Maths Problem Solving Techniques

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Maths Problem Solving ©Techniques ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• m . u

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Eight problem solving strategies and practice activities for the primary . te classroom. o

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Written by David Stephenson. Illustrated by Rod Jefferson. © Ready-Ed Publications - 1996.

Published by Ready-Ed Publications (1996) P.O. Box 276 Greenwood W.A. 6024 Email: info@readyed.com.au Website: www.readyed.com.au COPYRIGHT NOTICE Permission is granted for the purchaser to photocopy sufficient copies for non-commercial educational purposes.

However, this permission is not transferable and applies only to the purchasing individual or institution.

ISBN 1 86397 110 6

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

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Problem solving should be approached as a challenge to achieve a workable solution.

A good problem solver has the innate or learned ability to internalize problem solving techniques and sub-skills. When faced with a problem, real or abstract, he or she can then consciously select the most appropriate skill or skills to solve the problem.

Some people are said to be “born problem solvers” just as some people are “born athletes”. While this may be true, our experience has also shown that students can markedly and dramatically improve their ability in problem solving after they have worked through, applied and practised the skills presented in this book.

Of greater concern is that, as educators, we are often indifferent to the variety of processes that can be used to arrive at the correct solutions. Students come into the educational setting possessing different sub-skills, different mental processing patterns, even a different perception of their own ability to solve problems. Students need to be provided with the encouragement to try different approaches, and the opportunity to fail without censure and retribution. It is only by the ongoing trial of probable solutions that correct/best answers can be located. Encouraging this experimental approach is the greatest assistance that can be provided to a student.

Good problem solvers have experienced early success and encouragement, thus reinforcing their belief in themselves as ‘good problem solvers’. Generating encouragement and reinforcement that students are good problem solvers can lead to surprising results. After all, “you are what you think you are”.

Overview of the Problem Solving Techniques

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This book has been created to assist educators faced with the difficult task of presenting, explaining, teaching and practising problem solving techniques and skills.

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a) a brief overview of eight problem solving techniques. Each technique is accompanied by a worked example that teachers should talk through with the class to give a full explanation of the sub-skills and thinking processes through which students may proceed. We realize that there are many more problem solving techniques than those presented here, and that problem solving techniques are never used in complete isolation. However, for the purpose of teaching we have broken the techniques into separate usable approaches to problem solving.

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This book provides:

b) four practice questions with each of the eight techniques. These should be used as teaching exercises, changing the numbers as required to consolidate the technique being taught.

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c) a nongraded mixture of 100 different problems requiring use of the problem solving techniques. These can be blackboarded, photocopied or pasted onto cards. Allow students to work through at their own pace, omitting problems initially if they feel ill equipped to solve them.

Students can return to those problems later on as their repertoire of problem solving skills increases.

The ability to solve problems at whatever level of complexity provides the solver with an intrinsic sense of self-worth. As educators, we should provide as many opportunities as possible to raise students' self esteem - providing them with these eight techniques to solve problems is a great commencement point.

Happy problem solving.

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Contents Problem Solving Techniques Overview

2

Personal Record Chart

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Strategy 1

Making a Diagram

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Guess and Check (Two Variables)

8

Guess and Check - cont. (Three Variables)

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Using a Table or Chart

13

Strategy 2

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Strategy 2 Strategy 3

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100 Problems - Results/Comments

Strategy 4

Compiling an Organised List

Strategy 5

Looking for a Pattern

Strategy 6

Kinaesthetic/Real Objects Approach

Strategy 7

Logical Reasoning

Strategy 8

Working Backwards - Building Up Information

16 18 20 22 25

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© ReadyEdPubl i cat i ons 100 Problems to Solve 28 •f orr evi ew pur posesonl y• Answers 62

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NAME .........................................................................

Personal Record Chart R 2R 3 R Guess and Check 1R 2R 3R r o e t s Using a Table or Chartr 1RB 2R 3R e o p o3R Compiling u an Organised List 1R 2R k S Looking for a Pattern 1R 2R 3R Kinaesthetic/Real Objects Approach 1R 2R 3R Logical Reasoning 1R 2R 3R Working Backwards 1R 2R 3R

Strategy 1 Making a Diagram 2 3

5 6 7 8

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51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

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Results/Comments ©100 ReProblems adyE-d Publ i cat i ons •f orr evi ew pur posesonl y•

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Strategy 1 Making a Diagram r o e t s Bo r e p ok u S

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Making a diagram is a simple technique that is a good starting point in teaching problem solving. “Doodling” while you think is useful! Information is continually documented, which helps to keep track of all discoveries and see patterns which might not be immediately obvious.

There are two diagrammatic techniques which are especially useful: a) Scaling - where the precise information can be converted to a scale diagram.

Solutions can be determined by the size, shape, amount etc. indicated by the resultant diagram. = 9 kilometres

e.g. If this is 3 kilometres,

how long is this?

b) Sketching - where the information is arranged visually to produce a pattern and a possible solution.

Example to work through

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In a game of Pacman, there are 15 rows of 10 power points. Calculate the number of power points Pacman devours as he travels the . . . . . . . . . . following route from the start point: up: 3, right 2, . . . . . . . . . . up 6, right 3, up 4, right 1, down 9, left 6. . . . . . . . . . .

Describe how far away Pacman lands from where . . . . . . . . . . he started. . . . . . . . . . .

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Teaching Strategies + Read all the problem through before commencing. + Do each instruction or step in isolation. Don’t try to predict outcomes too prematurely. + Be specific as you answer the questions.

Working the Solution + 3 + 2 + 6 + 3 + 4 + 1 + 9 + 6 = 34 + Ends four power points from the start.

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Strategy 1: Making a Diagram - Practice Sheets

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Practice Problem 1 At sunrise a sand flea began jumping up a 20 metre sandhill. Every hour he was able to jump up five metres, but then slid back three metres in the loose sand.

How long did it take the sand flea to reach the top?

© ReAnswer: ady EdPubhours l i cat i ons ________________ •f orr evi ew pur posesonl y•

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Practice Problem 2 ‘Scary’ the superstitious spider only walks along the cracks in between the slabs on a footpath. If each slab is one metre times one metre, how far would he travel if he went along two slabs, right one slab, along three slabs, left one slab, along three slabs, right two slabs and back five slabs?

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Answer: _________________ metres

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Strategy 1: Making a Diagram - Practice Sheets

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Practice Problem 3 In the dense jungle of Sweatyland a small tribe of natives lived in five huts. The jungle was so thick they could only cut a narrow path between the huts. How many paths would they have had to create so that each hut was joined to all of the others?

© ReaAnswer: dyEdPubl i cat i ons ________________ paths •f orr evi ew pur posesonl y•

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1 km .8 km

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Practice Problem 4 Lanky Runner entered a round-the-streets marathon. What is the shortest distance he has to run to go through all the checkpoints and return to the starting line? (He must stay on the streets.)

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.25 km

.25 km

.5 km

.2 km

1 km

Start

Answer:__________________km

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Strategy 2 Guess and Check (Two Variables)

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As the title suggests, this strategy arrives at a verifiable solution by hypothesizing possible answers, checking back to see which fits the problem, and modifying answers from the results of previous checks. In this strategy the teacher’s role is to guide students towards:

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a) a probable or likely starting point; b) working in the right direction with large enough gains to quickly solve the problem.

I.e.The guessing game: I'm thinking of a number between 1 and 1000.

Can you guess it in 10 tries?

The most successful students achieve this by eliminating as many numbers as possible with each guess, e.g. 500 lower, 250 higher.

Example to work through

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• Teaching Strategies Billy has 56 marbles in his collection. If he had 14 more catseyes than bullets, how many of each marble did he have?

+ Decide what you are trying to find out.

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+ Read the question for clues of where to start, i.e. a) In this question, the number of catseyes and bullets is ‘given’: No. catseyes = bullets + 14; No. of bullets = catseyes - 14. b) There are more catseyes than bullets.

