Brain Benders Series: Book 2 - Ages 9-11 by Teacher Superstore

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Ready-Ed

Acknowledgements for Front Cover i.

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

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Publications

The purchasing educational institution and its staff have the right to make copies of the whole or part of this book, beyond their rights under the Australian Copyright Act 1968 (the Act), provided that: 1.

The number of copies does not exceed the number reasonably required by the educational institution to satisfy its teaching purposes;

Copies are made only by reprographic means (photocopying), not by electronic/digital means, and not stored or transmitted;

Copies are not sold or lent;

Every copy made clearly shows the footnote, ‘Ready-Ed Publications’.

educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL) under Act. For details of the CAL licence for educational institutions contact: Copyright Agency Limited Level 19, 157 Liverpool Street Sydney NSW 2000 Telephone: (02) 9394 7600 Facsimile: (02) 9394 7601 E-mail: info@copyright.com.au

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

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Except as otherwise permitted by this blackline master licence or under the Act (for example, any fair dealing for the purposes of study, research, criticism or review) no part of this book 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 at the address below.

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

ISBN: 978 1 86397 780 7 2

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

Reproduction and Communication by others

6 6 7

Animals in the Zoo The Lily Pond Answers

8 8 9

Six Squares Nine Squares Answers

10 10 11

The Apple Thief Farmer Smith’s Troughs Answers

12 12 13

Friends in a Line Answer

30 31

Farmer Brown’s Farm Answer

32 33

Arohana’s Marbles Answer The Wine Barrels Answer

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The Jellybeans The Eggs Answers

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

34 35 36 37

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The Computer Thieves Answer

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Occupations Answer

Pocket Money Answer The Race Answer

Who’s Wearing What? Answer Merry-Go-Round Answer

18 19 20 21

Pages in a Book Answer

42 43

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Out to Dinner Answer

Brain Busters

Connecting Cables to Houses Answer

44 45

Weighing the Marble Answer

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26 27 28 29

Brother and Sister Answer

48 49

Derryn and James Answer

50 51

Egg Timer Answer

52 53

Filling the Bath Answer

54 55

Teachers’ Notes What is this book about? This book contains twenty-eight photocopiable mathematical problems. The problems have been written and presented to suit a range of abilities and ways of thinking and learning in middle primary school. Problem solving is an important part of the mathematics curriculum and this book has been designed to help students become familiar with, and put into practice, a range of problem solving techniques. The strategies which this book encourages students to use and develop are: guess and check, looking for patterns, drawing pictures and modelling objects, listing and eliminating possibilities, filling in grids, using timelines and making assumptions and estimates and judging the reasonableness of them. The problems are also designed to highlight the importance of reading mathematical language carefully.

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Why have we written this book? We have both been primary school teachers and are aware of the kind of support materials that busy teachers need. Over the years, we have collected the mathematical problems that appear in this book, and have shared many of them with our peers. It is their promptings that have brought about the publishing of this collection. The problems that appear in this book will add flavour and interest to a mathematics programme. They will create discussion and debate and stimulate mathematical thought. It is our belief that children exposed to such problems as the ones in this book, will develop greater powers to solve problems, investigate information and make decisions inside as well as outside of the classroom. Most of these problems do not have immediately obvious answers. Their solutions might well include group discussion, or time to think them over at school or at home.

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What is different about this book? In spite of the fact that the types of problems that we have published have existed for a long time, are enjoyed by children and are an integral part of the curricula, it seems that not all teachers are using them. We think that one of the reasons for this is that they have not been made available in a ‘ready-to-use’ format. We have published this book in what we believe is a ‘ready-to-use’ format: using large print for creating overhead transparencies, mainly limiting one problem to one page, and providing answers which focus on the step-by-step methods which children are likely to use to solve the problems. We are aware that there are more sophisticated and sometimes shorter explanations of answers to some of the problems, but we have chosen to explain the answers in ways that we think children will best understand them.

