All You Need to Teach: Problem Solving Ages 5-8

Solving Problem

Pr

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Turning problems

All you need to teach . . . is a comprehensive series for

smart teachers who want information now so they can

get on with the job of teaching. The books include

background information so teachers can stay

up-to-date on the latest pedagogies, and then

interpret that information with practical activities and

ideas that can be immediately used in the classroom.

The step-by-step lessons in All you need to teach . . . Problem Solving will strengthen your students’ logical and creative thinking skills. With the strategies taught in these lessons, your students will be able to tackle just about any problem!

Inside you’ll find these strategies:

• Locate key words• Look for a pattern• Assume a solution• Create a table or chart• Make a drawing• Work in reverse• Find a similar but simpler problem• Make a model• Think logically.

Also ava i lab le :All you need to teach . . . Problem Solving Ages 8–10

All you need to teach . . . Problem Solving Ages 10+

All the tools a smart teacher needs!

Solving Problem

T e a c h i n g T i p s L e s s o n p L a n s W o r k s h e e T s T a s k c a r d s a n s W e r s

Peter Maher

Ages 5-8

Prob Solv Front Cov 5-8Sept2012.indd 1 26/09/12 9:42 AM

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Solving Problem

into solutions

Turning problems

Ages 5–8

Peter Maher

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First published in 2004 by

MACMILLAN EDUCATION AUSTRALIA PTY LTD15–19 Claremont Street, South Yarra 3141Reprinted 2006, 2007, 2009, 2011, 2012

Visit our website at www.macmillan.com.au

Associated companies and representatives throughout the world.

Copyright © Peter Maher/Macmillan Education Australia 2004All You Need to Teach Problem Solving Ages 5–8

ISBN 13: 978 0 7329 9766 3ISBN 10: 0 7329 9766 6

Edited by Vanessa LanawayDesign by Trish HayesIllustrations by Stephen King

Printed in Australia at On-Demand, Port Melbourne, Victoria

Copying of this work by educational institutions or teachersThe purchasing educational institution and its staff, or the purchasing individual teacher, may only reproduce pages within this book in accordance with the Australian Copyright Act 1968 (the Act) and provide the educational institution (or body that administers it) has given a remuneration notice to Copyright Agency Limited(CAL) under the Act.

For details of the CAL licence for educational institutions contact:

Copyright Agency Limited Level 15, 233 Castlereagh Street Sydney NSW 2000Telephone: (02) 9394 7600Facsimile: (02) 9394 7601Email: [email protected]

Reproduction and communication for other purposesExcept as permitted under the Act (for example, any fair dealing forthe purposes of study, research, criticism or review) no part of thisbook may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher.

DeDication:To Tom and Frieda, who taught me that no problem is too difficult to overcome.

Problem Lwr 5-8 imprint.indd 2 5/07/12 10:29 AM

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How to Use tHis Book ................................................................................... 4scope and seqUence cHart ........................................................................... 5

All the teAching tips You need ........................................................... 6creating a sUccessfUl proBlem solving environment ................. 7tHe nine proBlem solving strategies .................................................. 11mini-poster: tHe nine proBlem solving strategies ..................... 16mini-poster: wHen i proBlem solve i mUst rememBer tHese tHings ................................... 18

All the lesson plAns And Worksheets You need .................. 19 1. Light Bars ..................................................................................................... 20

2. Digit Shift .................................................................................................... 22

3. Storey Street ............................................................................................... 24

4. A Weighty Problem .................................................................................. 26

5. This Card Is ... ........................................................................................... 28

6. Colour the Trains ....................................................................................... 30

7. What Comes Next? .................................................................................. 32

8. Teddy Traffic Lights .................................................................................. 34

9. First to 50 .................................................................................................... 36

10. One Die Roll ............................................................................................... 38

11. Mrs Murphy’s Class .................................................................................. 40

12. Birds and Dogs and Beetles .................................................................... 42

13. Money Matters .......................................................................................... 44

14. I’m a Lumberjack ....................................................................................... 46

15. Sarah, James and Tom ............................................................................. 48

16. Cubes, Cubes, Cubes ............................................................................... 50

All the tAsk cArds You need ............................................................. 52

All the AnsWers You need ................................................................... 61

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All You Need to Teach Problem Solving Ages 5–8 is the first in a series of three books designed to help teachers develop the capabilities to strengthen logical and creative thinking skills in the students under their care. This book caters for teachers of students in the first three years of schooling and is in four parts.

All The Teaching Tips You Need presents the strategies and techniques that need to be developed and applied by students to solve the range of problems in the books. It also suggests ways to implement a successful problem-solving program in the classroom.

All The Lesson Plans and Worksheets You Need contains 16 lesson plans with accompanying blackline masters. The lesson plans outline the theoretical background of the problems and suggest the best manner to present them to the students. The blackline masters give students the opportunity to draw and describe the strategies and working they used to solve the problems.

All The Task Cards You Need contains 16 task cards designed to be photocopied and laminated. Each card presents a variation or extension of the problem found on the blackline master of the same number. The task cards provide an ideal way to assess the development of each student’s problem solving capabilities.

All The Answers You Need offers solutions to both the blackline masters and the task cards.

These lesson plans, blackline masters and task cards are designed to be practical, intellectually stimulating and to contain high motivational appeal.

As your own problem solving capabilities grow, your ability to successfully teach problem solving will be similarly enhanced. Problem solving also offers the opportunity to have an enormous amount of fun in the classroom. Mathematics is a discipline that offers a number of opportunities for excitement and stimulation. The All You Need to Teach Problem Solving series offers the potential to clearly demonstrate this fact.

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What is Problem solving?Problem solving is the application of previously acquired skills and knowledge to an unfamiliar situation. Numerical equations presented in words as story or worded problems are often mistaken for problem solving. A typical example of what is not problem solving would be the transference of 5 cents + 5 cents + 10 cents + 10 cents into:

Four children emptied out their pockets. Kate found five cents. Jake found five cents. Jason found ten cents and Sarah found ten cents. How much money did the four children find altogether?

This is not problem solving because the technique required to solve this story problem (addition) is easily identified and requires little creative thought.

Questions that transfer numerical problems into a practical context are an essential part of any effective mathematics program. It is vital that students are constantly shown why they are learning mathematical skills.

A related problem-solving question could be:

In how many different ways can 30 cents be made in our money system?

This question requires the student, in a logical manner, to search for a strategy to solve the problem and to apply the previously acquired skill of addition and their knowledge of the coin denominations available in Australian currency. The story problem offers a context. The problem solving exercises the student’s ability to work flexibly, creatively and logically.

Why teach Problem solving?It can be argued that problem solving should be the most effective and significant aspect of any mathematics course. As adults, both at work and at home, our everyday lives are filled with situations that demand flexible thinking and creativity. The role of both parents and teachers is to turn dependent children into independent people who are capable of functioning in a society that demands resilience, intelligence, a high emotional quotient and tractability. Such traits are best fostered through the development of an ability to problem solve.

It therefore follows that all students will benefit from regular exposure to problem solving in schools. Problem solving should not lie solely in the domain of the most intelligent and capable students, which is, regrettably, often the case. Although the more intelligent students may achieve best on problem solving tasks, regular problem solving should feature strongly in every student’s learning experiences.

creAting A successFul proBleM solving environMent

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classroom atmosPhere

It is essential for students to believe in their own capabilities and to have healthy self-esteem. They need to understand that to have a go, even if their attempt is wrong, is far preferable to not attempting a question at all, and to realise that making mistakes plays a vital role in the learning process.

Students should come to realise that intellectual challenges should be viewed as opportunities to demonstrate how much they have learned and how bright they are becoming. Problem solving should come to be seen by students as not just important, but a great source of fun.

This positive classroom atmosphere can best be engendered by teachers who are confident in their own ability to teach problem solving. It is as true for the teacher as for the student: Practice may not necessarily make perfect, but it will lead to improvement. Improvement leads to increased success and greater self-confidence. Success leads to enjoyment!

raising the bar

The capabilities of young students are often quite remarkable. They come to school today with far greater confidence and knowledge than any previous generation. In a classroom with a positive atmosphere and a school that celebrates the love of learning, students will not only rise to intellectual challenges, they will thrive on them.

The concept of raising the bar refers to the idea that students should be extended until their full intellectual capabilities are reached, regardless of supposed appropriate year level standards. Every student deserves the opportunity to strive for his or her intellectual best. Problem solving is an excellent adjunct for the teacher to assess such potential. Present the students with challenges and step back to observe the outcomes. I can assure you that, in the appropriate learning environment, most students will exceed expectations.

timetabling Problem solving

A problem solving approach to teaching mathematics should be adhered to in each classroom for the reasons outlined on page 7. Whenever lessons involving ‘core material’ are conducted, every attempt to include open-ended questions should be taken.

If, for example, in a Year 2 classroom the concept of 2 + 2 digit column addition is being taught, the following question should be introduced to extend the concept:

Given four digits, say 2, 4, 7 and 1, what is the biggest possible 2 digit + 2 digit sum?

I believe that it can be strongly argued for a lesson a week to be devoted to strengthening problem solving skills. This four day a week core material, one day a week problem solving ratio would complement each component of the mathematics course very well. This is especially the case if the questions posed for the problem solving sessions could be related to the core material topic under review at the time.

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the student as a concrete oPerator

Developmental theory shows that the great majority of primary school-aged students, especially the very young, benefit significantly from the manipulation of concrete materials when dealing with mathematical concepts.

