the activities in book2 maths on the go book...

Mental

Computation

Measurement

Chance and Data

Space

Number

Book

2

Rob V i ngerhoets

Book

2

Book

2

Also availableMaths on the Go

Mat

hs on t

he Go

Rob Vingerhoets

More ideas from the best-selling author of Maths on the Go Book 1 !

Book

1ISBN 0732978807 ISBN 9780732978808

About the authorRob Vingerhoets is an experienced primary maths teacher and author. He is a popular and engaging presenter of maths ideas for busy teachers and understands the demands of a full curriculum. His books show how it is possible to squeeze more maths into less time!

The activities in Maths on the Go Book 2:• are easy to organise and implement• require minimum equipment, or none

at all• are easily adapted across a range of

year and age levels • cover the important content strands:

Mental Computation, Measurement, Space, Number, and Chance and Data

• can be readily used as part of any maths program

• are perfect for revision or extension.

Make the most of every minute of the teaching day . . .• 30 minutes before morning recess• 20 minutes between library and lunch• 5 minutes at the end of the day.

5to

45 m

inute

mat

hs ac

tivitie

s for all primary levels

5

5 to 45 minute activities for all primary levels • Mental Computation • Measurement • Space • Number • Chance and Data

Rob Vingerhoets

Book 2

AcknowledgementsTo Mick Ymer, a great friend and a terrific maths person, for the ideas behind ‘Date Maths’ and ‘Closest to 1’ and other ideas he gladly allowed me to use for this book — something I really appreciate, as they were just too good not to appear in someone’s book somewhere.

To those teachers in Australia and New York who personally requested — or whose classes inspired — the sort of activities found here, and who were only too happy to allow me to trial new activities with their students; in particular, Adrian Dilger and Lorraine Kennedy for ‘We’re Going on a Shape Hunt’, ‘Thinking Linking 1’, ‘Thinking Linking 2’ and ‘Two Coins on a Ruler’.

As always, to my wonderful wife Marg, for her support and advice; and for not running away and leaving me to write another book on my own!

First published in 2006 by

MACMILLAN EDUCATION AUSTRALIA PTY LTD627 Chapel Street, South Yarra 3141

Visit our website at www.macmillan.com.auAssociated companies and representatives throughout the world.

Copyright © Rob Vingerhoets/Macmillan Education Australia 2006

Maths on the Go Book 2

ISBN 978 1 4202 0931 0ISBN 1 4202 0931 0

Publisher: Sharon DalgleishCover design and illustration: Cliff WattText design: Domenic LauricellaEditors: Tricia Dearborn, Laura DaviesPrinted in Australia by Gillingham Printers Pty Ltd

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Mental Computation

1 Mind-Reader Number, space, logic, recall Years 0–6 10–15 ✗ 7

2 How Much in Number, logic Years 2–6 10–15 ✓ 9 My Pocket?

3 Heads or Hips? Number Years 2–6 5–15 ✗ 12

4 What is the Question? Number, working backwards Years 2–6 5–10 ✗ 14

5 The Number Formerly Number Years 2–6 all day ✗ 15 Known As . . .

6 Date Maths Number Years 3–6 10–15 ✗ 16

7 And the Biggest Number, operations Years 3–6 15–25 ✗ 18 Answer Is . . .

8 In the Ballpark Number, division and Years 4–6 20–30 ✓ 21 multiplication

Measurement

9 If You’ve Got It — Applying common units of Years 1–2 25–30 ✓ 24 Measure with it measurement

10 Steppin’ Out Applying standard units of Years 3–5 25–30 ✓ 26 measurement

11 Like a Pendulum Making predictions, identifying Years 0–6 10–20 ✓ 29 Swings trends and patterns, hypothesising

12 Triangular Areas Working out area of Years 5–6 25–35 ✓ 32 right-angled and equilateral triangles

Space

13 Tell Me Ten Things Shape recognition, attributes Years 0–6 10–15 ✗ 34

About . . . of shapes, spatial terminology

14 Left, Right, Spatial and directional Years 0–6 10–25 ✓ 36

Compass and Clock terminology

15 We’re Going on a Recognition of 2D and Years 0–6 20–25 ✗ 38

Shape Hunt 3D shapes

16 The Big Space Quiz 2D and 3D shapes, angles, Years 4–6 20–30 ✓ 42

directions, spacial terminology

Introduction 5

Contents

Activity Focus Level Duration Equipment/ Page (minutes) Worksheet

Activity Focus Level Duration Equipment/ Page (minutes) Worksheet

Number

17 The Human Place value, odd and even Years 0–6 10–20 ✓ 44

Number Line numbers, cardinal and

ordinal numbers

18 The Magic 8 Number Years 0–2 20–30 ✓ 47

19 Pattern Block Addition, shape, fractions, Years 1–2 20–35 ✓ 49

Numbers logic

20 Pick It Out Place value, operations Years 0–6 30–45 ✓ 52

21 Connect the Numbers Operations Years 2–6 25–35 ✗ 56

22 Grab a Zero Operations, properties of Years 3–6 20–30 ✓ 61

operations, multiplying by tens

23 Match-Up to 1 Adding and/or subtracting Years 4–6 30–45 ✗ 63

simple fractions, decimals

and percentages

24 Closest to 1 Comparing and ordering Years 5–6 20–30 ✓ 65

fractions

Chance and Data

25 Thinking Linking 1 Sorting/classifying people Years 0–2 10–15 ✗ 67

by attributes

26 Thinking Linking 2 Sorting/classifying people Years 0–2 10–15 ✗ 69

by attributes, collecting

simple data

27 The Human Sorting/classifying people by Years 0–3 10–15 ✓ 71

Birthday Graph attributes, collecting/

interpreting simple data

28 Four Corners Probability Years 1–6 10–15 ✓ 74

29 Two Coins on a Ruler Probability, data interpretation Years 2–6 15–20 ✓ 77

�

IntroductionMore than a little bit of water has passed under the mathematical

bridge since Maths on the Go was published back in 2001, and I have to

say that this time it’s been a bit of a dual continental effort. Good thing I’m

‘bi-hemispherical’, as Dame Edna would put it.

