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y They must be given opportunities to show the “upper limits” of what they know and understand.

The approach in this book allows for both of those requirements.

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At the same time, competence in mathematics should mean that students are able and willing to use mathematics in settings outside the maths lesson. This book supports that by providing a wide, cross-curricular context for the activities it contains.

The activities provided in this book are based on realistic situations which school students of age 9 - 10 could expect to be familiar with or to face.

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Therefore, the aim of having students “able to deal readily and efficiently with common situations requiring the use of mathematics” is addressed.

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.curriculum.edu.au/catalogue/

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Contents 2 4

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Rationale Teachers’ Notes Guess and Check Teachers’ Notes Number: Student Information Page Number: Have a Guess Measurement: The Longest Throw Measurement: An Errand Error Space: Skeleton in a Shoebox Chance & Data: Lucky Counters Chance & Data: Favourite Flavours Make a List Teachers’ Notes Number: Student Information Page Number: Disco Fever Number: Fashion Crisis Number: A Sweet Problem Chance & Data: Roll the Dice Measurement: Planning for Plants Space: Which Way? Find a Pattern Teachers’ Notes Number: Student Information Page Number: Number Patterns Measurement: Time After Time Measurement: Growing Beans Space: Try Tiling - 1 Space: Try Tiling - 2 Chance & Data: Take a Raincheck - 1 Chance & Data: Take a Raincheck - 2 Solve an Easier Version Teachers’ Notes Number: Student Information Page Number: Music Madness Number: Siren Solution Number: Swimming on the Bus Space & Measurement: Make It Easy on Yourself Measurement: Fancy Fences Chance & Data: Lucky Dip Draw a Diagram or Table Teachers’ Notes Space & Measurement: Student Information Page Space & Measurement: Shelf Space Space & Measurement: A Rubbish Problem Chance & Data: Red Pen Blue Pen Work Backwards Teachers’ Notes Number: Student Information Page Space & Measurement: Stop the Clock Logical Reasoning Teachers’ Notes Number: Student Information Page Chance & Data: What Number Am I? Space, Chance & Data: Fancy Dress Space: Follow the Map Answers

21 22 23 24 25 26 27 28 29

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Teachers’ Notes This book contains a collection of activities for students aged 9 - 10 to enable them to develop their Mathematical problem-solving skills. Strategies for solving problems are introduced individually and explained, so that these can be taught explicitly, practised and then applied to problems which relate to the students’ school or life experience.

The book is devised as a course in mathematical problem solving and should be worked through sequentially, rather than “dipped into”. It is divided into seven sections, one for each of the problem-solving strategies to be taught. There is no need for teachers or students to have prior familiarity with any of the strategies presented. At the beginning of each section there is an explanation of the strategy to assist teachers and students, and then students are guided through a ‘scaffolding’ approach to eventual independence. Although the tasks become progressively more complex in each section, all students should be able to manage the initial example and thus become familiar with the strategy. Subsequent problems could then provide extension and/or enrichment activities which are meaningful and interesting.

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Examples are provided for each of the strands of Number, Space, Measurement, and Chance & Data. The Working Mathematically strand is embedded in all the activities through the problem-solving approach used, where students need to choose and use operations, make decisions about their “plan of attack” and strategies to be employed.

All of the activities in the book address the sub-strand 3.3 (Using Problem Solving Strategies), of the Working Mathematically strand in the student outcome statements. Most are at Level 3 and this is indicated at the foot of each page by the notation WM 3.3. Level 4 has no outcomes specified for this strand.

Strand: Working Mathematically 3.3: Related Outcome

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• Students pose mathematical questions prompted by a specific stimulus or familiar context and use problem-solving strategies which include those based on representing key information in models, diagrams and lists.

However, as mathematical operations in Number, Space, Measurement and Chance & Data are also being practised, the relevant outcome statement for these will also be given for each activity.

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Teachers may therefore use the guideline footnotes to assist with levelling students after the activity has been completed. However, care should be taken when reading the outcome statement as not every element contained in the statement may be addressed by the activity. Completing the activity may not satisfy all facets of the outcome and more evidence may be needed.

On the other hand, if teachers are specifically looking for activities to support a sub-strand and/or level which has been identified in their planning, then the footnotes can be scanned for an appropriate activity.

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The activities provided represent excellent opportunities for teachers to see how students apply operational skills learned in Mathematics and they would be ideal to use as part of an assessment portfolio if required.

The specific outcome/s linked to the activity on that page will appear in full at the foot of each page to assist teachers in this.

Teachers may even wish to use these activities in curriculum areas other than Mathematics, as many of the problems could easily fit into a Society & Environment, Health or Technology & Enterprise program, for example.

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Guess and Check

Teachers’ Notes: Guess and Check

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This strategy is an excellent way to introduce students to a problem-solving approach to mathematics. Many of them will find that they have employed this strategy informally without recognising it as such, and therefore it will seem familiar and within their “comfort zone”. Validating “Guess and Check” as a method will be a terrific confidence boost and should help students feel able to approach the other strategies with a positive attitude. At the same time, the students will learn and practise sensible guessing and understand the importance of accuracy in the “check” part of the strategy.

As its name suggests, the students first of all guess an answer to the problem and then use that answer to check whether all requirements of the problem have been met. If not, then the answer is adjusted and checked again.

Teachers should stress that it is extremely unlikely that a student’s “guess” is going to be correct - that is not the point - and that the guess provides a starting point, which is the key to all problem-solving.

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Name: _____________________

Guess and Check: Number

Student Information Page When approaching problem solving, the main problem can often be figuring out “where to start”!

Sometimes the easiest and most sensible way is to simply take a guess at what you think the answer might be and then check to see if that’s possible. You’ve probably done something like that yourself. If someone told you that there were twice as many blue chairs as grey chairs in your classroom, and that altogether there were 33 chairs, could you tell them how many blue and how many grey chairs there were?

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r o e t s Bo r e p ok u You could soon find out by the “Guess and Check” method. Your guess is S probably going to be closer than you think, although it’s not likely it will be spot on.

Firstly, think about what you know:

You know one number has to be double the other and that they both have to add up to 33. Therefore, the easiest thing to do would be to start with a number for the grey chairs, double it for the blue, and then see if they both add up to 33. So, you won’t guess that there were 4 grey chairs, because then there must have been 8 blue, and together that makes only 12. Likewise, you won’t guess 50 for the grey as that means 100 blue, which is 150 altogether.

Already you have an idea of a range inside which the answer may lie.

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Here is an example of how this could have been done: 1st guess 8 16 24

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It can be useful to draw up a grid, based on what you know, and then use it to help you find the solution. As you get the answer for each of your guesses, decide whether your number of grey chairs needs to go up or down.