For many students, the converse of this argument has dramatic results. The more abstract a concept or question is, the less likely that a student will understand it and, hence, be able to solve it. The message for primary teachers is clear: provide concrete materials whenever your students attempt problem-solving tasks to achieve the best possible results.

structuring the lessons

Each problem to be solved should be preceded, where appropriate, by a background discussion about the context of the problem and the previously acquired skills or knowledge necessary to successfully complete the task. For example, a question asking ‘How many rectangles can be found on a foursquare court?’ should prompt a discussion of the game and then move on to the concept of a rectangle and how a rectangle can be formed by using pre-existing rectangles. It should also be pointed out that a square is merely a special type of rectangle (opposite sides equal and four right angles).

The problem should be read out aloud by either the teacher or a competent student. The students should be asked to select key words, underline them and write them down. At this point some of the brightest students will be eager to get into the problem. Let them do so. For others in the class, a discussion of appropriate strategies that could be employed is valuable. Soon many other students will be ready to begin.

For those still in need of guidance, it may be necessary to commence an appropriate strategy together. It may take some time, but by following this ploy, each student will be on the right track. Not all will finish the problem – some may only make a start. However, this is a step in the right direction and should be praised.

Some students prefer to problem solve individually, some enjoy the cut and thrust that cooperative group learning brings. Either approach is suitable, providing that within the group, each participant has a role to play.

It should also be noted that some problems lend themselves better to group work. Fermi problems such as: ‘How many pies will be eaten in Australia today?’ are best ‘solved’ by sharing ideas, estimates and general knowledge.

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reflection

Following the completion of each blackline master, gather the class for shared reflection. Encourage students to describe the strategies they have used and to outline the mathematics contained in their technique. Many problems can be solved with more than one strategy, as this reflection will demonstrate. This time will be especially valuable when problems were solved in a hit and miss fashion by students who were unable to recognise a pattern. Celebrating differences is a very healthy classroom activity and should be highlighted at this time.

This will also enable you to offer students praise and encouragement for their efforts. Keep their self confidence and mathematical self-esteem as high as possible. Praise their attempts at all times, even if they may be misdirected. Bear in mind that problem solving is intellectually challenging both for children and adults alike. Remember that any attempt is far preferable to no attempt at all.

Ask students the following types of questions:

D What helped you understand what the question was asking you to do?

D What strategy or strategies did you use in attempting to solve the problem?

D Have we used this/these strategies anywhere before?

D When else might you use this strategy?

D How did you feel when you solved the problem?

D Do you think that your problem solving skills are getting better?

D Do you think Mum or Dad might be able to solve the problem with your help?

D Did you find it useful to work with a partner on the problem?

D Could you make up a problem of your own like this one?

D Was the problem as difficult as it first appeared?

You may also consider encouraging students to record their progress in a journal. As well as providing a useful teacher reference, this can help students to see what they have learned.

fast finishers

The students who are likely to enjoy problem solving the most will be those who are more mathematically able. Because of their innate capabilities they will be the first to finish their work and will need more. Please do not give these students extra drill and practice examples to do. This is far more likely to dull their enjoyment of the subject and dim their creativity. Make space in your room for fast finishers and offer them a corner full of problem solving tasks, games or puzzles to encourage their love of challenges.

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Which strategy to use

There are relatively few types of problem solving questions. As a consequence, the more a student practises these strategies, the more comfortable they will become with problem solving in general. The strategies are similar across the three books in this series for a very sound reason – they are as relevant to a five-year old as they are to an adult. The two key strategies of problem solving are locating key words and looking for a pattern. These are fundamental to almost all problem solving tasks. The other strategies, while still relevant to all learners, are more question specific. The sooner key strategies can be exercised and practised, the better the problem solving skills will become.

For some questions, once students understand what needs to be done, just one strategy will be sufficient. For other questions, more than one strategy will need to be used. Sometimes a range of different strategies may be appropriate. In some instances, two totally different strategies may successfully solve the same problem in two logical and equally creative ways.

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locate Key Words

The instructions involved in a problem must be understood before students can begin to attempt the question. Often a student will not attempt a problem because no strategy is immediately apparent. Getting started can often be the toughest part.

The technique of underlining and then writing down the key words in a question (committing something to the page) is an excellent way for students to gather their thoughts and to make a start. This strategy will be emphasised in every problem in this series.

Take the question: ‘In how many different ways can 30 cents be made in our money system?’

Underlining and then writing down ‘How many ways 30 cents’ can focus a student’s thinking. This précis of the problem can also make it seem easier to handle. Once they understand the problem, direct the students to work in a logical manner while offering suggestions for the students to build upon. Point out that a sensible way to begin would be to make the total as quickly or economically as possible – in this example by using just two coins: 20 cents + 10 cents.

Ask whether this is the only way to make 30 cents with two coins, then ask the students why we should start with the biggest coin possible. Then move on to all the ways that 30 cents can be made with three coins:

20 cents + 5 cents + 5 cents and 10 cents + 10 cents + 10 cents.

Ask why we can no longer use a 20 cent coin. Then move on to all the ways of making 30 cents with four coins starting with the biggest coin: 10 cents + 10 cents + 5 cents + 5 cents.

The final two solutions of 10 cents + 5 cents + 5 cents + 5 cents + 5 cents and

5 cents + 5 cents + 5 cents + 5 cents + 5 cents + 5 cents should follow logically.

Such success and such a logical approach to the problem obviously require that students understand the question and can identify the problem’s key components.

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assume a solution (guess and checK)This approach encourages the student to have a go and is particularly useful in problems involving variables. By assuming a solution, patterns often appear, suggesting a way to success. Using the problem:

A farmer keeps horses and ducks on his farm. One day the farmer notices that his animals, together, have 10 heads and 34 legs. How many horses and how many ducks does the farmer own?

A first assumed solution may be that there are as many horses as ducks, i.e. five of each type of animal. This fulfils the head category but not the legs category.

5 horses = 20 legs. 5 ducks = 10 legs. Heads = 10 but the legs only equal 30.

Most students should realise that there are either too many ducks or too few horses in this attempted solution. A second assumed solution may be 6 horses and 4 ducks. Again, this fulfils the head component of the problem, but offers too few legs.

6 horses = 24 legs. 4 ducks = 8 legs. Total number of legs = 32.

The student should realise that this assumed solution is closer to the mark, therefore they are on the right track. The next answer of 7 horses and 3 ducks is, of course, the correct one.

Although this approach is not the easiest or the fastest way of solving the problem, at least the student is putting something on paper. Even if they don’t get the right answer, at least they have made some progress.

looK for a Pattern

This strategy, used effectively, is often the shortcut to success and can greatly simplify problems that may initially appear very difficult. Take the following problem for example:

A farmer keeps horses and ducks on his farm. One day the farmer notices that his animals, together, have 10 heads and 34 legs. How many horses and how many ducks does the farmer own?

If a student’s first attempt was one horse and nine ducks, giving 22 legs as the result, a second attempt of two horses and eight ducks, giving 24 legs as a result, may lead to the discovery that each exchange of a horse for a duck adds two legs to the total.

The second attempt at the solution is still ten legs shy of the target, therefore five exchanges need to be made.

The discovery of this pattern has saved time and effort. It can be argued that pattern recognition is the most apparent trait in the brightest problem solvers. This fact is often frustrating for both teachers and students. The students who can’t see patterns readily are the ones who most need short cuts.

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create a table or chartTables and charts can often be useful in making potential patterns more perceivable to students.

A question such as ‘What is the 43rd odd number?’ could be solved easily and quickly by using a chart in the following manner:

The pattern 2n – 1 (where ‘n’ = the ordinal number) may be recognised by many students using this technique. Thus, the 43rd odd number is 2 x 43 – 1, equalling 85.

For younger students, the table or the chart should be provided by the teacher. The student’s task should be to fill it in rather than to create it. In later primary years when the students have had exposure to the structure and nature of a table, it is appropriate to ask them to form the structure themselves.

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make a DrawingDrawing is particularly useful for beginning problem solvers, for students who are visually oriented in their thinking and learning styles, or when applied to certain spatial questions. For example, the problem: ‘I cut a log into four equal pieces. How many cuts did I make?’ can be solved effectively by drawing the log and inserting the cuts appropriately. The pattern of ‘number of cuts = number of pieces – 1’ should then emerge.

Making a drawing is also an excellent way of getting students started. In a similar way to verbalising and discussing the structure of a question, drawing helps to crystallise what needs to be done.

When asked to solve, for example, problems involving 2D or 3D shapes, or questions regarding perimeter and area, it certainly helps to sketch the shape. As a result, dimensions, faces, vertices and edges become more apparent and often shed significant light on the problem.

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Ordinal Number Even number Odd number

1st 2 1

2nd 4 3

3rd 6 5

4th 8 7

5th 10 9

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FinD a Similar but SimPler ProblemThis is a particularly valuable strategy when applied to what can appear to be very complex problems. Take the following question:

35 + 17 – 35 – 17 + 5 = ?

This may initially appear to be quite difficult and well out of the range of any lower-primary aged student. However, by replacing the given numbers with much smaller values, while remaining faithful to the structure of the problem, we could get, for example:

3 + 1 – 3 – 1 + 5 = ?

This much simpler question will help the problem solver to discover that the 35s and the 17s in the original problem simply cancel each other out, like the threes and the ones in the simpler question, leaving five as the answer.

