Timed Maths Problems

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Timed Maths Problems

Problem Solving Maths for 8y-E10 old ©R ead dPyear ubl i ca t i ons •f orr evi e w pur posesonl y• students . te

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Stimulating ‘word’ problem solving activities organised according to anticipated solution time.

o c . che e r o t r s super Written by Jane Bourke. Illustrated by Melinda Parker.

Published by Ready-Ed Publications (2001) PO Box 276 Greenwood Perth Australia 6024 E-mail: info@readyed.com.au

Web Site: www.readyed.com.au

COPYRIGHT NOTICE Permission is granted for the purchaser to photocopy sufficient copies for non-commercial educational purposes. However this permission is not transferable and applies only to the purchasing individual or institution.

ISBN 1 86397 200 5

Timed Maths Problems Introduction

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The ‘Timed Maths Problems’ book presents a range of problem solving techniques in a gradually more complex way as each section of the book is encountered. This enables problems to be grouped according to the time intended for an activity to be completed; 5 minutes, 10 minutes or 15 minutes. Naturally, these times are arbitrary and will range widely depending upon the abilities of the students, but the opportunities exist to extend students by presenting problems as a challenge to be completed within the specified time.

The Teachers’ Notes section gives an outline of the various strategies that the students will use as they attempt the problems.

The problems in this book are ideally suited to a maths learning centre set up in the class room. The problems can be copied, cut up and placed in boxes with students selecting a problem from the appropriate box, depending on how much time they have.

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Contents An Introduction................................................................................................................ 2 Teachers’ Notes ............................................................................................................. 4

Five Minute Problems .............................................................................................. 7

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Making a List................................................................................................................... 7 Guess and Check ......................................................................................................... 10

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Find a Pattern ............................................................................................................... 13 Solve an Easier Version of the Problem ....................................................................... 16

Logical Reasoning ........................................................................................................ 19

Ten Minute Problems ............................................................................................ 22

Making a List................................................................................................................. 22

© ReadyEdPubl i cat i ons Logical Reasoning ........................................................................................................ 26 •f orr evi ew pur posesonl y• Guess and Check ......................................................................................................... 29

Create a Diagram ......................................................................................................... 24

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Find a Pattern ............................................................................................................... 31

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Working Backwards ...................................................................................................... 32

Mixed Fifteen Minute Problems ..................................................................... 34-38

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Answers ................................................................................................................ 39-40

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Teachers’ Notes The problems in this book require the use of the range of strategies detailed in these notes. Teachers may choose to introduce the activities by explaining the strategies before assigning the problems. With this in mind, the initial use of each section in this book incorporates explanation and examples for students to consider before attempting the problems themselves.

Making a List: This strategy involves examining all the possibilities for a solution by listing the various elements in the problem. The technique can be used when information has to be gathered and checked in order to cover a variety of possibilities.

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Example: Sam has an orange, a banana and yoghurt in his lunch box. List all of the different combinations he could eat them in.

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orange, banana, yoghurt banana, orange, yoghurt yoghurt, orange, banana yoghurt, banana, orange banana, yoghurt, orange orange, yoghurt, banana

There are six combinations.

Guess and Check: This is a good strategy to use when introducing children to

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• Example: A farmer has 55 cows and sheep in total. If he has nine more cows than sheep, problem solving. As its name suggests the children guess an answer to the problem, use the guess to reach an approximate answer and then attempt the problem. When an answer is obtained the guess is modified so that an answer which is closer to the correct one can be gained. how many of each does he have?

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The first guess might be 25 cows. If this is true then there would be 16 sheep, making a total of 41 animals altogether. This guess is too low so a higher guess can be made until the student works out that there must be 32 cows and 23 sheep.

A table can be used to check guesses.

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Number of cows

Number of sheep Total

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Find a Pattern: This requires the problem solver to find a pattern in the information given. This must then be continued on to find the answer.

Example: The answer is

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Solve an Easier Version of the Problem: This strategy is similar to Finding a Pattern. The student finds the solution to a complex problem by working out an easier version and then applying the same rules to the harder version.

Example: There are 30 people at a meeting. Everyone shakes hands with each person once. How many handshakes take place?

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

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Logical Reasoning: This strategy helps students to develop skills in deductive reasoning by allowing them to use what they already know to solve the problem. Students develop a hypothesis and then check their answer as opposed to guessing the answer.

Clues should be written down in the grid as shown.

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Cowboy

Pirate

Neil

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no

Jennifer

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yes

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Example: Jennifer and Neil went to a fancy dress party. The boy wore a cowboy suit. The girl wore a pirate suit. What did Neil go as?

Create a Diagram: This strategy requires the students to draw a diagram of the

© ReadyEdPubl i cat i ons Example: •f orr evi ew Julie pu r p ses on l y has twoo identical garden beds that• form a problem which can then be used to provide a solution. It is particularly useful with problems relating to area.

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cross. Bed 1 is 14 metres long and Bed 2 is 3 metres wide. What is the distance around the edge of the garden.

Bed 2

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Working Backwards: This strategy works best when a problem is stated so that the final outcome is clear. It is necessary to determine the range of events that occurred that produced the result.