2nd guess 3rd guess 4th guess Answer . te o 14 10 12 Grey chairs 11 c . che e r 28 20 24 Blue chairs o (double) 22 r st super 42

30

36

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33

Understands mathematic conjectures as more than simply a guess, makes straightforward tests of conjectures and discards those that fail the test.

Calculates with whole numbers, drawing mostly on mental strategies to add and subtract two-digit numbers and for multiplications and divisions related to basic facts.

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Name: _____________________

Guess and Check: Number

Have a Guess Use the “Guess and Check” procedure to find the solution to this problem: Your new class has 30 students in total. There are four more boys than girls in the class. Use the “Guess and Check” method to find out how many boys and how many girls are in the class. Use the grid below to help you - remember, write in what you know first.

r o e t s Bo Answer r 3rd guess 4th guess e p o u Girls k S Boys

2nd guess

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1st guess

Total

The solution is 13 girls and 17 boys.

How many guesses did you need? _____

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It is your turn to help the Library assistant put away returned books on the shelves. She tells you that there are 7 more fiction books than non-fiction books to be replaced, and that altogether, 55 books need replacing. How many of each are there?

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Now have a try at solving the next problem by yourself. Use the grid below and write in what you know before you have your first guess.

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2nd c guess 3rd guess 4th guess Answer e her r o Fiction st super Non-fiction Total

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Understands mathematic conjectures as more than simply a guess, makes straightforward tests of conjectures and discards those that fail the test.

Calculates with whole numbers, money and measures, drawing mostly on mental strategies to add and subtract two-digit numbers and for multiplications and divisions related to basic facts.

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Name: _____________________

Guess and Check: Measurement

The Longest Throw You can use the “Guess and Check” strategy to solve practical problems of Measurement as well. The only difference is that you are working with length, mass, area or volume and capacity instead of just with numbers.

Try using the strategy to solve this:

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Solve the problem using the “Guess and Check” strategy.

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Your school sports day includes an event of “longest throw”, where each person throws a softball as far as they can. At a training session for the sports, one of your friends threw the ball 50 centimetres further than you. The combined distance of both of your throws was 49.5 metres. What was the distance of each of your throws?

Use this space to draw up a grid. Work out what headings you will need.

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Understands mathematic conjectures as more than simply a guess, makes straightforward tests of conjectures and discards those that fail the test.

Calculates with whole numbers, money and measures.

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Name: _____________________

Guess and Check: Measurement

An Errand Error Your teacher sends out two students on an errand to the office. She begins to wonder where they have got to, when one student then returns. Your teacher asks what’s happened to the other student and is told that he’s “just coming”.

The teacher asks everyone to watch the clock until he arrives, and he gets there 1 minute and 30 seconds later. “Do you realise,” asks the teacher, “that a total of 13 minutes and ten seconds of learning time has been lost between the two of you?” You put your hand up and ask how that could be because the two of them have been gone less than ten minutes, but your teacher replies that she is adding both friends’ lost time together.

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r o e t s Bo r e p ok u Can you work out how much time was lost by each student?

S Remember, draw up a grid for your guesses, entering what you know first.

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WM 3.2 WM 3.3 M 4.19

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Understands mathematic conjectures as more than simply a guess, makes straightforward tests of conjectures and discards those that fail the test.

Students measure area by counting uniform units including where part-units are required, and measures length, mass, time capacity and angle, reading whole number scales.

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Name: _____________________

Guess and Check: Space

Skeleton in a Shoebox To solve this next problem, you will need to use the “Guess and Check” strategy, but instead of writing numbers or using measurements, you will be building - each of your constructions will be a “Guess” until you make one that works. Just like when you were working with numbers, though, your “guesses” will get closer and closer and you will have some idea of how to start.

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You need to build a skeleton of a rectangular prism (that is, just the edges) which will fit inside a shoe box. You may use materials available such as straws, matchsticks, popsticks, etc. You may join these but you may not cut them and you cannot measure the shoe box first. Your aim is to make the best model, which takes up as much of the shoebox as possible, with the lowest number of pieces.

Record each of your tries - what you used, how many pieces, what was wrong - in the working space below. Use extra paper if you need it.

First try: Materials:

© ReadyEdPubl i cat i ons orr evi ew pur posesonl y• Problem: •f Number of pieces:

Materials:

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Number of pieces: Problem:

Third try: Materials:

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Number of pieces: Problem:

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Understands mathematic conjectures as more than simply a guess, makes straightforward tests of conjectures and discards those that fail the test.

Students attend to the shape, size and placement of parts when matching, making and drawing things, including making nets of 3D models which can be seen and handled and using some basic conventions for drawing them.

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Name: _____________________

Guess and Check: Chance & Data

Lucky Counters The “Guess and Check” strategy can be used to help determine how likely something is to happen, or not happen, or whether one thing is more or less likely to occur than another.

Your class sometimes uses counters to play maths games. You like using red counters (it’s your lucky colour), but you know that there are 20 more blue counters than red ones in the pack of 100.

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r o e t s Bo r How many chances ine ten do you have of getting a red counter? p ok u Explain your answer below.

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WM 3.2 WM 3.3

Understands mathematic conjectures as more than simply a guess, makes straightforward tests of conjectures and discards those that fail the test.

C&D 3.23 Places events in order from those most likely to happen to those least likely on the basis of numerical and other information about the events.

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Name: _____________________

Guess and Check: Chance & Data

Favourite Flavours For this next problem, you will need to “Guess and Check” to get the data (information) for a graph.

You need to use A4 1 centimetre graph paper to graph the results of a survey about students’ choices between three ice cream flavours. Each centimetre mark on the vertical axis of your graph should stand for 2 people. You decide how wide the columns should be.

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r o e t s Bo r Altogether 64 students tooke part in the survey. p o u 16 students said chocolate was their favourite, and this was k the least popular S choice. Of the rest, 4 more chose caramel than chose bubble gum.

Use the space below to problem-solve and then prepare the graph of the results on your graph paper.

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WM 3.2 WM 3.3

Understands mathematic conjectures as more than simply a guess, makes straightforward tests of conjectures and discards those that fail the test.

C&D 3.26 Displays frequency and measurement data using simple scales on axes and some grouping.

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Make a List

Teachers’ Notes: Make a List This strategy involves identifying all possibilities for a solution by listing them systematically. Sometimes this will need to be done just to find out how many different options are possible, and at other times the strategy is employed in order to have the information upon which to base a decision.

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The making of systematic lists enables information to be checked to ensure that all possibilities are valid and that all permutations have been included. It also helps to develop the logical and ordered thinking necessary for effective problem solving.

Students should find that list-making will be incorporated with other strategies as they become more adept at problem solving.