+ As there are more catseyes than bullets, make the first guess about the number of catseyes, and make the estimated number higher than half of the marbles:

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Half of 56 marbles = 28, therefore catseyes = 30? catseyes = 30, bullets = 30 - 14 = 16.

If there were 30 catseyes, the total number of marbles would be 46, 10 less than the actual total. + Halve the difference between the two totals, and add it to the previous guess: 30 catseyes + 5 = 35. bullets = 35 -14 = 21.

Total marbles equals 56.

Therefore, the second guess is correct. Billy has 35 catseyes and 21 bullets in his collection.

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Strategy 2: Guess and Check - Practice Sheets (Two Variables) Practice Problem 1 A farmer has 62 sheep and cows in total. If he has eight more cows than sheep, how many of each does he have?

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Answer: sheep__________________

Practice Problem 2 A baker bakes 36 doughnuts and tarts each day. If he makes six more doughnuts than tarts, how many of each does he bake?

© ReadyEdPubl i cat i ons Answer: doughnuts__________________ tarts__________________ •f orr evi ew pur posesonl y•

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Practice Problem 3 A stamp collector collects French and English postage stamps. She had 212 stamps in her collection, with 62 more French than English. How many of each did she have?

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Answer: French _________________ English _________________

Practice Problem 4 In a school of 568 students, there were 26 more girls than boys. How many boys were there?

Answer: _________________ boys

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Strategy 2 Cont.

Guess and Check (Three Variables)

r o e t s Bo r e p ok u S Example to work through

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This technique of guess and check can be extended to include many variables and subsets of information. In these examples, more information needs to be documented and checked.

At the zoo there are 90 animals, including birds, reptiles and mammals. If there are eight more reptiles than marsupials, but two more marsupials than birds, how many of each are there?

Teaching Strategy

+ Create a chart.

+ Guess again. + Check if correct.

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36

28 (8 less) 26 (2 less)

= 90

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Marsupials Birds Total © ReadyEdReptiles Pub l i cat i ons Guess the sizes of the groups. I.e.: 32 24 (8 less) 22 o (2 less) = 78 Total animals is r only 78.i • f o r e v e w p u r p o s e s n l y • Need 12 more, i.e. 4 more of each.

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Strategy 2: Guess and Check - Practice Sheet (Three variables) Practice Problem 1 All of the 42 children at Hilltop Primary School had to choose a sport to play. They had a choice of netball, football and volleyball. Six more chose football than netball, while three more chose netball than volleyball. How many chose each sport?

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

l Guess

Volleyball

Check Re-Check

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Re-Guess

Netball

Re-Guess

Re-Check

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Answer: football ____________ netball ____________ volleyball ____________

Practice Problem 2

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Re-Guess Re-Guess

Sheep

Cows

Pigs

Check

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Answer: sheep _____________ cows _____________

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A farmer has cows, pigs and sheep on his farm. He counted 106 animals in total, with six more sheep than cows, but five less pigs than cows. How many of each does he have?

Re-Check Re-Check

pigs _____________

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Strategy 2: Guess and Check - Practice Sheet (Three variables) Practice Problem 3 John Dough the baker cooked 67 biscuits each morning. He only made three flavours.

If he made six more chocolate than mint, but eight more mint than liquorice, how many of each flavour did he make?

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

Guess

Mint

Liquorice

Check

Re-Guess

Re-Check Re-Check

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Re-Guess

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Answer:Chocolate _____________ Mint _____________ Liquorice ____________

Practice Problem 4

Guess

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Claire

Sharon

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Three friends pooled all their pocket money. Kate and Claire had $28 altogether. When Sharon and Kate counted, they had $22, while Claire and Sharon had counted $26 together. How much has each person got?

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Check

Re-Check Re-Check Re-Check

Answer: Kate ____________ Claire _____________ Sharon _____________

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Strategy 3 Using a Table or Chart r o e t s Bo r e p ok u S Example to work through

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This strategy requires students to set out the information given on an orderly chart, graph or table. When set out, the data should suggest a pattern or part of a solution that can be completed by filling in the missing information. The teacher’s role is to assist students in the creation of the most appropriate headings to display all the information provided.

Jane and Ben both go to the same dancing centre that operates seven days a week. Jane goes every third day while Ben attends every second day. If they meet at the school on Sunday, when would they meet again?

Teaching Strategy

© ReadyEdPubl i cat i ons S M T W T F S + Work out which • f orr evi ew pur posesonl y• way to present

+ Read all the problems, be aware of unstated factors (e.g. operates only during the weekdays?)

J

B

J

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+ Create symbols for the information and fill in the table. + Answer: They would meet again on Saturday.

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B

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the table - days or visits.

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Strategy 3: Using a Table or Chart - Practice Sheets

Jenny

Johnny

Total

6 7

8

= 14

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Practice Problem 1 Jenny and Johnny are six and eight years old respectively. As you can see, the sum of these birthdays add up to 14. How old will Johnny be when the sum of their ages is 28?

© Re ady EdPub l i c t i ons Answer: _________________ years olda •f orr evi ew pur posesonl y•

a) Lids on?

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b) Lids and flavours?

_________________ _________________

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Practice Problem 2 A factory worker is bored with his job, so he decides to make it more interesting while filling his 30 bottles of cordial. He decides to put the lid on every fourth bottle, the flavour in every fifth and the label on every sixth bottle. If he started at the 1st bottle, how many bottles would have:

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c) Flavours and labels?

_________________

d) Lids, flavours and labels? _________________

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Strategy 3: Using a Table or Chart - Practice Sheets

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Practice Problem 3 In his secret code book, Ima Snoop kept the answers to his code on a special page.

The page number was a three digit number, with one of the digits being a three. The total of all the digits was ten. The book contained less than 160 pages. What page was the code on?

© Re ad y E Publ i cat i ons The code was ond page__________________ •f orr evi ew pur posesonl y•

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Practice Problem 4 While making a line of 50 cookies, a forgetful baker creates the following pattern. She ices every third cookie, puts cherries on every fifth and places chocolate chips on every seventh along the line.

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How many cookies are complete with all three things?__________________ Ready-Ed Publications

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Strategy 4 Compiling an Organised List r o e t s Bo r e p ok u S Example to work through

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While similar to the previous technique (using a table or chart), this technique is normally used when there is far more information to present. The information needs to be set out more systematically to present the array of probable solutions. Students need to be encouraged and allowed to create lists and jottings, rather than carrying all the permutations in their heads.

When five friends meet they each shake hands with each other. How many handshakes are exchanged?

Teaching Strategies

+ Read the problem carefully. Imagine what will happen.

© ReadyEdPubl i cat i ons Look for r repeats, 1 with 2w is thep same asp 2 with 1. e •outf o r ev i e ur os sonl y•

+ Ascertain how many variables there are, e.g. five people - but only four actions as you don’t shake hands with yourself.

with 2 with 3 with 4 with 5

2 with 3 2 with 4 2 with 5

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Answer = ten handshakes

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3 with 4 3 with 5

4 with 5

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Strategy 4: Compiling an Organised List - Practice Sheets Practice Problem 1 Rosemary has a choice of how to get to school each day. She can walk or cycle to her friend's house, then be driven or catch a bus to the station, and then use a scooter or catch a train for the last section to school. How many different combinations of travel could Rosemary use on her way to school?

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Answer: _________________ combinations

Practice Problem 2 Four boys, Danny, David, Don and Duane, always line up first at the canteen. How many different combinations of order could they line up in?

© Re ady EdPub l i cat i ons Answer: _________________ combinations •f orr evi ew pur posesonl y•

How many days before he began to repeat a combination?

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Practice Problem 3 For afternoon tea, Freddie Fang is able to have three things: a cake or biscuit, an apple or banana, and a glass of milk or a thickshake.

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Practice Problem 4 In a tennis tournament, six players are left. In order to find the winner, everyone has to play everyone else. How many games would this take?