Example: Sebastian has saved $30 in his account this week. Each week he saves $5 more than the week before. How much did Sebastian save three weeks ago?

If Sebastian saved $30 this week he must have saved $25 last week and $20 the week before. This can be written into a table.

Money saved: Weeks ago:

$30

$25

$20

$15

$10

This week

one

two

three

four

From the table we can see that Sebastian must have saved $15 three weeks ago.

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Student Record Sheet 5 Minut e Problems: Mak ing a List Minute Making 1 2 3 4 5 R R R R R

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5 Minute Problems: Guess and C heck Check 1 2 3 4 5 6 R R R R R R

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5 Minu te Problems: Logical Reasoning Minute 1 2 3 4 5 6 R R R R R R

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10 Minute Problems: Making a List 1 2 3 4 5 R R R R R

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5 Mi nute Problems: Find a Patter n Minute Pattern 1 2 3 4 5 6 R R R R R R

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10 Minu te Problems: Lo gical Reasoning Minute Logical 1 2 3 4 5 6 R R R R R R 10 Minut e Problems: Guess and Check Minute 1 2 3 4 5 R R R R R

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10 Mi nute Problems: Working Backwar ds Minute Backwards 1 2 3 4 5 R R R R R 15 Mi nute Problems: Mixed Minute 1 2 3 4

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Student Information Card

Making a List

5 Minute Problems

This strategy can be used to list all possible answers. You can use this technique when you need to gather and check all the information contained in the problem.

Example: There are three balloons in a bag (one yellow, one red, and one green). In how many different combinations can they be drawn out of the bag?

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Answer:

Strategy: Making a List

1. Carl works in a bike shop. He is getting the front window display ready and has a mountain bike, a racer, and a tandem bike which he must place from left to right. How many ways could he arrange the bikes?

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Strategy: Making a List

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2. Shelly had a choice of banana cake, mud cake, carrot cake and apple slice at afternoon tea. She decided to try a piece of each. How many different orders could she eat the cakes in?

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Strategy: Making a List

3. Jonathan, Bill, Jane and Sue are about to run in a race.

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How many different outcomes to the race are possible? ............................................

Strategy: Making a List

4. George's Restaurant has soup or salad as entrees, roast or fish as a main course and ice cream or jelly for dessert. How many different combinations can be ordered?

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Strategy: Making a List

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5. Alicia goes to a video shop and rents four videos - one new release, one family film, one comedy and one drama. She is going to watch them all in one day. How many different sequences can she use to watch the videos? ......................................................................... ......................................................................... .........................................................................

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Strategy: Making a List

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6. Kelly has to pick up messages from different people. The people are William, Neil, Rachel and Liam. How many different routes can Kelly take to collect the messages? ......................................................................... .........................................................................

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Strategy: Making a List

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7. In a chess competition there are six players. To find the winner everyone has to play everyone else. How many games would this take?

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Strategy: Making a List

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8. Jordan’s teacher likes to vary the school subjects each morning. On Monday the subjects were taught in the following order: Maths, Science, English and Geography. If the teacher used a different combination every day, how many days would it be before the same combination would be used again?

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Student Information Card

Guess and Check

5 Minute Problems

You can use this strategy for many different types of problems. As the name suggests, you need to look at the problem and make a guess at the answer. Use this guess to attempt the problem, then modify it so that a closer answer can be gained. The guess and check problems from Number 1 to 5 use two variables. The level of difficulty increases when more variables are used such as in problems 6, 7 and 8.

Example: Simon watched a class of young children ride past his house. He knew there were 25 students and counted 60 bike wheels. Some of the children were on bikes and some were on tricycles. How many were on each type of bike?

Guess 10 15 25

Wheels 20 45 65

Guess 12 13 25

Wheels 24 39 64

Guess 15 10 25

Wheels 30 30 60

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Bikes Trikes Total

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Strategy: Guess & Check

1. Joanne bought twice as many plums as apples.

If she bought 21 pieces of fruit altogether, how many plums did she buy? ........................................................................

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2. Mark swam a total of 42 kilometres in two days. On the first day he swam ten kilometres less than the second day. How many kilometres did Mark swim on the second day? ......................................................................... ......................................................................... .........................................................................

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Strategy: Guess & Check

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3. Fiona made twice as many chocolate brownies as ginger snaps and the total number of biscuits baked was 39. How many ginger snaps did she bake? ........................................................................ ........................................................................

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Strategy: Guess & Check

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4. Marcus has a jar containing 165 jelly beans. He knows that this number is the sum of three consecutive numbers, e.g. 13 + 14 + 15.

Find the three numbers that add up to 165.

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Strategy: Guess & Check

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5. Katrina and Shane were playing tennis. Katrina won twice as many games but Shane served three times as many aces as Katrina. The number of aces served was twenty and a total of fifteen games were played. Work out the number of aces and games won for each person.

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Strategy: Guess & Check

6. Annie, James and Chloe went berry picking.

Annie picked six less than James who picked twice as many Chloe. If a total of 44 berries were picked altogether, how many did each child pick?

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Strategy: Guess & Check

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7. Jack, Luke and Samantha were competing in the diving championships. At practice Jack and Luke made a combined total of 30 dives and Luke dived six more times than Samantha who made half the number of dives that Jack did. Find out how many dives each person made by using the table below to check.