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Name: _____________________

Make a List: Number

Student Information Page This strategy is easy to use and helps you to “think straight”. Sometimes a problem does not involve any real calculating, but there is just too much information for you to handle in your head – so, you write it down.

Making a list means you can think about each part in turn without worrying that you will forget the parts that came before.

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Let’s say you wanted to buy a birthday present for your friend and your Mum said you could buy one box of Lego and one toy car. When you went to the toy shop, you found 5 different Lego sets and 4 different toy cars to choose from. How many different ways could you make a present of a toy car and a Lego set?

It would be very hard to try and think of all the ways you could do it without writing them down. But there is nothing difficult to actually work out. You only have to make sure you haven’t missed any items out.

Here’s how the “Make a List” strategy helps.

You need to find a system to write down all the possible ways you can do it and stay “in the rules”. The rule this time is to have one Lego set and one toy car.

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Start by listing all the Lego sets and toy cars separately: Pirate Pete Porsche Bob the Builder Mercedes Jack Stone Torana Dinosaur Commodore Space Vehicle

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Now start with the first Lego set and match it to each toy car in turn: Pirate Pirate Pirate Pirate

Pete Pete Pete Pete

with: with: with: with:

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Porsche Mercedes Torana Commodore

TOTAL: 4 combinations

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Now do the same for the next Lego set: c e he r Bob the Builder with: Porsche, Mercedes, Torana, Commodore.

Total = 4. o t r s super You should be able to see that for each one of the Lego sets, there will be 4 different ways you could make the present.

You know there are 5 different Lego sets, so there will be 5 lots of 4. Altogether, there are 20 different ways you could make a present from the toys at that shop.

Can you work out how many there would be if you found another different Lego set? Use the back of this sheet to make a list.

WM 3.2, N 3.12

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Name: _____________________

Make a List: Number

Disco Fever Try this one on your own now.

You are trying to decide what to wear to the school disco which is coming up soon. Your friends are coming over to help you decide and you get out all the pieces of clothing that you are going to choose from. Your mum comes into your room and says that there are loads of different outfits you can make from the four tops, two skirts, pair of jeans and pair of pants that you have laid out on your bed. It doesn’t seem like very much to you, so your mum suggests you work out how many different outfits are possible. She says that if it is less than 10, she’ll buy you something new.

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r o e t s Bo r e p ok u S Do you get any new clothes?

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List 1 ______________________________

List 2 ______________________________

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Combinations: (start with the first item from List 1)

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Recognises, describes and uses patterns involving operation on whole numbers, and follows and describes rules for how terms in a sequence can be linked by multiplication or an addition or subtraction based strategy.

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Name: _____________________

Make a List: Number

Fashion Crisis!

You are disappointed that you don’t get your new clothes, but then you point out to your mum that there isn’t really the number of outfits you worked out because one of the tops is red and white striped, and one of the skirts has a purple floral pattern, so you couldn’t possibly wear those two together. Also, when your friends come over, one of them tells you that she has a top exactly like one of yours - not the red and white one, and she is planning to wear it to the disco. Of course, that means you won’t be able to wear yours. Mum is starting to wish she hadn’t said what she did! Will you get something new now?

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r o e t s Bo r e p ok u List the combinations S that you can make.

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WM 3.2, 3.3 N 3.12 Recognises, describes and uses patterns involving operation on whole numbers, and follows and describes rules for how terms in a sequence can be linked by multiplication or an addition or subtraction based strategy.

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Name: _____________________

Make a List: Number

A Sweet Problem The strategy of making a list can also be useful for finding out the number of possible ways things can be placed in order. For instance, if you had a lollipop and an ice cream, no doubt you can see that there are two possible orders in which you can eat them - although only one sensible way, if you don’t much like melted ice cream!

r o e t s Bo r e You can check by p making a systematic list. Start byo writing the first item u k first and then write the other two in two different orders, like this: S Mars Bar, snake, lollipop Mars Bar, lollipop, snake

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However, if you had a Mars Bar, a giant snake and a lollipop, would there then be three possibilities for the order in which you could eat them?

That’s two possible ways. Now write the second item first and do the same thing: Snake,

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So, you can see that there are actually six possible orders.

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© ReadyEdPubl i cat i ons That’s four possibilities. Now the third item first: •f orr evi ew pur posesonl y• Snake,

Now try the question below. Use the same listing system that you did for the lollies, but be careful, because now there are 4 items - you may be surprised how much difference that makes!

. te o c You are colouring a border on your work and want to. make a repeat ccolours e pattern with theh red, yellow, blue and green.

In how many r e o r different orders could you use colours? st s r uthe pe

WM 3.2, 3.3 Recognises, describes and uses patterns involving operation on whole numbers, and follows and describes rules N 3.12 for how terms in a sequence can be linked by multiplication or an addition or subtraction based strategy.

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Name: _____________________

Make a List: Chance & Data

Roll the Dice You can use list-making to give you the information you need to determine how likely something is to happen. Systematic lists will enable you to “work out” answers that will seem amazing to people trying to do it in their head.

To solve the next problem, just make a list as you did before. The only difference is that this time you have to think of what all the possible results are, before you list them. Then you look at your information to decide how likely it is that one particular thing will happen.

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r o e t s Bo r e ok- then you can Take your time, think of p one thing at a time, and write it down u check back to see if you’ve thought of everything.

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You are playing a board game with a friend, using two dice. He needs a three to finish and you need an eight. It is your turn. What are your chances of throwing an eight?

(Hint: Start by thinking: What if I throw a one with the first die? Think of all the possibilities. Then: What if I throw a two with the first die, and so on).

If you don’t get your eight, has your friend got the same chance as you had of getting his three?

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WM 3.3 C&D 3.26 Interprets and makes numerical statements of probability based on lists of equally likely outcomes and using fractions and percentages.

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Name: _____________________

Make a List: Measurement

Planning for Plants In your Science lessons, you are going to be working in pairs growing alfalfa seeds and experimenting with the amount of water, fertiliser and light each patch receives. Each pair will have their own patch of seeds measuring 36 cm². Your teacher has asked each pair to make their rectangular patch a different rectangle to those of other pairs - that is, vary the lengths of the sides, while still having 36 cm². You and your partner decide to have a square of 6 cm x 6 cm.

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r o e t s B r e oodifferent patches If you only measure in whole centimetres, how many p could be made? u k S

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Understands and applies directly length, area and volume relationships for shapes based on rectangles and rectangular prisms.

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Name: _____________________

Make a List: Space

Which Way? You are planning to go to the shops after school with your friend and want to go to the ice cream kiosk, the video game parlour and the newsagent.

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Use coloured pencils to draw all the different routes you could take to visit all these shops. You should draw arrows to show the direction you are taking each time.

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Uses distance, direction and grids on maps and plans and in descriptions of locations and paths.