Maths On The Go Book 2 is based on the premise that students need

to be engaged in mathematics, and on my firm belief that maths doesn’t

have to hurt. The book was largely written in New York City, and is a

combination of ideas and activities I had used in Australia, those I had

seen first hand (Mick Ymer) and those I steadily put together over time

while working in various grades. The New York activities (from Brooklyn

and Queens elementary schools in the Big Apple) included ones I

thought of on the spot, others I worked on in my head over time, some I

witnessed or read about and then modified, and many others that arose in

response to teacher requests for me to model lessons on a range of topics.

What all the activities in this book have in common is that all of

them have been tried and tested on kids in living, breathing classrooms

— and, luckily, they all worked well! Some needed some improving, but

the audiences I tried them on were only too willing to give me (brutally)

honest critiques on my new activities.

None of the activities are hypothetical or theoretical — they are

all soundly practical and mathematical. What the activities also have

in common is a steadfast commitment to engage students in their

mathematics. It simply isn’t sufficient to merely teach your students

mathematics, working them through a set of rules to get through a mound

of content: students must be able to relate to maths. They must be able to

use it to make sense of their everyday lives.

We, as teachers, must set up the mathematical experiences so that

students discover the maths. The vast majority of primary school students

already believe that you, the teacher, are wise beyond belief: you don’t

need to convince them you’re clever. Let them do the discovering — don’t

take this opportunity away by providing them with the answers and/or the

methods.

Maths On The Go Book 2

As for the original Maths on the Go, this edition also contains activities

that cover the strands of Measurement, Space, Number, and Chance and

Data. There is also a specific section in this book on Mental Computation.

It has been placed before the other sections as I believe that having

well-honed mental computation skills (and this doesn’t just mean automatic

response) is absolutely critical to students’ success in their mathematical

and everyday lives. And the best part is that it can be plain, good fun.

�

The activities span the range of Years 0 to 6, and the duration of

the activities varies from 5 to 45 minutes. Most of the activities require a

minimum of equipment, if any. Some include a worksheet; others don’t.

The majority of the activities presented here can be used repeatedly

throughout the year. They are not one-offs, but activities students will

improve in and benefit from each time you use them — and the great

thing is that you will find students in your class asking for them regularly.

For each activity there is an indication of the main focus, the age

level/s the activity is suitable for, the duration of the activity, a list of

required equipment (if any), objectives, organisation (whole class, small

groups, etc.), guidelines for implementing the activity, and variations that

allow for enrichment or taking it back a step or two.

Hints on Using Maths on the Go Book 2

• For many of the activities, it is actually not essential that the whole class

participate or that every individual get a turn. In fact, to keep activities

dynamic and involved it is often better to have half the class or seven

or eight students participate in a given activity: ‘Steppin’ out’, ‘Closest to

1’ and ‘Thinking Linking’ are good examples in this book.

• It’s a good idea to have available many copies of the list of students in

your class. When only a limited number of students will be involved

in an activity, you can allay students’ fears of never getting a turn by

writing the name of the activity at the top of the class list and ticking

off the names of those involved today. Tell students you will choose

different people next time you do this activity.

• Many of the activities presented here can and should form part of your

regular classroom routines. For example, you could decide that every

Tuesday and Thursday morning during Term 2, between this time and

that time, the class is going to play ‘Mind-Reader’. This would be apart

from and in addition to the daily maths lesson.

• Other activities detailed in the book can be part of your day-to-

day lesson format. Many of them are ideal for warm-up activities,

particularly the mental computation activities. Simply rotate them on a

regular basis so that they remain fresh and challenging.

• Finally, many of the activities presented here represent ‘full-on’ maths

lessons and can be readily incorporated into your regular maths

program. Read through them, make a judgement about where they

might fit into your program/planning and then give them a go!

Objective

To develop in students the capacity to think logically and draw conclusions from information presented to them.

Organisation

This activity can be conducted in table groups; or do it with the whole class and have individuals raise their hand when they think they know the answer.

Introduction

When played in groups, ‘Mind-Reader’ is a good cooperative group activity as students must discuss probable or likely solutions and reach a consensus. It’s good for mixed ability groups, as students enjoy the fact that all groups receive similar type clues.

Procedure

Have students sit in their table groups, or in groups of four or five.

Tell students you are going to give each group some clues to a number. After all the clues have been provided, the group needs to discuss possible answers, or the definitive answer, and reach a consensus.

When 30 seconds to a minute have elapsed (depending on the age and abilities of the class/group), ask a group member to nominate the number they believe matches all the clues.

Go over the clues in terms of the number suggested by the group. If all clues match, the group receives the coveted ‘It’s a match!’ accolade. (Frequently more than one number can meet all the criteria/clues.)

This activity can also be done with the whole class, with individuals raising their hands when they think they have the right answer, or being selected to give a response. I prefer the group approach as it promotes sharing of knowledge, communication, listening skills, reaching consensus and teamwork.

‘Mind-Reader’ is a very good once- or twice-a-week activity. You need only give one ‘Mind-Reader’ per group — although students will ask for more. It can be used as a warm-up session, as an activity any time during the day, or as a regular routine.

�

FOCUS Number, space, logic, recall

AGE LEVEL Years 0 to 6 (5- to 12-year-olds)

DURATION 10 to 15 minutes

EQUIPMENT None

Maths on the Go Book 2 MENTAL COMPUTATION

Activity 1 Mind-Reader


�

A sample gameRemind students to wait until you have given them all five clues. Tell them if someone in the group is sure they know the answer, that person should share their answer with the rest of the group and say why they think it is right, so everyone in the group understands.