Jack Luke

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Strategy: Guess & Check

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8. Joey, Steve and Charlie were playing Nintendo. Joey and Steve reached 24 levels between them, Charlie and Joey reached 28 levels between them and Charlie and Steve’s levels added up to 38. How many levels did each person reach? Use the table below to check.

Joey

Steve

Charlie

Guess 1

Check

Guess 2

Re-Check

Guess 3

Re-Check

Guess 4

Re-Check

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Student Information Card

Find a P attern Pattern

5 Minute Problems

This strategy is very useful in saving the amount of time it takes to work out a problem. Often the way to solve a problem can be by identifying a pattern that occurs, making it easy to predict what will happen next. Tables can be used in this strategy to help you find possible patterns.

Example: Madison has started collecting soccer cards. On Tuesday she collected two cards and on Wednesday she received four cards, while on Thursday she collected eight cards. If she continues at this rate, how long will it be before she has over 100 cards?

A table can be used to identify a pattern: 1

Cards

2

2

3

4

5

6

7

4

8

16

32

64

128

After 7 days she will have 128 cards.

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Day

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Strategy: Find a Pattern

1. Fred delivers papers to all the houses in his street. He starts at his house which is number 46 and then does his neighbour’s house which is 44. The last house on that side is nineteen houses away. What number is it?

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Strategy: Find a Pattern

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2. The 419 bus that goes from the school to the shops has nine stops in between. The first stop is numbered seven and the next is fourteen.

What is the number of the stop at the shops?

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Strategy: Find a Pattern

3. Pete the painter is improving on his painting speeds. On the first day he painted 10 fence posts, on the second day he painted 15 posts, on the third day he painted 20 posts. How many posts will he paint on the tenth day if this rate keeps up?

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Strategy: Find a Pattern

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4. Rod delivers newspapers after school. He has noticed that one street has a strange order for numbering the houses. The first house is numbered one and the second house in numbered three. The third house is numbered six and the fourth house is numbered ten.

Number the houses according to the pattern.

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5. Carrie is growing broad beans in her garden. She has recorded the growth rates for each day as follows. After one day the shoot grew one centimetre, and by the second day the plant had increased by two centimetres. On the third day the plant was five centimetres tall. If this growth rate continues how tall will the plant be on the eighth day?

Day Height

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Strategy: Find a Pattern

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6. Matthew’s older brother shapes surfboards which Matthew sells on the weekends. On the first weekend he sold three surfboards, the next weekend he sold six surfboards and the following weekend he sold nine surfboards. After seven weekends how many surfboards will he have sold altogether if he continues to sell them at this rate? ....................................................................................................................................

Weekends

3

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Surfboards

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Strategy: Find a Pattern 5 7. Antoinette is starting a health club and is offering a special discount on new membership. The offer involves each new member introducing two new members in the month after they join. At present she is the only member in her club. If the plan works how many members should she have after six months?

Number ofE New Members Total Number ofs Members © Rea dyd Publ i c at i on 1 2 3 •f orr evi ew pur posesonl y• Months

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Strategy: Find a Pattern

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8. Carlos is training for the Olympics. His first training session lasted for one hour and he increases each training session by ten minutes every time. After fifteen sessions how many minutes will he spend training?

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Student Information Card

Solve an Easier V ersion Version

5 Minute Problems

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This strategy is similar to the one you used in ‘Finding a Pattern’. Look for an easier version to the problem and then apply the same rules to the harder version.

Example: There are 32 people in a tennis tournament. Each player plays in the tournament until they lose a game. How many games will be played before the champion is known?

Try solving a similar problem with only three players: Players Z, Y and X. Z plays X and X wins.

Then X plays Y and X wins again, becoming the champion. A total of two games were played. Z and Y didn’t play each other as they lost their games.

A slightly harder example would include looking at a tournament with eight players. Three games are played in round one with four winners. These four winners play each other in two games resulting in two finalists who then play each other in the final. A total of seven games were played.

From the two examples we can see that the number of games is always one less than the number of competitors, so in a tournament of 32 people, 31 games would be played.

Strategy: Solve an Easier Version

1. At Christmas the Givalot family exchange presents altogether. There are sixteen members in the family and each person gives a present to every other family member. How many presents are there altogether?

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Strategy: Solve an Easier Version

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2. Fifteen students are playing table tennis.

Each student must play all of the other students.

How many games are played all together? ......................................................................... ......................................................................... .........................................................................

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Strategy: Solve an Easier Version

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3. Gina has built a brick tower that is fourteen bricks high. She has decided to paint her tower red, including the top. How many sides of the bricks are painted altogether? ........................................................................ ........................................................................

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Strategy: Solve an Easier Version

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4. The mountain train travels 30 kilometres each way between the base and the mountain top. It makes the round trip three times every day.

How many kilometres does the train travel during the month of September?

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Strategy: Solve an Easier Version

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5. Anne pulled 45 weeds from her garden in ten days. At this rate, how many

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would she pull out in 100 days? ..................................................................................