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Find a Pattern

Teachers’ Notes: Find a Pattern Much higher order mathematical thinking involves applying patterns which are understood and known to be true, and mathematicians are used to looking for and “seeing“ patterns as part of their approach to their work.

Thus, developing a facility for patterning is an important facet of developing numeracy.

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r o e t s Bo r e Students will no doubtp be used to exercises in which they have to identify, and o u k then possibly continue, patterns in number or shape. However, in many of S these exercises, merely finding the pattern is the point.

In this section, students practise finding patterns in order to solve a problem or reach a solution. It is the application of the pattern which is important.

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Name: _____________________

Find a Pattern: Number

Student Information Page You have probably seen questions in Maths books which ask you to find the next number or the next shape in a pattern. You would find it easy to do this one: Continue the pattern: 10, 20, 30, ________________________

r o e t s Bo r e p ok u Spattern which went square, circle, triangle, square, Or if it was a shapes Teac he r

Once you know the pattern, you don’t even need to know every number to be able to fill in a gap further along the pattern. If you had to say what the 8th number in the above pattern would be, you could say it would be 80 because you know 8 lots of 10 is 80.

Complete this pattern now:

Circles would be the 2nd, 5th, ___, ___, ___, Triangles would be the ___, ___,___, ___,

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circle, triangle, etc, you can see that squares would be the 1st, 4th, 7th, 10th shapes and so on.

© ReadyEdPubl i cat i ons a) What shape would be in the 21 place? _______ •f orr evi ew pur posesonl y• st

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These are both quite simple patterns because only one thing is being done at a time. The numbers are increasing by ten every time, and the shapes stay in the same repeated order.

Sometimes, though, number patterns can be a bit harder to figure out because more than one thing is being done to them – different operations, which just means adding, subtracting, multiplying and dividing.

. te otaken c In this number pattern for you to try, two different operations. have ch place each time, in order to get to the next number. Butr ite is a pattern e o r because the same two things have been done.

See if you can work it out: st s up er 2, 5, 11, 23, 47, ____________

b) What do you think the pattern is? __________________________________ _________________________________________________________________ WM 3.3 N 3.12 Recognises, describes and uses patterns involving operation on whole numbers, and follows and describes rules for how terms in a sequence can be linked by multiplication or an addition or subtraction based strategy.

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Name: _____________________

Find a Pattern: Number

Number Patterns Here’s an even more difficult one for you to try. This time, there are still two different operations on the numbers each time, but one of the operations does not stay exactly the same – but there is a pattern in how it changes! a)

10, 11, 11, 10, 8, 5, 1, ____________

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Find the pattern, using the space below for your working.

The key to solving number pattern problems or puzzles is finding the relationship between the numbers. Once this relationship has been found, you can use the pattern to work out answers that would take a lot of time.

Think (and look) back to when you made a list of all the possible ways you could make your repeat pattern on the border of your work, if you had 4 different colours. What if you had 5 colours? Or 6? Or your whole packet of Textas? You should be able to realise that you would need a very, very long list and it would take you an awfully long time.

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So, perhaps there is an easier way! Let’s have a look at the pattern which comes from this.

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When you had 2 things, the number of possible orders was: When you had 3 things, the number of possible orders was: When you had 4 things, the number of possible orders was: If you had 5 things, the number of possible orders would be:

2 6 24 120

. tthe number of possibilities found each time by theonew number If you multiply e c . of things, you will find how many new possibilities there are. cout e h r e o t r 2 things: 2 possibilities, x 3 (new number ofr things) = 6. s s u e p 6 possibilities, x 4 (new number of things) = 24.

24 possibilities, x 5 (new number of things) = 120.

b)

If you had 6 things, how many possibilities would there be? ___________ Imagine how long it would have taken to make a list of all of those!

WM 3.3 N. 3.12

Recognises, describes and uses patterns involving operation on whole numbers, and follows and describes rules for how terms in a sequence can be linked by multiplication or an addition or subtraction based strategy.

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Name: _____________________

Find a Pattern: Measurement

Time After Time The siren for the start of school sounds at 8:50 am each morning. You notice that the clock in your classroom shows 8:53 am, and then when you come back in from lunch at 12:50 pm, the clock is showing 12:55 pm. At 10 minutes to 3, the clock shows 4 minutes to 3.

Next day, when you get to school, your teacher is getting the clock off the wall to reset it. You tell him that if he waits until the siren goes, he can set it back an exact 15 minutes. Your teacher is very impressed and asks how you can be so sure.

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r o e t s Bo r e p ok u Can you explain?

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Recognises, describes and uses patterns involving operations on whole numbers and follows and describes rules for how terms in a sequence can be linked by multiplication or an addition or subtraction based strategy.

Measures by counting uniform units including where part-units are required, and measures length, mass, capacity, time and angle, reading whole number scales.

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Name: _____________________

Find a Pattern: Measurement

Growing Beans You are sprouting beans in a jar in your classroom and recording the rate of growth and the amount of water needed. On Monday, you set your bean in blotting paper in the jar, mark the side of the jar to show the level you need to top the water up to each day, which is 200 ml, and fill the jar to this level. You will use a small measuring cup which holds 100 ml to top up your water each day.

r o e t s Bo r e p ok u S When you get to school on Monday morning, you ask if you can have the half litre container to top up your water.

Why do you need it? Explain.

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On Tuesday, you don’t use any water as you can see that the level has barely fallen, but on Wednesday, you use 10 ml of water to bring the level back up to the mark. On Thursday, you need 20 ml to bring the level up, and on Friday, 40 ml. You take your jar home and continue to top up the water.

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Recognises, describes and uses patterns involving operations on whole numbers and follows and describes rules for how terms in a sequence can be linked by multiplication or an addition or subtraction based strategy.

Measures by counting uniform units including where part-units are required, and measures length, mass, capacity, time and angle, reading whole number scales.

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Name: _____________________

Find a Pattern: Space

Try Tiling - 1 Have you ever looked at a pattern on carpet or on tiles on the floor or the wall? If you have, you may have been able to see that patterns are sometimes made by turning squares or circles a certain way and then joining them up.

Look at the picture of this square tile:

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r o e t s Bo r e p ok u If several of these S tiles were placed next to each other in a particular way, a pattern could be made:

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You should be able to see what would happen if more tiles were added to keep the pattern going.

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Can you use the same tile, but arrange several of them differently to make a new pattern? Remember, you cannot change the design of the tile, but you can turn each tile any way you wish. Use the space below to make your pattern. You can try out other arrangements on the back of this page.

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Recognises rotations, reflections and translations in arrangements and patterns and translates, rotates and reflects figures and objects systematically to produce arrangements and patterns.

