More Timed Maths Problems

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Timed Maths Problems ©R eadyEdPubl i cat i ons

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Written by Jane Bourke. Illustrated by Terry Allen. © Ready-Ed Publications - 2002.

Published by Ready-Ed Publications (2002) 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 453 9

More Timed Maths Problems Introduction This book follows on from the very popular ‘Timed Maths Problems’ of a few years ago.

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It 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 opportunity exists 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 classroom.

The problems can be copied, cut up, laminated and placed in boxes with students selecting a problem from the appropriate box, depending on how much time they have.

Student Outcome Statements

© R e a d y E d P u b l i c a t i o n s Working Mathematically: Uses Problem Solving Strategies 3.3.

Uses problem solving strategies that those based ons selecting key n information and •f or r evi einclude wp ur po eso l y• representing it in models, diagrams and lists.

The problems in this book relate to the following outcome statements:

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Equations 3.13: Applying Numbers 3.14: Written Computation 3.16:

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Chance and Data: Interpreting Data 3.27:

Evident when the student ...

Uses patterns and follows rules to solve word and number problems.

Uses patterns to solve number problems.

Chooses appropriate calculations to solve word problems.

Calculates word and number problems using multiplication, addition, subtraction and division.

Evident when the student ...

Interprets straightforward one and two-way tables.

Arranges data in a meaningful way in order to solve a word problem.

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Number: Number Patterns 3.12:

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Referenced from Mathematics - a curriculum profile for Australian Schools (1994).

Published by Curriculum Corporation St Nicholas Place 141 Rathdowne St Carlton Victoria 3053 Page 2

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

Five Minute Problems

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Making a List ............................................................................................................................. 7

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Guess and Check ................................................................................................................... 10

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Find a Pattern .......................................................................................................................... 13

Solve an Easier Version ........................................................................................................... 16 Logical Reasoning ................................................................................................................... 19

Ten Minute Problems

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Making a List ........................................................................................................................... 22 Create a Diagram .................................................................................................................... 24

Guess and Check ................................................................................................................... 29

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Find a Pattern .......................................................................................................................... 31 Working Backwards ................................................................................................................ 32

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Mixed Fifteen Minute Problems

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Mixed Strategies ...................................................................................................................... 34 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, each section in this book incorporates an 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 sandwich and yoghurt in his lunch box. List all of the different orders he could eat them in.

There are six different orders.

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

solving. As its name suggests, the children guess possible combinations in 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.

© ReadyEdPubl i cat i ons Example: A• farmer has 55 cows and sheep in total.

If p he has nine more cows than sheep, f o r r e v i e w p u r o s e s o n l y • 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

25

34

32

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25

41 (too low)

59 (too high)

23

55 (correct)

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. Patterns might be based on number qualities, repetition of shapes, repetition of words or spelling, for example.

Example: The answer is:

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.............................. , .............................. . ,

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Solve an Easier Version of the Problem: This strategy is similar to Find 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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Example: Jennifer, Neil and Amanda went to a costume party. They had a red mask, a green mask and a black mask to choose from. Jennifer’s favourite colour is green. Amanda hates black and Neil took the leftover mask. Who wore each colour?

Green

Red

Black

Jennifer

yes

no

no

Neil

no

no

yes

Amanda

no

yes

no

© 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

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

which can then be used to provide a solution. It is particularly useful with problems relating to area.

3m

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 6

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

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5 Minu te Problems: Logical Reasoning Minute 1 2 3 4 5 6 7 1 0 Minute Problems: Making a List 1 2 3 4 5 10 1

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© ReadyEdPubl i cat i ons •f o rr evi ew puar p osesonl y• Minute Problems: Cr eate Diagram Create 2

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

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r o e t s Bo r e p o 7 u k S 5 Min ute Problems: Solve an Easier V ersion Minute Version 5 Mi nute Problems: Find a Patter n Minute 1 2 3 4 5 6

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15 Mi nute Problems: Mixed Minute 1 2 3 4 5 6 Page 6

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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 is an apple, a banana and a peach in a bag. In how many different orders can they be drawn out of the bag?

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Answer:

Strategy: Making a List

1. Mark works in a car yard. He has to wash the cars once a week and then park them back in the display room. Currently, there are four cars in the display room - an Audi, a Honda, a Mercedes and a Renault. The cars are parked in a row. In how many different orders can he arrange them?

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Strategy: Making a List

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2. Louise has won a holiday in a raffle.

She will be visiting these destinations: Singapore, Bali and Thailand. She can choose any order in which to visit the countries. How many choices does she have?

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Strategy: Making a List 3. It is Sports Carnival day and the four teams at Smart School are all hoping to win the flag. The teams are the Purple Panthers, the Blue Bananas, the Green Grouches and the Orange Owls.

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Smart School wants to print the results of the carnival on the front page of the school newsletter but isn’t sure which team will come first, second, third or fourth. Because they want to send the newsletters out straight after the carnival they have decided to make a copy of every possible result and then only use the page with the real result. How many different versions of the newsletter will the school need to prepare?

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Strategy: Making a List

4. On the last day of school Miss Nice has let the students choose what activities they would like to do. Here are their choices: 1. Before recess:

Play computer games OR watch a video.

Play board games OR read a book.

Go to the pool OR go to the beach.

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© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• How many possible combinations could they choose 2. Before lunch: 3. After lunch:

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from for the day’s activities?