Give them their clues.Clue 1: It’s a two-digit number.Clue 2: It’s an odd number.Clue 3: The tens digit is even.Clue 4: One of the digits is twice as big

as the other one.Clue 5: The sum of the two digits is 9.

Repeat the clues. (With younger classes, it is likely you will need to repeat the clues at least three times.)

The group now gets into a huddle so as to reach consensus on the number. If students are not unanimous, it should be a case of majority rules.

Call for a spokesperson or choose one of the group randomly to say the number and justify the choice by juxtaposing the number with the clues. For example, a spokesperson for Group 1 suggests the number is 63. Is 63 a two-digit number? Yes. Is it an odd number? Three is an odd number, so yes. Is the numeral in the tens place even? Six is an even number, so yes. Is 6 twice as big as 3? Yes. When I add 6 and 3, do I get 9? Yes. It’s a match!

You will find that the remaining tables will all have attempted Group 1’s ‘Mind-Reader’. I encourage this by informing the whole class that if Group 1 misses, I’ll go to Group 2 for their answer, and so on.

Here are some ‘Mind-Readers’ to get you started. Make up your own, and have your students create some to give to other groups.

Clue 1: I am a number between 1 and 10.

Clue 2: I am supposed to be a lucky number.

Clue 3: I am bigger than 5 and less than 9

Clue 4: I only have straight lines in my number.

Clue 5: I am the number of days in one week.

Clue 1: I am a two-digit number.

Clue 2: I am an odd number.

Clue 3: Both my digits are odd.

Clue 4: One of my digits is 3 times the other one.

Clue 5: When you add my two digits together you get 12.

Clue 1: I am a three-digit number.

Clue 2: I am an odd number.

Clue 3: My first and last digits (ones and hundreds) are the same

Clue 4: The digit in the middle is even.

Clue 5: When you add my three digits it totals 8.

Clue 1: I am a three-digit number.

Clue 2: I am an even number.

Clue 3: All of my three digits are even.

Clue 4: The digit in the tens is twice the digit in the hundreds. The digit in the ones is twice the digit in the tens.

Clue 5: The sum of my digits is 14.

I don’t want to spoil ‘the fun’ and challenge of you yourself working these out but, if necessary, here are the answers—7, 39, 323, 248

Variations

‘Mind-Reader’ also works well for geometrical challenges. For example, for a trapezium:

Clue 1: I am a four-sided shape.Clue 2: My top and bottom sides are parallel.Clue 3: The top side is smaller in length than the bottom side. Clue 4: The left- and right-hand sides are the same length but smaller than the top and bottom

sides. They are not parallel.Clue 5: There are no right angles.

�


Objective

To develop in students the ability to think logically and apply these logical thoughts in formulating effective questions.

Organisation

A whole-class activity.

Introduction

‘How Much In My Pocket’ is one of those activities that you could and should play on a regular basis, for example once every two weeks. It also makes a very effective warm-up or mini-lesson.

Procedure

Take some notes and $1 and/or $2 coins, total them up, and place them in your pocket. I normally double-check the amount and commit it to memory before entering the classroom

The total doesn’t really matter, but it’s good to vary the amount so students don’t always know you have between $5 and $10, or $10 and $20.

Inform the class that they have eight questions they can ask to try to narrow down the range of amounts it could be. (Eight questions works well for Years 4 and 5; seven works well for Year 6; and nine or ten is good for Years 2 and 3.) After that, they must guess a specific amount.

Quickly put the following on the board:

HOW MUCH IN MY POCKET?

1. 2. 3. 4.

5. 6. 7. 8. I now put the challenge into context by stretching my arms out to the

sides and telling students that my outstretched arms represent the range of specific amounts I could have in my pocket. I could have $1 (wiggle left hand); I could be doing the grocery shopping after school and have $200 (wiggle right hand); or it could be any amount in between. Students have to find out, and they can only ask eight questions before they have to guess the amount.

FOCUS Number, logic



EQUIPMENT Dollars totalling various amounts (real notes/coins if you can)

Activity 2 How Much in My Pocket?


10

Particularly with Years 2 to 4 students, now is the time to emphasise that students should not be guessing specific amounts until all the questions have been used up. Make it clear that there is a big difference between asking a question that can give a lot of information and rule out many numbers in one go and making a guess that eliminates only one number. I illustrate this by using my outstretched arms and saying that questions that get rid of many numbers in one go — for example, ‘Is it less than $50?’ — have this effect (I bring in my arms about a quarter of the distance). Asking me a specific amount like ‘Is it $10?’, on the other hand, has this effect (I move my arms in a minuscule amount). If students ask me if it is a specific amount eight times, they have little chance of working out the amount and it will forever remain a mystery!

Each student gets one question only, and each question must be one that you can answer with ‘Yes’ or ‘No’. Students should turn around and discuss with someone else the question they are thinking of asking, to check that it is a good question that can get rid of many numbers in one go.

Tell students also that you will write up on the board next to the ‘Yes’ or ‘No’ response exactly what they find out from their question. This is very important, no matter what Year level, as students are likely to drop out of the activity if they cannot recall the clues or vital information they have gained from initial questions. It also enables you to review what they have learned after every three or four questions.

A sample gameQuestion 1: Do you have more than $50?

Answer 1. No. (It is < $50.) Illustrate how effective this question was by stretching your arms out and bringing them in by a substantial amount. Reassure students that getting a ‘No’ answer doesn’t mean it’s a bad question. Record what students have learned (It is < $50) on the board. Use the ‘less than’ (<) and ‘more than’ (>) symbols; simply put them into words as you use them.