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How might you use this book? Teachers can use these problems in a variety of ways. Some teachers have found it effective to give their students a problem at the end of a mathematics lesson as a starter discussion for the following day. Students are often sufficiently interested in the problems to discuss them at home. The most important thing for teachers to realise, is that if the problems are at the right level for their students, then they will not be solved immediately but will require some thought and possibly some discussion and debate. At the back of the book we have created a Brain Buster section which includes more difficult mathematical problems. You may use the problems which appear in this section as you wish. They could, for example, be used to extend more able students or to occupy fast finishers. We hope that you and your students enjoy solving these problems.

Barry Brocas and Brenda Bicknell 4

Curriculum Links NSW Working Mathematically (stage 3) Number (stage 3) Patterns and Algebra (stage 3) Measurement (stage 3) Space and Geometry (stage 3)

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WA Appreciating Mathematics (1) Working Mathematically (3) (4) (5) Number (6) (7) (8) Measurement (9) (10) (11) Space (15) (16) Algebra (19)

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NT Spatial Sense (band 2) Movement and Data Sense (band 2) Number Sense (band 2)

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Number Algebra Measurement Space

SA Essential Learnings Measurement (standard 3) Number (standard 3) Pattern and Algebraic Reasoning (standard 3) Spatial Sense and Geometric Reasoning (standard 3)

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VIC Victorian Essential Learning Statements Number (level 4) Space (level 4) Measure (level 4) Structure (level 4) Working Mathematically (level 4)

Brain Bender 1

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

The Jellybeans Ainsleigh ate one hundred jellybeans in five days. Each day she ate six more than she ate the previous day.

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How many jellybeans did Ainsleigh eat on the first day?

_____ jellybeans

Hint: Use a guess and check approach. Guess how many she ate on the first day. Then adjust guesses accordingly.

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

The Eggs

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If three hens will lay three eggs in three days, how many eggs will six hens lay in six days?

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_____ eggs 6

Brain Bender 1

Answer

The Jellybeans

Students should use a guess and check approach.

One possible approach is to have a guess and adjust the figures until all five days total a hundred. For example, if a student guesses that Ainsleigh ate one jellybean on the first day, they will be thirty-five jellybeans short:

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1 + 7 + 13 + 19 + 25 = 65

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This is an average of seven jellybeans short each day. So seven should be added to each number to get: 8 + 14 + 20 + 26 + 32 = 100

So the answer is: Ainsleigh ate eight jellybeans on the first day.

Some more able students may be able to understand, if taught, that the middle number has to be the average of the five numbers, because the difference between consecutive numbers is constant. So the middle number must be twenty. It is then easy to reach the answer.

Brain Bender 2

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Students use computation to solve this problem.

The Eggs

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Answer

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If only the number of hens doubled and the number of days remained the same, the answer would be six eggs – double the number of eggs.

Brain Bender 3

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Animals in the Zoo

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Hint: There are many possible solutions to this problem.

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A small zoo has enough fencing to surround five animals but not enough to separate them. However, by digging three straight trenches, (drawing three straight lines on the diagram below) the animals can be separated from each other. The animals cannot move from their places as shown below. Where could the trenches (lines) be placed?

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

The Lily Pond

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A water lily on a pond doubles in size each week. If it takes ten weeks to cover the whole pond, how many weeks would it take to cover half the pond?

Brain Bender 3

Answer

Animals in the Zoo

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Students will arrive at a solution through trial and error. There are many solutions. Two possibilities are shown below.

Brain Bender 4

The Lily Pond

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Students work backwards to solve this problem.

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Answer

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Since the water lily doubles in size every week and it covers the whole pond after ten weeks, it must have covered half of the pond after nine weeks.

Brain Bender 5

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Six Squares Place seventeen sticks so that they form a pattern of six squares as shown below. Now take away five sticks so that only three squares are left.

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Note: Every stick remaining must be part of a square.

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Nine Squares

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Use twenty-four sticks to make the pattern below. Remove four sticks so that you take away four squares and leave only five squares. Each stick that is left must be part of a square.

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Brain Bender 5

Six Squares

Answer

Most students will adopt a trial and error approach to this question.

Method 1: Trial and error

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Method 2: A reasonable guess is that by removing five sticks, three squares are taken away, so one square can be removed by taking away two sticks. The second square can be removed by taking another two sticks away and the third square can be removed by taking one stick away.