Strategy: Making a List

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5. When Geoff, Noel, Denis, Carl, John and Michael arrived at the playing field, each boy shook hands with every other boy.

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How many handshakes took place?

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Strategy: Making a List

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6. Elsie is dropping off birthday party invitations to her friends. She has to drop one off to Ella, Rosie, Mia, Parker and Ginger.

If Ella and Mia live in the same house, how many different routes can Elsie take to deliver the invitations to each person’s house?

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Strategy: Making a List 7.

Miss Purple had to organise the refreshments for morning tea. Ten people wanted coffee, including Miss Purple, five people wanted tea, eight people wanted cola, three people wanted orange juice and six people wanted a glass of water. The six people who asked for water had also asked for one other type of drink.

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© ReadyEdPubl i cat i ons If everyone had at least one how many •f orr evi ew pu r p os edrink, so n l ypeople • were at the morning tea?

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Strategy: Making a List

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Fabulous Funland is about half an hour away from Boringtown by bus. A bus leaves Boringtown every fifteen minutes to arrive at Fabulous Funland. If the first bus leaves at 8.00am, how many buses will arrive at Fabulous Funland before 10.00am?

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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: Jack went to the zoo and found that the cats had been thrown in with the orangutans!

He counted 17 animals and then he counted 44 legs.

How many cats were there?

1st Guess 10 x 4 7x2 17

Legs 40 14 54

2nd Guess 8x4 9x2 17

Legs 32 18 50

3rd Guess 5x4 12 x 2 25

Legs 20 24 44

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Cats Orangutans Total

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Strategy: Guess and Check

1. Irene won 24 ribbons at the school sports carnival.

She had twice as many “First” ribbons as “Second” ribbons.

How many of each ribbon did she have?

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2nds

Strategy: Guess and Check

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Denis went mountain climbing and climbed a total of 264 metres in two days. On the first day he climbed 66 metres more than the second day. How many metres did Denis climb on the second day? ..............................................................................

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Strategy: Guess and Check 3.

Denise and Alan spent all day playing backgammon. They played less than 48 games.

Alan won exactly three times as many games as Denise. Denise won more than eight games. Alan lost at least eleven games. How many games did they play altogether and how many did each person win?

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Strategy: Guess and Check

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4. Sarah was selling sausage rolls and party pies at the school fair. She sold twice as many sausage. rolls as party pies. The total number of items she sold was 84. How many did she sell of each?

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Strategy: Guess and Check

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5. Chrissie and Shane were planting flowers. Chrissie planted twice as many petunias as Shane, however Shane planted three times as many rosebushes as Chrissie. The total number of petunias planted was 36 and the number of rosebushes planted was 12.

How many petunias and rosebushes did each person plant?

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Strategy: Guess and Check 6. Casey, Ari and Joe went fishing off the jetty. Casey caught two more fish than the number of fish Joe caught. Ari caught six less fish than the number Casey caught. Altogether the group caught 55 fish.

How many fish did each person catch?

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Strategy: Guess and Check

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7. Jane, Sarah and Emily were participating in the school Read-a-thon.

Jane and Sarah read a total of 50 books between them. Emily read the most books, reading four more than Jane and eight more books than Sarah. A total of 81 books were read.

Use the table below to work out how many books each girl read.

Jane Sarah

ofd Books Read © Number Rea yEdPubl i cat i ons •f orr evi ew pur posesonl y•

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Strategy: Guess and Check

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8. Grant, Justine and Darren were racing car drivers at the Grand Prix. Grant and Justine made a combined total of 110 laps. Darren and Grant together completed a total of 93 laps while the combined number of laps that Justine and Darren completed was 95. Using the table below, work out how many laps each driver completed.

Grant

Justine

Darren

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 time to work out a problem. Often a problem can be solved 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.

Day

No. of Pamphlets Time taken

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30

Tue

40

35

Wed

50

40

Thu

60

45

Fri

70

50

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Example: Jessie has started delivering pamphlets after school each day. On Monday she delivered pamphlets to 30 houses and it took 30 minutes.

On Tuesday she delivered 40 pamphlets in 35 minutes. On Wednesday she delivered 50 pamphlets in 40 minutes. If she continues at this rate, how many pamphlets will she deliver on Friday and how long will it take?

A table can be used to identify a pattern:

Strategy: Find a Pattern

1. Abby and Maddy are playing cards.

Out of every three games that they play, Abby wins two.

They have played a total of 33 games. Use the table below to work out how many games each girl has won.

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Number of games played

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6

9

12

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Willy Worm is trying to crawl up the wall. Every time he crawls 10 cm he immediately slides backwards by 5 cm.

The wall is 50 cm high. It takes him exactly one minute to crawl 10 cm.

How long will it take him to reach the top?

Use the table below to help you work out the answer.

Time (minutes) Distance travelled (cm)

1

2

3

4

5

6

7

10 cm - 5 cm

Actual distance 5 cm

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Strategy: Find a Pattern 3. Fiona is training for the Rottnest Island swim-through.

She has been increasing the amount of laps she swims each day.

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On Day 1 she swam 10 laps, Day 2 she swam 12 laps, the third day she swam 15 laps and on Day 4 she swam 19 laps. If she continues at this rate, on what day will she complete 64 laps?