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Name: _____________________

Find a Pattern: Space

Try Tiling - 2

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Design your own tile which can be turned around and joined with others the same design to make a new pattern. Remember, you can use squares, circles, rectangles, ovals, or any other shape to make your pattern, but each tile should be the same on its own. Use the space here to try out ideas and then draw your design on art paper and colour it.

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Recognises rotations, reflections and translations in arrangements and patterns and translates, rotates and reflects figures and objects systematically to produce arrangements and patterns.

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Name: _____________________

Find a Pattern: Chance & Data

Take a Raincheck - 1 As well as displaying information about how much, how many or how often, graphs can also show patterns which are easily seen, especially if they are made up from information which is gathered over a period of time.

Graphs of weather information, such as temperatures and rainfall, are a very good example of this. If graphs of the rainfall for a particular area are made every year and kept, then after a few years, a definite pattern will be able to be seen.

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r o e t s Bo r Look at this rainfall graph for one year: e p ok u S

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Jan

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1. Which month was the driest? ______________________________________ 2. Which was the wettest? ___________________________________________

. t erainfall was there in July compared to June?co 4. How much more . c e _________________________________________________________________ her r o t s sup r 5. What can you say about the rainfall for e the months April and November?

3. How many millimetres of rainfall fell in September? ___________________

_________________________________________________________________ However, if you were to look at the rainfall graphs for the same place for the previous five years, then you would be able to read patterns from the graphs, and those patterns would provide different information.

Now complete the questions on the next page.

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Name: _____________________

Find a Pattern: Chance & Data

Take a Raincheck - 2 Look carefully at this graph and by looking for the patterns, you will be able to answer these questions.

90 80 70 60

40

1st year 2nd year 3rd year

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4th year 5th year

30 20 10

© ReadyEdPubl i cat i ons 1.Which• month shows the most change from year to s yearo (that means that f o r r e v i e w p u r p o s e n l y • the rainfall is most unpredictable in that month)?

0

Jan

Feb

Mar

April

May

Jun

Jul

Aug

Sept

Oct

Nov

Dec

2.Is August always the wettest month?

4.Find a month and year which was unusually wet:

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3.Which month would you choose to have a beach holiday?

5.Which of these months shows the most consistent amounts of rainfall over a five year period? (Circle one.)

o c . che September October Januar Januaryy e r o t r s s r u e p 6.Compare the first year’s rainfall with the fourth year’s rainfall. What similarities can you find?

WM 3.3 C&D 3.27 Reads and makes sensible statements about trends and patterns in the data in tables, diagrams, plots, graphs and summary statistics and comments on their data collection processes and their results.

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Solve an Easier Version

Teachers’ Notes: Solve an Easier Version This technique is similar to the strategy of “Finding a Pattern”. The students find the solution to a complex problem by working out an easier version and then applying the same rules to the more difficult problem.

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Example: There are 30 players at a football game. Every player shakes hands with each of the other players once. How many handshakes take place?

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Students could first work out how many handshakes would occur with a group of five and then look for a pattern to apply to the more difficult problem.

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The key lies in simplifying the variables - whether it be distances or size of spaces, numbers, or amounts of time - so that the student can be confident of the type of calculation which needs to be done. The variables should be simplified to the point where students will be sure their answer is correct, as this will the give them the confidence that their approach is appropriate.

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Name: _____________________

Solve an Easier Version: Number

Student Information Page Sometimes problems look harder than they really are just because of the size of the numbers or measurements involved, and then that can make you think that it’s impossible to solve! You feel you just can’t begin to think about it.

For instance: 1000 trucks brought 70 tonnes each of wheat to the port. The wheat had to be emptied into silos which each held 2500 tonnes. How many silos would be needed to hold all the wheat from the trucks?

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r o e t s Bo r e pcalculator and hope you’re doing okit right, what if we Before you reach for your u change the size S of the numbers - and solve an easier version first, so that you will have a better idea of whether you are doing it right?

10 trucks brought 7 tonnes each of wheat to the port. The wheat had to be emptied into silos which each held 25 tonnes. How many silos would be needed to hold all the wheat from the trucks?

Easy! 10 trucks X 7 tonnes each = 70 tonnes of wheat altogether.

Each silo holds 25 tonnes. 25) 70 = 2 with 20 remainder.

You will find that 2 silos won’t be enough.

© ReadyEdPubl i cat i ons So you would need 3 silos.

•f orr evi ew pur posesonl y• Now try the first problem again, following the same steps:

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Each silo holds 2500 tonnes. _________= ________ So you would need _________ silos.

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1000 trucks X 70 tonnes = _________ tonnes of wheat altogether.

. t e ____________ o difficult problem? c . c e Did you need it for both parts of the calculation? _____________ h r er o t s super

Did you use your calculator to help you work out the answers to the more

WM 3.3 N 3.15

Calculates with whole numbers, money and measures (at least multipliers and divisors to 10) drawing mostly on mental strategies to add and subtract two digit numbers and for multiplications and divisions related to basic facts.

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Name: _____________________

Solve an Easier Version: Number

Music Madness The Music Teacher takes recorder lessons three times each week. At each lesson, she brings spare recorders for students who forget to bring their own.

After each lesson the teacher has to clean these recorders in a sink. She puts 5 litres of water into the sink each time. How much water is used in a year to wash all the recorders?

r o e t s Bo r e phere: ok Write that calculation u S

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Teac he r

This time, try solving an easier version by changing the amount of time.

Could you do it if you had to find out how much water was used in a week?

© ReadyEdPubl i cat i ons Now you may need to find out how many weeks there are in a school year •f o rr e i e wp ur ponoschool ses onl y• - remember, there arev no recorder lessons holidays!

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Write your final calculation here:

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A total of _________ litres of water is used to wash all the recorders in a year.

N 3.15

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Calculates with whole numbers, money and measures (at least multipliers and divisors to 10) drawing mostly on mental strategies to add and subtract two digit numbers and for multiplications and divisions related to basic facts.

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Name: _____________________

Solve an Easier Version: Number

Siren Solution Now have a go at solving this one on your own: How many times does the siren (or bell) sound at your school each term?

How will you make it into an easier version?

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r o e t s Bo r e p ok u S here: Write that calculation

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Write your final calculation here:

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© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• Is there anything you need to find out?

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The bell or siren is sounded _________ times each term.

N. 4.3

Calculates with whole numbers, money and measures (at least multipliers and divisors to 10) drawing mostly on mental strategies to add and subtract two digit numbers and for multiplications and divisions related to basic facts.

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Name: _____________________

Solve an Easier Version: Number

Swimming on the Bus The next two pages are for you to work out the problems all by yourself.

Remember: Write down what you did to solve an easier version first.

Show your calculations.

Find out anything you need to know.

Write the final answer in a sentence.

Teac he r the pool.