Answer: _________________ games

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Strategy 5 Looking for a Pattern Often mathematics is about probability and prediction. By using this strategy (an extension of the tabling and organised list strategies) and visually presenting the information, a pattern often emerges. If a pattern can be established, it becomes relatively easy to predict what comes next. Once a pattern is able to be verified and checked it can be applied at any stage throughout the problem.

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This strategy underlines the fact that problem solving strategies are interrelated and interwoven. Students will be using at least two or three subskills simultaneously as they approach the higher levels of problem solving.

Example to work through

Two flies bred at an alarming rate - on day one there were two flies, on the second day there were six flies, on the third there were 14 flies, on the fourth day there were 26 flies. How many flies would there be by the eighth day and the tenth day?

© ReadyEdPubl i cat i ons Teaching Strategy • f o rr evi ew pur posesonl y• + Read the problem thoroughly.

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+ In this case the number appears to grow by a factor of four. + Project this fact through to day 8 and day 10.

Complete the chart Day

1

2

3

4

5

Starting total

2

6

14

26

Grows by

4

8

12

6

14

26

6

7

8

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+ Organise the information given into a grid to show up any patterns.

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+ Answers: Day 8 = 114 Day 10 = 182

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Strategy 5: Looking for a Pattern - Practice Sheets Practice Problem 1 Study these patterns carefully and fill in the next three numbers.

V 1, 3, 5, 7, 9, _____, _____, _____.

V A, C, E, G, _____, _____, _____.

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V 6, 10, 8, 12, _____, _____, _____.

Practice Problem 2 Supply answers that follow these word patterns.

V Ant, Butterfly, Caterpillar, __________, __________, __________.

V John, Ken, Leon, __________, __________, __________.

V Denmark, Egypt, France, __________, __________, __________.

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V 2, 4, 7, 11, _____, _____, _____.

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V Allan, Cindy, Edward, Gloria, __________, __________, __________.

Practice Problem 3

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Peter the footpath painter slowly increases the number of cement slabs he can paint each day.

On day one he paints three, on day two he paints five, on day three he paints 8, day four he paints 12 and so on.

On what day will he paint more than 32? _________________ How many days will it take to paint over 150 slabs in total? _________________

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Practice Problem 4 Supply the next three numbers or words in these sequences.

V 1000, 520, 280, 160, _____, _____, _____.

V a, at, all, arms, __________, __________, __________.

V 3, 6, 11, 6, 19, _____, _____, _____.

V 3, 6, 4, 7, 5, _____, _____, _____.

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Strategy 6 Kinaesthetic/Real Objects Approach r o e t s Bo r e p ok u S Example to work through

Use six matches to create four equilateral triangles.

Teaching Strategy

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Often it is difficult for students to deal in the abstract and visualize patterns or solutions.

By providing real materials that can physically symbolize an element of a problem, students can move, manipulate and present solutions to problems more easily: 3 - D doodling!

+ Use any manipulatives that are of equal size and length to avoid inserting an extra dimension to the problem. + Define the term 'equilateral'.

© ReadyEdPubl i cat i ons Result •f orr evi ew pur po sesonl y•

+ Simple try, e.g.

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+ Try adding a triangle, e.g.

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Used five matches, only two triangles.

Now seven matches, too many.

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+ Suggest different sizes, e.g.

One correct solution. All four triangles didn’t need to be equal size, only equilateral.

+ While this answer also creates other shapes it nonetheless, creates four equilateral triangles - 2 large and 2 small. Find other solutions.

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Strategy 6: Kinaesthetic/Real Objects - Practice Sheets Practice Problem 1 Using a set of matches, complete the following tasks.

V Use five matches to create two triangles.

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V Use four matches to create three triangles.

V Use seven matches to make three triangles.

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V Use seven matches to make seven triangles.

Practice Problem 2 At the Agricultural Show, six cranky bulls each had to be provided with their own yard.

Unfortunately there were only twelve sections of steel fence, and these couldn’t be sawn or shaped. Show how the fence could be arranged to make six separate yards.

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Practice Problem 3 At the soft drink factory they only had eight litre and six litre containers. Workers were allowed to drink four litres of soft drink per day, but they had to measure the four litres exactly. How would you measure out four litres exactly?

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Practice Problem 4 Set out eight counters in a line. In four moves, jumping over two and only two piles each time, leave four piles of two counters.

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Strategy 7: Logical Reasoning r o e t s Bo r e p ok u S Example to work through

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This strategy is also called the if/then approach. Logical reasoning forms the basis of a great many problem solving techniques. In this strategy, statements or information can be used or extrapolated to create the next part of the solution (unlike guess and check where it is an all or nothing statement). Each piece of information should confirm the previous hypothesis or lead to a new hypothesis, hence the if/then label. Students steadily progress through the statements. Often, eliminating an option can be as useful as solving a piece of the puzzle.

At the sports carnival, five students ran in a mixed 100 metre race. Use the clues below to complete the table.

Answer 1st

+ John was the first boy to finish.

John

© ReadyEdP ubl i cat i ons Emma Dereck finished behind Joan.

Dereck • f o r r e v i e w p u r posesonl y• Pam raced her best to beat Joan.

+ Emma finished in the position before Dereck. + +

2nd 3rd 4th 5th

Pam

+ John came after Pam and before Joan.

Joan

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Who was ahead of Emma?

Teaching Strategy + Set up the labels and categories. Work out all the names. + Read each clue thoroughly. + Do not always start with the first clue because others may provide more information, e.g. John being the first boy to finish doesn’t mean he finished first, although he may have. + Make a statement and use the clue to confirm or dismiss it. + Use named counters if required and see if the order fits the clues, i.e.:

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Clue 2

Emma

Clue 3

Joan

Clue 4 Clue 5

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Pam Pam

Dereck Emma

Joan John

Emma Joan

Dereck Dereck Emma

Dereck

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Strategy 7: Logical Reasoning - Practice Sheets Practice Problem 1 Match the student with the sport, using the clues outlined below.

Students: James, Karen, Charles, Terri, Lee Sports: Soccer, Basketball, Tennis, Softball, Cricket. + + + + +

James likes to practise shooting, however kicking is not allowed in this sport.

Throwing and catching aren’t always part of the team game Charles plays.

Lee often uses her own glove and can get out more than once each game.

Terri finds it difficult to practice her sport without others.

Karen has developed a great passing shot in her sport.

James

Sports

________________

Karen

________________

Charles

________________

Terri

________________

Lee

________________

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Fred

_________________

Felicity

_________________

Phil

_________________

Freda

_________________

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Practice Problem 2 At the Pizza Palace, Fred, Felicity, Phil and Freda all decided to order a different type of pizza. They chose prawns, ham, cheese, and anchovies.

Use the following clues to match the pizza choice to the person. + Fred hates anything from the sea. + Phil is allergic to cheese. + Felicity is a vegetarian and won't eat any meat. + Freda hates anything salty.

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Strategy 7: Logical Reasoning - Practice Sheets Practice Problem 3 Santa Claus has mixed up the name labels for five Christmas bears.

Dancer

Prancer

Dasher

Slasher

Rudolph

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Use these clues to name the bears correctly. + Rudolph and Slasher are smiling. + Dancer has a big nose. + Dasher and Rudolph have big eyes.

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This is a plan of a new shopping centre. The signwriters have been asked to correctly identify and label each shop. Can you help them?

1) The video shop is between the chemist and the grocer.

2) The butcher is three shops away from the grocer.

3) The chemist has a corner entrance, while the music shop also has a side entrance.

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Strategy 8 Working Backwards Building Up Information r o e t s Bo r e p ok u S

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Example to work through

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This strategy is useful for problems which are presented by a range of events that have occurred. Each stage or piece of information affects the next stage. Students start at the end point and work backwards to ascertain the original situation, as if they were rewinding a video tape.