Day

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Laps

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Strategy: Find a Pattern

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4. Christopher is jogging around his neighbourhood. The houses have been painted in a pattern. The first house is orange, the second house is green, the third and fourth houses are orange and the fifth is green. The next three houses are orange. Complete the pattern using the houses below. How many houses are green? .................................

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Strategy: Find a Pattern

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5. Crazy Charlie is selling lemonade to make extra pocket money.

He set up his stall at 9am and by 10am he had sold five glasses.

In the next hour he sold ten glasses. Between 11am and midday he sold 15 glasses.

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If he continues to sell at this rate, how many glasses would Crazy Charlie have sold: a) between 2pm and 3pm?..................... b) between 4pm and 5pm?.....................

Time Glasses

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10am

11am

Midday

5

10

15

1pm

2pm

3pm

4pm

5pm

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Strategy: Find a Pattern 6. Jessie and Layla have started a book club. They need members so they have each told a friend to come along to the next meeting. It was decided that each week each member, including the new members from the week before, would bring a friend.

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How many weeks will it be before there are at least 100 members?

Weeks

7.

1 2

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Members

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Strategy: Find a Pattern

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Rocket Rod is a rally driver. He completed his first lap in 30 minutes. He has noticed that he is finishing each lap 30 seconds quicker than the one before it.

How long did it take him to complete his 6th lap?

Laps

Time (minutes)

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Strategy: Find a Pattern

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8. Can you work out what comes next in each of the patterns below?

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a) 3, 6, 9, ______, ______, ______, ______,

b) 1, 2, 5, 10, 17, ______, ______, ______, ______, c) A, Z, B, Y, ______, ______, ______, ______,

d) 1, 4, 9, 16, 25, ______, ______, ______, ______,

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Student Information Card

5 Minute Problems

Solve an Easier V ersion Version

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

Example: There are 48 people in a chess competition. Each player stays in the competition 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 1, 2 and 3. Player 1 plays Player 2 and Player 1 wins. Then Player 1 plays Player 3 and Player 1 wins again, becoming the champion. A total of two games were played. Player 2 and Player 3 didn’t play each other as they lost their games.

A slightly harder example would include looking at a competition with eight players. Four 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 are 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 48 people, 47 games would be played.

Strategy: Solve an Easier Version

1. In Miss Clever’s class children swap their stories to read to each other. There are 37 students in the class. What is the least number of swaps that need to occur before every child has someone else’s story?

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Strategy: Solve an Easier Version

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2. Exactly 24 students at Smart School are involved in an ongoing badminton competition. Each student must play against all of the other students and then the points are added up to determine the overall winner.

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How many games are played altogether? .................................................................................... .................................................................................... ....................................................................................

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Strategy: Solve an Easier Version

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3. Johnny has a book that contains 250 pages. On how many pages will the number 2 appear at least once? ..................................................................... .....................................................................

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Strategy: Solve an Easier Version

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4. Jacinta works on a boat that ferries people to an island resort each day. The island is twelve kilometres from the mainland. She makes the return journey three times a day, six days a week.

© ReadyE dPubl i cat i ons How many kilometres by boat will she have travelled after six weeks?

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Strategy: Solve an Easier Version

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5. There were 20 students competing for the fastest student award. Rather than hold one race, the Phys. Ed. teacher wanted to see who was the fastest over a number of races. Each of the students raced every other student and times were recorded.

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How many individual races took place?

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Strategy: Solve an Easier Version

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6. One hundred students are competing in a quiz competition.

After each round, the two lowest scoring students are knocked out of the competition. In the final round there are only two players left. How many rounds will take place before there is a winner?

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Strategy: Solve an Easier Version

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7. A plane leaves the Capital Airport every fifteen minutes.

Exactly how many planes fly out of this airport in one week?

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Strategy: Solve an Easier Version

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8. Carl works in a supermarket and is in charge of making the displays at the end of the aisles. He has arranged tins of tomato soup in a triangular shape.

There are 20 cans on the bottom row. How many cans have been used in the entire display?

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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: Rod, Rebecca and Brett each had a pet. Rod doesn’t like dogs, Brett’s pet can fly.

Which pets belong to which person?

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Rebecca

Brett

canary cat

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poodle

Strategy: Logical Reasoning

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1. Tim, John and Andrew each like a particular flavour of milkshake. The flavours are caramel, spearmint and chocolate.

Tim will only have chocolate.

Andrew doesn’t like spearmint.

© ReadyEdPubl i cat i ons caramel chocolate spearmint •Timf orr evi ew pur posesonl y•

Use the table below to work out which flavour each boy prefers.

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Andrew

Strategy: Logical Reasoning

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2. Natalie, Scott and Kathryn have each taken a book out of the library. The books they chose were “Black Beauty”, “Harry Potter” and “Great Cricketers”.

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Natalie loves reading about wizards.

The boy is a keen sportsman.

Work out who chose each book.

Black Beauty

Harry Potter

Great Cricketers

Natalie Scott Kathryn Ready-Ed Publications

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Strategy: Logical Reasoning 3. Jackson, Elsie and Rosie all take part in different activities after school, such as violin lessons, ballet and swimming.

Elsie has a fear of water.

Jackson can play several songs on the violin.

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Which activity does each child do?

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Jackson Elsie

ballet

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violin

Strategy: Logical Reasoning

4. Jason, Evan and Gabrielle live in a different country to each other. They are from Australia, New Zealand and Canada. Where does each child live?