This strategy was also used in the earlier cited problem of what is the 43rd odd number, with the assistance of using a chart. The chart helped identify a simpler version of the same problem, so the pattern 2n – 1 became clear.

This example emphasises that a particular problem may well be solved in more than one way or even by using more than one strategy concurrently.

work in reverSeThis strategy is relevant to a specific type of problem, usually numerical and in many parts. Take the following problem for example:

I gave four of my lollies to Matthew. Then I gave five lollies to Claire. Then I gave six lollies to Caitlin. I then had two lollies left for myself. How many lollies did I have in the first place?

This can best be solved by working in reverse, using addition rather than the subtraction used in the question.

Thus, my two lollies, when added to the four, five and six already distributed, means that 17 must have been available in the first place. Superimposing this answer into the problem, an excellent ploy at any time, confirms this as the correct solution: 17 – 4 – 5 – 6 = 2.

Another example of using this ploy would be in the question ‘Which number when doubled and doubled again gives 20 as the result?’

Again, this is best solved in reverse. The opposite operation to doubling is halving and so we can see that when 20 is halved and halved again, we find the solution of 5.

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think logicallyMany problems with numerous potential solutions can be solved by using deductive reasoning to identify and eliminate impossible options. Games like ‘Guess Who’, ‘20 Questions’ and ‘Mastermind’ are examples of activities that utilise this strategy. For example:

I am a two-digit number. My digits are different. My tens digit, when added to my ones digit, equals 10. What is the highest number I could be?

This question asks for a logical analysis to be employed which involves deduction and the elimination of impossible options until only one solution, 91, remains.

Step 1: A two-digit number _ _

Step 2: Both digits different – eliminates a significant number of options.

Step 3: Digits sum to ten – leaves the options of 19, 28, 37, 46, 64, 73, 82, 91.

Step 4: The highest value is 91.

The game of 20 questions, when applied to numbers, is another excellent example of an activity that asks its participants to use logic to eliminate impossible answers.

Starting with the clue ‘I am a 3 digit number’, questions such as:

Am I even?

Am I bigger than 500?

Am I in the three-times table?

Do I have repeated digits?

Are all of my digits odd?

eliminate many of the initial 900 possible answers. By keeping a record of the questions that the students ask and the range of the possible answers, the teacher can focus the students’ attention on the task at hand as well as demonstrate the power of this problem solving strategy.

make a moDelThis strategy is often employed by concrete thinkers whose innate spatial sense may not lend itself to tackling geometric or space-oriented questions in any other way.

Given the net of a rectangular prism, and asked to colour the face which, when folded, would be opposite B, constructing a rectangular prism out of paper will lead to the solution.

Making a model is often time consuming but, when all else fails, it is far better to obtain a lengthy solution than no solution at all.

The fact that the youngest students are usually very concrete in their thinking means that the use of concrete material is not only useful, it is often essential. Until basic number facts develop to a near automatic level, concrete material in problem solving, such as connecting blocks, is a wonderful adjunct to learning. This enables the student to make a model related to the question and to see patterns far more readily.

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16All You Need to Teach Problem Solving Ages 5–8 © Peter Maher/Macmillan Education Australia.

This page may be photocopied by the original purchaser for non-commercial classroom use.

1. Look for the important words in the question

D Write them down.D Put a circle

around them.D Underline them.D Make sure I know

what to do.

3. Have a goD Try an answer.D Does the

answer make sense?

2. Look for a pattern

D Can I see something happening over and over again?

D Will this help me get an answer?

THE N INE PROBLEM SOLVING STRATEGIES

4. Make a table or chartD Will something like this

help me?D Will it help me to see a

pattern?

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6. Work backwardsD Can I start at the end

of the question to help work it out?

D Will my answer work?

7. Try an easier problemD What is the same about

this easier problem?D Will this help me?D Can I see a pattern?

8. Make a modelD Will a model

made out of paper help me?

D Will a model made out of blocks help me?

9. Think carefullyD Can I tell

something about the answer straight away?

D Can I see some answers that will not work?

5. Make a drawingD Can I draw

something about the problem?

D Will a picture help me?

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D Read the question carefully.

D Look for the important words.

D Make sure I know what

to do.

D Look for a strategy to help me

find the answer.

D Check to see if my answer

is correct.

D Tell someone what I have

found out.



WHE N I P R O B L EM SO L V E I M US T R E M EM B E R T H E S E T H I N GS

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wAll the

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leSSon PlanS anDworkSheetS

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wLesson Plan 1

BackgroundCalculators display numbers in a digital format by using light bars. Standard calculators have a display with eight cells, and seven light bars in each cell. The light bars illuminate to form numbers. For example, all seven bars light up to form the number eight.

resources:

D a calculator

D iceblock sticks or MAB longs

orientationExplain to students how light bars form digital numbers (see ‘Background’ above). Draw diagrams to show the hidden cells and what they contain. The students should then enter the digits from 0 to 9 on a calculator, counting the light bars used to create each number.

Use iceblock sticks or MAB longs to represent the light bars and ask students to create the digits 0 to 9 in digital format. Then ask students to make the number 8 in digital format. 8 is the ‘king number’ on a calculator because it uses every possible light bar.

Ask the students to remove one light bar and create the digit 0, then to replace it and return to 8.

Remove one bar and create 6, then replace it and return to 8.

Repeat this until all ten digits have been made from the 8.

guided discovery with BLM 1Read the first question with the class. Help students to locate the key words: Use calculator – draw 0 to 9.

Be aware of the common reversals of 3 and 7, and the confusion of 6 for 9 and 2 for 5.

Read the second question and locate the key words: number of light bars used. Point out that it is only on a calculator that the 7 has four light bars. On all other digital displays (can you name some?) the 7 doesn’t have an overhang and, thus, only uses three light bars.

Explain that different numbers can be made on a calculator with the same number of light bars, for example, 2 and 5, 0 and 9.

Read the next question and locate the key words: Four light bars – biggest digital number.

Read the last question and locate the key words: Four light bars – smallest digital number.

In a similar way, ask the students to complete the chart working through five, six, seven and eight light bars.

Patterns will form. The digital form of 1 is valuable when finding large numbers because it only takes up two light bars. 0 and 8 are valuable when finding smallest values because they use many light bars.

Further expLorationTask Card 1

Read the question with the class: ‘Some different digits use the same number of light bars. 4 and 7 both use four light bars. Use four light bars to make as many different numbers as you can. Now write these numbers in order from smallest to largest. Do the same thing using five light bars. Do the same thing using six light bars.’

Locate the key words: Four light bars – as many different numbers as you can. Smallest to largest. Same thing, five, six light bars.

Patterns form again in this activity and raise a discussion of our place value system. Why is it that the number 14, which uses six light bars, enables us to create 41 without even using our iceblock sticks? Why does this also work for 17 and 71?

This task card activity can be extended to include larger numbers of light bars that will effectively reinforce the concepts under review.

StrategiesD Locate key wordsD Look for a patternD Create a table or chart D Make a model

Light Bars

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Strategies D Locate key words D Look for a pattern D Create a table or chart D Make a model

All You Need to Teach Problem Solving Ages 5–8 © Peter Maher/Macmillan Education Australia.This page may be photocopied by the original purchaser for non-commercial classroom use. 21

Name Date BLM 1

Use your calculator to help you draw the digital numbers 0 to 9.

0 1 2 3 4 5 6 7 8 9

Write the number of light bars used to form each digital number.

Do the same for five, six, seven and eight light bars.

Light BarsLight Bars

Use four light bars to make the biggest digital number possible.

Use four light bars to make the smallest digital number possible.

number

light bars

number

light bars

Number of light bars 5 6 7 8

Biggest digital number

Smallest digital number

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wLesson Plan 2

BackgroundOur Hindu-Arabic system of numeration has place value embedded within it. Therefore, unlike in ancient numeration systems such as Egyptian Numerals and, to a lesser extent, Roman Numerals, the placement of a digit in a number determines the value of the entire number.

159 and 915, for example, use the same three digits and yet, because of place value, represent two very different numbers.

orientationExplore the concept of place value with students. Demonstrate the way in which the position of a digit in a number changes the value of the digit. For example, although 21 and 12 are made up of the same digits, the placement of the digits makes them entirely different amounts. Have students make as many different one- and two-digit numbers as possible using only the digits one and two.

guided discovery with BLM 2Read the question with the students. Locate the key words: Six JAN plates, all use 1, 2 and 3.

Suggest that students try a similar, but simpler problem. For example, use 1 and 2 rather than 1, 2 and 3. This leads to the possible answers 12 and 21. Note how these are in numerical order.

Now begin the problem by finding the smallest possible number that could appear on the numberplate: 123.

Ask students to find the next smallest number (132). This strategy will help students to locate patterns and will train them to record their results in a logical manner, instead of selecting answers at random, which makes it less likely that all possible answers will be found.


This is a similar combinations problem, but with a fixed value (the colour black) and colours replacing numbers, offering further challenges.

The students need to recognise that the skills developed in BLM 2 are readily transferable to this task card.

Read the question with students: ‘Traffic lights on Planet Zog are very different to traffic lights on Earth. Black is always at the top and means “Fly Safely”. The other colours are orange, meaning “Go up”, blue, meaning “Go down”, pink, meaning “Light speed”. Orange, blue and pink can be second, third or fourth on the traffic light. Use coloured counters to show what the traffic lights on Zog might look like.’