In an all-in wrestling championship, all the wrestlers entered the ring together.

Within three minutes, half were thrown over the top ropes and eliminated. In the next four minutes, half of those remaining in the ring were eliminated. By the ten minute mark, the number in the ring had dropped by half again. Fifteen minutes after the start of the competition, half of these remaining competitors had been tipped over the rope, and in the last five minutes, one more wrestler was tossed to the floor, leaving only one wrestler standing - the winner!

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

How many wrestlers entered the ring at the start of the match?

Teaching Strategy

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end of match = 1 wrestler 5 minutes before end = 2 wrestlers

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15 minutes into the match = 4 wrestlers

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+ Start with the winner and double at each recognisable time lapse:

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10 minutes into the match = 8 wrestlers 4 minutes into the match = 16 wrestlers 3 minutes into the match = 32 wrestlers

+ 32 wrestlers entered the ring.

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Strategy 8: Working Backwards - Practice Problems Practice Problem 1 In Mary's gymnastics group there were 36 girls. 1/6 of the girls were born in the first quarter of the year and half of the remaining girls were born in April and May. 3/5 of the remaining birthdays are in September, and two students were born in October and November. All the other girls have birthdays in December.

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How many December birthdays?

_________________ birthdays © Answer: Rea dyEdPDecember ubl i c at i ons •f orr evi ew pur posesonl y•

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John

Jenny

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Practice Problem 2 In a book balancing competition, six contestants fared as follows: + John built his stack five higher than Jenny's stack. + Jerome built his stack three higher than John's. + Joan built her stack two more than Jerome's, while Joseph was six higher than Joan. + If Joe stacked 27 books, this being two books taller than Joseph’s stack, show how high each stack was.

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Joe

Joan

Joseph

How many books would be needed altogether? _________________

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Strategy 8: Working Backwards - Practice Problems

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Practice Problem 3 If a 1/3 of 1/2 of 1/4 of 1/6 = 3, what would be the starting number? _______________

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chocolate

_________________

mint

_________________

butterscotch

_________________

caramel

_________________

bubblegum

_________________

honey

_________________

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Practice Problem 4 At the icecream factory the quantities of each of the six flavours are decided mathematically. The total amount of chocolate, mint and butterscotch is always half of the days production.

They produce fifteen more tubs of chocolate than mint, and fifteen more tubs of mint than butterscotch. Of the remaining icecream half is honey, with the rest being split 1 /3 : 2/3 between caramel and bubblegum. If there are fifteen tubs of caramel icecream, how many tubs of each of the other flavours are made?

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100 Problems to Solve r o e t s Bo r e p ok u S

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The problem solving activities on the following pages are designed to provide consolidation of the techniques that students have encountered as they have worked through the preceding pages.

The 100 problems contained in this section can be photocopied and pasted on to card to allow students to be working on different problems during the mathematics lesson.

The problems are non-graded and non-sequential, and many are a combination of two or more problem solving techniques.

The Personal Record Chart on Page 4 should be made available so that students can plot their progress and achievement as they complete the activities in this book.

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Problem 1

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Taken for a Ride The inner city trains are very efficient, and can travel everywhere. However, your ticket only allows a maximum of six stations, including pick-up and stop-off. Using the map below, find eight different routes which you could take from Home Station to Office Station.

Newtown Station Home Station 1

2

3 Oldtown Station

Boystown Station 4 Sometown Station 5 6

7 Office Station

My-town Station

Answer:_________________

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Problem 2

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Bumper Ride Four friends visited the ‘Lot-o-Fun’ Park. They decided to ride in the jumbo hovercrafts.

Bert always drives while Tim sits next to Bev, who sits behind the driver. Jenny sits in the remaining seat.

Draw where they would all sit.

Problem 3

Snack Time After school, Janet, Jane, Joan and Jenny raided Jane’s parents’ fridge for snacks.

They only found milk, orange juice, a banana and bread, and each chose one thing. + Jane never chooses liquid. + Janet loves anything yellow. + Jenny is allergic to dairy products.

Who had what?

_________________

Joan

_________________

Jenny

_________________

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Problem 4

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

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Swimming Solution At the school swimming carnival, Mike Missdit was confused about what order the four teams, Blue, Green, Gold and Red, finished up in. He knows that either Red or Green came first. Knowing this, how many different combinations of the final points order could there be?

Answer: _________________ combinations

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Problem 5

r o e t s Bo r e p ok u S Answer: _________________ days

Problem 6

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Row-away Rodney ‘Rower’ Reed is attempting to row around the island of Zete against a current.

His pattern of rowing is day one - 30 kilometres, day two - 28 kilometres, day three 26 kilometres ... less each day as he tires. Additionally, each night the current takes him back eight kilometres. How many days would it take to row the 120 kilometres around the island?

Humble Pie In training for the ‘Pie Eating Contest’, Big Bill eats five pies the first day, ten the next, and so on, adding five more each day as he becomes better trained. How many days would Bill train until he used up all his 330 practice pies?

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Problem 7

Answer: _________________ days

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Enough Dough George Gutz loves doughnuts. If he eats one a day for every day in March, April and May, how many would he eat?

Answer:_________________ doughnuts

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Problem 8 Booked Out Four friends, Larry, Harry, Karri and Sharri, entered the book exchange and purchased the following titles: Goosebumps, Lost at Sea, Black Beauty and Around the World in 80 Days. + Larry loves nautical stories; + Harry doesn’t like scary stories;

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+ Karri doesn’t read animal books;

+ Sharri chooses books with long titles, as they are ideal for book reviews.

Who chose what?

Harry _________________

Sharri _________________

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Karri

_________________

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Larry _________________

Problem 9

Cat-Astrophe Three friends, Jim, Joan and Jerry, have come to pick up their cats from the vets. The cats are called Fats, Rough and Tuff. The cat carriers are labelled with the owners' last names - Able, Broad and Coombes.

Using the clues below, help the vet work out which cat goes in which carrier, and which carrier belongs to each person.

1. Tuff belongs to the Broads.

2. Jim is before Jerry and Joan in the class attendance roll.

3. Joan’s name is not Broad, but the name can be applied to her cat!

Last names Cats Jim

_________________

_________________

Joan

_________________

_________________

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_________________

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Problem 10

_________________

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Jumbled Up At a jumble sale Janet Jumper collected a large armful of clothes. She kept a third for herself, gave half of the remaining clothes to her husband, and her three children were given the rest. But the three children - Joey, Jennifer and Johnny - argued.

Johnny ran off with three times as many clothes as Joey, and Joey and Jennifer were left with three garments each.

How many garments did Janet start with?

Answer:_________________ garments

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Problem 11

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Problem 12

Numbers Up? If 1/2 of a number is 6 more than 1/3 of the number, what is the number?_________________

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What's My Number?

A quarter of the number is three less than a third of the number.

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Problem 13

+ + + + +

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What Am I? I’m a three digit numeral whose digits total 15.

I’m divisible by three.

I'm an odd number I’m larger than 50% of 1000 but smaller than 60% of 1000.

My first and last digits have the same value.

What number am I?_________________.

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Problem 14

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Problem 15 Shopping Where?

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Who’s Who? A group of friends are standing in line at the cinema. From the clues given below, work out who is who. + John and Jenny always stand together. + Terry is taller than Sherri but shorter than Tim. + Anita always likes to stand on one end.

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Using the clues below, name each store. + Andy's is 2 shops away from Don's and 2 away from Cools. + Cool's is taller than Moles but shorter than Central Office. + Kate's cafe is a corner shop. + Andy's is the smallest building.

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Problem 16

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In the Dog House Match the dogs to their kennels.

+ Butch and D’for don’t have water bowls. + Fifo and D’for have decorated kennels. + Tiny can’t be housed next to Butch. + Max is kept two kennels from Butch and next to D’for. + Max likes his water bowl.