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Strategy: Solve an Easier Version

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6. At a party 49 people are playing statues. If one person gets out every time the music stops, how many times will the music stop before only one person is left in the game? ........................................................................

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Strategy: Solve an Easier Version

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7. There are 15 students in Miss Julio’s class.

She has asked the children to swap their work for marking. If all the children swap their work, what is the least amount of swaps to ensure that all students have someone else’s work?

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Strategy: Solve an Easier Version

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8. Allbright School is holding a Quiz Day. There are 12 contestants and each contestant must compete in a round with each of the other contestants with the scores being added up at the end. How many rounds will there be altogether?

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Student Information Card

Logical Reasoning

5 Minute Problems

You can use logic to help you solve problems. The simple example below shows you how you can use a grid to match up clues. This helps you to provide an explanation that can be checked rather than just guessing and checking.

Example: Matt, Ben and Carla each had a pet. Matt doesn’t like dogs, Carla’s pet can fly.

Which pets belong to which person?

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Ben

Carla

Bird

Guinea Pig

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Dog

Strategy: Logical Reasoning

1. Rover, Fido and Rex like different brands of dog biscuits. The brands are Doggy Delights, Yum Yums and Pooch’s Heaven. Can you work out which dog likes which biscuit?

3 Fido only eats Yum Yums.

3 Rex hates Doggy Delights and Yum Yums.

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Strategy: Logical Reasoning

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2. Jane, Edna and Waldo are going to dinner. The meals they order are: a steak, chicken and vegetarian lasagne. Who ate what dish?

3 Jane hates meat.

3 The male ordered the chicken. steak

chicken

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Jane Edna Waldo

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Strategy: Logical Reasoning

3. Marcus, Sam and Katherine love sport. One likes football, one likes tennis and the other likes water polo. Who likes which sport?

3 Katherine likes all water sports.

3 Marcus enjoys football. football

Marcus Sam

water polo

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Katherine

tennis

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Strategy: Logical Reasoning

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4. Liam, Nicole and Natalie are going on holidays. One is going snow skiing, one is going fishing and the other is going on a cruise. Who is going on which holiday?

3 The boy is going fishing 3 Natalie hates skiing.

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Nicole

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Strategy: Logical Reasoning

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5. Amanda, Rachel and Chloe each have their favourite subject. The subjects are French, maths and art. Who likes which subject?

3 Rachel likes all languages other than English.

3 Chloe hates mathematics.

French

Maths

5

Art

Amanda Rachel Chloe

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Strategy: Logical Reasoning

6. Darren, Angela and Stella are cooking dinner for their parents. Who is cooking what?

3 Darren is making a cheesecake.

3 Angela is not doing the soup.

Soup

Darren Angela

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Stella

Main

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Strategy: Logical Reasoning

7. The Flyers, the Lakers and the Warriors are the top three basketball teams in a competition. What order are the teams in?

3 The Flyers have won more games than the Lakers.

3 The Warriors are in second place.

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Strategy: Logical Reasoning

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8. Mitch, Dee and Chris have a shared vegie patch in the back garden in which they grow their favourite vegetables. Which vegetable is each person growing?

3 Mitch is growing a leafy vegetable.

3 Dee only eats carrots.

Spinach

Cucumber

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Carrots

Mitch Dee Chris

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Student Information Card

Making a List

10 Minute Problems

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R Remember, when using this strategy you need to list all the possibilities so that they can be counted. Refer back to the Student Information Card for ‘5 Minute Problems’ if you need more help.

Strategy: Making a List

1. Steve has a choice of how he can travel to his friend Mike’s house. He can either walk or ride to the station, then catch either a bus or train to the next station and then either walk to Mike’s house or get picked up by Mike’s mum and driven to their house. How many ways can Steve choose from?

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2. Rod had to interview five people for a job.

They were Bill, Paul, Zoe, Kelly and Jim. In how many different orders can he conduct the interviews? ......................................................................... ......................................................................... .........................................................................

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Strategy: Making a List

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3. Libby has to include the following subjects in her timetable for Monday. They are English, maths, science, art and history. How many different ways can she write up her timetable? ........................................................................ ........................................................................

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4. Carolyn must choose one art activity for each term. Her choices can be made from the following list: Term 1: sculpture, collage, pottery or ceramics Term 2: drawing, weaving, screen printing or painting.

List the choices she can make.

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Strategy: Making a List

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5. Mitchell the magician does the following tricks for his act: a card trick, disappearing act, magic rabbits, sawing a person in half and reading someone’s mind.

How many different combinations can Mitchell have for his act?

10

........................................................................ ........................................................................ ........................................................................

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Student Information Card

Create a Diagram

10 Minute Problems

Diagrams and graphs are useful for presenting the problem in an easier way. They are particularly useful for problems dealing with area and perimeter. They can also help to clarify information.

Example: It takes Ellie two minutes to saw a log into two pieces. How long would it take for her to saw the log into eight pieces?

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Teac he r

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From the diagram it can be seen that only seven cuts are needed and not eight.

Therefore the sawing would take 14 minutes.

R You can use grid paper to create diagrams for some problems of this type.