Question 2: Do you have less than $20?

Answer 2. No. (It is > $20 but < $50.) Illustrate how effective this question was by again bringing in outstretched arms by a significant amount.

Question 3: Is it $25?

Answer 3. No. (It is not $25.) Prepare yourself for the single, specific amount question, even after all the initial explanation and illustration. Illustrate how ineffective this question was by bringing in your arms a centimetre or two. You won’t need to say much!

Question 4: Is it more than $30?

Answer 4. Yes. (It is > $30.) Bring in the arms even more. Now is a good time to go over what students have found out through their questions so far. We know it’s less than $50 and more than $30. There are actually two bits of information we don’t need that might distract us. Which bits of information are they? Students should tell you that we no longer need the information acquired from questions 2 and 3. Remove them from the board. Getting rid of superfluous information is a good word problem strategy, so encourage this.

11

Question 5: Is it an odd number?

Answer 5. Yes. (It is odd.) Bring in your arms by half the remaining distance as this is exactly what this question achieved — halving the possible numbers.

Question 6: Is it between $40 and $50?

Answer 6. No. (It is not between $40 and $50.) Show with arms, and review what is known: odd and less than $40 but greater than $30.

Question 7: Is it between $30 and $37?

Answer 7. Yes. (It is between $30 and $37.) Your hands should now be nearly touching. Ask students what numbers are left that it could be. Since it is between $30 and $37 and odd, it could be $31, $33 or $35.

Question 8. Is it $35?

Answer 8. No. (Not $35.) Move your hands even closer together. All allowed questions have now been asked, and students must now guess the amount. I sometimes choose a student here, or better still put it to a class vote.

As it turns out, 15 have voted for $33 and 11 think it is $31. So it’s $33 by a margin of 4 votes. I bring out the money and count it, out loud: ‘I have a $20 note and a $5 note (= $25: students will do the adding up) and another $5 note (= $30) and a $1 coin (= $31) and a $2 coin (= $33). Well done!’

Variations

Not all examples will play out as well as the one above. In the first one or two challenges you conduct with students (particularly in Years 4 and below), you will invariably get some inappropriate, illogical and one-number-specific questions. Persevere. Students will learn from the experiences, and they will get better. It generally only takes one child to ask one good question to generate good questions from others. Limiting it to one question per student stops students from leaving the questions up to just a few. If students are struggling after four or five questions, allow them to have a discussion with a person alongside them. As they get better, allow them one fewer question, or try some decimals, for example in Years 5/6 set the amount at $27.60.


12


Objective

To have students perform addition/subtraction/multiplication operations mentally in order to determine whether their answer is under or over a given number.

Organisation


Introduction

This simple, enjoyable activity makes a very good warm-up or regular mental computation activity (say, once every two to three weeks).

Procedure

Start with all students standing.

Think of a number. Twenty works well for Year 2 students, 50 is a good number for Years 3 and 4, and 100 for Years 5 and 6 students.

Tell students you are going to tell them a number. For this example, say you have Year 3 so the number is 50. You are then going to give them a problem that they need to work out in their heads. If they believe the answer to the problem is over 50, they should put their hands on their heads. If they believe the answer to the problem is under 50, they should put their hands on their hips.

Inform students that they will have a finite amount of time to complete the mental calculation. Vary the time according to the Year level. When about two-thirds of the class have that look in their eyes that says they know the answer, I start a 10, 9, 8, . . . countdown.

Tell students that they must not place their hands on head or hips until the countdown is complete. This helps stifle the temptation to simply copy another student. Once placed, the hands cannot then move from head to hips, or vice-versa.

Before giving students the first problem, explain that you know that many of them will simply watch someone they regard as being very good at maths or mental computation and copy what that person does. Explain that you will be asking a random number of students with their hands in the correct position to explain how they made their calculation. If a student can’t give an explanation, they will be asked to sit down. Students soon realise there is no way around this but to attempt the calculation.

Any student with hands on the incorrect area — obviously having made the mental computation incorrectly — is asked to sit down, but sincerely invited to continue to work out all following problems.

I usually set five or six problems, and the students who remain standing after those (usually two to three) are heartily congratulated for their effort.

FOCUS Number



EQUIPMENT None

Activity 3 Heads or Hips?

13

A sample game (for year 3)Inform students that 50 is the magic number today.

First problem: What is 3 x 17? Give students approximately 30 seconds, then commence the countdown. When you reach zero, they must place their hands on their hips or head. Those who have not completed the calculation should sit, as should those with their hands in the incorrect position (hips). Of those remaining, select two or three students and invite them to explain their strategy for completing the mental calculation. For example, Declan says that he multiplied 3 x 20 and got 60, then took off the 3 x 3, which is 9, and 60 – 9 = 51; or Cathie says 3 x 10 = 30 and 3 x 7 = 21 and 30 + 21 = 51.

Second problem: What is half of 96 plus 3? Same procedure. You will find that the seated students will all be doing the problem — congratulate and encourage this.

Continue on for one or two problems such as: What is 3 x 16 + 11? What is 5 x 14 – 22?

Final problem: What is 4 x 13 – 2? When playing this game for the first time, it can be fun to finish with a problem where the answer is exactly 50, or whatever magic number you’re using. Warn students before the final problem that there are to be no questions. Watching as students’ hands move up, then down, then up, then down, as they try to work out where their hands should be can be quite amusing. I have seen some very creative responses, and it’s a great way to eliminate a number of students if there are many remaining by the final problem.

Variations

Vary the magic number as students become more proficient at their mental computation.

Decrease the time allowed for making the calculation (being mindful of trying to give all students a feeling of success in the activity).