The girl lives in the desert.

Evan has never been to Canada.

5

© Read yEdPu bl i cat i o ns Australia Canada New Zealand Jason •f orr evi ew pur posesonl y• Evan

w ww

. te

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Gabrielle

Strategy: Logical Reasoning

o c . che e r o t r s super

5. Donelle, Tarlie and Fiona went shopping for clothes. A skirt, dress and a pair of shoes were bought. Work out which girl bought what item.

Tarlie is able to wear a shirt with her new item.

Fiona bought some boots.

Donelle

Tarlie

5

Fiona

shoes skirt dress

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Ready-Ed Publications

Strategy: Logical Reasoning

5

6. Simon, Evan and Natasha all went to the football final with their dads, Mr Smith, Mr Jones and Mr Brown. Who does each child belong to?

Simon did not arrive with Mr Brown.

Mr Smith has no sons.

Mr Smith Simon Evan

Mr Brown

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

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Teac he r

Natasha

Mr Jones

Strategy: Logical Reasoning

5

7. Denise, Barbara and Alan were the final three in a skateboarding competition.

Alan reached second place.

Denise did better than Barbara.

What was the final result?

© ReadDenise yEdPub l i cat i ons Barbara Alan •1stf orr evi ew pur posesonl y• 2nd

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3rd

Strategy: Logical Reasoning

o c . che e r o t r s super

8. Samantha, Carrie and Charlotte all get to school at the same time but each girl travels to school in a different manner.

Work out how each child gets to school using the clues below.

Carrie had a puncture last week on the way to school.

The bus does not go near Charlotte’s house. by bus

by bicycle

5

by car

Samantha Carrie Charlotte

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Student Information Card

10 Minute Problems

Making a List

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

ew i ev Pr

Teac he 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. Michael has opened a new bank account. The bank teller has told him he can choose his own personal identification number (PIN). He can choose any order of the following digits for his PIN: 3, 7, 8, 2.

10

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

•f orr evi ew pur posesonl y• What is the largest number he could pick?

How many different orders can he pick from?

What is the smallest number he could pick?

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

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

Strategy: Making a List

o c . che e r o t r s super

2. Abby, Bill, Caroline, Damian and Emily were attending a rock concert. They were sitting in the front row, seats 1-5.

10

How many different orders could they sit in?

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

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Strategy: Making a List

10

3. Vivienne is rolling two dice. How many possible combinations could she roll? .............................................................................. .............................................................................. ..............................................................................

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

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

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Teac he r

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

Strategy: Making a List

4. Perry was waiting at the bus stop. He counted all the legs of the people that walked past him. He also counted all the legs of the dogs that people were walking. He counted a total of 64 legs.

How many combinations of dogs and people could he have seen?

List them below.

10

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

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. te

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If every second person had one dog how many dogs and people must have passed Perry? ...................

Strategy: Making a List

o c . che e r o t r s super

5. Franco has three pairs of shoes (red, black and navy), two pairs of board shorts (striped and plain) and two shirts (one t-shirt and one collared shirt).

10

How many different outfits could he wear?

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

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Page 23

Student Information Card

10 Minute Problems

Create a Diagram

○

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: Jean is cutting a cake for her guests. There are 20 people who would like a piece.

How many cuts does she need to make? ○

○

○

○

○

○

○

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

Completion of the diagram will show that only 10 cuts, rather than 20 individual cuts, are needed to divide the cake into 20 pieces.

○

○

○

○

○

○

○

○

○

○

○

○

○

○

○

○

○

○

○

ew i ev Pr

○

○

Strategy: Create a Diagram

1. Mrs Barrett’s class has the students arranged in groups: the Lions, the Tigers, the Hawks and the Eagles. How many different ways can the groups be arranged in her class?

Lions

10

Tigers

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

•f orr evi ew pur posesonl y• .............................................................................. ..............................................................................

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Eagles

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Hawks

teacher

Teac he r

○

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

Strategy: Create a Diagram

o c . che e r o t r s super

10

2. Abby, Maddy, Jessie, Lara and Poppy all live on different islands. There is a ferry service connecting each island to all the other islands. Draw the ferry routes on the diagram.

How many separate routes are there altogether? ............................................................................... ............................................................................... ...............................................................................

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Ready-Ed Publications

Strategy: Create a Diagram

10

3. Five boys live in a row on one street. Nigel’s house is the last one on the street. Nigel lives next door to Paul. Paul lives two houses down from Cameron.

Denis lives in between Paul and Cameron. One of Cameron’s next door neighbours is Alan.

Show the order in which the boys live.

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

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Teac he r

Nigel’s house

Strategy: Create a Diagram

4. Jeremy and Patrick are in their speedboat on the river.

Markers are placed three kilometres apart on the stretch of water they are in. They started at the first marker and have now passed twelve markers.

10

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

Exactly how far have they travelled? ...............................

Strategy: Create a Diagram

o c . che e r o t r s super

5. Barry the builder built a wall 48 metres high. Next to that wall he built another wall half as high. The third wall was half as high as the second wall and the fourth wall he built was half as high as the third wall.

Barry continued in this fashion and built a total of six walls.

10

How tall was the sixth wall? ..................................

Ready-Ed Publications

Page 25

10 Minute Problems

Logical Reasoning

Strategy: Logical Reasoning

10

1. Belinda, Nicole, Danielle, Renee and Natasha each play a musical instrument. Work out which girl plays each instrument using the clues below.