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The two possible answers are:

Brain Bender 6

Nine Squares

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Most students will adopt a trial and error approach to this question.

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Answer

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Since the removal of four sticks takes away four squares, we need to look for situations where the removal of one stick removes one square. There are four of them. We are left with the following arrangement.

Brain Bender 7

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

The Apple Thief While three security guards were guarding an orchard, a thief slipped in and stole some apples. On the thief’s way out he met the three security guards, one after the other. To each of the guards in turn he gave half of the apples he had and then two more. The thief managed to escape with only one apple.

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How many apples did the thief steal?

_____ apples Hint: To solve this problem, work backwards.

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Farmer Smith’s Troughs

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Farmer Smith’s farm has eight troughs on it and it is in the shape shown below. He wants to divide his farm into four identical paddocks with two troughs in each paddock. He is going to use only straight fences. Show how he could do this by drawing only two lines on the plan of the farm.

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Brain Bender 7

The Apple Thief

Answer

Students need to work backwards to solve this problem.

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So the answer is: the thief stole thirty-six apples.

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To his one remaining apple, add the two that he gave to the third security guard. This equals three apples. This must be half the number of apples he had before he gave half of his apples to the third guard. So when he met the third guard he had six apples. Add two to six and double it to get sixteen, which is the number of apples he had when he met the second guard. Add two to sixteen and double it to get thirty-six, which is the number of apples he had when he met the first security guard.

Brain Bender 8

Answer

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Students can reach an answer through a trial and error approach.

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The two lines shown subdivide the land into four parts, equal in size and congruent in shape. Each part has two troughs.

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Farmer Smith’s Troughs

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Brain Bender 9

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Frog in the Well

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A frog is at the bottom of a thirty metre well. Each hour it climbs three metres and then slips back two metres. How many hours does it take for the frog to reach the top of the well? Drawing a vertical time-line may help you to solve this problem.

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_____ hours Hint: Think about what happens when the frog gets close to the top of the wall. 14

Brain Bender 9

Answer

Frog in the Well

Students may want to draw a vertical time-line to solve this problem.

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However, after 27 hours, the frog will have climbed a total of 27 metres and during the next hour it will reach the top of the well. Then it will not slip down.

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After one hour, the frog will be one metre up the well, because it will have climbed three metres up the well wall and slipped two metres back down. After two hours, the frog will be two metres up the well wall. Looking at the pattern so far, it is tempting to conclude that the frog will take 30 hours to reach the top of the well.

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Brain Bender 10

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Out to Dinner A family of four (father, mother, brother and sister) sit around a table in a restaurant for dinner. Each person eats a main meal. The meals that they order are chicken, steak, fish and vegetarian. They also each have a drink of juice. The drinks that they order are orange, tomato, apple and grapefruit juice.

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Use the clues below to find out who is sitting in which seat, what they are eating and what juice they are drinking.

Clue 3

Clue 4

Clue 5

1 4

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2 3 4 Hint: Fill in the grid provided to find the answer. 16

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Clue 2

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The sister sits in seat one. The person who eats the chicken is not sitting in seat one or seat four. The person in seat three who drinks tomato juice and eats steak is not the mother, because the mother eats chicken. The person who eats the vegetarian meal and has grapefruit juice sits opposite the person who eats steak. The brother sits in an even numbered seat opposite the person who has an orange juice.

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Brain Bender 10

Answer

Out to Dinner

Students will arrive at the answer by completing the grid provided as they read the clues.

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Main Meal

Drink

Sister

vegetarian

grapefruit juice

Mother

chicken

orange juice

Father

steak

tomato juice

Brother

fish

apple juice

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Seat

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Brain Bender 11

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

The Computer Thieves A police officer questions three thieves (Davis, Giles and Mills) after they are discovered with some stolen computers. The thieves try to confuse the police officer by giving conflicting statements. Each thief makes two statements. Only one of the thieves tells the truth in both statements while the other two thieves each give one true statement and one false statement.

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The statements are: Davis: “Mills passed them out. Giles broke the window.” Giles: “Davis passed them out. I broke the window.” Mills: “I put them in the van. Giles passed them out.”