Give students who really struggle with mental computation a calculator. I have done this on a number of occasions, and the other students have been excellent in realising that this is a fair thing to do. I realise that this precludes the student or students from using mental computation, but for some students recalling the number and operations, pressing the correct buttons, understanding the answer and then determining where to place their hands is challenge enough.


14


Objectives

To have students work backwards, starting with the answer and attempting to develop (mentally) a problem/question that will produce that answer.

To have students appreciate that maths is not always about one correct answer.

Organisation


Introduction

This is a very effective and worthwhile activity to use with students on a regular basis (say, once every one or two weeks). Each time you play it, students will become more creative and think more broadly about numbers and their applications.

Procedure

Think of a number — any number! Let’s say you have some Year 4 students in front of you, and choose 24.

Tell students the answer is 24. What could the question be? Before seeking responses, I tell students it’s just like working backwards: you know the answer is 24 — now think of a question that will get you to the answer. Advise students that every single person in the class is going to be asked.

Be positive about all correct responses, but particularly encourage those questions that are creative and reflect broader thinking about number. Possible questions for the answer ‘24’ could be: ‘What is 6 x 4?’; ‘What is 3 x 10 – 6?’; ‘How many hours are there in a day?’; ‘What is half of 48?’. Specify that each response must begin with a question word.

Make sure you ask every student. If a student does not have a question ready to go, I make no fuss and simply tell them that I’ll come back again later— and I always do. This is one of those open tasks where every student can contribute at their own level: as long as a response is mathematically accurate it is accepted and noted.

Don’t worry if students are a little rusty the first time you conduct this activity. You will invariably get many a ‘What is 23 + 1?’ type question, but persevere, they do get better. By the third or fourth time — particularly if you have been very positive about those responses where students have thought broadly — you will get some very insightful and enlightening questions.

FOCUS Number, working backwards



EQUIPMENT None

Activity 4 What is the Question?

1�


Objective

To have students look for alternative ways of orally identifying a number.

Organisation

A whole-class activity. No special arrangements necessary.

Introduction

This activity came about one day when I was writing the numbers 1 to 9 on the board and inadvertently left out the number 6. When one of the students pointed this out, I replied that I had done it on purpose. I’d woken up this morning and decided that I really did not like the number 6 at all, and that the number formerly known as 6 was banned from being used in any classroom that I was in that day. A girl near the front of the room asked the obvious: ‘So we can’t say or write the number 6?’ I immediately covered my ears in mock horror (as did a number of students — which was really very funny), screeching ‘Don’t say that number — it’s banned!’

Procedure

Simply choose a number that you are intending to ban from all use for a day or a set portion of the day. You and your students must now avoid saying or writing that number (including in calculators) and must therefore find alternative ways of saying or writing it. For example, if the banned number is 9, alternatives could be: 10 – 1; 3 x 3; the number between 8 and 10; the last single digit number; 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1; how many lives a cat is supposed to have, and so on.

Each alternative can only be used in the classroom once, and cannot be repeated.

Variations

You may wish to cover all manifestations of the banned number with self-stick notes, tape or paper. I have seen a room in which the number 7 was covered up on the number line, on the 1 to 100 chart, on the clock, in the date — everywhere!

Postscript: The first day I banned a number — the number 6 — one of the girls from the classroom where the ban had occurred came running up behind me as I was leaving the school, tugged on my jacket and said, ‘Mr V. It’s after school, so — six, six, six, six, six’ and ran off laughing. I had a chuckle as well.

Activity 5 The Number Formerly Known As . . .

FOCUS Number


DURATION All day, or as long as you can handle the pressure of not being able to say the number formerly known as . . .

EQUIPMENT None

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Objective

To develop mental computation skills and the ability to think logically to solve problems.

Organisation

A whole-class activity that can be conducted in pairs, in small groups or individually. I discourage the use of pens and paper.

Introduction

This activity can effectively be used once a week or every two weeks. It can produce some frustration born of lack of success in solving some of the problems, but this frustration can be the source of some very inventive and creative solutions. Too much frustration is not a good thing, though, so be ready to give a clue or hint — but only when absolutely necessary.

Procedure

Place today’s date on the board, for example 10/04/07.

Set up the board so it looks like this:

10/04/07

1. = 1 11. = 11

2. = 2 12. = 12

3. = 3 13. = 13

4. = 4 14. = 14

5. = 5 15. = 15

6. = 6 16. = 16

7. = 7 17. = 17

8. = 8 18. = 18

9. = 9 19. = 19

10. = 10 20. = 20

Inform students that their challenge, collectively, is to make an equation for each of the numbers 1 to 20 — using only the numbers in today’s date.

Students can change the order of the numbers. For example, the 4 and the 0 can be 40. The 1 and 4 can be used as 14. The 1, 0 and 0 can be 100. However, if there is only one 4, say, you can only use 4 once in any single equation — 4 ÷ 4 x 1 = 1 would not be valid.

To encourage creativity, I tell Year 5/6 students that any three-number equation that is mathematically correct will earn them 1 point, but any five-number

FOCUS Number



EQUIPMENT Stopwatch (optional)

Activity 6 Date Maths

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equation will get them 3 points. If they use all six numbers in their equation, they earn themselves 5 points. When the class has completed all the equations, or as many as they can, total up the number of points. Keep a record of the score. Beating this score becomes the class’s challenge on the next occasion you do this activity.

Each student can provide only one equation. This prevents students from leaving the work for other students to do. It becomes more of a team-orientated activity, and spreads the responsibility and accountability around.

To enhance the team effort notion, I tell students that a useful strategy may be to look ahead and have an equation ready for number 7 or number 12 or number 20. In this way, when I call for an equation for that number someone in the room has it ready to go. This is particularly relevant if you choose to have students perform this activity against the clock (see ‘Variations’).