Strategy: Create a Diagram

1. Zoe is going to jog around some streets as practice for the cross country race. She will run down Smith St for 50 metres, turn left onto Jones Rd and run for 30 metres, left into Julie Ave and run 65 metres and then left into Elizabeth Lane and run for 26 metres. How far does she run? ........................................................................

10

© ReadyEdPubl i cat i ons ........................................................................

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........................................................................

Strategy: Create a Diagram

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2. Veronica bounced a ball from the roof of the school which was 16 metres high. The first bounce reached up to 8 metres, the next bounce was half as high again. How high would the fifth bounce have been? ......................................................................... ......................................................................... .........................................................................

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Strategy: Create a Diagram

10

3. Kate enjoys painting. She must cover the top half of her canvas blue, for the sky. The height of the whole canvas is 60 cm and the width is 24 cm.

What is the area that will be covered in blue? ........................................................................ ........................................................................

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........................................................................

Strategy: Create a Diagram

10

4. In Jim’s gymnastic class six students were used to make a human pyramid. Three people were on the bottom, two on the next level and one person on top. Jim wants to make a giant human pyramid with seven layers.

© ReadyEd P ub l i ca t i o ns How many students will he need? ...................

How many people will onn the bottom •f orr evi ew p ur po se sbeo l y •layer?

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Strategy: Create a Diagram

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5. Kristy, Louis, Barry, Johnno and Macy are camping in the country. Each tent is joined by a path to all the other tents.

How many paths are there altogether?

10

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10 Minute Problems

Logical Reasoning

Strategy: Logical Reasoning

10

Bill

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films

sport

drama

comedy

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Teac he r

1. Bill, Wayne, Megan, Julie and Cynthia enjoy watching different channels on pay TV. The channels they watch are the news, films, sports, dramas and comedies. Who likes which channel?

3 All of the females hate sports.

3 Megan only enjoys films.

3 Cynthia laughs a lot during her shows.

3 The news is watched by Wayne.

Wayne Megan Julie

Cynthia

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• Strategy: Logical Reasoning

10

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Jill, Amanda, Nathan and Betty are going to a concert. One will go by car, one by bus, one by train and the other by tram. How does each person get to the concert?

3 The male hates public transport.

3 Jill lives next to a bus stop.

3 Amanda enjoys trains.

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bus

train

tram

Nathan Betty

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Strategy: Logical Reasoning

3. Robert, Liam, Sarah and Jane are going on a picnic. Each will bring one item.

The items are bread, ham, salad and juice. Who brings which item?

3 3 3

10

Both girls are strict vegetarians. They will bring the salad and juice.

Sarah loves salad.

Liam baked fresh bread yesterday and has decided to bring it.

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Liam

salad

juice

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Robert

ham

Sarah Jane

© ReadyEdPubl i cat i ons •f orr evi ew pur pStrategy: oses onl y• Logical Reasoning

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lipstick

blush

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4. Rachel, Stacey, Alice and Monica are going to buy make-up. The make-up includes lipstick, blush, eyeshadow and nail polish. Who buys what make-up?

3 The female with the shortest name needs blush.

3 Rachel does not need eyeshadow or lipstick.

3 Stacey bought lipstick yesterday. eyeshadow

polish

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Monica

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Strategy: Logical Reasoning

5. Paul, Jenny, Chloe and Michael are Christmas shopping. They buy presents, food, wrapping paper and decorations. Who buys what item?

3 3

10

Both boys enjoy buying presents, however, Michael has agreed to buy the food.

Jenny knows someone who will sell her decorations cheaply.

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Jenny

wrapping paper

decorations

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Paul

food

Chloe

Michael

© ReadyEdPubl i cat i ons •f orr evi ew pur p osesonl y• Strategy: Logical Reasoning

Guitar Piano Violin

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Angie

Brian

Cathy

Denis

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6. Angie, Brian, Cathy, Denis and Ellen each play a musical instrument. Who plays what instrument?

3 Denis plays a stringed instrument.

3 Cathy plays the guitar.

3 Brian doesn’t play the recorder 3 Ellen’s instrument does not start with a ‘p’, ‘r’ or a ‘b’.

Ellen

Bass Recorder

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Student Information Card

Guess and Check

10 Minute Problems

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Teac he r

R Remember, that to use this strategy, you make an accurate guess at the answer by rounding numbers to the nearest ten or hundred.

Strategy: Guess & Check

1. At the zoo there are ostriches and deer in the same enclosure. Jarrad counted 60 heads and 160 legs. How many animals of each type were in the enclosure?

10

©R eadyEdPubl i cat i ons Guess •f orr evi ew pur posesonl y• Ostriches Total

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Deer

Strategy: Guess & Check

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10

2. 106 students play sport on a Friday afternoon at Olympus School. The sports are cricket, hockey and basketball. The number of people playing cricket is six more than that playing hockey and the number of people playing basketball is five less than the number playing hockey. How many students play each sport? ......................................................................... ......................................................................... .........................................................................

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Strategy: Guess & Check

3. Sasha, Aaron and Christian were collecting golf balls. Together the number of balls Sasha and Aaron found was 63. The total number of Sasha and Christian’s golf balls was 57 and the number of balls found by Aaron and Christian added up to 40. Work out how many balls each person found by using the table below.