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Each person did exactly one of the following three tasks: break the window, pass the computers out, put them in the van. Who put the computers in the van?

Drawing several grids like the one below to explore the many possibilities might help.

Put them in the van

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Passed them out Broke the window

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Davis

Giles

Mills

Hint: Make an assumption and test it. For example, assume that both Davis’ statements are true. 18

Brain Bender 11

Answer

The Computer Thieves

Students should adopt a guess and check approach to answer this problem.

r o e t s Bo r e p ok u S Davis Mills Giles

Put them in the van

Passed them out Broke the window

x x

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Students should make assumptions then check that their assumptions match the information given. They could begin by assuming that Davis’ two statements are correct.

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Brain Bender 12

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Occupations

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1. Dahl and Ellis are neighbours and take turns at driving each other to work in their cars. 2. The bus driver drives a bus to work each day. 3. The police officer lives at the Police Station. 4. The car salesperson has tried unsuccessfully to sell Cuff a car. Cuff, who walks to work, says he does not need one. 5. Brown lives in his own house. 6. The teacher plays golf with Cuff on the weekends. 7. The teacher is female. 8. Dahl’s wife takes yoga classes.

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Clues

Abel, Brown, Cuff, Dahl and Ellis each have one of the following occupations: a bus driver, a police officer, a car salesperson, a teacher and a plumber.

Use the grid below to help you to find the occupation of each of the five people.

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Salesperson Teacher

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Plumber

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Police Officer

o c . c e h Abel: _________________________________ r er o t s super Brown: _________________________________ Cuff :

_________________________________

Dahl:

_________________________________

Hint: Place a cross in the columns that show what occupations these people CANNOT be first. 20

Brain Bender 12

Answer

Occupations

Students should fill in the grid provided to help them solve the problem.

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Bus Driver

Dahl

Ellis

Salesperson

Teacher

Plumber

x x

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Cuff

Police Officer

•

Brown

The first, second and third clues mean that neither Dahl nor Ellis are the police officer or the bus driver. Xs should be placed in those four boxes.

© ReadyEdPubl i cat i ons •o The fourth clue means that Cuff is o nots the car salesperson •f r r e v i e w p u r p e s onl y• or the bus driver or the police officer. Xs should be placed in those three boxes.

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The sixth clue means that Cuff is not the teacher. So Cuff is the plumber. One tick and four Xs should be placed in the bottom row.

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The fifth clue means that Brown is not the police officer. So Abel is the police officer. One tick and four Xs should be placed in the left column.

. te o • So Brown is the bus driver. A tick and two more Xs should be c . placedc inh Brown’s column. e r eeighth o t • The seventh andr clue together mean that Dahl is not s s r u e p the teacher. The grid can now be completed. So the answer is: Abel is the police officer, Brown is the bus driver, Cuff is the plumber, Dahl is the salesperson and Ellis is the teacher.

Brain Bender 13

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Pocket Money Gemma and Louise have just been given their pocket money. Gemma thinks that she might have less pocket money than Louise so she suggests that they put their money together and share it. Louise, who knows how much each of them have been given, agrees, as long as Gemma can guess how much Louise has been given. She gives Gemma the following information:

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“If you give me one dollar, I will have twice as much as you. If I give you one dollar we will each have the same amount.” How much pocket money did each of them get?

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Use the grid to help you solve the problem.

Pocket Money Gemma’s Louise’s Money Money

If Gemma Gave Louise $1 Gemma’s Louise’s money money

If Louise Gave Gemma $1 Gemma’s Louise’s money money

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Gemma $_____ Hint: To solve the problem you need to guess and check. E.g., assume that Gemma has $2 and then test your guess in the table. 22

Brain Bender 13

Answer

Pocket Money

Students should complete the grid to solve the problem.

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Gemma’s Money 2 3 4 5

Louise’s Money 4 5 6 7

If Gemma Gave Louise $1 Gemma’s Louise’s money money 1 5 2 6 3 7 4 8 (4 x 2 = 8)

If Louise Gave Gemma $1 Gemma’s Louise’s money money 3 3 4 4 5 5 6 6

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Gemma will have as much as Louise, if Louise gives her one dollar. So Louise must have two dollars more than Gemma. Guess and check some combinations.