Help students to locate the key words in the question: Black on top. Orange, blue and pink in any order.

Ask the students to complete the first traffic light in the order given in the question:

Black, orange, blue, pink

Then ask students for a logical second light. The more randomly they search for answers, the less likely they are to get all the answers.

Black, orange, pink, blue is a logical second answer because it retains the positions of both black and orange and completes all the possible answers that have orange in the second position.

Ask students what might come next, and ask them to give reasons.

StrategiesD Locate key wordsD Look for a pattern D Find a similar but simpler problem

Digit Shift

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Strategies D Locate key words D Look for a pattern D Find a similar but simpler problem


Name Date

Jan wants a new number plate for her car. There are six JAN number plates available. They all use the numbers 1, 2 and 3.Write down all six number plates Jan could choose from.

Digit ShiftDigit ShiftBLM 2

JAN

JAN JAN

JANJAN

JAN

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wLesson Plan 3

BackgroundThis activity exercises the students’ abilities to bond to the number ten and to explore addition in a creative way.

resources:

D concrete materials to construct a model

orientationDemonstrate that ten can be made up in many ways by using addition, for example: 1 + 9, 4 + 6, 2 + 4 + 3 + 1, and so on. Have each student come up with a number of different addition equations that bond to ten.

guided discovery with BLM 3Read the question with students. Help students to locate the key words: Each house one more storey. Make a model. Draw it.

Provide appropriate concrete material for the students to manipulate. Ask them to make a model of the street. What patterns do they see forming?

With the class, read the next part of the problem, and locate the key words: Number of storeys – first nine houses – three different ways.

a) Left to right: 1, 3, 6, 10, 15, 21, 28, 36, 45. Respond that this is correct, but is not the best way. Emphasise that it is the strategies used, not the answer, that is important.

b) 1 + 9, 2 + 8, 3 + 7, 4 + 6, + 5. This bonds to ten and is quicker and easier.

c) 1 + 2 + 3 + 4 , 5 + 6 + 9, bonds to ten then to 20, then add 7 to 8 for another 10 + 5.

d) Ask if the students could bond to nine: 1 + 8, 2 + 7, 3 + 6, 4 + 5, 9 = 5 groups of 9 = 45.

The numbers from one to nine are consecutive and so can be separated into groups of double fives, because five is the median of the set of numbers.

1 + 9 = 5 + 5, 2 + 8 = 5 + 5, 3 + 7 = 5 + 5, 4 + 6 = 5 + 5. Then add on the other five, giving nine groups of five.

The students may come up with their own creative strategies as well.


Read the question with the class: ‘On the right side of Storey Street the houses double in size. The first house has one storey, the second house has two storeys, the third house has four storeys, and so on. Make a model of the right side of Storey Street. Find how many storeys altogether are on the first five houses on this side of the street.’

Help students locate the key words: Houses double in size. Model of first five houses. Number of storeys.

While the students are connecting their blocks together, ask them if they can see any patterns or if they could find the number of storeys in the houses in any faster way.

Ask them what would be the quickest way of adding up 1 + 2 + 4 + 8 + 16

Can we bond to 10 in any way? 2 + 8, 4 + 16, + 1 is much easier.

How could we use a calculator to find the answer?

If they work from left to right, students may see the pattern that the sum is always one less than the number of storeys in the next house.

Ask students to predict the sum of the first six houses on this side of the street. What about the first seven houses in the street?

StrategiesD Locate key wordsD Look for a patternD Make a drawing D Make a model

Storey Street

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Strategies D Locate key words D Look for a pattern D Make a drawing D Make a model


Name Date BLM 3

What you need: connecting blocks

On the left side of Storey Street, each house has one more storey than its next-door neighbour. The first house has one storey, the second house has two storeys, the third house has three storeys, and so on.Make a model of the left side of Storey Street by using connecting blocks. Draw what you have made.

Find three different ways of finding out the number of storeys, altogether, on the first nine houses.

1

2

3

Storey Street

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wLesson Plan 4

BackgroundA is heavier than B. B is heavier than C. Therefore, A must be heavier than C. This logic is obvious to an adult, but to a young student, this needs to be proven concretely. Through the manipulation of concrete materials, the activities found in this lesson will foster the development of logical analysis.

resources:

D classroom items for weighing

D scales

orientationAdvise the students that we can draw conclusions about certain things by using our experience and our common sense. Show them two items that are clearly different in weight, and ask them to nominate which is heavier. Gradually develop the concept by comparing items increasingly similar in size, to help students understand the basic relationship between size and weight.

guided discovery with BLM 4With the class, read the question. Help students to locate the key words: Three things to weigh. Two items at a time. Compare masses.

Use the strategy of finding a similar but simpler problem to show them that when we deal with numbers we can also use these skills:

7 is bigger than 5

5 is bigger than 2

So, it must be the case that 7 is bigger than 2.

Ensure that the students select three objects that have similar masses, or at least are not obviously different. Ask them to draw their conclusion before the final weighing, then get them to prove the validity of their conclusion by using the scales.


With the class, read the question: ‘Find four things that you own at school. Use scales to compare their masses, two at a time. Once you have done this, lay them out in order from lightest to heaviest.’

Help students to locate the key words: Four things. Compare masses two at a time. Order from lightest to heaviest.

During the activity, ask the students to draw conclusions progressively as they discover accurate pair comparisons, such as:

A is heavier than B

B is heavier than C

C is heavier than D

So, what can we say about A compared to D? What can we say about B compared to D? What can we say about C compared to A?

StrategiesD Locate key wordsD Find a similar but simpler problem D Think logically

A Weighty Problem

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Name Date BLM 4

A Weighty ProblemA Weighty Problem

What you need: items for weighing scales

Find three things in your classroom that you would like to weigh.

What are they?

Take two items at a time. Use a set of scales to compare their masses.

Fill in the missing words:

is heavier than

is heavier than

So must be heavier than

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wLesson Plan 5

BackgroundDeveloping probability concepts enhances logical thinking skills. Using the structure of a deck of playing cards to broaden understanding of probability is classic problem solving. The game of This Card Is ... is an excellent way of demonstrating this fact and a very enjoyable mathematical exercise.

resources:

D deck of playing cards

orientationIn this activity students will learn to play the game of This Card Is ... . Remove all the picture cards and jokers from a deck of playing cards. Tell students that the ace has a value of one, giving a total of 40 cards left at face value. Show the class how the 40 cards can be separated and grouped in different ways, for example:

D 10 clubs, 10 diamonds, 10 hearts and 10 spades.

D 20 black and 20 red cards

D four of each number

D 12 cards with four letters in their number name

D 20 even and 20 odd cards, and so on.

Have students form a single line in front of you, and place a deck facedown on the table. State a ‘fact’ concerning the top card in the deck, such as ‘This card is a 5’. The student can respond either ‘Yes’ or ‘No’. If their response is correct, the student goes to the back of the line and stays in the game. If they are incorrect, the student sits down. The game continues until only one player remains.

guided discovery with BLM 5Prior to using the blackline master, ensure that students are familiar with the rules and structure of the game. Explain that the statements found on this sheet are just as if they were a part of the game previously played.

Read the question, then locate the key words: Answer ‘yes’ or ‘no’.

Through practice, students will come to realise that what seems to be a sensible answer won’t always be correct. For example, the statement ‘This card is bigger than eight’ prompts the sensible response of ‘No’. However, students will come to realise that there is a chance that the card may be bigger than eight. If you feel the students are ready, discuss the concepts of 20% or 8 compared to 32 or 8/40.


Ensure students are well practiced in This Card Is ... before completing this task card activity.

Read the question: ‘Pretend that you are the teacher in a game of This Card Is ... . Place a deck of cards in front of you, and make a statement about the next card you will pick up.

Make up five statements that should get ‘yes’ as the answer. Make up five statements that should get ‘no’ as the answer. Now make up three statements that could get either ‘yes’ or ‘no’ as the answer.’

Answers will vary for this activity but make sure that you ask the students why they have selected the statements that they have. Also, encourage them to vary their statements as much as they can.

StrategiesD Locate key wordsD Think logically

This Card Is...

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Strategies D Locate key words D Think logically

What you need:deck of playing cards with picture cards removed

Pretend that you are playing This Card Is ... . Answer ‘yes’ or ‘no’ to the following statements:

This card is a 5

This card is smaller than 3

This card is bigger than 9

This card is an even number

This card is either a 4 or a 9

This card is black

This card is bigger than 4

This card is a card with 3 letters in its number name


Name Date BLM 5

This Card Is...This Card Is...

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wLesson Plan 6

BackgroundFinding combinations is an excellent way for students to develop their powers of clear thinking. Young students tend to take a random approach to questions that present numerous possible answers or components. This can mean that, while they do find some solutions, they are unlikely to find them all. As teachers, we can develop students’ logical perspective as they attempt questions such as these.

resources:

D four students chosen at random

D red, blue and yellow pencils

orientationBegin the lesson by asking four students to form a line. Explain to the class that this order, from left to right, of, for example, Adam, Brittany, Cleo and Davis, is not the only possible order in which the four students could have lined up. Another option could have been Cleo, Adam, Brittany and Davis or Davis, Cleo, Brittany and Adam, and so on. Ask the class to estimate the number of different arrangements there could be for these four students (there are 24 ways).

guided discovery with BLM 6Read the question with the class. Help students to locate the key words: Red, blue, yellow, six trains, all different. Only one colour on any engine or carriage.

