For Junior Primary
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Maths Problem Solving for Juniors © ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•
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Book 2
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Problem solving strategies . teand practice activitieso c for the classroom. c e.
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Written by Lynne 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 87526 825 1
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Problem Solving Maths Book 2
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Problem Solving Maths Book 2
Contents ACTIVITIES
Page
Developing Logical Thinking
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6 7 8 9 10 11 12 13
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Which Watch? Toys, Toys, Toys Feeding At The Zoo Artists At Work Packed Away Sporting Fun Playing Trio Family Conference
Using Number Concepts To Develop Logical Reasoning Reading Record Drive Time School Days Cash For Containers Canteen Orders Doing The Shopping
15 16 17 18 19 20
© ReadyEdPubl i cat i ons Treasure Island 22 Patterns To Copy 23 • •f o rr evi ew pur posesonl y Shapes To Copy 24
Developing Visual Imagery
Feeding The Pigs Models To Build
Numbers In Code Number Mazes - 1 Number Mazes - 2 Amazing Totals Turtle Shell Totals Magical Mushrooms Twelve Trains Bug Patterns
32 33 34 35 36 37 38 39
ANSWERS
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Patterns In Squares Patterns In Circles Shape Patterns Matchstick Patterns
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Pattern Perception
25 26
28 29 30 31
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Problem Solving Maths Book 2
Teachers’ Notes This Problem Solving Maths set represent a response to current syllabus trends and the considerable emphasis which is placed on the development of problem solving skills in primary school students. Essentially activities are designed to interest and stimulate children in the 6 - 9 years age range. They are presented as Blackline Masters which are able to be photocopied for use in the classroom.
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IMPLEMENTING PROBLEM SOLVING MATHS ACTIVITIES
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A variety of factors will be considered by a teacher when he/she plans a unit of work including, aspects such as the previous problem solving experience of the students, their level of ability and the resources available. The use of an appropriate teaching strategy is often an area of concern when undertaking to implement a different emphasis in a new syllabus. The following considerations may assist: * Time taken to solve the problem prior to presenting it to students often proves to be valuable. It provides the opportunity ...
- to be fully aware of the requirements of the problem; - to attempt the solution using a variety of problem solving strategies; - to reduce the problem into stages with appropriate hints for students to use, rather than giving an answer; - to identify possible alternative solutions and their acceptability; - to evaluate the suitability of the activity to the particular class, group or individual.
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* Initially problem solving activities could be worked through step-by-step with the students. This provides a framework which individuals can use when required to work more independently. Further, it may reduce the frustration at times encountered by mathematically able students or creative thinkers who experience difficulty with reading. During these initial sessions a teacher is able to demonstrate a plan or procedure such as the one below which students can follow, e.g. S Step 1 Read and discuss the problem. S Step 2 Think about it. S Step 3 Ask questions about it. S Step 4 Try and use a strategy. S Step 5 Check the answer. * Working through a number of problems in a structured, step-by-step approach allows students to become familiar with a range of problem solving strategies: - estimating - simplifying the problem - organising information into lists, table, etc. - drawing pictures or diagrams - identifying patterns - predicting - thinking logically - evaluating the strategy used, the solution - checking results - working backwards
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For convenience the booklet has been divided into five sections. In each section a particular aspect of problem solving is in focus, although it is recognised that a great deal of overlap exists. It is felt that more effective use would be gained by selection of activities from each section rather than working through the package in page number sequence. Generally, within each section a number of similar activities have been provided in order that skills can be introduced and consolidated. Sections are headed by a brief summary of the skill being covered in the pages to follow.
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Problem Solving Maths Book 2
Section 1 Developing Logical Thinking The activities in this section encourage the ability to think clearly and to reason logically. A number of activities require students to organise their information in table format, i.e. matrix logic problems.
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Procedure for Matrix Logic Problems
Names are listed down the left side of the table. Across the top of the grid the features are listed, e.g. position, colour, sport, etc. The information is tabled by ticking the appropriate box.
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1. 2. 3.