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r o e t s Bo r e The 72 Year Fives are all going to the swimming poolo for swimming p u k lessons. The lessons are held every day for two weeks. Each child’s S parents have to pay 50 cents for each one-way trip on the bus to and from How much will it cost for bus fares for all the Year Fives for their swimming lessons?

Explain your working out:

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WM 3.3 Calculates with whole numbers, money and measures (at least multipliers and divisors to 10) drawing mostly on N 3.15 mental strategies to add and subtract two digit numbers and for multiplications and divisions related to basic facts.

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Name: _____________________

Solve an Easier Version: Space & Measurement

Make It Easy on Yourself

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1. If your school playing field had a 400 metre running track painted around it, how many of your steps would it take to walk three times around it?

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2. How much time would you use up in two weeks if you were sent to the office with a message four times each day?

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Students calculate with whole numbers, money and measures, drawing mostly on mental strategies to add and subtract two-digit numbers and for multiplications and divisions related to basic facts.

Selects appropriate attributes, distinguishes perimeter from area and time from elapsed time, and chooses units of a sensible size for the descriptions and comparisons to be made.

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Name: _____________________ Solve an Easier Version: Measurement

Fancy Fences The last question for this section will take you longer to reach the final answer, as there are a lot of steps involved. Remember to make the first calculation easy, and write down everything as you go.

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Explain your working out:

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Work out how much red, yellow, blue and green paint you will need to paint both sides of this fence with each paling painted in a repeat colour pattern - that is: one red, one yellow, one blue, one green and so on.

One side of one paling uses 25 ml of paint.

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Your school wants to build and paint a fence from the same palings to go around the Pre-Primary sandpit. The sandpit area is a square with each side measuring 3 metres. Each paling is 10 centimetres wide. The paint costs $9 per litre, but if you buy a four litre tin, it costs $28. The school wants to ask the Parents’ Committee for the money for the paint.

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o c . che e r o t r sif the fence is painted in sup How much will it cost the Parents’ Committee r e the same colour pattern?________________

How much would it cost if they reduced the number of colours?

(Hint: Think about how many palings and how much paint are in a metre!)

Show your working on a spearate piece of paper.

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Selects appropriate attributes, distinguishes perimeter from area and time from elapsed time, and chooses units of a sensible size for the descriptions and comparisons to be made.

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Name: _____________________Solve an Easier Version: Chance & Data

Lucky Dip At the local supermarket, there is a novelty dispenser which has a handle that can be in four positions.

You put in 50 cents and turn the handle one notch in a clockwise direction to get your novelty out.

When the handle is in the “9 o’clock” position, the next person to put in 50c and turn the handle to the “12 o’clock” position will receive a bonus novelty two for the price of one!

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for yourself and your sister.

If you check the machine every Saturday, how many novelties would you expect to be able to collect in a year?

Explain how you calculated your answer.

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r o e t s Bo r e p o u k Your mum has told you that if the handle is at “9 Suse the machine, and get a novelty o’clock” you may

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Places events in order from those least likely to those most likely to happen on the basis of numerical and other information about the events.

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Draw a Diagram or Table

Teachers’ Notes: Draw a Diagram or Table Many people find that a visual representation helps to organise thoughts - you literally ‘can see what’s happening’. Tables help to organise information when the information needs to be recorded under more than one heading or when a matrix is required.

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Diagrams are especially helpful when solving problems of measurement and space - and of course, graphs and pie charts are a form of organised diagrams. Because of this, the problems included to teach this strategy have combined elements of Measurement and Space and so are presented as integrated tasks.

This strategy links in to other problem-solving approaches. In solving some kinds of problems, the diagrams are in fact just “Making a List” in graphic form. The students should be helped to understand this.

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Name: _____________________

Draw a Diagram or Table: Space & Measurement

Student Information Page

Another way of organising your information and dealing with the problem “bit by bit” is to draw a diagram to help you – sometimes words and numbers are not what you need.

In the first task, you will need to draw a diagram to help solve the problem.

However, if you think back to the “making a list” strategy, you should be able to see that you are still making a list, it’s just that this time it is a list of diagrams, not words.

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r o e t s Bo r e p ok Triangle Puzzle u S Make some puzzles for younger students in your school by cutting a piece of A4 card into triangles. You need a total of five triangles. One of the triangles

must be half of the whole area of the paper. The children solve each puzzle by putting the triangles back into the A4 rectangle.

Here is one way you could do it:

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How many different puzzles are you able to make? Use the space below to make your “list of diagrams”.

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Selects, describes and compares figures and objects on the basis of spatial features, using conventional geometric criteria.

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Name: _____________________

Draw a Diagram or Table: Space & Measurement

Shelf Space

Sometimes drawing diagrams is just the easiest way to find a solution, when you need to “get a picture” to know if the answer will work.

You need to pack eight cardboard boxes onto a shelf. Boxes may not stick out over the edge.

The size of the shelf is a rectangle 5.5 metres long and 60 centimetres wide.

r o e t s Bo r e p o u k The sizes of the boxes are: S 1 m x 0.5 m 1 m x 25 cm 60 cm x 20 cm 40 cm x 25 cm 1.5 m x 40 cm

75 cm x 30 cm 25 cm x 25 cm 1.2 m x 25 cm

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Use the working space, or use graph paper, to work out how you could store the boxes on the shelf. (There may be more than one way!)

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Visualises and makes models of 3D shapes and arrangements and interprets and produces conventional mathematical drawings of them.

Takes purpose and practicality into account when selecting attributes, units, and instruments for measuring things and uses the relationship between metric prefixes to move between units.

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Name: _____________________

Draw a Diagram or Table: Space & Measurement

A Rubbish Problem

Your teacher has announced a class competition to find the best place for the paper rubbish bin, so that it is as close to as many people as possible.

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Where would this be in your classroom?

Use whatever you need to help you. Draw the layout in the box below.

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Uses distance, direction and grids on maps and plans and in descriptions of locations and paths.

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Name: _____________________ Draw a Diagram or Table: Chance & Data

Red Pen Blue Pen Venn diagrams are an ideal type of diagram to draw if you are trying to sort information in order to find items which suit certain rules, or which have characteristics in common with other items, or if you want to find “the odd one out”. You could discover what you needed to know through list-making, but organising the information in a Venn diagram in these cases is much simpler and faster. It also makes it far easier for other people to read the information.

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r o e t s Bo r eclass. Three of the boys There are 24 students inp your have only a blue biro o and two of the girls have only a blue. None of the girls has only a red biro, but u k one boy has only aS red. Everyone else has a blue and a red. There are four more boys than girls in the class.

Your teacher wants to give a sticker to everyone who has both colours.

How many stickers will he/she need for the boys and how many for the girls?