Ask the students to colour the first engine red. Can any of the carriages now be red?

Ask the students to colour the carriages on that train. What might they look like?

Move on to the next train. Ask the students to colour this engine red too. Therefore, what must these carriages look like?

Ask the students to complete the question, emphasising the need for a logical approach to the remaining trains. Ask the class what patterns they see forming as they work through the colouring exercise. Also ask them why they think the problem is being solved in this way.


Read the question with the class: ‘Mrs Mane is going to the races. She has a black hat, a black dress and black shoes, a white hat, a white dress and white shoes. Use black and white counters to show the eight possible outfits that Mrs Mane could wear to the races.’

Help students to locate the key words: Black hat, dress and shoes. White hat, dress and shoes. Eight different outfits.

Suggest that students start with an outfit that is all black. Then suggest that they make an outfit that is all white. Suggest that four outfits will have a black hat and that four will have a white hat.

Now invite students to finish the task.

StrategiesD Locate key wordsD Look for a pattern D Make a drawingD Think logically

Colour the Trains

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Name Date BLM 6

What you need: red, blue and yellow pencils

The Sunny Hill railway has bought six new engines and 12 new carriages.The painter can use red, blue and yellow paint to paint the new trains. He must make sure that they all look different.He can use only one colour on any engine or on any carriage. Each train must have a red, blue and yellow section.Colour the six trains.

Colour the TrainsColour the Trains

Strategies D Locate key words D Look for a pattern D Make a drawing D Think logically

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wLesson Plan 7

BackgroundThis activity draws upon the students’ creativity, knowledge of basic number facts and understanding of what a sequence is and how it is structured. The lesson plan concentrates on the strategy of looking for a pattern.

resources:

D calculator

orientationBegin the lesson by explaining that a sequence is a number pattern and, as such, it must have a rule. For example, the sequence 1, 4, 7, 10 ... has a rule of + 3.

Some sequences have more than one rule. For example, the sequence 1, 4, 9, 16 ... has the rule of multiplying numbers by themselves (squaring) or + 3 then + 5, then + 7, and so on.

Most sequences never end, which is signified by three dots following the final term given. The three dots can be read as ‘and so on’.

Each member of a sequence is called a term. Thus, in the sequence 10, 15, 20, 25 ... the first term is 10, the third term is 20 and, because it has a rule, we could work out that the tenth term would be 55.

guided discovery with BLM 7Read the question with the class. Locate the key words: Six different sequences that start with 1 and 2.

Encourage the students to be creative with their thinking. Do we need to consider these digits as being 1 and 2? Could they make 12, or be the start of a larger number still? Do sequences always increase in size?

For some students, concrete materials will offer invaluable assistance with this task.


This task card presents only the first term of a number pattern. Again, encourage students to be creative and to produce as many different sequences as they can. For each sequence, ask them to give the rule of the pattern they have formed. Again the use of concrete materials should be offered and encouraged.

Read the question with students: ‘On a piece of paper, draw ten screens like the one you see here. Draw a 5 in the far left cell on each screen. This is the start of ten different number patterns. Can you use your calculator to make ten different patterns that start with 5?’

Then help students to locate the key words: 10 sequences starting with 5.

StrategiesD Locate key wordsD Look for a pattern D Think logically

What Comes Next?

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Name Date BLM 7

What you need: calculator

Enter the digits 1 and 2 into your calculator.This is the start of a sequence.Fill in the calculator screens below.Make six different sequences that start with 1 and 2.

What Comes Next?

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wLesson Plan 8

BackgroundThis activity extends students’ understanding of likelihood and builds upon fractional knowledge.

resources:

D 2 red, 2 yellow and 2 green teddy countersD a lidded boxD one red, yellow and green disc (can be found in

a set of attribute blocks)

orientationTeach students to play Teddy Traffic Lights. Show the teddies to the class, then place them in the box, put the lid on and shake it. Space the red, yellow and green discs around the classroom. These represent traffic lights. Ask students to predict which coloured teddy will emerge from the box, and to move to that coloured disc. If their prediction is correct, they stay in the game. After a teddy is removed, leave it aside. Before the second selection, show students the remaining teddies then ask them to stand at a traffic light. Ask students why they are standing where they are and how many teddies of their chosen colour are left in the game. The game continues until no one is left in or until all teddies are removed.

Once the students are familiar with the game, they can be introduced to the fractions involved. For the first choice each student has a two in six chance of staying in the game and a four in six chance of leaving the game. After the first selection, the odds change, and continue to do so throughout the game.

guided discovery with BLM 8Read the question with the class. Help students to locate the key words: Play two games. Predict each colour one at a time. Record the colour of the teddy. Write down the chance.

Before each teddy is drawn, ask students why they have made their choices. Check the fractional chance before each draw, assessing students’ fractional understandings.

Explore these questions with the class:

Was the winner the luckiest or the smartest?

At which point in the game did players have a 50/50 chance of staying in?

Why did no one leave the game when the sixth teddy came out?

The first teddy removed was red. Did this increase or decrease the chance of the next teddy being red?

Was there a pattern in the way the teddies were drawn? If not, why not?

Note that the chance of staying in the game is one in 90, no matter what combinations of colours emerge.


Students will need plenty of practice playing Teddy Traffic Lights before undertaking this task card.

Read the question with the class: ‘In a game of Teddy Traffic Lights there were two blue and two yellow teddies. Place a green counter next to the sensible predictions and a red counter next to those that are not sensible.’

Help students locate the key words: Predictions. Green if sensible. Red if not sensible. This activity asks students to analyse the predictions of a student in an imaginary game. Students assess whether the predictions were sensible or not sensible for each of the six draws. Ask each student why they have made their judgement.

StrategiesD Locate key wordsD Look for a pattern D Create a table or chartD Think logically

Teddy Traffic Lights

34

A sample game follows:

COLOUR OUT CHANCE

1. Red 2 out of 6

2. Yellow 2 out of 5

3. Green 2 out of 4

4. Yellow 1 out of 3

5. Green 1 out of 2

6. Red 1 out of 1

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Play two games of Teddy Traffic Lights.

Strategies D Locate key words D Look for a pattern D Create a table or chart D Think logically


Name Date BLM 8

Teddy Traffic LightsTeddy Traffic LightsWhat you need: red, yellow and green teddy counters; a lidded box

Turn What colour What colour What was the do you did come chance that predict will out? this colour come out would come of the box? out?

1

2

3

4

5

6

Turn Predict Colour Colour Out Chance

1

2

3

4

5

6

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wLesson Plan 9

BackgroundThe game of First to 50 extends addition skills and develops students’ ability to use a calculator.

resources:

D calculator

orientationIn this activity, students will learn to play First to 50. Have students form pairs, taking turns to add numbers less than ten to the total. The first player to reach 50 wins the game. For example:

Player A: + 7 = 7

Player B: + 6 = 13

Player A: + 8 = 21

Player B: + 5 = 26

Player A: + 9 = 35

Player B: + 7 = 42

Player A: + 8 = 50 and wins the game.

This example represents two players who are, with the exception of the final move, playing randomly. It is also possible that the players have little idea of the effect of their addition prior to hitting the = button.

Ask students to analyse the final move. Tell the class that the best players use a strategy to win.

Have students pair off and see what they can discover. The first key discovery should be that whenever a player is forced above 40, they are in a losing position because from there, only one move is required to get to 50. Eventually, through trial and error, finding a pattern and by using logic, students should come to realise that the trick is to keep increasing the total to a multiple of ten. For example:

Player A: + 5 = 5

Player B: + 5 = 10

Player A: + 7 = 17

Player B: + 3 = 20

Player A: + 4 = 24

Player B: + 6 = 30

Player A: + 9 = 39

Player B: + 1 = 40

Player A: + 8 = 48

Player B: + 2 = 50. Player B wins.

guided discovery with BLM 9Read the question with the class, and locate the key words: Force me on to 10, 20, 30, 40.

Practise bonding to ten prior to completing this sheet. Ask the students why this strategy will work.

This game can be extended in many ways. The numbers used can be increased or decreased and the total can be varied. For each variation, a new strategy will need to be developed. For example, with a target of 30, and a range of numbers from 1 to 6, the object is to force the opponent onto multiples of 7 less than the target: 23, 16, 9 and 2. The strategy will always be to force the opponent onto a multiple of one more than the highest addition allowed.


Read the question with the class: ‘Play a game of First to 20. Take it in turns with a partner to add a number, until one of you takes the total to 20. You can use the numbers 1, 2, 3 or 4.’

Locate the key words: First to 20. Use 1, 2, 3 or 4.

Monitor the students’ progress as, hopefully, they win by using the strategy of forcing their opponent on to 5, 10 and, eventually, 15.

StrategiesD Locate key wordsD Look for a pattern D Assume a solution

First to 50

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Strategies D Locate key words D Look for a pattern D Assume a solution


Name Date BLM 9

First to 50First to 50

MY TURN YOUR TURN TOTAL +3 10 +8 20 +1 30 +6 40 +2 50

Let’s play a game of First to 50.

I’ll go first.Try to force me on to 10, then 20, then 30, then 40.

See if you can beat me!