Once a box is ticked all other boxes across from, and above or below, the feature are crossed.
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e.g. A Running Race
Names Joe Anne Liz Max Patrick
Position 1st
2nd
3rd
4th
5th
© ReadyE dPu bl i c at i ons •f orr evi ew pur posesonl y• N.B. Further information is entered as clues are used to solve the problem.
Table Logic Problems
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In Table Logic Problems diagrams must show the correct seating around a table. Students need to know that: * Partners always sit opposite each other. * People sit facing the table. (This is important when working out who sits to the left or right of a person.) * A person’s first (given) name is written on the top line, surname on the bottom. * Answers may appear to be different from your solution. Make sure that each person has the correct partner and the correct person to the left and right. e.g. Ross Miles
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John Smith
David Long
Pam Thomas CAN BE SHOWN AS---
David Long Ross Miles
Pam Thomas John Smith
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Name: ................................
Understanding and applying mathematics.
Which W atch? Watch? R
Every day for a fortnight Mrs Smith’s class had swimming lessons. During the lessons the children wearing watches left them in a special box.
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© ReadyEdPubl i cat i ons Use the clues to help you work out: •f orr evi ew pur posesonl y•
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Which watch belongs to Kate? * It has a round face. * It is not a digital type. * It has a plain band.
Colour the band in bright colours.
Kate’s watch is numbered Write her name in the box near her watch.
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Which watch belongs to Sam? * It has a rectangular face. * It has a black band. * Sam chose it because he cannot read clock-face time.
Sam’s watch is numbered Write his name in the box near his watch. 3. Which watch do you like?
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Write two clues about it.
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Problem Solving Maths Book 2
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Toys, T oys, T oys Toys, Toys
Understanding and applying mathematics.
John, Tim, Anna and Erin have been playing together. Use the clues to match the children to their own toys.
John’s toyhas buttons to push. does not have lots of pieces. gives you a score when the game is over.
Erin
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2. Erin’s toy* can be played by one person * has pieces which fit together * makes a picture.
. 3. Anna’st toye * * *
Anna
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Tim
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John
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4. Make up some clues about Tim’s toy. * * * Ready-Ed Publications
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Name: ................................
Understanding and applying mathematics.
Feeding A t The Zoo At
bear
snake
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penguin
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crocodile
Your family has just arrived at the zoo. It is feeding time for some of the animals. You would really like to go straight to the animals which are due to be fed.
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Use these clues to help you find out where to go first.
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The ......................... has just been fed.
2. Which animal isn’t fed today? * It lives on land. * It lays eggs.
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© ReadyEdPubl i cat i ons Draw a plan of the zoo here mark your route on it. •f orr evi ew pur poand se sonl y•
1. Which animal has just been fed? * It has four legs. * It lives only on land. * It is covered in fur.
o c . * It has no legs.c e her r o t s su The ............................ is not fed today. per 3. Which animal is next to be fed? * It lives on land. * It has two legs. * It lays eggs.
The .................................’s cage is where you will go first. Page 8
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Understanding and applying mathematics.
Artists A t W ork At Work
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Alex, Kate, Emily and Brett enjoy art lessons at school. One child uses only markers. One child uses only black pencils. One child uses only paint and one child uses only crayons.
© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• Enter the information on the grid below. Find out what each child uses.
1.
Emily loves to use bright colours but does not like to use markers. Brett and Emily never have paint on their hands. Alex takes care of his brushes. Kate thinks black pencils are boring.
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Markers
2. 3. 4.
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Pack ed A way acked Away
Problem Solving Maths Book 2 Understanding and applying mathematics.
Jill, David, Tom and Chris all go for a swim in the pool.
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When they come back to their gear after a swim, they discover that David’s little brother has put their towels into his toy box.
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Use these clues to find out which towel belongs to each child. Enter the information on the table below.
1.
Tom’s towel is his favourite colour. He does not like brown.
© ReadyEdPubl i cat i ons David’s towel matches his blue bathing costume. •f orr evi ew pur posesonl y•
2.