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

drums

guitar

flute

Belinda Nicole

Danielle Renee

ew i ev Pr

Teac he r

Belinda can’t play the drums, flute or harp.

Nicole is very good on the drums.

Natasha plays a stringed instrument.

Renee has been playing the piano since she was six.

harp

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

Natasha

Strategy: Logical Reasoning Noel, Lesley, Carl and Emily are meeting at the football game. One person is walking there, one is driving, one is riding a bicycle and one is catching the bus.

w ww

How does each person get there?

Lesley has ridden her bike several times to the game.

Carl did not drive to the game.

Emily lives right near a bus stop.

. te

10

m . u

2.

o c . che e r o t r s super car

bus

walked

bicycle

Noel Lesley Carl Emily

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Strategy: Logical Reasoning

10

3. At the beginning of term, Sarah, Denis, Trisha and Aaron each chose an activity to do in their spare time. Can you work out which child does each activity?

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

pottery

reading

Sarah Denis

Trisha Aaron

hockey

ew i ev Pr

Teac he r

Both girls are interested in arts and crafts.

Denis is a very good sportsman.

Sarah has made a vase and several bowls.

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

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Jean has no sisters.

John has one son.

James is older than Rebecca but younger than Fiona and Jean.

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o c . che e r o t r s super

Mother James

Father

Oldest

10

m . u

4. Fiona, Rebecca, James, John and Jean all belong to one family. Can you work out which family member is which by using the clues?

Middle

Youngest

John Fiona Rebecca Jean

Ready-Ed Publications

Page 27

Strategy: Logical Reasoning 5. Michael, Bernard, Tex, Terry, Jane and Kylie are performing at the school concert.

10

Tex is on immediately after Michael, but before Kylie. He is one of the last two acts.

Terry is one of the first three performers, but doesn’t appear before Jane.

Bernard performs in between Terry’s song and Michael’s act.

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

Work out the order in which the students are appearing.

1st

Bernard

3rd

4th

5th

6th

ew i ev Pr

Teac he r

Michael

2nd

Kylie Tex

Terry

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

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Myrtle always has plenty of snow on the lawn at Christmas time.

Edna does not live in New Zealand or Wales.

Daisy has caught a ferry from her country to the country in which Beatrice lives.

Daisy is Welsh but Beatrice is not Canadian.

. te

o c . che e r o t r s super

Myrtle Canada

10

m . u

6. Myrtle, Edna, Daisy, Gertrude and Beatrice all met on the Internet and they all live in different countries. These countries are Canada, Australia, New Zealand, Ireland and Wales. Work out where each lady lives.

Edna

Daisy

Gertrude

Beatrice

Australia New Zealand Ireland Wales

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Ready-Ed Publications

Student Information Card

Guess and Check

10 Minute Problems

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ew i ev Pr

Teac he r

Remember, that to use this strategy, you make a guess at the answer, and then adjust your next guess depending on whether the answer needs to be higher or lower.

Strategy: Guess and Check

1. Each of the letters stands for a different number. Find the number that each letter represents by working out the sum.

10

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FBB -ns J J= ______ F= ______B= ______ ©R ea dyEd Publ i cat i o F• •f orr evi ew pur posesonJl y

Strategy: Guess and Check

o c . che e r o t r s super

2. Nicole was having a party and bought some soft drink for her friends. At the supermarket, cola was 90c each and lemonade was on special at 60c. Nicole spent a total of exactly $10.20 on the drinks. How many cans of each drink did she buy?

Hint: She bought at least five cans of cola.

10

Use the table below to help.

Cans of Cola Cans of Lemonade Total Ready-Ed Publications

Page 29

Strategy: Guess and Check 3. Morrie, Mitchell and Shane went fishing. Together the number of fish Morrie and Mitchell caught was 77. The total number of Morrie and Shane’s catch was 82 and the number of fish caught by Mitchell and Shane added up to 89. Work out how many fish each boy caught by using the table below.

Morrie

Guess 1

Mitchell

10

Shane

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

Check

Guess 2

Re-Check

Guess 3

Re-Check Re-Check

ew i ev Pr

Teac he r

Guess 4

Strategy: Guess and Check

4. Rosetta had a jar full of 267 marbles. She divided the total number of marbles into three piles. Two of these piles were for her friends Antonio and Stefan. She gave herself a pile of marbles that contained one more marble than Stefan’s pile and one less marble than the number in Antonio’s pile.

How many marbles does each child now have?

10

© ReadyEdPubl i cat i ons Stefan•f orr evi ew pur posesonl y• Antonio

w ww

Total

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m . u

Rosetta

Strategy: Guess and Check

o c . che e r o t r s super

5. Matthew walked into the science lab and noticed that there were two different kinds of stools for students to sit on. Some of the stools had four legs while some had only three. He counted a total of 87 legs and also noticed that the the number of three-legged stools was greater than the number of four-legged stools.

10

How many of each type of seat were in the lab? ...................................... three-legged stools four-legged stools Total

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Find a P attern Pattern

Student Information Card

10 Minute Problems

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Teac he 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.

Strategy: Find a Pattern

1. A new cafe has been opened at the end of Freddy’s street.

On the day of opening Freddy counted two customers.

On the second day, he counted twice as many customers plus one more. On the third day he counted twice as many customers as the second day plus two more.