What you need: calculator©

Macmillan Education Aus

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One Die RollwLesson Plan 10

BackgroundThis activity introduces students to the concept of probability and enhances their number sense through rolling a six-sided die. Before starting the activity, students should have the experience of rolling one die, observing the results and eventually concluding that all six numbers have an equal chance of being rolled. This activity requires participants to apply these facts to unfamiliar situations often with more than one variable and requiring higher-level logical thinking skills.

resources:

D one die

orientationIn this lesson, students will learn how to play One Die Roll. Have students choose to stand on either your left or right, forming two lines with a corridor between them for you to roll the die. Each line represents an outcome. The students stay in the game as long as their predicted outcome is the actual outcome.

For example, before rolling the die, say ‘Stand to my left if you think the number will be even. Stand to my right if you think the number will be odd.’

Some other directives could be:

Left = a number less than 3. Right = 3 or more.

Left = a number with an e in its number name. Right = a number with no e in its number name.

Left = 1, 2, 3 or 4. Right = 5 or 6.

Left = a number with 3 or 4 light bars in its digital form. Right = a number with 2, 4 or 6 light bars.

Left = a number that is your age. Right = a number that is not your age.

Left = a number bigger than 2 + 2. Right = a number that is 2 + 2 or smaller.

Left = a number that is half of 10. Right = a number that is not half of 10.

guided discovery with BLM 10The blackline master offers the students the opportunity to demonstrate their understandings of chance through a mock game of One Die Roll.

Read the question with the class, then help students locate the key words: Tick left or right column.

Ensure that the students reflect before each judgment is made. The use of a pencil and a piece of paper will assist with considered decision making. Many students will prefer to write down the possible outcomes and the numbers that will determine their choice on each occasion. Other students will be satisfied to consider the comparative probabilities mentally. Both options should be seen as equally appropriate.


With students, read the question: ‘Pretend that you are the teacher in a game of One Die Roll.

Write down three instructions that will give both sides of the room the same chance. Write down three instructions that will favour one side over the other. Now write down three instructions that will only give one side of the room a chance.’

Locate the key words: Three instructions both sides the same chance, three favouring one side and three giving only one side a chance.

Encourage students to draw on their understanding of probability to determine which instructions are most likely to achieve the desired result.

StrategiesD Locate key wordsD Think logically

One Die Roll

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Strategies D Locate key words D Think logically


Name Date BLM 10

For each roll, tick what you think is more likely to happen — Outcome A or Outcome B.

One Die RollOne Die Roll

Roll Outcome A Outcome B1, 2 or 3 is rolled

1 or 2 is rolled

The number is lessthan 5

The number is even

The number is a halfof 8

The number has 4letters in its name

The number is oddor a 4

The number is lessthan 7

1

2

3

4

5

6

7

8

4, 5 or 6 is rolled

3, 4, 5 or 6 is rolled

The number is 5 orgreater

The number is odd

The number is not ahalf of 8

The number doesn’t have4 letters in its name

The number is a 2 or a 6

The number is 7 ormore

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wLesson Plan 11

BackgroundThe activity, ‘Mrs Murphy’s Class’, enables the students to exercise several problem solving strategies, including making a model, by actually acting out the mathematics contained within the problem.

resources:

D 12 playing cards, randomly chosen

D 12 students, randomly chosen

orientationExplain to the class that the number 12 can be shared in many different ways. Ask 12 students to come to the front of the class and give them one card each. Explain to students that we can share 12 cards with 12 people because they will get one card each and no cards will be left over. Ask what other groups will work in the same way when sharing 12 things.

Attempt to distribute the 12 cards among 11, ten, nine, eight, seven, six, five, four, three, two and eventually one student to demonstrate the fact that 12 has the factors of one, two, three, four, six and 12.

guided discovery with BLM 11Read the problem with the class. Help students to locate the key words: Pairs, groups of three, four, six. More than 15.

An excellent way to begin is to present the students with a similar, but simpler problem. Ask the students if, for example, there were 12 students in the class, how could this number be grouped? Encourage the students to draw 12 stick figures and to group them by circling into as many different equal groups as possible. Then get them to physically stand and group themselves into one, two, three, four, six and 12 equal groups. The blackline master can now be attempted in a similar way. Ask the class for a suggestion to explore. The suggestion that the number of students must be even is a great start. Why is this the case? Have students test a prediction of 20 through physical modelling, drawing stick figures and so on. The chart should then be filled out progressively.


Read the question with the class: ‘When Mr Smedley’s class is split into pairs, there is always one student left over. When split into groups of three there are always two students left over. When split into groups of four there are always three students left over. There are more than 15 students in the class. How many students are in Mr Smedley’s class?’

Locate the key words: Pairs – one left over. Groups of three – two left over. Groups of four – three left over. More than 15.

Begin by asking the students what we already know about the number. Why, for example, must it be odd? If the number is odd, what is the first number to model? Why didn’t 17 work?

StrategiesD Locate key wordsD Assume a solution D Create a table or chart D Make a drawing D Find a similar but simpler problemD Make a model

Mrs Murphy’s Class

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Name Date

Mrs Murphy’s class loves to work in pairs, and in groups of three, four or six.There are more than 15 students in the class.How many students are in Mrs Murphy’s class? Fill in this chart to find the number of students that can make groups of two, three, four or six.

So the answer is


Strategies D Locate key words D Assume a solution D Create a table or chart D Make a drawing D Find a similar but simpler problem D Make a model

BLM 11


GROUPS

2 3 4 6 16 yes no yes no

17 18 19 20 21 22 23 24 25

Students

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wLesson Plan 12

BackgroundThis activity utilises students’ understanding of addition and encourages several problem solving strategies.

resources:

D connecting blocks

orientationBegin the activity with a discussion of the ways in which animals can differ – their outer coverings, the ways they move, the ways they give birth, and so on. Remind students that animals can also differ in the number of legs they possess. Ask for examples of animals with two, four, six, eight and ten legs.

guided discovery with BLM 12Read the question with the class. Help students to locate the key words: Birds, dogs, beetles. 12 legs altogether.

Encourage students to connect 12 blocks to represent the 12 legs. Then direct them back to the blackline master to see that there are five possible types of answers. Explain that the collection could contain all three types of creatures, or just two or possibly the same creature over and over again.

Ask the class for a possible solution and examine its accuracy. Does it use all 12 legs? Is it grouped into twos, fours and sixes? Should we start with the creature with the greatest or smallest number of legs? Logic suggests that the collections should follow a pattern and so it is sensible to start with the greatest number of legs.


To successfully solve this problem, students need to be logical in their approach and work through the task in a methodical fashion.

Read the question with the class: ‘Hannah has pet birds, dogs, butterflies, spiders and scorpions. In one part of her backyard she counted four creatures with 20 legs altogether. What creatures might Hannah have seen?’

Help students to locate the key words: Birds, dogs, butterflies, spiders, scorpions. Four creatures, 20 legs.

Ensure that students have enough connecting blocks to complete the task. Remind them that their solution must have four animals and 20 legs. Discuss the number of legs on each type of creature. Encourage the students to start with the creature with the greatest number of legs and work through in a numerical format. Do not expect the students to get all of the answers. It is the process used to reach the answers that matters.

StrategiesD Locate key wordsD Look for a pattern D Assume a solution D Make a modelD Think logically

Birds and Dogs and Beetles

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Name Date


Billy loves animals, especially birds, dogs and beetles.One day, he counted all of the legs on the animals in his pet collection. There were 12 legs altogether.Use connecting blocks to show what his pet collection might look like. Then write down your answers.

2 creatures:3 creatures:

4 creatures:

5 creatures:6 creatures:


Strategies D Locate key words D Look for a pattern D Assume a solution D Make a model D Think logically

BLM 12


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wLesson Plan 13

BackgroundThis activity utilises students’ knowledge of addition and the coins in our money system.

resources:

D one example of each Australian coin

orientationIntroduce students to the coins in the Australian currency system. Show them how different combinations of coins can make the same amount, for example, $1 can be made with a one dollar coin, two fifty-cent coins, five twenty-cent coins, and so on. Ask students which coins would be required to make up 30 cents in the simplest way possible. How about 70 cents? Help students to become familiar with the values of different coins.

guided discovery with BLM 13Read the question with the class, then help students to locate the key words: $1.05. Equal numbers of 20, 10 and 5 cent coins.

At this point in the lesson ask students for a suggestion that fulfils the rules and structure of the question. One of each coin is an excellent start. The table on the sheet should then be filled in, in the following manner:

Call for another suggestion. Keep filling in the table until the correct solution is found.

Help students to identify a pattern in the table. How much money does the total increase by whenever a new coin is added to the 20, 10 and 5 cent list? If the answer is 35 cents, could we have counted in lots of 35 cents to find the answer faster?


Read the question with the class: ‘Jackson has five coins in his pocket. Each coin is different. Together, the coins add up to $3.75. What coins are in Jackson’s pocket?’

Help students to locate the key words: Five different coins adding up to $3.75.

Ask the students to list all of the coins in our money system. Can we say that any of these coins must be in Jackson’s pocket? Why must there be a 5 cent coin? Why must there be a $2 and a $1 coin?

Encourage students to assume a solution, to add up the total and to tinker with the amount until the solution is found.

StrategiesD Locate key wordsD Look for a pattern D Create a table or chart

Money Matters

44

20 10 5 Total of Total number cents cents cents money of coins

Number 1 1 1 35 cents 3 of coins

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Name Date

Jackson has $1.05 in his moneybox.He has an equal number of 20 cent, 10 cent and 5 cent coins.How many coins does Jackson have in his moneybox? Use the table below to work out the answer.