You will need to use the “Guess and Check” strategy that you learned to help you get the answer, but if you arrange the information into Venn diagrams, you will have everything you need to know “at a glance”. It makes it easier to answer any more questions about the biros, such as:

© ReadyEdPubl i cat i ons How many red biros are in the •f or r e vi ew pclassroom?____________ ur posesonl y•

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How many girls own a blue biro? ______________

Girls

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WM 3.3 C&D 3.25 Displays and summarises data using frequencies, measurements and many-to-one correspondence between data and representation.

N 3.15 Calculates with whole numbers, money and measures, drawing mostly on mental strategies to add and subtract two-digit numbers and for multiplications and divisions related to basic facts.

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Work Backwards

Teachers’ Notes: Work Backwards This is a useful strategy to employ when an outcome is clear and known, but the range or sequence of events which produced the known result may be needed.

r o e t s Bo r e p oData examples in The resolving of these problems, much like the Chance & k u the “Draw a Diagram” section, usually involves the plotting of information so S that a number of questions concerning the events can then be answered.

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Thus, the type of problems which call for this strategy will typically involve calendars or other measures of time.

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Name: _____________________

Work Backwards: Number

Student Information Page You can use this strategy when you know the end result of what has happened, and you need to know details about what happened along the way, or at the beginning.

For instance: a) If a friend was increasing the number of stickers on his file by one each day, and you can see that he has seventeen now (Friday), you could work out when he started collecting them and how many he had last Thursday. You would count backwards from yesterday. Work it out in the space below.

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r o e t s Bo r e p ok u S As with all problem solving, think about how to organise your information. In

this case, the information is in numbers and days. Draw up a grid to help you, with the days of the week across the top.

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Use the same approach to find the answer to the question below.

b) As a fund-raising idea, the Year Sevens are working with Year Five students to make toffee. Each Year Five pays 50 cents and two students from each class have a turn each day. The Year Sevens are keeping a total of the money collected and change the total at the end of each day. They have $8.00 from your class so far. They come to your room and call out your name for today but you’ve already had a turn! Your name has not been ticked off, and the Year Sevens want to know which day you had your turn. You can’t remember, but you do remember seeing the collection total when you were cooking with them - and it was $3.00. Today is Wednesday.

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Can you check which day you had your turn? Make a grid, either on the back of this page or on another sheet of paper.

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Adds and subtracts whole numbers and amounts of money and multiplies and divides by one digit whole numbers, drawing mostly on mental strategies for additions and subtractions readily derived from basic facts.

Directly and indirectly compares and orders things by length, area, capacity, mass, time and angle, measures them by counting uniform units and uses standard scales to measure length and time.

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Name: _____________________

Work Backwards: Space & Measurement

Stop the Clock You are playing a Maths game with clocks in your classroom, and your teacher explains that each person in your group of six must take turns to move the clock hands forward by five minutes each five seconds, after she says “Go”. She says go and you start, but after a while someone from another group calls out that the hands of their clock are sticking and they can’t move them properly. “Stop, everyone!” calls your teacher, and then she asks you all to put the clock hands back to where they were when you started, so that the game can start again. Your clock shows 7.05 now, but no-one in your group noticed where they were when you started!

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two people have had two turns.

Use the clock faces to help you solve the problem.

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r o e t s Bo r e p ok u “Oh well,” your teacher says, “You’ll just have to work it out.” Can you?

S You know that four people in your group have had one turn and the other

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Recognises and sketches the effect of straightforward translations, reflections, rotations and enlargements of figures and objects using suitable grids.

Directly and indirectly compares and orders things by length, area, capacity, mass, time and angle, measures them by counting uniform units and uses standard scales to measure length and time.

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Logical Reasoning

Teachers’ Notes: Logical Reasoning Logical reasoning is a process by which information is noted in a systematic way so that further necessary information can be gained and likewise noted.

At times calculations will be necessary during the process in order to give rise to information.

r o e t s Bo r e p ok u This strategy has been introduced last, as so often deductive problem solving S involves the application of the other, previously learned strategies.

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Thus, the information is presented much in the manner of “clues”. If the answers to these clues are recorded properly, the solution will show itself.

By now, the students should be familiar with the idea of organising their approach and dealing with the problem “bit by bit” in systematic fashion.

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Ready-Ed Publications

Name: _____________________

Logical Reasoning: Number

Student Information Page As you have seen throughout this book, the secret to successful problem solving is starting with what you know and then thinking in an organised fashion to fill in what you don’t know. Sometimes that organised thinking means you need to draw a diagram or make a list, or you can use ways such as “Guess and Check” or working backwards in time. Sometimes, you will need to do more than one of these things.

r o e t s Bo r e p ok u The last section S of the book deals with “Logical Reasoning”. It’s a bit like being a detective, as you note down clues in an organised way, usually a grid,

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You should be getting a good idea by now of what you need to do in order to solve a particular kind of problem.

and check the information you’ve been given. Then as you work out each part and write it in, it helps you get to the final answer. Last of all, you check back to the question and make sure that your answer fits and makes sense. It is just another way of conducting ordered thinking so that your brain doesn’t boggle.

Try finding the following mystery number by following the clues. You will find it helpful to list the possible answers from the first clue and then cross off numbers as more clues are added:

© ReadyEdPubl i cat i ons The• number is r greater than butu less than 50.

Its is o an n even number, f or evi ew10 p r p os e l y •

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List your possible answers below:

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which can also be divided by 4. However, neither the numerals 2 nor 4 appear in the answer. If you add the two digits of the number together, it makes 9. What is the number?

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Understands the meaning, use and connections between the four operations on whole numbers, and uses this understanding to choose appropriate operations and construct and complete simple equivalent statements.

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Name: _____________________

Logical Reasoning: Number

What Number Am I? Here is a more complicated version of the same type of puzzle: The number has three digits. It is an even number, and the first and the last digits are the same. The second digit is the number you get when you add the first and last digit together. The number can be divided by 4 with a whole number result.

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Remember, note down all the possibilities from the first clue and then cross off what doesn’t fit when you read more clues. When you think you have the answer, check your answer against all the clues to make sure!

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. t e of this style for a friend. o Write a problem c . che e r o t r s super Now swap your sheet with a partner to see if they can solve it.

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Understands the meaning, use and connections between the four operations on whole numbers, and uses this understanding to choose appropriate operations and construct and complete simple equivalent statements.

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Name: _____________________

Logical Reasoning: Space Chance & Data

Fancy Dress

At your class fancy dress party, three of your friends went as cartoon characters. The taller girl went as Donald Duck and the twins went as Tweety (the shorter twin) and Sylvester. Amy and Lisa are from different families and Amy is shorter than Matthew and Lisa.