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Problem 17

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Problem 18

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Timetabled In Ross has just entered High School. He has the choice of three teachers for his Maths class, two for English, one for French and two for Sport. How many different combinations of teachers could he have, while still covering all 4 of his subjects?

Eat it again Sam Sam’s Swedish Smorgasbord offers a wide variety of food choice. Pete Pudgey eats at Sam’s every night. He tries a different combination of dishes each time. Pete always has one entree, one main dish and dessert. How many days can Pete eat at Sam’s before he repeats a meal combination?

© ReadyEdPubl i ca t i ons Main Sweets + Salmon + Steak Supreme + Semolina •f orr evi e wp ur pose onl y• + Scallops + +s Swedish Salad Strawberries Entrees

+ Spaghetti

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Problem 19

Answer: _________________ days

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+ Soup

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Fun of the Fair Mary Q. Contrary has been saving all of her pocket money for the annual fair. If she took $18 to the fair and spent everything on special theme cakes, what different combinations of cakes could she have bought?

Fab Fair Cakes + Aunty Anne Cake + Dangerous Dan Cake + Football Cake

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$3 $5 $4

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Problem 20

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Problem 21

Here’s Looking at You Two friends love videos.

They go to the same seven-day-a-week outlet. Trevor goes every three days to hire a movie, while Tony goes every five days.

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See You Soon Liza and Leonie both work part-time at Big Jack’s Burgers, which opens seven days a week. Liza works every Friday, while Leonie works every fourth day. If they worked together on Friday 1st November, when would they work together again?

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Answer:_________________ days

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Problem 22

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© ReadyEdPubl i cat i ons If they met on a Tuesday, how many days• would pass before they would f o r r e v i e w pur posesonl y• meet there again?

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Dough - Nut Terry Ific loves doughnuts and eats them every day. Unfortunately they are only sold in lots of 12. If he eats one doughnut every day for June, July and August, how many dozen doughnuts will he need to buy?

Answer:_________________

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Problem 23 Office Carpark The sales staff of Knowall Drapery always park in the same positions each day.

(Hint: Read all the clues before starting work.) + The Porsche is always against the wall. + The Lamborghini is never next to or in the same row as the Porsche.

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+ The Rolls always parks in the middle.

+ The Bentley is diagonal to the Porsche.

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+ The Jaguar is alongside the Rolls while the owner's V W is facing the Porsche.

Problem 24

Who’s Eaten What?

Six friends have all started eating doughnuts. Work out who owns each doughnut from the clues below.

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John ate about half a doughnut.

Jenny ate the least while Jeremy ate the most.

Joan and Jerome always sit together, as do John and Jeremy.

None of the girls, including Jan, ate over 1/2 a doughnut.

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+ + + +

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

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Final Seats At the football finals seven friends all had the same seat number, however each was a row behind the other. Can you fill in who sat where?

+ John sat three rows ahead of Jenny, who sat two rows behind Terry.

A1

__________________________

B1

__________________________

+ Andrew never sat in front of Jenny.

C1

__________________________

D1

__________________________

E1

__________________________

F1

__________________________

G1

__________________________

+ Julie always sat directly behind Jenny and somewhere in front of Andrew. + Carol and Dean always arranged themselves in alphabetical order between everyone.

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Problem 26

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Problem 27

I Started With...

If 1/4 of 1/3 of 1/5 of 1/2 = 3, what number did I start with?

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Pattern Building This pattern has been discovered in an Ancient Tomb. Can you fill in the missing symbols that would complete the pattern?

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Problem 28

Answer: _________________

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Missing Leaf In Luke’s latest Goosebumps book, the page with the solution to the mystery is missing! Which page has Luke lost?

The page has a three digit number with one digit being a two. The total of all three digits is twelve. If every book in the series has less than 140 pages, what page is the solution on?

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Problem 29 Round - a - bout Mrs Foster always visits the same four shops when she is doing her weekly shopping. Each week she takes a different route.

If she visits every shop once, and uses each path only once, how many different routes could she go?

Butcher

Bakery

Home

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Answer:_________________ different routes

Problem 30

Chemist

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Supermarket

Stand-Up Stan had a stand-in job setting up the stand at the ‘Stand-Up-Knockem'-Down’ stall.

He wanted to stand his 28 cans in a triangular shape.

How many would he put in the bottom row of the stand?

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Problem 31

Answer: _________________ cans

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Rounding It Out In a round robin competition everybody plays everyone else at least once. How many games would need to be played if there were six players?_________________ How many more games would be needed if three extra players entered the competition?________________

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Problem 32 Jenny is five years younger than Janet, and together they total 27 years old. In seven years time, how old will each person be?

Jenny _________________ _________________

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Janet

Problem 33

Whole Approach Dan Drainer was given the task of filling a leaky 100 litre drum. Trying as hard as he could, he poured in five litres a minute, but two and a half litres drained out every two

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Answer:_________________

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Problem 34

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Fairly Old The Fairly family are great mathematicians. When asked how old everyone was the family replied, “Our family of two adults and three children is exactly 123 years old altogether.

The father and son are 59 combined, which is five years younger than the mother and the twin girls combined. Mr Fairly is three years older than Mrs Fairly, while the son Mark is three years older than the girls.”

Can you work out everyone’s age?

Mr Fairly:

_________________

Mrs Fairly:

_________________

Mark:

_________________

Twins:

_________________

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Problem 35

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Feeding Time May B. Ware is in charge of feeding the animals at a small zoo. She places food into 22 enclosures holding one animal. While doing this, May counts a total of 64 animal legs. Calculate how many two and four legged animals there are.

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Problem 36

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Answer: 2-legged_________________ 4-legged_________________

Freckle Friends Fran was teasing Freda about how many freckles she had. Together they counted 219 freckles on their faces. However, Fran had twice as many freckles as Freda.

How many freckles did each girl have?

_________________

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Problem 37

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How many less freckles did Freda have?_________________

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Pets The Adams family has a total of eleven pets, all either birds or cats.

If they counted 36 legs in total, how many of each animal are there?

Answer: birds_________________

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cats_________________

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Problem 38 Find The Signs Find the mathematical signs that would make these number sentences correct.

3 3 3 3 3 = 3 4 4 4 = 4 6 6 6 6 6 = 2

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6 6 6 6 6 = 186

Problem 39 What's my Number?

+ I'm smaller than 33 lots of 33, but larger than 300 x 3. + My digits when added are divisible by three.

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Use +, -, x and ÷ (but not necessarily all in each problem!)

+ One of my digits is a zero

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© ReadyEdPubl i cat i ons + My is_________________ •number f or r evi ew pur posesonl y•

+ I am a three digit number divisable by thirty three.

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Guess This Can you guess the number I’m thinking of ?

+ It’s less than 9 x 9.

+ It’s an odd number, but not divisible by 3. + Its two digits added together equal 11. + The first digit is smaller than the second. + It’s larger than 1/2 of 100. + It is_________________

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Problem 41

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Problem 42

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Present and Correct Echo McAll celebrated his birthday yesterday. Of all the presents Echo received: 1 /4 were games; 1 /4 were clothes; 1 /8 were fluffy toys.

He also got two footballs and four books.

How many presents did he receive altogether?

All Angles ‘All Angles Camera Shop’ sells four-legged camera stands for $60 and three-legged camera tripods for $40. If the store sold 17 stands totalling 59 legs, how much money did they make from the sale of stands?

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Problem 43

Answer: $_________________

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Tripping Out The ‘Terrifying Tourist Trip’ takes a boat journey to the crocodile infested waters of Lake Dread. Each group is strictly five tourists to every guide, and each boat can only hold twenty people, including the boat's captain. How many boats would be needed to carry the 140 tourists booked to take the trip?

Answer: _________________ boats

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Problem 44

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Combining Colours At the “Stuffy Bear” factory, the bear assemblers have a choice of three colours for each of three parts: nose, tie and abdomen.

How many different combinations can they create whereby each bear has all the 3 colours? Draw and colour them.