Money Matters

Strategies D Locate key words D Look for a pattern D Create a table or chart

BLM 13

20 10 5 Total Total number cents cents cents money of coins

Number 1 1 1 of coins

Number of coins

Number of coins

Number of coins

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wLesson Plan 14

BackgroundThis activity exercises the students’ abilities to work in a spatial context and with fractions and enables them to assume a solution and look for a pattern.

resources:

D fraction kits that show equal pieces in the form of pizza slices, sandwich slices, bricks on a wall or birthday cake pieces

orientationUsing fraction kits, demonstrate to students that the top layer of the kit represents one whole thing after being cut into two equal pieces with just one cut. Each piece is called a half.

Then show the students the one-third pieces that have been created with the use of three separate cuts (the peace symbol). Ask the students if this is the only way that something can be cut into three pieces. Can it be done with just two cuts? Would these three pieces be equal in size?

guided discovery with BLM 14Read the first part of the question with the class, then help students to locate the key words: Three equal pieces. How many cuts?

Trial and error will enable the students to solve this part of the problem.

Read the next part of the question. Locate the key words: Four equal pieces. How many cuts?

Trial and error will, again, eventually lead to the correct answer. Encourage students to start looking for a pattern at this stage.

Read the next part of the question, and locate the key words: Pattern. Fill in table.

Discuss the pattern of the number of cuts being one less than the number of pieces, and extend as a general rule.


This activity presents students with both horizontal and vertical sectioning. Again, the need to look for a pattern features strongly in this task.

Read the question: ‘Ashlee cut her birthday cake into four equal pieces with just two cuts. Can you see how she did this? Draw Ashlee’s cake and show where the cuts were made.’

Locate the key words: Two cuts. Four pieces.

Trial and error should produce the correct answer.

‘Ashlee made two more cuts and found that she now had eight pieces of cake. What does the cake look like now? Draw Ashlee’s cake and show where the cuts were made.’

Locate the key words: Two more cuts. Eight pieces.

Ask students whether their answer was the only possible solution. Two cuts made four pieces and four cuts made eight pieces of cake. Can they see a pattern and predict what nine cuts might produce?

StrategiesD Locate key wordsD Look for a pattern D Assume a solution D Create a table or chart

I’m a Lumberjack

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Name Date

Ashlee is a lumberjack. She needs to cut a fallen tree into three equal pieces. How many cuts does she need to make?

Ashlee needs to cut a piece of wood into four equal pieces. How many cuts does she need to make?

Can you see a pattern? Fill in this table:

I’m a Lumberjack

Strategies D Locate key words D Look for a pattern D Assume a solution D Create a table or chart

BLM 14

I’m a Lumberjack

Number of cuts Number of equal pieces

1

2

3

4

5

10

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wLesson Plan 15

BackgroundThis activity encourages students to work in reverse. This strategy lends itself well to the type of multi-stage numerical question found below.

resources:

D 20 playing cards and three students, randomly chosen

orientationLine three students up. Give 20 playing cards to the first student (A). Then ask this student to give five of the cards to the second student (B) in the line. Then ask student A to give four cards to the third student in the line (C). Then ask the class how they could work out how many cards student A had without counting the cards in his/her hand.

4 + 5 = 9, 9 + 11 = 20. Student A must have 11 cards. Thus, the inverse operation has been used to solve the problem, working from the end of the line to the front, moving in reverse order to the way the question was structured.

guided discovery with BLM 15Read the question with the class. Help students to locate the key words: Sarah gave $10. James gave $5. All the same money. How much did Sarah have?

Trial and error will, of course, eventually lead to the correct solution, particularly with the assistance of concrete materials. However, it is the technique of working in reverse that lends itself best to this question.

The last stage of the question tells us that Tom had $5. If all of the children had the same amount of money, it logically follows that James has $5, confirmed by the fact that $10 – $5 = $5. This means Sarah must have $5 too, after giving away $10, so she must have started with $15.


As with BLM 15, this problem could be solved with trial and error by assuming a solution and testing it, but working in reverse should be the encouraged technique.

Read the question with the class: ‘Sarah, James and Tom collect pony stickers. Sarah has twice as many stickers as James. James has twice as many stickers as Tom. Tom has ten stickers. How many stickers do James and Sarah have?’

Help students to locate the key words: Sarah twice James. James twice Tom. Tom has ten.

Ask the students if the information about Tom’s collection can be used to help find the number of stickers that the other two have. If James has twice ten, he must have 20. What fraction of Sarah’s stickers does this represent? How can we work this out?

StrategiesD Locate key wordsD Work in reverse

Sarah, James and Tom

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Name Date

Sarah gave James $10.Then James gave Tom $5.Then all three children had the same amount of money.How much money did Sarah have to start with?


Strategies D Locate key words D Work in reverse

BLM 15


Show your working:

Now Tom has $

So James must have $

And Sarah must have $

To start with, Sarah must have had all the money.

So, Sarah must have started with $

Now go back to the original problem and see if this answer works.

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wLesson Plan 16

BackgroundExploring the attributes of cubes leads to valuable arithmetic and geometric discoveries. The correct name for a cube is hexahedron, coming from two Greek words meaning ‘six faces’. The cube is one of five regular polyhedra, which are constructed from the same polygon (many corners). The cube, a regular polyhedra, looks the same no matter how oriented. It contains eight vertices or corners and 12 edges, is a wonderful construction tool and makes a spectacular model in a large format.

resources:

D MAB blocks

D connecting blocks

orientationBegin the lesson with a discussion of the properties of the cube. Point out that, in a sense, the cube is a 3D square. Ask students where they would find cubes in the environment. Answers such as sugar cubes and some tissue boxes spring readily to mind. Bring out some MAB blocks to show the students that the unit block is a cube, as is the 1000 block. Other base MAB cubes, if the school has any, form a very impressive cube tower when stacked on top of each other. The students should be encouraged to count the number of blocks on the bottom row of the 1000 block, then the number on the side and finally the number in the block’s height. What do we notice? Turn the block around so that the students can appreciate that the cube remains looking the same no matter which way it is facing. Why does this happen?

Show students that connecting blocks, such as multilink, click cubes or centicubes are, in fact, cubes themselves. No matter how they are turned they look exactly the same. The blocks are each one unit long, one unit deep and one unit high.

guided discovery with BLM 16The blackline master activity asks students to apply their knowledge of cubes to creating models.

Read the question with the class and help students to locate the key words: One block is a cube because ... Eight connecting blocks to make a cube. Draw it.

Then read the next question and identify the key words: 18 connecting blocks. Make a cube. If can’t, why not?

Note that 18 has been selected because this will make a two-layered rectangular prism with length and depth of three but with a height of only two. Students often mistake this for a cube.


Read the question with the class: ‘A cube can be made with 27 connecting blocks. Prove this. A cube can be made with 64 connecting blocks. Prove this.’

Help students to locate the key words: 27-block cube. 64-block cube.

This will reinforce the underlying structure of the cube. Remind the students that this regular polyhedra must contain the same number of blocks in its length, depth and height.

Ask the students to estimate the side lengths appropriate for 27- and 64-block cubes and to check the cubic nature of their finished product.

StrategiesD Locate key wordsD Look for a pattern D Make a drawing D Make a model

Cubes, Cubes, Cubes

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Name Date


Pick up one connecting block. This is a cube because

Now use eight connecting blocks to make another cube. Draw what you have made.

Can you make a cube out of 18 connecting blocks?If not, why not?

Cubes, Cubes, Cubes

Strategies D Locate key words D Look for a pattern D Make a drawing D Make a model

BLM 16

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wAll the

You Need

taSk carDS

taSk carDS

52

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53

proBleM solving tAsk cArd 1


Dazzling Digitals

Planet Zog

All You Need to Teach Problem Solving Ages 5–8 © Peter Maher/Macmillan Education Australia.This page may be photocopied by the original purchaser for non-commercial classroom use.

Traffic lights on Planet Zog are very different to traffic lights on Earth.Black is always at the top and means ‘Fly safely’.The other colours are:Orange - ‘Go up’Blue - ‘Go down’Pink - ‘Light speed’Orange, blue and pink can be second, third or fourth on the traffic light.Use coloured counters to show what the traffic lights on Zog might look like.

What you need: D black, orange,

blue and pink counters

Some different digits use the same number of light bars. 4 and 7 both use four light bars.Use four light bars to make as many different numbers as you can.Now write these numbers in order from smallest to largest.Do the same thing using five light bars.Do the same thing using six light bars.

What you need: D calculator D pencil and

paper

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Equal Weight?

What you need: D four items

to weighD scales

All You Need to Teach Problem Solving Ages 5–8 © Peter Maher/Macmillan Education Australia.This page may be photocopied by the original purchaser for non-commercial classroom use.54

Find four things that you own at school. Use scales to compare their masses, two at a time.Once you have done this, lay them out in order from lightest to heaviest.

Double TroubleOn the right side of Storey Street the houses double in size. The first house has one storey, the second house has two storeys, the third house has four storeys, and so on.Make a model of the right side of Storey Street.

How many storeys altogether are on the first five houses on this side of the street?

What you need: D connecting

blocks

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55



What’s Next?

What you need: D black

and white coloured counters


Pretend that you are the teacher in a game of This Card Is ... . Place a deck of cards in front of you, and make a statement about the next card you will pick up.Make up five statements that should get ‘yes’ as the answer.Make up five statements that should get ‘no’ as the answer.Now make up three statements that could get either ‘yes’ or ‘no’ as the answer.