4.
Chris’s towel is not the colour that Tom doesn’t like.
5. 6.
Tom’s favourite colour is that of Jill’s beach bag.
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Jill’s mother bought her a new bright red beach bag.
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Jill David Tom Chris Try to make up a problem like this for others to try. Page 10
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Name: ................................
Understanding and applying mathematics.
Sporting F un Fun
Match the children to the sport. Each plays a different sport. Use the clues below.
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Soccer
1.
2.
Karen
Chris
Tracey
Tennis
Cricket
Basketball
Which sport does Chris play? He has been a member of his team for two years. He likes to practise throwing. He often scores goals in a game. Kicking the ball is not allowed in this sport.
Lisa
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Softball
Chris plays
© ReadyEdPubl i cat i ons Which sport does Lisa play? • f orr evi ew pur posesonl y • Lisa plays She spends a lot of time practising
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throwing. She needs to be able to run quickly. Lisa took a great catch last week. She sometimes doesn’t get to bat in a game.
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Which sport does John play? John doesn’t like ball kicking sports. He likes to play on grass. He practises his shots each day. Catching is not a part of this game.
4.
Which sport does Karen play? Goal scoring is not part of this sport. Karen is a good fielder. She always uses her own glove. She sometimes gets out more than once in a game.
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John plays
Karen plays
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Problem Solving Maths Book 2
Playing T rio Trio
Understanding and applying mathematics.
Trio is a great card game. The children in this class play it often.
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name)
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(First
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Show where each child sits for this game. You will need to follow the rules for ‘table logic’ activities. Your teacher will remind you about them.
(Surname)
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1. Shane Monks is Amy’s partner. 2. Val is Amy’s brother's partner. 3. David Bird is Amy’s brother. 4. Val’s last name is Rule. 5. Val is sitting to the left of Shane. Page 12
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Family Confer ence Conference R
1.
Understanding and applying mathematics.
This is an important family meeting. It is not a game. The members of the family have things to discuss. Where does each person sit?
Amy into Blake.
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3.
Kevin Kennedy is happy with the results of the discussion.
4.
Phillip is Blake’s father.
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Problem Solving Maths Book 2
Section 2 Using Number Concepts T o To Develop Logical R easoning Reasoning
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This section places emphasis on the application of number and measurement skills in logical thinking activities. Appearing on the top right corner of many of the worksheets is a calculator symbol indicating that a calculator is a useful aid in completing the activities on the page. Students require a basic level of understanding and skills in the use of a calculator.
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Problem Solving Maths Book 2
Name: ................................
Reading R ecor d Recor ecord
Making sense of no. problems: problems involving 'real life'.
Last week Ross enjoyed reading an exciting book. It took him 2 hours to finish. He read each day except for Saturday.
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On Friday and Sunday he read for the same length of time.
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On Monday he read for 10 minutes.
4. 5.
On Tuesday he read for 3 times as long as he did on Monday.
© ReadyEdPubl i cat i ons Onf Thursday hei read forp 5u minutes longer than he did on • o r r e v e w r p o s e s o n l y • Tuesday.
On Wednesday he read for half the time he did on Monday.
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Reading Time Monday
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○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○
○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○
○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○
Friday Saturday Sunday
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○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○
○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○
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Drive Time
Making sense of no. problems: problems involving 'real life'.
Each weekday Mr Johnson drives to work and back home. On his way to work he collects a workmate, Joe, who lives 4km east from Mr Johnson’s house. He then travels north for 3km to arrive at work. From work he travels straight home which is a distance of 5km.
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Joe 's H ou se
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© ReadyEdPubl i cat i ons This• plan shows of the given above. f o rr esome vi ew pinformation ur pose so nl y• Johnson H ouse
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How far does Mr Johnson travel ...
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Add any other details or lines which will help you to answer the questions below.
How many kilometres extra does Mr Johnson drive to collect Joe ... - each day? - in one week?
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School Days
Problem Solving Maths Book 2 Making sense of no. problems: problems involving 'real life'.