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

Guess 1

10

Aaron

Christian

Check

Guess 2

Re-Check

Guess 3

Re-Check Re-Check

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Guess 4

Strategy: Guess & Check

4. At the market cucumbers cost 70c each and pumpkins cost 50c each. Artie bought some cucumbers and pumpkins paying a total of $5.80. How many pumpkins must he have bought? Use the table to help you guess and check.

Hint: He bought less than 12 vegetables altogether!

10

© ReadyEdPubl i cat i ons •off orr evi ew pur posesonl y• Number cucumbers Number of pumpkins

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Total

Strategy: Guess & Check

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5. Sandra waited at the bus stop for twenty minutes and saw 43 vehicles go past. She counted a total of 122 wheels and noticed that only motorbikes and cars went past. She also knows there were more motorbikes than cars.

Use the table below to work out the numbers of each vehicle that went past.

10

Cars Motorbikes Total

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Find a P attern Pattern

Student Information Card

10 Minute Problems

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Strategy: Find a Pattern

1. In Australian Rules football six points are given for a goal and one point for a behind. In the new season one of the teams scored the following amounts: 10 goals and 15 behinds, 12 goals and 13 behinds, 14 goals and 11 behinds.

10

© ReadyEdPubl i cat i ons scores be? ......................................................

•f orr evi ew pur posesonl y• What other pattern can you find? ....................

a) If this pattern continues, what will the next two

b)

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R Refer to the ‘5 Minute Problems’ Student Information card for a reminder on how to use this strategy to solve the problems in this section.

c) What will the score be in the eighth game?

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Strategy: Find a Pattern

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10

2. Andrew has to move house and is packing his books into five large boxes.

Each box is a different colour and he has numbered all of the books. He puts the first book into the red box, the second book into the green box, the third book into the purple box, the fourth book into the yellow box, the fifth book into the black box and then keeps going by adding the sixth book to the red box. a) Which boxes will contain these books?

9th....................... 23rd ......................... 44th ....................... 78th ...................... b) The books found in the black box are all multiples of what number? ....................... c) Andrew’s brothers and sisters have decided to add their books to the boxes. His sister Katie has numbered her books from 101 to 250. Which box will contain the books numbered: 192? ................... 223? ........................ 250? ...................... 165? .....................

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Student Information Card

10 Minute Problems

Working Backwards

Each stage of the problem affects the outcome yet only the result is known, so the problem can only be solved by going backwards.

Example: During a winter’s week Mr Baker’s class had several students away with the flu.

On Monday all of the students came to school. On Tuesday only half the students showed up. By Wednesday the class had decreased by half again and on Thursday only a quarter of the students who were there on Wednesday came to school. On Friday only one child was in class which was half the number that were there on Thursday. How many students are normally in Mr Baker’s class?

R By setting out the information in a table the problem becomes more clear.

Tuesday 16

Wednesday 8

Thursday 2

Friday 1

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Monday 32

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Strategy: Working Backwards

1. Mary, Lillian, Hope, Ella and Mabel were knitting rugs. So far Mary has used five more balls of wool than Lillian who has used twice as much as Hope. Ella has used 6 balls less than Hope and has used three times as much wool as Mabel. Mabel has only just started her rug and has used five balls of wool.

How many balls of wool has each person used?

10

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• Mary ....................... Lillian ...........................

Hope ...................... Ella ...............................

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Mabel .....................

Strategy: Working Backwards

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10

2. William went shopping with his savings and only had $6 left over. He spent three times this amount in a book shop and before that he was in a clothes shop. He paid four times the amount he spent in the book shop on some new clothes and half as much as he spent on clothes was spent in the sports shop which was six times the amount he now has left over.

Complete the table below and use it to answer the questions underneath.

Sports Shop

Clothes Shop

Book Shop

Change $6

a) How much money did William start out with? ............................................................. b) How much money was spent in the clothes shop? ..................................................... c) In which shop did he spend the least amount of money? ...........................................

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Strategy: Working Backwards

10

3. In Stevie’s class there were 36 students. One third of them lived in the town and half of the rest of the students came from nearby farms. A third of the remaining students came from the island off the coast and the remaining children were on overseas exchange. Use the table to help you answer the questions below.

Place Students

Town

Farms

Island

Overseas

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a) How many students lived on the island? ....................................................................

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b) How many students came from another country? ......................................................

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c) Which location was home to the most number of students? .......................................

Strategy: Working Backwards 4. Maria went to the zoo and saw twice as many lions as tigers. The number of kangaroos she saw was four times the amount of tigers and there were three times as many monkeys as kangaroos. The amount of animals in the nocturnal house was half the number of monkeys and the number of camels was equal to one third of the amount of animals in the nocturnal house.

Maria counted 18 animals in the nocturnal house.

Work out how many animals of each type were at the zoo. ........................................................................

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Strategy: Working Backwards

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10

5. Sharee arrived at the airport and noticed she had one hour to wait until her flight. She then found that her plane had been delayed by three hours. Later, another one and a half hour delay was announced and her plane finally left at 3 pm.

What time did Sharee arrive at the airport? ......................................................................... ......................................................................... .........................................................................