Problem 45

Sam’s Sports Store Sam stocks six different items in his ‘Special's Section’. During the week he made $156 from $29.00 sales of goods on special.

$18.00

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

Find ten different combinations of goods which add up to $156, e.g. 2 footballs; 2 rugby balls, 1 glove and 4 whistles = $156.

$27.00

$3.00

__________________________________

$16.00

$35.00

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Darting Around Dan Darter practises darts every day on his special dart board. His favourite game is 15.

How many different ways could Dan score 15 using all his three darts, assuming that all darts score and that no combinations are repeated?

8 2 3 7 6 5 4

Answer:_________________different ways Ready-Ed Publications

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Problem 47

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Problem 48

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Language Learner Kromer is very keen to learn Japanese. On the first day he learns one word. On the next three days he learns 3, 5, and 7 words, improving by two words each day. How many days will it take for Kromer to learn 100 Japanese words?

Doubling Pounds Penny Pinching loves to save money. She saved one cent on Saturday, two cents on Sunday, four cents on Monday and eight cents on Tuesday, doubling every day.

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How much would she save in total in one week?_________________

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Problem 49

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Could she afford a $20,000 car by the end of three weeks?_________________ (Saving the amount each day and adding it all together.)

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Doughnut Days Terry Tummy was president of the Doughnut Eating club. He and two club members, David and Dan, decided to hold a ‘Doughnutathon’ every day in the month of December. Terry eats twice as many doughnuts as David and Dan combined. David and Dan each eat 1 1/2 doughnuts per day. How many doughnuts does Terry eat? How many doughnuts will the trio have to buy for their December Doughnutathon?

Answers:Terry_________________ total doughnuts_________________

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Problem 50 Rabbit Riot Anna began an intensive breeding programme with her pet rabbits. In the first month there were 23 rabbits, the second month she had 28 more rabbits, the third month, an extra 35, and the fourth month, an extra 44.

How many months before she has 100 rabbits in total?

_________________

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How many months before she has 200 rabbits in total?

_________________

Problem 51

Goals Galore Jenny wants to break the club’s goal scoring record of 100 goals in the season.

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In how many months would she have more than 400 rabbits in total? _____________

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She scores eight goals in the first game, four in the second game, eight in the third and four in the fourth game, and this pattern continues throughout the season.

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Answer:_________________

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If there are 18 games each season, will she break the record?

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Goal Getter Kate Catcher plays for the ‘Giants’ Netball Team. She shot one goal in the first game, three the next, five the next and so on, improving by two goals each game. How many games will it take to reach 50 goals in total?

100 goals in total?

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_________________ _________________

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Problem 53

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Problem 54

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Cashing In After the football two friends were collecting cans and bottles for recycling. They collected 105 containers in total, finding twice as many cans as bottles. If cans are worth four cents, and bottles are worth two cents, how much money would the friends make?

Well Fed Pinky and Perky were Mrs Purr’s two favourite cats. Every night she would feed them 56 cat biscuits, making sure Pinky received twelve more biscuits than Perky.

How many biscuits did each cat get?

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Answer: Pinky:_________________ Perky_________________

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Book Detective Uptown Library has three times as many books as the Downtown Library and School Library combined. But Downtown Library has twice as many books as the School Library. One quarter of the way through the Downtown Library stocktake, the librarians had counted 108 books.

How many books does each library have?

Uptown Library

_________________

Downtown Library _________________ School Library

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_________________

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Problem 56 Even It Out Using the digits 1-16, set out this square so that each column, row and diagonal equals 34. We have started it for you.

7

2

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5

16

Problem 57 Gnikrow Sdrawkcab

If 1/3 of 1/6 of 1/2 of 1/10 = 7, what was the starting number?

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14

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Problem 58

Answer:_________________

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Counting Down and Across Remove one number from each row so that the total in each column and row is 20.

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3

4

3

5

5

6

6

4

10

2

2

2

5

7

2

3

4

6

6

4

2

3

4

3

3

4

2

5

5

6

3

4

3

7

3

3

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Problem 59

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Answer: poodles_______________

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Dog Tired At the ‘Happy Pooch Hair Salon’ Sandy earned $26 a day washing dogs. She was paid $3 per poodle and $5 per hound, and she washed 1/2 as many poodles than hounds. How many poodles and hounds did Sandy wash each day?

hounds_________________

Problem 60

It All Adds Up The Nutty Newspaper editor believes that advertisements are more important than news.

She has a set rule about the number of ads that must be on each page.

The first five pages must be 3/4 ads. The next five pages are 1/2 ads and 1/2 news. The next five pages are 1/4 advertisements, the following five are 1/8 ads, and so on.

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Problem 61

Answer:_________________

o c . che e r o t r s super

Arranging Fractions Arrange the following fractions so that each line of the triangle equals 1 1/2.

1

/6, 2/6, 3/6, 4/6, 5/6, 6/6

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If the latest issue of the Nutty Newspaper was 18 pages long, what fraction of the paper was devoted to ads?

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Problem 62

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

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Charging Around Ken the courier charges customers by the shape of their packages. He charges $10 per square box, $12 for cylinders and $4 for rectangles. On Monday he charged $100 for delivering 13 packages. What did Ken deliver?

Answer: squares_____________cylinders_____________ rectangles_____________

Problem 63

Cost Conversion Mr and Mrs Dunnit and their three children bought tickets to ‘Circus Complete’. If they spent $52.50 in total, and adult's tickets were twice as much as children’s tickets, how much did each ticket cost?

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© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• Answer: adult's_________________ children's_________________

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Problem 64

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Icing Happy Birthday Johnny took his mum and fifteen friends ice skating for his tenth birthday party. An adult’s ticket cost $8, and children’s tickets were half the price of an adult ticket. In groups of over ten people, every fourth child could get in for free.

How much did it cost for the group to go ice skating?

Answer:_________________

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Problem 65 Bank Balance Kate Accumulator can never remember how much money she has in the bank. She can, however, remember this formula - the number of dollars is twice as much as a quarter of 1000 and the number of cents is twice as much as a fifth of 100.

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

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How much money has Kate got in the bank?

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Answer:_________________

Problem 66

Breaking Down If 1/5 of 1/3 of 1/6 of 1/2 = 2, what was the starting number?

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Problem 67

Answer:_________________

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Tuned Up The school orchestra has 90 students. 30% wanted to play the drums. Of the rest, 1/3 wanted to play the flute. The remaining students were divided equally among the tuba, the trumpet, and the cello.

List how many students wanted to play each instrument.

Drums _________________

Flute

_________________

Tuba

Cello

_________________

_________________

Trumpet _________________

Page 50

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Problem 68 Removal Plans

3

3

3

4

2

6

3

2

3

1

1

3

5

6

2

2

3

6

1

7

7

3

1

1

7

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

Problem 69 All's Equal

Use the digits one to five once so that each line is equal.

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Teac he r

Remove one number from each row so that the remaining columns and rows all total 12.

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Problem 70

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Use the digits seven to eleven so that each line is equal.

o c . che e r o t r s super

Always Equal Arrange these numbers so that each line is equal: 700, 800, 900, 1000, 1100.

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Problem 71 Sunday Swimmers At Wet and Wild Waterworld, 190 people attended the Sunday session. 30 people went on the slides and 50 used the high board. Of the rest, four times as many stayed in the wading pool as went into the swimming lanes. How many people used each facility?

High board

_________________

_________________

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

Swimming lanes _________________

Wading pool _________________

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Slides

Problem 72

Age Old Problem A family of four total 100 years in age. The parent’s ages combine to be 80% of this total. The wife is only 3/5 of the husband's age. The children are 20% of the total with the oldest child being 70% of this. How old are the four people in this family?