10

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

If this rate continued, how many customers could be expected on the sixth day of business?

Customers

One 2

Two

Three

Four

Five

Six

5

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Day

What would the total number of customers be after the sixth day?...............................

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Strategy: Find a Pattern

o c . che e r o t r s super

2. Heather, Fiona, Rebecca and James were competing as a team in the Channel Swim. They each took turns in swimming sections of the course. At the start they each swam a section for 15 minutes, then decreased the time of the sections by one minute each time.

10

In the last section, they swam for a total of eight minutes each.

a) How many times did each person have to swim? .............. b) How many minutes did the entire swim take? ................ c) How many hours is this? ....................hrs ....................... mins.

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Page 31

Student Information Card

10 Minute Problems

Working Backwards

In these problems 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: The school bus was full as it left school one afternoon. At the first stop half of the students got off. At the second stop the number of students on the bus decreased by half again, and at the third stop another half of the students got off. By the fifth stop the last five students got off which was exactly half the number that were on the bus before the 4th stop. How many students were on the bus when it left the school?

By setting out the information in a table the problem becomes clearer.

4th stop 10

3rd stop 20

2nd stop 40

1st stop 80

ew i ev Pr

Teac he r

5th stop 5

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

Strategy: Working Backwards

1. Rebecca, Katie, Donelle, Fiona and Jane were training for the Olympic athletic team. Rebecca has been training for five weeks longer than Katie, who has trained for twice as long as Donelle.

Fiona has trained for one week less than the amount of time Donelle has trained and twice as long as Jane. Jane has only just started training and has been training for three weeks.

10

© ReadyEdPubl i cat i ons How many weeks has each girl been in training?

•f orr evi ew pur posesonl y• Rebecca ................... Fiona ...............................

Jane ................................

Donelle .....................

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Katie .........................

Strategy: Working Backwards

o c . che e r o t r s super

10

2. Damian was meeting a friend, Sebastian, at the beach. Right at the time they were due to meet, Sebastian called Damian on the mobile to tell him he was going to be half an hour late.

Damian waited half an hour and then received another call from Sebastian to say that he would be another 15 minutes. Sebastian finally showed up, taking three times as much time as he said he would during the last phone call.

It was exactly midday when Sebastian met Damian. a) What time had the two boys originally planned to meet? .............................................................................. b) For how many minutes was Damian waiting by himself? ..............................................................................

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Ready-Ed Publications

Strategy: Working Backwards 3. In Casey’s English Language School half of the total number of students were Indonesian. Out of the remaining numbers, half were from Japan. There were three times as many Japanese students as Malaysian students. The number of students from Thailand was exactly half the number from Malaysia. There was a total of 16 Malaysian students, which was four times the number of Vietnamese students in the class.

10

a) How many students came from each country?

Indonesia

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

Thailand

Vietnam

Malaysia

ew i ev Pr

Teac he r

b) How many students altogether were enrolled at the school? ........................

Strategy: Working Backwards 4. Sarah visited the Aquarium and counted some of the different sorts of marine life. She counted twice as many parrot fish as stingrays. The number of flounder was three less than the number of parrot fish. There were five times as many sea dragons as flounder. Sarah counted a total of 25 sea dragons.

10

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

How many animals did Sarah count altogether?

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stingrays

flounder

sea dragons

m . u

parrot fish

Strategy: Working Backwards

o c . che e r o t r s super

5. Marcelle, Nancy, Kelly, Natasha and Belinda have each started a Christmas Club account. Marcelle has saved $10 more than Nancy, who has saved twice as much as Kelly. Natasha has four times the amount that Kelly has saved and $5 less than Belinda. Belinda has $105 saved.

10

How much has each girl saved?

Ready-Ed Publications

Marcelle .......................

Natasha

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

Nancy

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

Belinda

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

Kelly

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

Page 33

Student Information Card

15 Minute Problems

Mixed 15 Min ute Problems Minute

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 minute sections.

Strategy: Mixed 1. Joe is the captain of a volleyball team. Six players are required to make up the team. He wants to rotate all the players after every two points so that they each play in all of the six positions and in all the different combinations that are possible.

15

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

On grid paper, draw boxes like this one to help you solve the problem.

1st two points

A

B

C

D

E

F

ew i ev Pr

Teac he r

Game plan:

a) How many times will Joe need to rotate the players before he repeats a game plan? ................................................................

b) How many points will need to be played before this happens?

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

•f orr evi ew pur posesonl y• Strategy: Mixed

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15

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2. Johnny, Lachlan and Greg all work as tradesmen. One is a bricklayer, one a carpenter and one a painter. They are each working on a different project.

These projects include building a bridge, renovating a church and constructing a school.

Work out who is involved with each project.

o c . che e r o t r s super

Johnny uses only water soluble paints.

The bridge is made entirely of steel and not wood or bricks.

Greg is not involved with the school but has built walls there in the past. bricklayer carpenter

Johnny

church

painter

bridge

school

Johnny

Lachlan

Lachlan

Greg

Greg

Johnny is a ...................................................and is working on the ........................................ .

Lachlan is a ..................................................and is working on the ........................................ .

Greg is a .....................................................and is working on the ........................................ . .

Page 34 Ready-Ed Publications

Strategy: Mixed

15

3. Billy, Tommy and Sam spent the weekend fruit picking in Farmer James’ orchards.

They picked a total of 320 fruits over both days. a) If three times the amount of fruit picked on Saturday was picked on Sunday, how many fruits did the boys pick on each of the days?