What you need: D deck of

playing cards

Race DayMrs Mane is going to the races. She has a black hat, a black dress and black shoes, a white hat, a white dress and white shoes.Use black and white counters to show the eight possible outfits that Mrs Mane could wear to the races.

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Number Patterns

Sensible Teddies

What you need: D calculatorD pencil and

paper

On a piece of paper, draw ten screens like the one you see here. Draw a 5 in the far left cell on each screen. This is the start of ten different number patterns.Can you use your calculator to make ten different patterns that start with 5?


In a game of Teddy Traffic Lights there were two blue and two yellow teddies. Place a green counter next to the sensible predictions and a red counter next to those that are not sensible. Prediction Actual Sensible or not ? 1. blue yellow 2. yellow yellow 3. yellow blue 4. blue blue

What you need: D 4 red

countersD 4 green

counters

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57



First to 20

Changing Chances

Play a game of First to 20. Take it in turns with a partner to add a number, until one of you takes the total to 20. You can use the numbers 1, 2, 3

or 4.


Pretend that you are the teacher in a game of One Die Roll.1. Write down three instructions

that will give both sides of the room the same chance.

2. Write down three instructions that will favour one side over the other.

3. Now write down three instructions that will only give one side of the room a chance.

What you need: D pencil and

paper

What you need: D calculatorD a partner

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Mr Smedley

Leggy Creatures


blocksD pencil and

paper

When Mr Smedley’s class is split into pairs, there is always one student left over.When split into groups of three there are always two students left over.When split into groups of four there are always three students left over.There are more than 15 students in the class.How many students are in Mr Smedley’s class?


What you need: D calculatorD pencil and

paper

Hannah has pet birds, dogs, butterflies, spiders and scorpions.In one part of her backyard she counted four creatures with 20 legs altogether.What creatures might Hannah have seen?

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59



Loose Change

Birthday Cake


paperD calculator

Jackson has five coins in his pocket.Each coin is different.Together, the coins add up to $3.75.What coins are in Jackson’s pocket?


Ashlee cut her birthday cake into four equal pieces with just two cuts. Can you see how she did this?Draw Ashlee’s cake and show where the cuts were made.Ashlee made two more cuts and found that she now had eight pieces of cake. What does the cake look like now?


paper

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How Many Ponies?

Connecting Cubes


paper


blocks

Sarah, James and Tom collect pony stickers.Sarah has twice as many stickers as James.James has twice as many stickers as Tom.Tom has ten stickers.How many stickers do James and Sarah have?


A cube can be made with 27 connecting blocks. Prove this.A cube can be made with 64 connecting blocks. Prove this.

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wAll the

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anSwerSanSwerS

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BLackLine Masters

BlM 1: light BArs

6 2 5 5 4 5 6 4 7 6

11

4

BlM 2: digit shiFt

JAN 123, JAN 132, JAN 213, JAN 231, JAN 312, JAN 321

BlM 3: storeY street

Refer to the notes found in the lesson plan.

BlM 4: A WeightY proBleM

Answers will vary

BlM 5: this cArd is ...

No; 4/40 chance

No; 8/40 chance

No; 4/40 chance

Even money; 20/40 chance

No; 8/40 chance

Even money; 20/40 chance

Yes; 24/40 chance

No; 16/40 chance

BlM 6: colour the trAins

Red, blue, yellow

Red, yellow, blue

Blue, red, yellow

Blue, yellow, red

Yellow, red, blue

Yellow, blue, red

BlM 7: WhAt coMes next?

Possible answers include:

1, 2, 3, 4, 5, 6, 7, 8

1, 2, 4, 8, 16, 32

1, 2, 1, 3, 1, 4, 1, 5

12, 13, 14, 15

123, 124, 125

1, 2, 3, 4, 4, 3, 2, 1

12, 11, 10, 9, 8

12, 24, 36, 48

12, 10, 8, 6, 4, 2

12, 9, 6, 3, 0

12, 8, 4, 0

BlM 8: teddY trAFFic lights

Answers will vary

BlM 9: First to 50

+ 7, + 2, + 9, + 4, + 8

BlM 10: one die roll

Either left or right

Right

Left

Either right or left

Right

Right

Left

Either right or left

BlM 11: Mrs MurphY’s clAss

There are 24 students in Mrs Murphy’s class. Any multiple of 12 will work for this problem.

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Number of light bars 5 6 7 8

Biggest digital no. 5 111 51 1111

Smallest digital no. 2 0 8 10

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BlM 12: Birds And dogs And Beetles

2 creatures: 6, 6 Beetle, beetle

3 creatures: 6, 4, 2 Beetle, dog, bird

4, 4, 4 Dog, dog, dog

4 creatures: 4, 4, 2, 2 Dog, dog, bird, bird 6, 2, 2, 2 Beetle, bird, bird, bird

5 creatures: 4, 2, 2, 2, 2 Dog, bird, bird, bird, bird

6 creatures: 2, 2, 2, 2, 2, 2 Bird, bird, bird, bird, bird, bird

BlM 13: MoneY MAtters

Jackson had nine coins in his moneybox.

BlM 14: i’M A luMBerJAck

BlM 15: sArAh, JAMes And toM

Sarah had $15.

BlM 16: cuBes, cuBes, cuBes

This is a cube because it has the same length, depth and height.

18 cubes will not connect to make a cube because the finished product will not be the same in all three dimensions.

task cards

tAsk cArd 1: dAzzling digitAls

4, 7 and 11. What strategy was used?

2, 3 and 5. What strategy was used?

0, 6, 9, 14, 17, 41, 71 and 111. What strategy was used?

tAsk cArd 2: plAnet zog

Black, orange, blue, pink

Black, orange, pink, blue

Black, blue, pink, orange

Black, blue, orange, pink

Black, pink, orange, blue

Black, pink, blue, orange

tAsk cArd 3: douBle trouBle

31 storeys

tAsk cArd 4: equAl Weight?

Answers will vary

tAsk cArd 5: WhAt’s next?

Answers will vary

tAsk cArd 6: rAce dAY

Black hat, black dress, black shoes

Black hat, black dress, white shoes

Black dress, white dress, black shoes

Black dress, white dress, white shoes

White hat, white dress, white shoes

White hat, white dress, black shoes

White hat, black dress, white shoes

White hat, black dress, black shoes

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Number Number of of cuts equal pieces

1 2 2 3 3 4 4 5 5 6 10 11

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tAsk cArd 7: nuMBer pAtterns

Possible answers include:

5, 6, 7, 8, 9, 10, 11

5, 7, 9, 11, 13, 15

5, 6, 7, 8, 8, 7, 6, 5

55, 56, 57, 58

5, 10, 15, 20, 25

5, 8, 11, 14, 17

5, 9, 13, 17, 21

5, 10, 20, 40, 80

5, 4, 3, 2, 1, 0

5, 6, 8, 11, 15, 20

50, 60, 70, 80

tAsk cArd 8: sensiBle teddies

Sensible

Not sensible

Not sensible

Sensible

tAsk cArd 9: First to 20

Answers will vary

tAsk cArd 10: chAnging chAnces

Answers will vary

tAsk cArd 11: Mr sMedleY

There are 23 students in Mr Smedley’s class. There are an infinite number of possible answers, each one less than a multiple of 12. For example, 23, 35, 47, 59, and so on.

tAsk cArd 12: leggY creAtures

10, 6, 2, 2 Scorpion, butterfly, bird, bird

10, 4, 4, 2 Scorpion, dog, dog, bird

8, 8, 2, 2 Spider, spider, bird, bird

8, 6, 4, 2 Spider, butterfly, dog, bird

8, 4, 4, 4 Spider, dog, dog, dog

6, 6, 6, 2 Butterfly, butterfly, butterfly, bird

6, 6, 4, 4 Butterfly, butterfly, dog, dog

tAsk cArd 13: loose chAnge

$2, $1, 50 cents, 20 cents and 5 cents

tAsk cArd 14: BirthdAY cAke

tAsk cArd 15: hoW MAnY ponies?

Sarah has 40 stickers.

James has 20 stickers.

Tom has 10 stickers.

tAsk cArd 16: connecting cuBes

The 27-block cube is a 3 x 3 x 3 structure.

The 64-block cube is a 4 x 4 x 4 structure.

64

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Solving Problem

Pr

ob

lem

Solv

ing

into solutions

Turning problems

All you need to teach . . . is a comprehensive series for

smart teachers who want information now so they can

get on with the job of teaching. The books include

background information so teachers can stay

up-to-date on the latest pedagogies, and then

interpret that information with practical activities and

ideas that can be immediately used in the classroom.

The step-by-step lessons in All you need to teach . . . Problem Solving will strengthen your students’ logical and creative thinking skills. With the strategies taught in these lessons, your students will be able to tackle just about any problem!

Inside you’ll find these strategies:

• Locate key words• Look for a pattern• Assume a solution• Create a table or chart• Make a drawing• Work in reverse• Find a similar but simpler problem• Make a model• Think logically.

Also ava i lab le :All you need to teach . . . Problem Solving Ages 8–10

All you need to teach . . . Problem Solving Ages 10+

All the tools a smart teacher needs!

Solving Problem

T e a c h i n g T i p s L e s s o n p L a n s W o r k s h e e T s T a s k c a r d s a n s W e r s

Peter Maher

Ages 5-8

Prob Solv Front Cov 5-8Sept2012.indd 1 26/09/12 9:42 AM

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Documents

All You Need to Teach: Problem Solving Ages 5-8