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Student Information Card

Mixed 15 Min ute Problems Minute

15 Minute Problems

R For this section students will need to choose the best strategy for solving each problem. All the strategies required for the following problems have been used in the ten and five minutes sections.

Strategy: Various

Teac he r

1. Miriam has taken three pairs of shorts, three caps and three t-shirts on summer camp. One pair of shorts is striped, the other plain and the third pair is blue. The t-shirts are white, green and red and the caps are black, blue and white. She aims to wear a different combination each day. How long will it be before she has to repeat a combination?

15

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© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• Strategy: Various

The fish they caught were trout, herring and sole.

Who used what bait and caught what fish?

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2. Shelley, Michael and Kane enjoy fishing. They use shrimps, pilchards and floats for bait.

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Trout can only be caught with floats.

The girl hates trout and herring because they have too many bones.

Michael bought shrimps on the way.

Shrimps Pilchards

Shelley

Floats

Trout

Herring

Sole

Shelley

Michael

Michael

Kane

Kane

Shelley used ............................................. and caught ......................................................

Michael used............................................. and caught ......................................................

Kane used ................................................ and caught ......................................................

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Strategy: Various

15

3. Frank’s Fishing Charter business is very successful. They took three people out on the weekend and caught 215 fish. a) If four times the amount of fish caught on Saturday were caught on Sunday, how many fish were caught on each day?

Saturday: .............................................

Sunday: ...................................................

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b) A mixture of sole, herring and cod were caught. The number of sole caught was four times the number of cod and there were five more herring than cod.

How many of each fish were caught?

................................. herring ...............................

cod ................................

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sole

© ReadyEdPubl i cat i ons Strategy: Various •f orr evi ew pur poses onl y •

4. Deanne, Katie and Thomas deliver parcels for a company. They use a bike, a car and a truck. The parcels they deliver are either small, medium or large.

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15

The girls don't like using the bike because it can only carry the smallest parcels.

Katie does not have a truck license which doesn’t matter because she can’t use the trolley to move the largest parcels.

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Deanne Katie Thomas

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large

bike

car

truck

Deanne Katie

Thomas

Deanne used ............................................ and delivered ...................................................

Katie used ................................................. and delivered ...................................................

Thomas used ............................................ and delivered ...................................................

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Strategy: Various

5. At Maxim’s Mediterranean Restaurant there are several choices available.

From the menu below, work out how many possible combinations could be made.

Starters Minestrone Antipasto

Entree $3 $5

Garlic Prawns Ravioli

Main

15

Dessert

$6 Lasagne $12.50 $4 Moussaka $8.50

Sorbet Fudge Cake

$2 $6

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a) Combinations: ............................................................................................................. ....................................................................................................................................

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b) What is the cheapest possible combination?.............................................................. ....................................................................................................................................

c) Which is the most expensive combination? ................................................................ ....................................................................................................................................

© ReadyEdPubl i cat i ons Strategy: Various •f orr evi ew pur pose sonl y•15

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6. Zoe, John, Marcus and Sean went to a fancy dress party. One went as an astronaut, one as a cowboy, one as a knight and one as a pirate. They each took either drinks, food, gifts or games. Who went in what costume and what did each one take?

The girl went as either a knight or a pirate.

John took the gifts.

Marcus took food and Sean was not the cowboy.

The pirate did not take food, drink or games.

The astronaut did not take food but took games.

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astronaut cowboy Zoe

knight

pirate

drinks

food

gifts

games

John Marcus Sean

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Strategy: Various

7. Steven, Richard, Jessica and Cassandra have their favourite lunch food and drink. The foods are salad, meat pies, hot dogs and chicken rolls. The drinks are chocolate milk, fruit juice, plain milk and soft drink. What did each person eat and drink?

3 3 3 3 3 3

15

One of the boys is vegetarian and hates milk.

Jessica loves pies.

Steven hates chocolate milk.

The soft drink and fruit juice don’t go with a pie.

Richard loves chocolate milk.

The chicken and fruit juice are loved by a girl.

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Steven

meat pies

hot dogs

chicken rolls

choc. milk

fruit juice

plain milk

soft drink

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salad

Richard Jessica

Cassandra

© ReadyEdPubl i cat i ons Strategy: Various •f orr evi ew pur poses onl y •

15

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8. Henry’s school has 336 pupils. One day Henry was standing at assembly and counted the pupils in a different way. He noticed that if you multiplied three consecutive numbers the answer was 336. Work out what the three consecutive numbers are.

Hint: Each number is less than 20. Use the table to check your tries.

Numbers

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Total

...............................

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............................

...............................

...............................

.............................

............................

...............................

...............................

.............................

............................

...............................

...............................

.............................

............................

...............................

...............................

.............................

............................

...............................

...............................

.............................

............................

...............................

............................... ............................... ...............................

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.............................

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...............................

.............................

............................

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Strategy: Various

Groups of 3

15

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Number in shop

3

6

3

4

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9. Carla’s Cafe was particularly busy one morning.

Carla noticed that people were only coming in as groups of three and that every time three customers came into the shop two others would leave. At 9.00 am there were three customers in the shop.

Another three customers came in shortly after and two other customers left. A third group of three customers walked in and again two left.

There are now 37 people in the cafe. Altogether how many groups of three have come into the cafe since 9.00 am?