© _________________ ReadyEdPubl i cat i ons Wife _________________ •f or r evi ew pur posesonl y• Oldest child _________________ Husband

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Problem 73

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Youngest child _________________

o c . che e r o t r s super

Customer Count Tina Tycoon’s business was going well. Each week there were more customers than the week before. In the first week Tina had 24 customers. In the second week 50% more customers came in, the third week’s total was 25% higher than the second week, and in the last week of the month Tina counted 20% more customers than she had in week three. How many customers did Tina have in total over the four weeks? ................ +

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

+ ................

+ ................ =

Answer:_________________customers

Page 52

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Problem 74

Teac he r

Ear It Is Lisa Lobe’s earring collection totals 112 pairs. If she has 16 more pairs of pierced earrings than clip-ons, how many of each sort are there?

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Answer: pierced _______________ clip-ons________________

Problem 75

Lost His Marbles Luke Floorwalker loved his prized marble collection. He had 353 marbles in total. If he had 20 more tombolas than catseyes, but 18 more catseyes than rainbows, how many did he have of each?

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

Answer: tombolas___________ catseyes_____________ rainbows___________

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Problem 76

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Card Collection Janine and Sharon collected Netball cards. Together they had 135 cards. If Sharon had twice as many as Janine, how many more cards did she have?

Answer:_________________

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Problem 77 Be A Sport Bea Fitt, the sports teacher, is organising her 81 students for the upcoming season.

The students have a choice of basketball, football and hockey. Twice as many students want to play football than hockey, and three times as many want to play basketball than football. How many students will be allocated for each sport?

Answer: basketball_____________

football_____________ hockey___________

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Problem 78

Fat Frogs At the National Bullfrog jumping championships, only three big, lazy frogs were entered - Blob, Fatso and Croaked. Blob only managed to jump eight centimetres further than Fatso, while Croaked managed six centimetres further than Blob. If the total distance of all three frogs’ leaps was 52 centimetres, how far did each frog jump?

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Answer: Fatso_____________Blob_____________

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Problem 79

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

o c . che e r o t r s super

Favourite Flavour The canteen staff always have trouble working out what drinks to order. Last Friday, 126 students bought drinks. Twice as many wanted milk as Pepsi, and half as many wanted milk than Coke. If the canteen had ordered 42 of each drink, how far off were they?

Ordered

Coke

Milk

Pepsi

42

42

42

Needed Off

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Problem 80 Chart Topper If you’re smart, find the rule and complete this chart.

19

r o e t s Bo r e p ok u S 26

18

41

37

44

49

Problem 81 Complete These Patterns

1, 3, 6, 10, _____, _____, _____.

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Teac he r

33

© ReadyEdPubl i cat i ons 3, 5, 4, 6, _____, _____, _____.

•f orr evi ew pur posesonl y• 1, 1, 2, 3, 5, 8, _____, _____, _____.

16, 10, 5, 1, _____, _____, _____.

J, F, M, A, _____, _____, _____.

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Problem 82

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C, F, I, _____, _____, _____.

o c . che e r o t r s super

Getting Fit Grant Grunting does his daily exercise by running around his kitchen table.

He records his results on a fitness chart: Day 1 Day 2 Day 3 Day 4 Day 5

25 circuits 21 circuits 27 circuits 23 circuits 29 circuits

Complete the pattern and work out how many circuits Grant did on Day 12_________________ How many circuits had he completed in total by day 12?_________________

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Problem 83 Nik-Naks At a White Elephant stall in the markets, Colin Collectable bought a large quantity of old books. Colin kept a quarter and gave a third of the remaining books to his wife Nona. The rest were divided between their three children, each receiving ten books.

How many books did Colin keep for himself?

_________________

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

If the books cost 20c each, what did he pay in total?

_________________

Problem 84

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Teac he r

How much change did he receive from $100? _________________

Stamp About?

Ross Strum and Luke Warm are continually comparing the size of their stamp collections. Together they have 3600 stamps. However, Luke has 1/4 more stamps than Ross. How many stamps do Luke and Ross have each?

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Answer: Luke_________________stamps Ross_________________stamps

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Problem 85

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Ballet On his twelfth birthday, T. P. Toes took his parents and two kid brothers to see the ballet.

The tickets cost $63 in total, and children were 1/2 the price of adults. How much did each ticket cost?

Answers: children_________________ adult_________________

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Problem 86

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

Answer: large_________________

small_________________

Problem 87

Sweet Teeth Luke loves sweets.

His two favourites are liquorice and lollipops. If liquorice is 15c a stick and lollipops are 20c each, what is the greatest number of sweets Luke could purchase for exactly $2.50?

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Teac he r

Bag ‘Em All Bob’s Beanbag Factory was paid $25 for small beanbags and $55 for large beanbags. At the end of the week Bob had $460 from sales. How many large and small beanbags had he sold?

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Answer:_________________ lollies

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Problem 88

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Out Of Pocket John works in a bakery putting sesame seeds on buns. He has saved $36.00 from his pay. Unfortunately, John can't resist the bakery's cakes.

If cream sponges are $8.00, large rock cakes are $5.00, and giant muffins are $1.00, how many different combinations of cakes could John buy with his $36.00?

Answer:_________________combinations Ready-Ed Publications

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Problem 89 + + + +

Guess It Again The number has three digits.

The sum of the three digits is eight.

It is a palindromic number (it reads the same backwards and forwards).

Two of the digits added together equal the third.

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

Problem 90 Started As

If 2/5 of 1/4 = 1, what would the starting number be?

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Teac he r

What is the number?_________________

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Problem 91

+ + + +

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Guess The Number I'm a four digit numeral.

My digits are in ascending order of value.

The sum of the first two digits is half the sum of the last two.

I don’t contain the digits one or eight.

+ I am_________________

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Problem 92 Non - Plussed Find the rule and complete this confusing chart.

23 11

13

r o e t s Bo r e p ok u S 6

17

19

21

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30

Problem 93

Trolling Along Trevor Troll’s favourite toys are trolls because he likes their name. Yesterday he spent exactly $31 at Troll City. If Green Dwarf Trolls are $4, and Orange Giant Trolls are $5, how many trolls did he add to his collection?

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Problem 94

Answer:_________________

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Quizzical Ima Guesser entered a local quiz show.

Ten points were awarded for correct answers, and fifteen points were taken off for incorrect answers. At the end of the show Ima’s score was 75. She had answered five questions wrong.

How many questions had she got correct?

Answer:_________________

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Problem 95

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Coloured Out Using five different colours (e.g. red, green, blue, black, yellow), colour every square so that each column and row only contains each colour once.

Problem 96

Pizza Pieces At Pete’s Pizza Palace they are always trying new ways of cutting their pizzas.

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• Using four straight cuts, divide this pizza into nine pieces, keeping an olive on each piece.

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Problem 97

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Using three straight cuts, divide this into seven pieces so that each piece contains an olive.

o c . che e r o t r s super

Short Cuts Michael's pet mouse has been trained to find cheese. Using the direction arrows, describe the shortest path to the cheese. How many squares would this take?

Shortest Route: __________

Cheese

_______________________ _______________________ _______________________ _______________________ _______________________

Start

No. of Squares: __________

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Problem 98 Starring Now try again so that the vertical and horizontal diagonals are the same colour, but no colour is repeated in a row.

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Using four colours, create a pattern so that every line of four only contains each colour once.

Problem 99

Sharing Out Here is a plot of land that has been divided into 16 squares. The land is to be equally divided between four children so that each child gets a piece of land the same shape and size. Can you find and colour five different ways to equally divide the land?

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Problem 100

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Dividing Equally A farmer has decided to share out his farm equally between his four children so that each gets a house (H) and a water dam (D). The land divisions must be the same shape and size to avoid jealousy. Using colours, show how the farm should be divided.

H

D

D

H

H

D H

D

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Page 61

Answers Strategy 1 Practice Problems 1. 9 hours (in 9th hour) 2. 17 metres 3. 10 paths 4. 5.5km Strategy 2 (2 Variables) Practice Problems 1. 27 sheep, 35 cows.