Cassie lives near her school. On Monday, Tuesday and Wednesday Cassie walked directly to school, which is a distance of 100 metres. On Thursday and Friday she called in and collected her friend, Jane on the way to school and dropped her off on the way home. Jane lives the same distance from school as Cassie. This doubled the distance Cassie walked to school.
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100 m e tr e s
C a s s i e 's H ouse
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S chool
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1. Complete this plan showing the information given. Show the distance to Jane’s house. Show the distance from Jane’s house to school.
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2. On Monday? 3. On Monday and Tuesday?
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How far did Cassie walk to school and back —
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How far would Cassie travel if she rode — 7. To Jane’s house? 8. Straight to school and back each day for a week? Ready-Ed Publications
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Name: ................................
Problem Solving Maths Book 2 Making sense of no. problems: problems involving 'real life'.
Cash F or Containers For
© ReadyEdPubl i cat i ons The• firstf time Simon took his containers he 33c. or r ev i ew pur p oreceived seso nl y•
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He took five items. Make a list of what Simon took to the depot.
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Simon’s teacher has talked a lot about conservation. Like his friends, Simon is doing something to save the environment. Every week, Simon takes the containers he has collected to the recycling depot near his house. He saves the money he is given for the containers.
The next time Simon went to the depot he collected 18c. He took nine items. Make a list of the containers Simon took.
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He took containers of one type only. Write the three possible collections he took.
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Canteen Or ders Orders R
Problem Solving Maths Book 2 Making sense of no. problems: problems involving 'real life'.
At Seaview School the parents run a canteen. The children are able to buy their lunch or a healthy Apples snack. 20 cents
r o e t s Bo r e p ok u S Sandwiches made to order. $1.00
Bananas 19 cents
Oranges 10 cents
Carrot sticks 5 cents a bundle
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Drew bought a sandwich and some fruit. His lunch cost $1.30. What fruit did he buy?
2.
Apricots 20 cents
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Grapes 30 cents
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She ordered a sandwich like Drew, but her lunch was $1.20. What did Kelly have for lunch?
Think of something different Kelly could have ordered with her sandwich and still paid $1.20.
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4.
If you had $1.30 to spend on your lunch at this canteen, what would you buy?
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Problem Solving Maths Book 2 Making sense of no. problems: problems involving 'real life'.
Doing The Shopping Mum spent $3.59 at the shops. Colour the items that she bought.
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Section 3 Developing Visual Imagery
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This section of the book includes activities which develop skills of copying, distinguishing features, and changing or reconstructing diagrams or models. Concrete activity is a vital component and should be encouraged as part of the problem solving process.
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Treasur e Island easure R
Shape and space: position and direction.
See if you can get to the buried treasure by following six different paths from the ship to the chest.
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Start here
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B. Go down
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C. Go down
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D. Go down
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E. Go down
then
F. Go down
then
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Patterns T o Copy To
Problem Solving Maths Book 2 Shape and space: position, direction, movement and angle.
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R Look carefully at the pattern in the box. Copy the pattern into the box on the right. Remember to use a sharp pencil and a ruler.
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These are known as spiral patterns. You may like to try to design one of your own.
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Shapes T o Copy To R
Problem Solving Maths Book 2 Shape and space: position, direction, movement and angle.
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Copy the three shapes in the box below.
© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•
These shapes are known as networks. The shape can be drawn without taking your pencil off the paper and without going over any line twice.
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Use this method to copy the networks in the space below.
o c . che e r o t r s uinp er Find some other networks tos put these grids.
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Name: ................................
Feeding The Pigs
Shape and space: position, direction, movement and angle.
The farmer has to feed his pigs. The pigs are kept in pens and he visits each pen only once. He starts at the shaded pen and walks through gates on the sides of each pen. (No diagonal moves.) R
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There are three possible routes he can take. Show them on the grid. The first has been started.
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The piglets are kept in smaller pens. The farmer feeds them in the same way. Show six different routes the farmer can take on the grids below.
2.