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So the answer is:y Louise has $7 Gemma has $5. © R ead Ed Pand ub l i ca t i ons •f orr evi ew pur posesonl y•

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Brain Bender 14

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

The Race Alf, Ben, Carl, David and Edgar often find themselves running against each other in track events. In the latest race, Carl was not first. David was two places behind Edgar, who was not second. Alf was neither first nor last and Ben was one place below Carl.

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Runners Carl

David

Edgar Alf

1st

2nd

3rd

4th

5th

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Place the runners in the order (1st, 2nd, 3rd, 4th, 5th) in which they finished the race. To help you solve the problem, complete the grid provided.

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Hint: Remember to mark which places the boys cannot cross the finishing line on the grid. 24

Brain Bender 14

Answer

The Race

Students should complete a grid to work out this problem.

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The first sentence allows us to eliminate Carl from finishing first. The second statement means that David cannot be first, second or fourth and that Edgar is not second. The last statement means that Alf was not first or last and Ben was not first. This means that Edgar is first. From this information, we can then deduce that David is third, Ben is fifth, Carl is fourth and Alf is second. So the answer is: Edgar = First Alf = Second David = Third Carl = Fourth

David

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Edgar Alf

x x

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Brain Bender 15

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Who’s Wearing What? Lauren, Hannah and Paris are dressed to go on a picnic with Grampa. Use the following clues to find out who is wearing what.

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Clue 1

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Clue 2 Clue 3 Clue 4 Clue 5 Clue 6

Hannah and Paris are both wearing shoes or are both not wearing shoes. Lauren is wearing a hat. Everyone who is wearing shorts has a hat. Everyone who has a hat is wearing shorts. Nobody is wearing shorts and shoes. At least two people are wearing shoes.

You might use the following grid to help you find out what each girl is wearing. Clue one provides two alternatives to test.

Alternative 1:

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Alternative 2:

Shorts

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Hat

Lauren Hannah Paris . te Shoes o c . c e r Hat h er o t s super Shorts

Lauren is wearing ________________________ Hannah is wearing ________________________ Paris is wearing ________________________

Brain Bender 15

Answer

Who’s Wearing What?

Students should fill in the grids provided to solve the problem.

Option A:

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Hannah

Paris

Lauren

Hannah

Paris

Hat

Shorts

Option B:

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Shoes

x P ©Shoes ReadyEd uxbl i cat i ons Hat •f o rr evi ew pur posesonl y•

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Shorts

The first clue allows both option A and option B, but the sixth clue eliminates option B. The rest of the clues lead to the following:

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Shorts

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Hannah

Paris

So the answer is: Lauren is wearing a hat and shorts and Hannah and Paris are wearing shoes.

Brain Bender 16

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Merry-Go-Round Thirty children ride on a merry-go-round. Every girl rides immediately behind a boy. Half of the boys ride immediately behind girls and the other half of the boys ride immediately behind boys. How many boys were on the merry-go-round?

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How many girls were on the merry-go-round?

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_____ boys

_____ girls

Hint: Guess and check. Draw a picture. Start with BGBG etc, and then change some boys to girls until the criteria is met. (Remember that because the merry-go-round is round, the front and the tail of the line will join.) 28

Brain Bender 16

Answer

Merry-Go-Round

Students should adopt a guess and check approach to answer this problem.

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Method 2: (for older or more advanced children) The 30 children are made up of the following groups: A: B: C:

Girls - they all ride behind boys. Boys who ride behind boys. Boys who ride behind girls.

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Method 1: Draw a picture. Start with BGBG etc, and then change some boys to girls until the criteria is met. (Remember that because the merry-go-round is round, the front and the tail of the line will join.)

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A and C must have the same number of children because if they were not the same, there would be a girl riding behind a girl. So now we have three sets of children, each with the same number.

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So there are ten girls and 20 boys.

Brain Bender 17

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Friends in a Line Jenny is standing in a line with her friends. There are two more friends standing ahead of her in the line, than there are standing behind her. There are three times as many of her friends in the line altogether (do not count Jenny) as there are behind her.