Invariably there will be one or two numbers in the teens that cause problems. When this occurs, I allow students to turn and talk with a partner to see if they can work something out together. What generally occurs (and I know it happens and students don’t think I know!) is that one of the students who has a possible solution but cannot be chosen as they have previously provided one now has the opportunity to pass their expertise onto someone else.

It’s especially likely the class (and you!) will get stuck when the date is from the first to the tenth of the month. After that (excluding the 20th and 30th), you have an extra digit to manipulate and it makes it significantly more possible to get an equation for every number from 1 to 20. For Years 3 and 4, I use dates from the eleventh on.

Variations

I regularly time students on this activity. It’s viable to set a challenge to Years 5 or 6 students of trying to beat 5 minutes if you have a date like 12/08/07.

This also makes a good working-in-pairs activity, with each pair trying to amass as many points for their equations as possible.

I have occasionally divided the class into four groups and given each group a selection of numbers, for example, Group 1 — numbers 1 to 5; Group 2 — numbers 6 to 10, etc. Although groups work separately, the objective is for the class to amass 20 equations.

If you’re looking for an engaging homework activity, set this as the task for students to take on individually at home, with permission to use siblings and/or parents for assistance when required.


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Objective

To develop mental computation skills, particularly rounding up or down, and the ability to recall and compare numbers.

Organisation

This activity makes for effective, cooperative group work. Seat students in groups of four, if possible. It will work with groups of five, or even six, but not quite as well.

Introduction

This is an engaging and challenging activity for getting students to estimate using rounding off skills. In Years 4 to 6, this activity works well for multiplication- and division-based problems and for Years 2 and 3 students it works well for addition and subtraction type problems. While this activity can work with students working individually, I have found that requiring a group to reach a consensus adds a valuable dimension to ‘And the Biggest Answer Is . . .’

Procedure

Presuming you have a Year 4 class and you want to focus on multiplication, place the following sums (or similar) on the board:

1. 27 x 83

2. 256 x 8

3. 87 x 23

4. 46 x 52

5. 78 x 32

Give each group in the class a letter, a colour or some other type of identifying name.

Direct students to the sums on the board and inform them that, working together in their small groups, they need to look at each of the five multiplication sums and choose the one that they, as a group, believe will produce the biggest product.

Let them know that it would be a waste of their time to ask you if they can use pen and paper, or to reach for pen and paper themselves. The whole of this activity is to be done mentally.

FOCUS Number, operations



EQUIPMENT None

Activity 7 And the Biggest Answer Is . . .

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Before you get the inevitable students crying out ‘I know! I know!’ and attempting to influence the other members of the group, assure all students that the only way that anyone can have any degree of surety is to use their rounding off (up or down) skills to make an estimate for the product for each of the five problems.

I generally put a practice example up on the board, something such as 63 x 29. Emphasise that you do not want an exact answer (you will invariably get one or two students doing ‘air sums’ with their fingers), but rather you want them to find an approximate answer — an estimation.

Someone may suggest rounding 63 to the closest 10, and doing the same for 29. This would mean rounding 63 down to 60, and 29 up to 30. We now have 60 x 30, which is 1800. This is the level of rounding ability that should be expected of your students.

Tell students in each group that they will have 5 minutes to reach a decision on which sum they believe will produce the greatest product — 1, 2, 3, 4, or 5. If the decision is not unanimous, the majority decision stands.

I count down the minutes, as this tends to keep the groups on task. I also look for any student either not contributing or not listening to the mathematical conversations. I encourage those students to at least listen in to the conversation and follow the thinking. I also remind the group, as a whole, that they should try to ensure that everyone is included in group work.

At the conclusion of the time, someone from each group writes down on a piece of paper the number of the chosen sum.

It is now a matter of going to each group and eliciting their decision. Don’t ask why the groups made their decisions at this point; simply record them on the board.

1. 27 x 83 Group B

2. 256 x 8 Group E

3. 87 x 23

4. 46 x 52 Group A, Group D

5. 78 x 32 Group C

Now go back and have each group explain their decision. Look for explanations such as ‘87 is a big number, but 23 is rounded down to 20, so it’s going to be 20 x 90 and that’s only 1800’.

Take out your calculator and inform students that the exact answers will now be placed on the board. It may look like this:

Exact answer

1. 27 x 83 Group B (30 x 80 is 2400) = 2,241

2. 256 x 8 Group E (250 x 4 is 1000 so 250 x 8 is 2000) = 2048

3. 87 x 23 (probably out of contention) = 2001

4. 46 x 52 Group A, D (50 x 50 is 2500) = 2392

5. 78 x 32 Group C (80 x 30 is 2400) = 2496

Group C now celebrates wildly!


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Students will probably ask you for another one. They really enjoy the challenge and semi-competitive nature of the activity. If you are doing multiplication or a set of division sums, ensure you choose four sums that produce results that are similar (have your calculator close by) and one that is discernibly out of contention.

Division problems regularly produce interesting conjectures, comments and estimations. Students often need reminding that the smaller the divisor, the more likelihood of a large quotient/answer.

Variations

In Year 3, you can use addition and subtraction and simple two-digit by one-digit multiplication problems.

For example, for addition:

1. 486 2. 38 3. 345 4. 616 5. 193 + 212 + 316 + 29 + 82 + 122 + 74 + 477 + 406 + 3 + 231

For subtraction:

1. 483 2. 624 3. 197 4. 1001 5. 317 - 235 - 294 -19 - 749 - 88

For multiplication:

1. 45 2. 54 3. 56 4. 65 5. 64 x 6 x 6 x 4 x 4 x 5

You can also use this format for addition of fractions/decimals/percentages:

1. 1−3 + 0.5 + 25% + 1−5

2. 0.4 + 3−4 + 1−10 + 30%

Again, the groups choose what they believe is the largest sum and then explain their choice.


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Objective

To develop mental computation skills, particularly rounding up or down for division and estimation problems.