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Models T o Build To R
Problem Solving Maths Book 2 Shape and space: properties of 3D shapes.
Use cubes to copy the shapes below. Record the number of cubes used. Build another model the same shape but twice the size, i.e. twice the length, twice the height and twice the width.
1.
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Model
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Make up a model of your own.
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Section 4 Pattern P er ception Per erception (Pages 27 - 30) This section acknowledges that recognising patterns is a significant strategy in problem solving. More complex problems require that a patterning strategy is used, e.g. Pet Parade - 3 cats are indicated as A B C.
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The use of a definite patterning strategy helps to organise different arrangements -
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1. ABC 2. ACB 3. BAC 4. BCA 5. CAB 6. CBA
Section 5 Pattern P ception Using Number Per erception er
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(Pages 32 - 38) The final section concentrates on identifying patterns in numbers. Some understanding of number concepts and recall of basic facts of addtion, subtraction, and multiplication is desirable but not essential. In fact, activities such as these may well encourage the establishment of the recall of facts.
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The calculator symbol appears on the activities appropriate to calculator use.
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Patterns in Squar es Squares R
Shape and space: properties of 2D shapes.
Look carefully at the pattern in each box. Copy the pattern into the box below. Colour the pattern.
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Try these and some of your own patterns on a sheet of grid paper. Page 28
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Shape and space: properties of 2D shapes.
Patterns in Cir cles Circles R
Look carefully at the pattern in each circle. Copy the pattern into the circle below. Colour the pattern.
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Try to make some of your own patterns. Use coins and a ruler.
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Shape P atterns Patterns
Problem Solving Maths Book 2 Shape and space: properties of 2D shapes.
Look at the shape patterns below. Make each using cubes. Continue the pattern with the cubes. Draw the next shape pattern on the grid.
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Matchstick P atterns Patterns
For these activities you will need some matchsticks. Look carefully at each pattern. Arrange your matchsticks to make each pattern.
© ReadyEdPubl i cat i ons Now you design matchstick patterns. • f o rcan r ev i ewsome pu r pose sonl y• Take the number of matchsticks shown. Use them to make a different pattern. Draw your pattern in the box.
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Shape and space: properties of 2D shapes.
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Ask a partner to make your patterns. Ready-Ed Publications
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Numbers In Code R
Problem Solving Maths Book 2 Making sense of no. problems: solving no. puzzles.
Codes are fun. This code uses the lines of a star to show numbers. For example 28 is written . Use this code to write the following numbers:
* your favourite number * your house number
* the number of children * your phone number in your class
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© ReadyEdPubl i cat i ons Unlock these padlocks. •f orr evi ew pur posesonl y•
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Number Mazes 1 R
Making sense of no. problems: solving no. puzzles.
Find a path from each animal to its lunch. The path must total 8.
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Number Mazes 2
Problem Solving Maths Book 2 Making sense of no. problems: solving no. puzzles.
The bee wants to reach the eggs. 1. Find a path that totals 10. 2. Find a path that totals 12. Use a different coloured pencil to show each path.
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Amazing T otals Totals
Making sense of no. problems: solving no. puzzles.
Begin at any number outside the maze. Find the paths to the centre adding the number as you pass through the spaces. Use a different coloured pencil to show each path.
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3.
Show a path which adds up to 19.
4.
Show a path which adds up to 25.
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Problem Solving Maths Book 2
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Turtle Shell T otals Totals R
Making sense of no. problems: solving no. puzzles.
For these puzzles the three numbers in a line should add up to the same total.
This is Sam Turtle. His shell shows you how to do the puzzle.
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This is Henry Turtle. Use the first five even numbers to solve the puzzle.
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Magical Mushr ooms Mushrooms R
Problem Solving Maths Book 2 Making sense of no. problems: solving no. puzzles.
In these puzzles the three numbers in a line should add up to the same total. Use the
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numbers 1 to 7.
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Problem Solving Maths Book 2
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Twelve T rains Trains R
Making sense of no. problems: solving no. puzzles.
Below is a Twelve Train. On its carriages is a group of numbers which add to 12.