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Teac he r

How many friends are ahead of Jenny in the line?

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_____ friends ahead of Jenny. Hint: Guess the number of friends standing ahead of Jenny, then use the information to check. It may help to draw Jenny and her friends in the line. 30

Brain Bender 17

Answer

Friends in a Line

Students should adopt a guess and check approach.

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

Behind Me

In front of me

Total Friends

1 person

3 people

3x1 = 3

2 people

4 people

3x2 = 6

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Teac he r

If students begin with the lowest possible number of friends who could be standing in the line, their second guess should be correct.

So the answer is: there are four friends ahead of Jenny in the line.

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Brain Bender 18

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Farmer Brown’s Farm Farmer Brown’s farm is made up of three square blocks. He plans to give each of his four children an equal share of the farm. He wants to divide the land so that each child will receive the same amount of land in exactly the same shape.

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Show how Farmer Brown can divide the farm.

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Hint: It may help to divide each square into four smaller squares. 32

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Brain Bender 18

Answer

Farmer Brown’s Farm

Students should use a guess and check approach to solve the problem.

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

Child 1

Child 3

Child 1

Child 2

Child 3

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Teac he r

It is clear that each child will receive three quarters of a square block. So divide each square into quarters. Since each child has to receive three of the little squares, it can be subdivided as shown below.

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Child 4

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Brain Bender 19

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Arohana’s Marbles Arohana dropped his marble jar and all the marbles spilled out. Arohana picked up some marbles and Rapana picked up twice as many. Arohana gave Rapana four marbles. Rapana gave Arohana twice as many marbles as Arohana then had.

r o e t s Bo r e p ok u SPicked up Gives Rapana Gives Arohana

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Teac he r

How many marbles did Rapana then have?

Arohana Rapana

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Hint: How many marbles did Arohana pick up? Guess and check. 34

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_____ marbles

Brain Bender 19

Answer

Arohana’s Marbles

Students adopt a guess and check approach.

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At first sight this problem might seem to not have enough information. Students need to guess how many marbles Arohana picked up.

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For example, suppose Arohana picked up six marbles. Then Rapana picked up twelve marbles. Arohana gave four marbles to Rapana so he then had two marbles and Rapana had sixteen. Then Rapana gave four to Arohana so Rapana had twelve. No matter what values you start with, Rapana will always finish with twelve marbles.

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Brain Bender 20

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

The Wine Barrels A wine merchant left thirty barrels of wine to be divided evenly among his three sons. Ten barrels were full of wine, ten were half- full and ten were empty.

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How can the wine be shared so that each son gets the same number of barrels and the same amount of wine without pouring any wine from one barrel to another?

1st son

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Teac he r

Use the table below to help you to solve the problem.

___Full barrel(s) ____Half- full barrel(s)

____Empty barrel(s)

2nd son ___Full barrel(s) ____Half- full barrel(s)

____Empty barrel(s)

barrel(s) ____Half- full barrel(s) ____Empty barrel(s) ©___Full Re adyEdPubl i cat i ons •f orr evi ew pur posesonl y•

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Hint: There is more than one solution. 36

Brain Bender 20

Answer

The Wine Barrels

There are a number of possible solutions to this problem.

Solution one:

Teac he r

1st son 2nd son 3rd son

Half (1/2s)

Empty (0s)

4 3 3

2 4 4

4 3 3

Full (1s)

Half (1/2s)

Empty (0s)

5 5 0

0 0 10

5 5 0

Full (1s)

Half (1/2s)

Empty (0s)

5 3 2

0 4 6

5 3 2

Full (1s)

Half (1/2s)

Empty (0s)

5 4

0 2

5 4

Solution two:

1st son 2nd son 3rd son

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r o e t s Bo r e p ok u S Full (1s)

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1st son 2nd son 3rd son

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. te o Solution four: c . che e r o t r s s r u e p 1st son 1 8 1 2nd son 3rd son

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

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Teac he r

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r o e t s Bo r e p ok u SBrain Busters

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Brain Bender 21

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Bank Accounts Three girls; Amy, Beatrice and Chloe, are discussing their bank accounts, which together add to less than $5000. Each girl makes two statements. Only one of the two statements made by each girl is true because they are trying to mislead each other.

r o e t s Bo r e p ok u Amy: S Chloe: Beatrice:

• I have $200 more than Beatrice (A1). • I have half as much as Chloe (A2).