Organisation

This activity makes for effective, cooperative group work. Seat students in groups of four to six.

Introduction

Students are not always successful in realising when an answer to a division problem is way off the mark. While this activity won’t solve all ills, it does help with the problem. I believe it works because students get to work together and apply strategies that are child-developed, child-explained and then, hopefully, mutually agreed upon by students as a group.

Procedure

Divide the class into groups of four to six. (Fours are better but may be difficult to accommodate.)

Nominate one person from each of the groups to be the recorder.

Tell students that you are going to place a division sum on the board. Each group will have the same sum to work on, and will have 45 seconds to estimate an answer to the sum. (This can be varied depending on the Year level and ability of your students.)

No pens, paper, calculators or any other devices, apart from the human brain, are permitted in working out the estimate. However, the recorder can use a pencil and paper to write down the group’s agreed upon response.

The nominated recorder within each group will be responsible for writing down the mutually agreed answer and announcing it to you, the teacher, when requested.

If a group or team cannot reach consensus, the majority rules.

Activity 8 In the Ballpark

FOCUS Number, division and multiplication



EQUIPMENT Calculator

Stopwatch (optional)

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Bonus points are available if a randomly selected member of a group can clearly and succinctly explain why his or her group chose a particular answer. All members of the group should be prepared to offer a justification for their answer. It is not the recorder’s job or prerogative to do this.

Place the following table on the board:

Problem Team Round 1 Round 2 Round 3 Round 4 Round 5 Total

1. 1

2. 2

3. 3

4. 4

5. 5

+ 1 = received a bonus point

Now explain the scoring system:

– within a given target range = 3 points

– just outside a given target range = 2 points

– reasonably close to the target range = 1 point

– not close to the target range = 0 points

Determine a range that is acceptable to you and that reflects the ability or developmental level of your class. I generally work out the exact answer (which obviously scores a 3!) and then give a range according to how difficult the problem presented is.

A sample game Write 85 ÷ 3 below the ‘Problem’ heading. Students’ 45 seconds starts

now. Either use a stopwatch or simply use a watch or clock to time the 45 seconds. I give 15 second countdowns and at the end of 45 seconds, I announce that students have 5 seconds to wrap it up and record the estimation.

Over to the ‘In the Ballpark’ scoreboard. Ask the recorder from Team 1 to provide his or her team’s agreed upon estimation. I record this in the Round 1 column next to the team number; for example, the Team 1 recorder registers 27 as the team’s response. Continue this process by asking all team recorders for their team’s response.

I now make a big show of entering the sum and announcing the exact answer. I then record this next to the original problem.

The next step is to determine the range and award an associated number of points. Team 1’s 27 is certainly well within the ballpark but there is an exact answer. This is entirely up to you, but I would give 2 points for 27. It’s a very good estimation, but the problem was a relatively non-threatening one. Team 2’s 28 earns 3 points — it’s within 1−3 and in the end you were not looking for an exact answer, just an accurate estimate. Team 3’s 25 is a good effort. Depends on your mood, really; I’m thinking 1 point. The points for Teams 4 (who estimated 28 1−3) and 5 (28) are straightforward. Place the scores for each team next to their estimate.


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Now randomly choose a team — let’s say Team 2. Select someone within Team 2 to explain how the group arrived at the estimation of 28. If the selected person is able to provide a clear, concise explanation — for example, ‘We knew that 25 x 3 = 75. It’s another 10 to 85 so we thought you could fit three more 3s in, and 25 + 3 = 28’ — that’s well worth a bonus point for Team 2.

I now usually ask if any other group used a different way to arrive at their answer/estimation. This sharing of strategies can be very valuable for students and insightful for you.

After Round 1, the scoreboard may look like the one below:


1. 85 ÷ 3 = 28 1−3

1 (27) 2

2. 2 (28) 3 +1

3. 3 (25) 1

4. 4 (28 1−3 ) 3

5. 5 (28) 3

If the Rounds 3 and 4 problems are a bit more difficult, you may need to give the teams 50 seconds and 5 seconds ‘wrap it up’ time. For Round 5, I generally give the teams 55 seconds plus the standard 5 seconds wrap-up time.

At the end of the game, the scoreboard may look like this:


1. 85 ÷ 3 = 28.33

1 (27) 2 (20) 3 (45) 2 (28) 2 (20) 3 + 1 13

2. 153 ÷ 8 = 19.1

2 (28) 3 + 1 (18) 2 (48) 0 (30) 3 (25) 1 10

3. 497 ÷ 12 = 41.41

3 (25) 1 (19) 3 (44) 2 (29) 3 + 1 (15) 1 11

4. 717 ÷ 24 = 29.8

4 (28 1−3 ) 3 (19) 3 (40) 3 (25) 1 (22) 2 12

5. 1111 ÷ 57 = 19.4

5 (28) 3 (21) 3 + 1 (41 5−12) 3 (28) 2 (30) 0 12


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Maths on the Go Book 2 MEASUREMENT

Objective

To develop an appreciation that we can use many different things to measure with, but that it is preferable to have a common unit of length if we want to compare lengths.

Organisation

Start with students on the floor in front of you. After the introduction to the lesson, students should work in pairs. Assign two or three pairs to a table or other area. Have ready on the table the materials they will be measuring with.

Introduction

This activity will help students grasp the reason behind using a common unit for measurement. The emphasis is on hands-on experience that students enjoy and benefit from.

Procedure

Have students sit on the floor. Show them the tape measure or metre ruler. Challenge them to tell you what they think the tape measure (or ruler) you have in your hand could possibly be used for. Expect a variety of responses, but there will generally be at least one student who will tell you that what you have there is some sort of device for measuring things.