9.00

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Use grid paper to work out your answer.

10a)Farmer Fred wants to plant some apple trees in the back garden. He has an area prepared that measure 5 metres by 8 metres.

He wants to plant each tree a metre apart and he wants all the plants to be at least a metre from the fence.

How many apple trees will he be able to fit in this area?

o c . che e r o t r s super b)

c)

.......................................................................

If the apple trees grow successfully, Farmer Fred will start an orchard in the new piece of land he has bought. The land measures 20 metres by 50 metres and again he would plant all the trees a metre apart. How many trees will he fit in the orchard? .......................................................................

Using what you already know, find the amount of apple trees he could fit in an area measuring 240 metres by 21 metres. .......................................................................

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ANSWERS:

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Page 7 1. Six ways 2. Eighteen orders Page 8 3. 18 possible outcomes 4. There are eight possible selections 5. 18 sequences Page 9 6. 8 ways 7. 30 8. 18 days Page 10 1. 14 2. 26 Page 11 3. 13 4. 54, 55, 56 5. Katrina: Ten games and five aces Shane: Five games and fifteen aces.

Page 12 6. Annie - 14, James - 20, Chloe - 10.

7. Jack - 16, Luke - 14, Samantha - 8.

8. Joey - 7, Steve - 17, Charlie - 21.

Page 13 1. Number 6 2. Shop Stop - 70 Page 14 3. 55 posts.

4. Pattern: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78.

5. 35 cm tall Page 15 6. 84 7. 127 members 8. 3 hours and twenty minutes.

Page 16 1. 240 presents 2. 210 games Page 17 3. 57 4. 5400 km 5. 450 Page 18 6. 700 km, 2800 km.

7. 8 8. 132 Page 19 1. Rover - Doggy Delights, Fido - Yum Yums, Rex - Pooch’s Heaven.

2. Jane - Vegetarian Lasagne, Edna - Steak, Waldo - Chicken.

Page 20 3. Marcus - football, Sam - tennis, Katherine - water polo.

4. Liam - fishing, Nicole - skiing, Natalie - cruise.

5. Amanda - Maths, Rachel - French, Chloe - Art. Page 21 6. Darren - Dessert, Angela - Main, Stella - Soup.

7. 1 - Flyers, 2 - Warriors, 3 - Lakers.

8. Mitch - Spinach, Dee - Carrots, Chris - Cucumber.

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Teac he r

Page 22 1. 16 routes 2. 80 orders Page 23 3. 80 ways 4. There are 16 choices. Check list.

5. There are 80 combinations.

Page 24 1. 171 metres 2. 50 cm Page 25 3. 720 cm squared 4. 28 with 7 on the bottom layer.

5. 10 Page 26 1. Bill - sports, Wayne - news, Megan - films, Julie - drama, Cynthia - comedy.

2. Jill -bus, Amanda - train, Nathan - car, Betty - tram.

Page 27 3. Robert - ham, Liam - bread, Sarah - salad, Samantha - juice.

4. Rachel - nail polish, Stacey - eye shadow, Alice - blush, Monica - lipstick.

Page 28 5. Paul - presents, Jenny - decorations, Chloe - wrapping, Michael - food.

6. Angie - Recorder, Brian - Piano, Cathy - Guitar, Denis - Bass, Ellen - Violin Page 29 1. 40 ostriches and 20 deer.

2. Cricket - 41, Hockey - 35, Basketball - 30 Page 30 3. Sasha - 40, Aaron - 23, Christian - 17.

4. 6 pumpkins, 4 cucumbers.

5. 25 motorbikes and 18 cars.

Page 31 1. a. 16 goals and 9 behinds, 18 goals and 7 behinds; b. The total number of scoring shots is equal to 25 in every game; c. 24 goals and one point.

2. a) Yellow, purple, yellow, purple. b) 5 c) green, purple, black, black.

Page 32 1. Mary - 47, Lillian - 42, Hope - 21, Ella - 15, Mabel - 5.

2. a - $132, b - $72, c - the bookshop.

Page 33 3. a) 4, b) 12 c) Farms.

4. Lions - 6, Tigers - 3, Kangaroos - 12, Monkeys - 36, Nocturnal - 18, Camels - 6.

5. 9.30 am Page 34 1. 27 days 2. Shelly - pilchards, sole, Michael - shrimp, herring, Kane - floats, trout.

Page 35 3. a. Saturday 43, Sunday 172, b. sole - 140; herring - 40; cod - 35.

4. Deanne - truck, large, Katie - car, medium, Thomas - bike, small.

Page 36 5. a) There are 16 possible combinations b) $17.50 - Minestrone, Ravioli, Moussaka and Sorbet. c) $29.50 - Antipasto, Garlic Prawns, Lasagne and Fudge Cake.

6. Zoe - knight, drinks, John - pirate, gifts, Marcus - cowboy, food, Sean - astronaut, games.

Page 37 7. Steven - salad, soft drink, Richard - hot dogs, chocolate milk, Jessica - pies, milk, Cassandra chicken rolls, fruit juice.

8. 6,7,8 Page 38 9. 13 10. a) 28 b) 931 c) 4780

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