2. 15 doughtnuts, 21 tarts.

3. 75 English, 137 French.

4. 297 girls, 271 boys.

100 Problems To Solve

2.

3. Fill 8 litres, pour into 6 leaving 2 litres, do this twice leaving 4 litres.

eg. 1, 2, 3, 7.

1, 2, 4, 7.

1, 2, 3, 4, 7.

1, 5, 6, 7.

1, 5, 4, 2, 3, 7...

r o e t s Bo r e p ok u S 4.

Strategy 2 (3 Variables) Practice Problems 1. 19 football, 13 netball, 10 volleyball.

2. 41 sheep, 35 cows, 30 pigs.

3. 29 chocolate, 23 mint, 15 liquorice.

4. Sharon $10, Kate $12, Clare $16.

Strategy 3 Practice Problems 1. 15 years old.

2. a) 7 lids, b) lids and flavour: 1 (No. 20), c) flavour and label: 1 (No. 30), d) label, flavour and lids: 0.

3. Page 136.

4. None.

2.

Jenny Tim

3.

Janet banana, Joan milk, Jane bread, Jenny orange juice.

4.

12 combinations

5.

7 days

6.

11 days

7.

92 doughnuts

8.

Larry - Lost at Sea, Harry - Black Beauty, Karri - Goosebumps, Sharri - Around the World in 80 Days.

3. Bears - (from left to right) Dancer, Dasher, Rudolph, Slasher, Prancer.

9.

Jim Able - Rough, Joan Coombes - Fats, Jerry Broad - Tuff.

4.

10. 45 garments

Strategy 7 Practice Problems 1. James - basketball Karen - tennis Charles - football Terri - cricket Lee - softball

Bert Bev

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1.

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Strategy 4 Practice Problems 1. 8 combinations.

2. 24 combinations.

3. repeats a combination on the 9th day.

4. 15 games.

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Strategy 5 Practice Problems 1. 11, 13, 15.

I, K, M. 10, 14, 12.

16, 22, 29.

2. Dog, Elephant, Frog.

Martin, Neil, Oswald.

Germany, Hungary, Ireland.

Ian, Kate, Mark.

3. 8th day, 9 days.

4. 100, 70, 55. any 5, 6, 7 letter words.

6, 27, 6.

8, 6, 9.

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11. 36 Music

Butcher

Chemist

o c . che e r o t r s super

Strategy 6 Practice Problems 1.

2. Fred - ham Felicity - cheese Phil - anchovies Freda - prawns

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

Video

Grocer

Strategy 8 Practice Problems 1. 4 December birthdays 2. John 14, Jenny 9, Jerome 17, Joe 27, Joan 19, Joseph 25, 111 books were needed altogether.

3. 432 4. Chocolate 45, Caramel 15, Mint 30, Bubblegum 30, Butterscotch 15, Honey 45.

12. 36

13. 555

14. Anita, Sherri, John, Jenny, Tim, Terry.

15. Kate's, Cool's, Moles, Andy's, Central Office, Don's.

16. Butch, Fifo, Max, D'for, Tiny 17. 12 combinations.

18. 18 days

Ready-Ed Publications

Answers 19.

Anne

Dan

1 2

F/ball

3 3

6 3

34. Father 45, Mother 42, Mark 14, Twins 11.

1 2

36. Fran 146, Freda 73, less 73.

11 1

21. 15 days.

38. Possible solutions: 3+3-3+3-3=3 4x4÷4=4 6+6÷6+6-6=2 6 x 6 - 6 x 6 + 6 = 186

Lamb

Bentley

VW

Jaguar

Rolls Royce

Porsche

57. 2520

58. Remove:

6 6 7 2 5

3

39. 990

24. Jerome Joan John Jeremy Jan Jenny

3

8 14

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37. 7 cats, 4 birds.

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5 15

6 16 9

35. 10, 4 legs, 12, 2 legs.

20. 29th of November

23.

7 13 12 2 10 4

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1 2

22. 8 dozen

56.

40. 56

59. 2 poodles, 4 hounds

41. 16 presents

60. 77/8 pages

42. 9 with 3 legs, 8 with 4 legs = $840. Other solutions are available.

61.

1

/6

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

26.

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27. 360

6

43. 9 boats

2

44. 6 combinations.

45. Possible answers include: ls 4, tr 1, bg 3, wh 3; fb 1, ls 1, tr 1, sb 1, bg 2, wh 5; fb 4, ls 1, bg 1, wh 2; fb 1, tr 4, bg 1, wh 1; sb 2, bg 5, wh 2.

46. 25 47. 10 days

5

/6

4

/6

49. Terry 6 per day 279 total

51. Yes

29. 16

52. 8 games, 10 games.

30. 7 cans

53. $3.50

31. 15 games, 36 games

54. Perky 22, Pinky 34

32. Jenny 23, Janet 18.

55. Uptown 1944, Downtown 432, school 216.

63. $15 adults, $7.50 children 64. $56

65. $500.40 66. 360

67. Drums 27, Flute 21, Tuba 14, Cello 14, Trumpet 14.

68. Remove:

3 3 5 7

7

69.

1 2

33. 27 minutes Ready-Ed Publications

/6

62. 4 squares 3 cylinders 6 rectangles

50. 4 months, 6 months, 8 months.

28. Page 129.

3

/6

o c . che e r o t r s super 48. $1.27 yes

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25. Several possible solutions e.g. A1 - John B1 - Terry C1 - Carol D1 - Jenny E1 - Julie F1 - Andrew G1 - Dean

3 5

7 4

8

9

10

11

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Answers 70.

e.g.

85. $18 adult $9 children

700

1000

900

98.

86. Possible combination: 14 small, 2 large.

800

R Y G

B

G B

R Y

R B

G

1100

71. High board 50, Slides 30, Wading 88, Lanes 22.

Y

90. 10

74. 48 clip-ons, 64 pierced

92.

27 23 29

16 11 7 6 8

Y

21 17 23

100.

H

H

19 15 21

Y B

R G

B G

Y R

99. Numerous solutions

D

13

10 17 13 19

76. Sharon 90, Janine 45, therefore Sharon has 45 more cards.

B

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73. 159 customers

91. Number of possible answers: e.g. 2355

G R

R = Red B = Blue G = Green Y = Yellow

89. 242

75. 137 tombolas, 117 catseyes, 99 rainbows

G B

r o e t s Bo r e p ok u S 88. 24

-

Y

B G

R

87. 14 at 15c, 2 at 20c = 16.

72. Father 50, Mother 30, Oldest 14, Youngest 6.

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R

Y

D

D

H

H

D

93. 4 at $4, 3 at $5 = 7 in total.

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

78. Fatso 10cm, Blob 18cm, Croaked 24cm.

79.

Coke 72 Off: -30

Pepsi 18 +24

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80.

Milk 36 +6

+ 14 19 29

94. 15 correct

95. Green = G Yellow = Y Black = Bl Blue = B Red = R

B Y Bl R G

R B Y G Bl G Bl B Y R Bl R G B Y Y G R Bl B

96. eg.

12 26 31 41 18 32 37 47

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4 18 23 33

30 44 49 59

81. 1, 3, 6, 10, 15, 21, 28.

16, 10, 5, 1, -2, -4, -5.

3, 5, 4, 6, 5, 7, 6.

1, 1, 2, 3, 5, 8, 13, 21, 34.

C, F, I, L, O, R. J, F, M, A, M, J, J (months).

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77. Basketball 54; Football 18; Hockey 9.

o c . che e r o t r s super 97. 5E, 2N, 1W, 2N, 1E, 1N, 2E, 2N, 1E, 1N, OR: 1N, 4E, 1N, 1W, 2N, 1E, 1N, 2E, 2N, 1E, 1N, = 18 squares.

82. Day 12 - 31 Total 336 83. Kept 15, cost $12, change $88.

84. Luke 2000, Ross 1600

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