• I have more than Amy (B1). • I have $1200 (B2).

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Teac he r

Read the statements below to help you work out how much money each girl has in her bank account

• I have $300 less than Amy (C1). • I have six times as much as Beatrice (C2).

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Amy:

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o c . cheBeatrice: e r o t r s Chloe: super

$___________

Hint: To solve the problem you need to make an assumption. For example, assume that A1 is true and A2 is false and work from there. If this doesn’t work, assume that A1 is false and A2 is true and so on. 40

Brain Bender 21

Answer

Bank Accounts

Students should use a guess and check approach and make assumptions as they work through the problem.

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Teac he r

If students assume that A2 is true, then C1 must be false, so C2 is true. This means that B1 must be false, so B2 must be true. So B has $1200, C has $7200 and A has $3600.

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These amounts though, add to more than $5000. So the assumption that A2 is true must be false, so A1 must be true. If A1 is true, then B1 must be false, so B has $1200 and A has $1400.

C2 must be false or the total would be more than $5000. So C1 is true and C has $1100.

A1 F T

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B1 F F

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The students may find it helpful to set the statements out as shown below and write in the truth value of each statement. The starting assumption is in bold. C1 F T

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C2 T F

Brain Bender 22

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Pages in a Book A book is made of folded sheets of paper. Each sheet of paper makes four pages of the book. If the book was pulled apart, one of the sheets of paper would have page 68 on the left-hand side and page 131 on the right-hand side.

Page 68

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Page 131

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Teac he r

How many pages are in the book altogether?

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_____ pages Hint: To solve this problem, work out the number of pages that are missing. Ask yourself, “How many pages come before page 68?” “How many pages are there between page 68 and 131?” and “How many pages are there after page 131?” It may help to make a small book with a few pages to help you answer these questions. 42

Brain Bender 22

Answer

Pages in a Book

Students will arrive at the answer by making a model of a book.

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

Method 2: Ask students to make a small book from, say, a few pieces of paper folded in half, and number each page. Get your students to take note that the sum of the page numbers on each side of every sheet is the same, and is one more than the number on the last page. Relate this to the given problem and this gives a total number of 198 pages: 68 + 131 – 1 = 198

Page 68

Page 131

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Teac he r

Method 1: Ask students to fold a sheet of A4 paper and label the numbers on it. The back of page 68 is page 67. So 67 pages have come before page 68, therefore 67 pages must be after page 131. So the total number of pages is 198 (i.e. 131 + 67).

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Brain Bender 23

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Connecting Cables to Houses Five houses are connected to one another using an old-fashioned cabling system in which every house has to be connected to every other house.

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1. Below are five houses connected to each other using the old-fashioned cabling system. Count the number of cables needed. Hint: Use a highlighter pen to help you count.

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_____ cables Hint: (Question 2) draw or model the ten houses using counters or straws. 44

Brain Bender 23

Answer

Connecting Cables to Houses

To answer the second question students may need to model the houses and cables using counters and straws, or may be able to see that a pattern has emerged in question one.

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Question 1: Students should count the cables in the diagram. There are ten of them.

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Teac he r

So the answer is ten cables.

Question 2: Students may be able to recognise that nine cables are required to connect the first house to all of the other houses. Only eight cables are needed to connect the second house to the other houses because it is already connected to the first house. Similarly, to connect the third house to the system, only seven cables are required and so on. So the figures which students need to add up are 9+8+7+6+5+4+3+2+1= 45

So the answer is: forty-five cables for ten houses.

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Brain Bender 24

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Weighing the Marble Suppose you have eight marbles, one of which is slightly lighter than the other seven, which all weigh the same.

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Using a balance scale, explain how you can identify the lighter marble by using the scale only twice.

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Brain Bender 24

Answer

Weighing the Marble

Students should try alternative lines of investigation when they realise that their initial attempts are not productive.