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

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

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b) A mixture of pears, nectarines and oranges were picked. The number of pears picked was twice the number of nectarines and ten times the number of oranges.

How many of each type of fruit were picked?

ew i ev Pr

Teac he r

pears .................................... nectarines .............................. oranges .............................

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

15

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None of the men drive a black car.

One of the women bought a silver MG. Pierre bought a red Jaguar and Felicity bought a V W.

Oscar

. te

VW

Louise Felicity

m . u

4. Oscar, Louise, Felicity and Pierre have all bought new cars. The cars include a V W, an MG, a Jeep and a Jaguar. Each car is a different colour: black, silver, red and yellow.

Work out who bought what.

o c . che e r o t r s super MG

Jeep

Jaguar

black

silver

red

yellow

Pierre

Oscar bought a ................... ...................... . Felicity bought a ...................

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

Louise bought a................... ...................... . Pierre bought a.....................

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

Ready-Ed Publications

Page 35

Strategy: Mixed 5. At the Razzle Dazzle Show, there is a stall that allows children to pick and choose what they would like to put in their showbag.

They can choose from the following groups, picking one thing only from each group.

Food Crisps 90c Lollies $1.80

Books

Cards

15

Novelties

r o e t s Bo r e p ok u S Harry Potter $4 Goosebumps $2.75

Basketball $1.75 Football $1.25

Paint Set $4.50 Magic Set $6.25

Teac he r

a) How many different showbag combinations are possible? ........................................

ew i ev Pr

b) What is the cheapest showbag on offer?

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

c) What is the most expensive combination you could make? ...............................................................................................

© ReadyEdPubl i cat i ons Strategy: Mixed •f orr evi ew pur poses onl y•1 5

6. Gino, Andre, Rita and Satomi all went to Vacation Care during the summer holidays. Work out how old each child is and which activity each one is involved in.

. te 6

m . u

w ww

Andre was not playing inside.

Gino is younger than Satomi but older than Andre.

Neither girl is interested in computers.

The 12 year old is involved in a craft activity.

The youngest child is painting.

o c . che e r o t r s super 8

10

12

Painting

Craft

Outdoors

Computer

Gino Andre Rita Satomi

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Ready-Ed Publications

Strategy: Mixed 7. Julian loves to read and spends a lot of time in the school library. He enjoys reading a large variety of books. Work out how many books of each kind he has read this year.

15

The number of science fiction books Julian has read is less than six.

Julian has read seven times as many horror stories as science fiction novels.

He has read twice as many sports magazines as history books.

The number of sports magazines read is one less than the number of horror stories.

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

The number of comics read is three more than the number of computer books.

He has read the same amount of history books as he has read comics.

Julian has read a total of 71 books and magazines.

Teac he r

Use the Guess and Check strategy in the table below.

ew i ev Pr

Computer books Horror stories

Sports magazines History books Comics Science fiction Total

w ww

8. Dave is the drummer in a band. His band was playing at a rock concert and Dave decided to count how many tickets were sold. He counted 11 880 tickets. He noticed that if you multiplied four consecutive numbers together the answer was also 11 880. What are the four consecutive numbers?

. te

Hint: Each of the numbers is less than 15. Use the table to check your tries.

Numbers

o c . che e r o t r s super

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

x ................

x

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

x ..............

= ................

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

x ................

x

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

x ..............

= ................

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

x ................

x

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

x ..............

= ................

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

x ................

x

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

x ..............

= ................

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

x ................

x

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

x ..............

= ................

Ready-Ed Publications

15

m . u

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

Page 37

Strategy: Mixed

15

9. Cleo the cake maker was having a crazy day. She decided to cut two of the cakes into uneven shapes as she thought some people might only want a small piece and some might want a larger piece. However, she knew everyone would want a piece with a cherry on top!

She cut cake 1 into 7 pieces using only three straight cuts.

She cut cake 2 into 12 pieces using only five straight cuts.

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

Cak e 1 Cake

ew i ev Pr

Teac he r

Draw the cuts Cleo would have made. Hint: Use a pencil and don’t forget to have a cherry on every piece!

Cak e 2 Cake

© ReadyEdPubl i cat i ons Strategy: Mixed •f orr evi ew pur poses onl y•1 5

How many days will it take for Susie to complete the swim?

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10. a). Susie is attempting to swim the Coral Channel solo. Her pattern of swimming is to swim 25 kilometres the first day, 24 kilometres the second day, 23 kilometres the third day and so on, less each time as she tires.The current also sends her back each night by three kilometres.

The channel is a total distance of 250 kilometres.

o c . che e r o t r s super

b). The Channel Ferry sails across the Coral Channel once every three days and makes a return trip the following day. The ferry passed Susie on the first day, making its return trip on the second day of Susie’s swim. How many times in total would the ferry pass Susie? ..............................................................................

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Ready-Ed Publications

ANSWERS: 5 Minute Problems Pages 7 - 9: Making a List 1.

2.

3.

4.

5.

6.

7.

8.

24 orders.

6 orders.

24 versions.

8 possible selections.

15 handshakes.

24 routes.

26 people.

7 buses.

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

Pages 10 - 12: Guess and Check

Teac he r

1st ribbons - 16, 2nd ribbons - 8.