2 3 r o e t s Bo r e p ok u There are manyS other groups of numbers which total 12. Write a different group of numbers that total 12 in each train. 1.
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How many different combinations can you find which total twelve? Page 38
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Bug P atterns Patterns
Making sense of no. problems: solving no. puzzles.
The numbers on the bugs in each line follow a pattern. Work out the pattern from the two examples given, then complete the line by printing the missing numbers.
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Problem Solving Maths Book 2
ANSWERS P 6 P 7
6. Kate 1. Sam 1. Handheld computer toy - 2nd picture. 2. Jigsaw - 1st picture. 3. Lego - 4th picture 1. bear 2. snake 3. emu Alex - paints Kate - markers Emily - crayons Brett - black pencils Jill - brown towel David - blue towel Tom - red towel Chris - green towel John - tennis Chris - basketball Karen - softball Tracey - soccer Lisa - cricket
P 8 P 9 P 10 P 11
P 12
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Shane Monks David Bird
Val Rule
Amy Bird P 13
Total 10 2 4 1 1 2 or 2 3 1 1 3 Total 12 2 4 0 3 3 or 2 3 1 3 3 or 2 3 2 0 3 2
P 35
1. Highest total 32 4 + 7 + 6 + 10 + 5 = 32 2. Lowest total 9 2+3+0+2+2=9 3. Path adding to 19 4+7+4+2+2 or 2+7+1+4+5 or 3+3+6+2+5 or 2+7+4+4+2 4. Path adding to 25 4 + 3 + 6 + 10 + 2 or 2 + 5 + 6 + 10 + 2 or 3 + 3 + 4 + 10 + 5 or 2 + 7 + 1 + 10 + 5
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Phillip Kennedy
Amy Kennedy
Blake Kennedy
P 15
Mon - 10 mins Tues - 30 mins Wed - 5 mins Thurs - 35 mins Fri - 20 mins Sat - zero Sun - 20 mins 1. 12km, 60km 2. 2km, 10km 1. 100m, 100m 2. 200m 3. 400m 4. 400m 5. 600m 6. 1 400m or 1 km 400m 7. 100m 8. 1000m or 1km 1. 2 glass bottles, 3 cans 2. 3 plastic bottles, 6 cans 3. 60 cans or 4 glass bottles or 15 plastic bottles 1. orange, apricot or grapes 2. sandwich, apple or grapes, or 2 oranges 3. grapes and carrot sticks Colour all except sausages. A. Go down 1 then 3 B. Go down 2 then 3 C. Go down 1 then 4 D. Go down 2 then 4 E. Go down 1 then 5 F. Go down 2 then 5
P 16 P 17
P 18
P 36 Sam's shell total is 9 Tom's shell total is 15
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Henry's shell total is 18
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Total - 12
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572 matches 3542 matches 906 matches
Total - 15
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*answers not to scale
P 32
4
6.
1. 8, double the model 64 2. 10, double the model (64 + 16) 80 3. 6, double the model (32 + 16) 48 4. 10, double the model 80 5. 5, double the model 40
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P 23
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P 38 Other groups of numbers 12 0 0 10 1 1 9 1 2 8 4 0 8 2 1 7 5 0 7 3 2 7 2 2 6 5 1 6 4 1 6 3 2 5 7 0 5 5 2 5 4 3 5 3 3 4 7 1 4 6 1 4 4 4 4 4 2
adding to 12 are: 0 11 0 9 0 9 0 8 1 8 0 7 0 7 1 6 0 6 1 6 1 6 0 5 0 5 0 5 1 4 0 4 1 4 0 4 2
1 1 3 3 2 4 3 6 4 3 2 6 5 4 8 6 5 4
0 1 0 1 2 1 1 0 2 3 2 1 1 2 0 2 3 3
0 1 0 0 0 0 1 0 0 0 2 0 1 1 0 0 0 1
P 39 1. 2. 3.
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8, 7, 3, 3 11, 9, 5, 6 10, 6, 2, 6
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