Who went as which character, and who are the twins?

r o e t s Bo r e p ok u S Donald Duck Tweety Sylvester

Matthew Amy Lisa

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If you organise the information contained in the clues into a grid, the solution becomes easier. Consider each person for each character and place a tick in the box when you have checked the clues and are sure of each one.

© ReadyEdPubl i cat i ons •f o r ev i e ur pao ses onl y• Your teacher hasr asked you tow helpp work out seating arrangement for five Now try this one. Instead of a grid this time, you may find a diagram is more helpful.

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students in your class and has given you some information about where they should sit. The desks will be arranged in one row across the room. Jackson must sit at one end and is left-handed. Thomas can sit next to Emma but not next to Amy. The girls may not sit together. Josh needs to be as close to the centre as possible.

Draw the five desks and place the correct names on them.

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Name: _____________________

Logical Reasoning: Space

Follow the Map Look at the map of the campsite below, and read the clues to answer the question: Who is staying in which tent?

Label each tent with its occupant/s. Use the space below to try your ideas.

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The tent closest to the showers has only one occupant.

Rhianna and Megan have to walk past three tents to get to the showers.

There are three girls in the tent furthest from the showers.

Chelsea, Tara and Tegan have a rubbish bin just near their tent.

Aaron and Jake’s tent is opposite Steven and Samad’s tent.

Callan, Owen and Regan share the biggest tent.

The teacher has a tent to herself.

David and Zac are brothers and Chelsea and Megan are cousins.

Thomas and James do not share a tent.

No-one shares a tent with anyone they are related to.

David, Thomas, James and Zac are all closer to the shower block than Dominic and Josh and are as close to the tap as Steven and Samad.

Shower Block

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Interprets order and proximity based on a set of instructions. Uses knowledge of direction and position to determine points on a map and uses logical reasoning to solve problems.

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Guess and Check

Answers Page 25 Because by Saturday, the jar would have needed 80 ml, and then needed 160 ml on Sunday. By Monday, the jar would be dry and would need filling and then topping up again later, using a total of 320 ml for the day.

Page 26 Answers will vary but tile tessellations should be checked to ensure rotations are accurate - although this will most likely be apparent from the resulting pattern. Students should be assessed for complexity of design.

Pages 27 One year graph: 1. February; 2. July; 3. 70ml; 4.

30 ml; 5. Almost the same.

Pages 28 Five year graph: 1. September; 2. No - some years it was July; 3. February; 4. November of the fifth year; 5. October; 6. Answers will vary.

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Make a List

Page 14 You would then have 6 Lego sets, each of which could make 5 combinations with each of the 5 cars, so that makes 30 combinations altogether.

Page 15 No, because there are 16 possible combinations.

Page 16 No, there are still 11 possibilities.

Page 17 There are 24 possible orders in which the colour pattern could be drawn.

Page 18 You have a 5 in 36 chance of throwing an eight.

Your friend has only a 2 in 36 chance of throwing a three.

Page 19 The possibilities are: 1 x 36; 2 x 18; 3 x 12; 4 x 9; 6 x 6; plus converse 9 x 4; 12 x 3; 18 x 2; 36 x 1.

Page 20 Students must show six different routes.

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Page 7 24 non-fiction and 31 fiction books need replacing.

Page 8 You threw 24.5 m and your friend threw 25 m.

Page 9 The first student was away for 5 mins. 50 secs. and the second for 7 mins, 20 secs.

Page 10 Answers will vary, but students need to show that they adjusted the materials used in a logically sequential way and constructed regular rectangular prism in skeleton form. Models should be checked in a shoe box.

Page 11 There are 40 red and 60 blue counters; therefore there is a 40 in 100, or a 4 in 10 chance of getting a red counter.

Page 12 16 chose chocolate; 22 chose bubblegum; 26 chose caramel.

Solve an Easier Version

Page 31 28 silos.

Page 32 600 litres per year (15 x 40 weeks).

Page 33 Answers will vary, but students should show that they have identified the number of times it sounds each day, then calculated the weekly and/or termly rate to arrive at the solution.

Page 34 $720 Page 35 1) Students should walk the actual steps to discover how many of their steps are needed for a shorter distance, such as 10 metres, then calculate the steps for 100 m, then 400 m, then 3 x 400 m.

2) Once again, students should discover the time needed for one message, then multiply that by four, and then by 10 for the two weeks.

Page 36 There are 12 metres of paling. Divided between the four colours, this makes 3 metres of fencing for each colour (i.e. each colour is equivalent to one side). One metre of fencing contains 10 palings, and each paling uses a total of 50ml of paint. Therefore one metre uses 500ml, so three metres uses 1500ml, or 1.5litres. Total paint needed is 6 litres. If four colours are used, the Parents’ Committee would have to buy a total of 8 one litre tins, at a cost of $72. If three colours are used, 2 one litre tins of each colour could be bought, at a cost of $54, and all the paint would be used. The cost would be the same for two colours (3 one litres of two colours). However, if only one colour is used, then one 4 litre tin and two one litre tins could be bought for a total cost of $46.

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Page 22 a) triangle; b) The number is doubled and then 1 is added.

Page 23 a) The pattern: add 2, take 1, add 2, take 2, add 2, take 3, add 2, take 4 and so on; b) 720.

Page 24 The clock is gaining one minute every two hours.

At 8:50 on the first day, it was three minutes fast, so 24 hours later it will have gained a further 12 minutes and thus be 15 minutes fast.

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Answers cont.

Solve an Easier Version cont.

Page 37 There is a one in four chance of finding the knob positioned at 9 o’clock each week. Therefore, of 52 Saturdays in the year, you could expect to find the knob at 9 o’clock on 13 of them. As the machine would then dispense two novelties, you could expect to have gained 26 novelties in a year.

Matthew Amy Lisa

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Sylvester X

Lisa and Matthew are twins. b) The best solution is Jackson, Amy, Josh, Thomas, Emma. Another possible solution is Jackson, Thomas, Emma, Josh and Amy - this is less desirable as Josh is not as central.

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Page 39 Students’ diagrams should show variations based on an initial diagonal bisection. (the bisection could be in either direction) Their diagrammatic “list-making” should show that they have used subsequent bisections of the triangles. The diagrams should show five triangles arranged into a rectangle, and there should be at least three variations.

Page 40 Students’ solutions should show the shelf and each box drawn to scale, and should show that they understand the relationship of cm:m. There are no solutions with all boxes placed end to end.

Page 41 Students’ solutions will vary and there will need to be dialogue for the student to explain their reasoning. Diagrams should appear to be in approximate (informal) scale.

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Work Backwards

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Page 45 The clock was set at 6:25 before the game began.

Logical Reasoning

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Page 47 The number is 36.

Page 48 The number is 484. Students’ own problems should be checked after being solved by a classmate, to make sure all clues are valid and the solution logical.

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