99 metres.

Total number of games played = 44. Denise won 11 and Alan won 33.

Sausage rolls - 56, party pies - 28.

Chrissie: 24 petunias, 3 rosebushes; Shane: 12 petunias, 9 rosebushes.

Casey - 21, Ari - 15, Joe - 19.

Jane - 27, Sarah - 23, Emily - 31.

Grant - 54, Justine - 56, Darren - 39.

Pages 13 - 15: Find a Pattern

Abby - 22 games, Maddy - 11 games.

11 minutes.

Day 10.

Pattern: 29 orange, 7 green. a) 30 glasses, b) 40 glasses.

7 weeks 27 minutes and 30 seconds. a) 12, 15, 18, 21 b) 26, 37, 50, 65 c) C, X, D, W d) 36, 49, 64, 81

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

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Pages 16 - 18: Solve an Easier Version 1.

2.

3.

4.

5.

6.

7.

8.

. te

19 swaps.

552 games.

89 pages.

2592 km.

380 races.

50 rounds.

672 planes.

210 cans.

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1.

2.

3.

4.

5.

6.

7.

8.

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1.

2.

3.

4.

5.

6.

7.

8.

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Pages 19 - 21: Logical Reasoning 1.

2.

3.

4.

5.

6.

Tim - chocolate, John - spearmint, Andrew - caramel.

Natalie - Harry Potter, Scott - Great Cricketers, Kathryn - Black Beauty.

Jackson - violin, Elsie - ballet, Rosie - swimming.

Jason - Canada, Evan - New Zealand, Gabrielle - Australia.

Donelle - dress, Tarlie - skirt, Fiona - shoes.

Simon - Mr Jones, Evan - Mr Brown, Natasha - Mr Smith.

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Page 39

Logical Reasoning cont.

7. 1st - Denise, 2nd - Alan, 3rd - Barbara.

8. Samantha - bus, Carrie - bicycle, Charlotte car.

Mixed 15 Minute Problems

10 Minute Problems Pages 22 - 23: Making a List 1.

2.

3.

4.

24 different numbers; 8732; 2378.

120 orders.

21 combinations.

15 different combinations; He counted 16 people and 8 dogs.

5. 12 outfits.

1. a) 80 times, b) 160 points.

2. Johnny: painter, bridge; Lachlan - carpenter, school; Greg - bricklayer, church.

3. a) Saturday - 80, Sunday - 240; b) pears 200, nectarines - 100, oranges - 20.

4. Oscar - yellow Jeep, Louise - silver MG, Felicity - black V W, Pierre - red Jaguar.

5. a) There are 16 possible combinations. b) $9.40 - Crisps, Goosebumps, Football Cards, Paint Set. c) $13.80 - Lollies, Harry Potter, Basketball Cards, Magic Set. 6. Gino - 10 years old, computer; Andre - 8 years old, outdoors; Rita - 6 years old, painting; Satomi - 12 years old, craft.

7. Computer books 7 Horror stories 21 Sports magazines 20 History books 10 Comics 10 Science fiction 3 8. 9 x 10 x 11 x 12 = 11880.

24 ways.

10 routes.

Nigel, Paul, Denis, Cameron, Alan.

33 km.

1.5 metres.

Pages 26 - 28: Logical Reasoning

1. Belinda - guitar, Nicole - drums, Danielle flute, Renee - piano, Natasha - harp.

2. Noel - car, Lesley - bicycle, Carl - walked, Emily - bus.

3. Sarah - pottery, Denis - hockey, Trisha painting, Aaron - reading.

4. Father - John, Mother - Jean, oldest - Fiona, middle - James, youngest - Rebecca.

5. 1st - Jane, 2nd - Terry, 3rd - Bernard, 4th Michael, 5th - Tex, 6th - Kylie.

6. Myrtle - Canada, Edna - Australia, Daisy Wales, Gertrude - New Zealand, Beatrice Ireland.

ew i ev Pr

Teac he r

Pages 34 - 38: Mixed

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

Pages 24 - 25: Create a Diagram 1.

2.

3.

4.

5.

4. 42; 8 parrot fish, 4 stingrays, 5 flounder and 25 sea dragons.

5. Marcelle - $60, Nancy - $50, Kelly - $25, Natasha - $100, Belinda - $105.

w ww

Pages 29 - 30: Guess and Check 1.

2.

3.

4.

5.

J - 9, F - 1, B - 0.

6 cola @ $5.40; 8 lemonade @ $4.80 Morrie - 35, Mitchell - 42, Shane - 47.

Stefan - 90, Rosetta - 89, Antonio - 88.

Four-legged stools - 12, three-legged stools - 13.

. te

9.

m . u

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

o c . che e r o t r s super

Page 31: Find a Pattern

10. a) 20 days; b) 10 times.

1. Day three - 12, day four - 27, day five - 58, day 6 - 121. Total: 225.

2. a) 7 times each; b). 368 minutes; c). 6 hours and 8 minutes.

Pages 32 - 33: Working Backwards

1. Rebecca - 17 weeks, Katie - 12 weeks, Donelle - 6 weeks, Fiona - 5 weeks, Jane - 3 weeks.

2. a) 10.45am. b) 75 mins.

3. a) Indonesia - 52, Japan - 24, Thailand - 8, Vietnam - 4, Malaysia - 16; b) 104.

Page 40

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