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Set two marbles aside. Put the remaining six on the balance scale. Two things can occur:

Teac he r

1. They are evenly balanced, in which case the lighter marble is one of the two marbles set aside. If this happens, put the two marbles that have been set aside, on the balance and find the lighter one.

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2. The six (three on each pan) are not evenly balanced. Take the three containing the lighter marble and put one aside and the other two on the balance. If they balance, the lighter marble is the one set aside. If they don’t balance, the lighter marble can be determined from the weighing.

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Brain Bender 25

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Brother and Sister A boy, when asked the age of himself and his sister replies:

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“Three years ago I was seven times as old as my sister, two years ago I was four times as old, last year I was three times as old, and this year I am two and a half times as old”.

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Last year This year

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How old are the children this year?

Brother’s Age:___________ Sister’s Age: ___________ Hint: Guess and check. 48

Brain Bender 25

Answer

Brother and Sister

Students adopt a guess and check approach to solve this problem.

r o e t s Bo r e p ok u S The answer is: this year the brother is ten and the sister is four.

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Teac he r

Students should use the grid provided to help them solve the problem. They should guess the sister’s age and the brother’s age 3 years ago and work forward. They should alter their guesses accordingly.

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Brain Bender 26

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Derryn and James Derryn is 15 years old and James is 17 years old. The sum of the digits of their two ages is (1+5+1+7)= 14. 1. How old were they when this last happened? To help you solve this problem, fill in the grid provided.

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

Sum of digits

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Derryn

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Derryn’s Age:_______ James’ Age: ______

. tewill this happen again and what will be their ages? 2. How many times o c Assume that neither of them reaches a three-digit age. . che e r o t r s super

_______ times Hint: Guess and check. To solve question 2 look for a pattern in your answer to question 1. 50

Brain Bender 26

Answer

Derryn and James

Students should fill in the grid provided and see if they can recognise patterns between numbers.

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2. This will happen five more times as shown in the table below. Students should note that their ages increase by nine years each time. The following is a list of all the possibilities.

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1. Filling in the grid should help students arrive at the answer that the last time the sum of the digits of their two ages was fourteen was when Derryn was six and James was eight.

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33 42 51 60

35 44 53 62

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Brain Bender 27

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Egg Timer 1. What is the easiest way to time the boiling of an egg for eleven minutes using only a three-minute egg timer and a seven-minute egg timer? __________________________________________________________________

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__________________________________________________________________

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__________________________________________________________________

_________________________________________

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Hint: You can start the timer(s) before you begin to boil the egg. 52

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_________________________________________

Brain Bender 27

Egg Timer

Answer

Students should use a guess and check approach.

Answer to Question 1

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Answer to Question 2

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Teac he r

Start the two timers at the same time. When the threeminute timer runs out, start boiling the egg. When the seven-minute timer runs out (four minutes later) start the seven-minute timer again. When that finishes, the egg has boiled for eleven minutes.

To boil an egg for eight minutes start the two timers together and restart the three-minute timer when that timer finishes. When it finishes the second time start boiling the egg.

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There was one minute left in the seven-minute timer, so ÂŠ ad y Ed ub i ca t i ons whenR it fie nishes flip itover, andP when it fil nishes the second time the egg has boiled for eight minutes. â&#x20AC;˘f orr evi ew pur posesonl yâ&#x20AC;˘

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Brain Bender 28

How do you rate this Brain Bender? Brain Boring!

Easy on the Brain!

Brain Bending!

Super Dooper Brain Bending!

Filling the Bath

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

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Teac he r

When the taps on a bath are fully turned on, the hot tap can fill the bath in twelve minutes and the cold tap can fill it in four minutes. If both taps were turned on together, what is the shortest length of time it would take to fill the bath?

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Brain Bender 28

Answer

Filling the Bath

Students solve a simpler version of the problem first.

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After two minutes the bath will be half full from the cold tap and one sixth full from the hot tap. So it will be two thirds full in two minutes. So it will be full in three minutes.

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Teac he r

After twelve minutes we will have run four full baths of water (three cold and one hot). So if it takes twelve minutes to fill four full baths then it will take three minutes to fill one bath.

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

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