I normally then ask how that student knew this, and if they could please explain how the tape measure/ruler actually works. This makes for some interesting answers and subsequent demonstrations, but is well worth doing, if only for getting valuable insights into students’ thinking.

I then ask something along the lines of, ‘If this is how we measure things today, how do you think they measured things before they invented the tape measure or ruler?’ This question can lead to many a weird and wonderful suggestion. My advice is to try to keep a straight face and give each response your due consideration. Eventually, a student is likely to suggest that in the past they may have used their hands or feet or arms. If this suggestion is not forthcoming from students, you may need to bring them along to this point by pointing out the practical problems associated with many of their suggestions, while things like your hands or feet are always (hopefully) there with you.

I now direct students’ attention to the tables with various materials gathered there. I go through the materials and tell students that they can choose one type of material

FOCUS Applying common units of measurement



EQUIPMENT Counters, crayons, markers or any other classroom item you have a large number of, and that vary from comparatively short to longer in length

Tape measure, preferably a retractable one, or a metre ruler

Activity 9 If You’ve Got It — Measure with It

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and/or their hands or feet, and that with a partner they should measure the following (written on the board):

1. the length of the table

2. the length of the mat

3. the length of a tub

4. the length of a marker

5. the length of your foot

Explain that one person should measure while the other partner counts how many. They should swap roles when they measure the next item.

For Year 1 students, five items is sufficient. For Year 2 students, seven or even eight items work well.

Place the name of each of the items to be measured on the top of a separate large sheet of paper and write the headings ‘Partners’, ‘Material/body part’ and ‘How many?’

As each partnership has finished measuring an item, record their names, what material or body part they used to measure, and how many of that material or body part the object measured.

If the opportunity arises, I get both partners to measure the same object using the same body part. Liana may measure the length of the table using her foot and Mai counts 7. Mai then uses her foot and Liana counts 8. How could this be?

After 15 to 20 minutes, I have students sit back on the floor and take them through the information recorded on the charts, without any comments or explanations. The chart showing the measurements for the table may look like this:

Partners Material/body part How many?

Liana and Mai Foot 7 (for Liana) 8 (for Mai)

Vincent and Adrian Large paper clips 33

Anton and Lindsay Foot 9 (Lindsay)

Jessica and Hassan Ice-block sticks 13

Walter and Yessenia Large paper clips 31

I then ask them to tell me all the things they notice. What I am looking for is observations such as the following: the measurements are different (depending on the material being used to measure); the larger the material being used, the smaller the number (of units); even though both partners used feet, it took more of Mai’s feet to measure the table, so Mai’s foot must be smaller than Liana’s; even when people use the same material, they can get different measurements (why?); for long items, using larger materials works better (why?); using the same material — for example, ice-block sticks — for measuring all the things would make it easier to compare their lengths.


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Objective

To have students understand that a common, standard unit is needed when measuring and comparing measurements.

Organisation

Start with students seated at their tables. After the introduction, students should work in pairs.

Introduction

I begin this activity by placing my right foot with a dramatic thump on a table near the front and centre of the room. I then ask students to consider how many of these (my right foot, that is) it would take to cross from one side (or end) of the room to the other. Alternatively, you can take off your shoe (if you can stand the hands-over-the-nose reaction from students) or simply walk around between tables drawing attention to the length of your feet.

I tell students that in the past the foot was a common unit of length and that their immediate task is to estimate how many steps — walking heel-to-toe — they believe it will take me to walk across (or the length of) the classroom.

Procedure

Once you have introduced the activity, ask students to record their estimation on a piece of paper.

Now begin your crossing of the classroom. You may start the count of one, two, three etc. for each step, but students will definitely continue it. At about the one-third point, I stop and ask students if any of them want to change their estimation. On average, about a third of the class will modify their estimation in accordance with the number of steps you have already taken and the distance remaining to be covered.

When you reach the other side, students will already have a good idea of whether their estimation was accurate or not. Ask who was within 10 steps of the actual number of steps recorded. Who was within 5 steps? Anyone within 3 steps? Anyone within a step? Ask these students if they used any strategy to determine their estimation.

FOCUS Applying standard units of measurement



EQUIPMENT Tape measures or metre rulers — enough for one for each pair of students

One worksheet (p. 28) for each pair

Activity 10 Steppin’ Out

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Now select a student, have them show their foot to the class, then ask students to make a new estimation of how many steps (heel-to-toe) it will take the selected person to cross the room. Hopefully, students will adjust their estimations in accordance with the knowledge of how many steps you took and the fact that the student has smaller (or larger) feet than you have.

Go through the same procedure, stopping the student about one-third of the way across the room and allowing students the opportunity to change their estimation if they wish to. If time permits, call on a second student volunteer to take the walk. The student estimations invariably improve as they bring prior knowledge into their calculations.

Three people have now walked the room and in each instance feet were used as the common unit. Three different people and three different amounts of steps. So how wide (long) is the classroom really? Do we use my number of steps or those of student 1? We need a standard common unit. Now show students a metre ruler. Hold it horizontally and vertically. Pose the question, ‘How many of these do you think I can place end-to-end across the room?’ Since the ruler is 1 metre long, have students record their estimations in metres.

You should find that all student estimations are within 10 metres of the actual measurement and that the vast majority of the class is within 5 metres. Why is it that so many of the class are within a close range? Allow about 2 minutes’ discussion time. Usually, some students will state that metres are bigger (units of measurement) than feet, so it’s a lot easier to estimate.

The numbers of steps compared to the number of metres to cross the room will verify this. The bottom line is that metres are great for measuring big things!

Is the opposite true? Are feet good things to measure the length of a pencil or an eraser or your little finger? Introduce the centimetre, and the fact that 100 of these make up a metre.

Now place students in pairs. Give each pair a copy of the worksheet and a tape measure (preferably) or metre ruler. Ask them to estimate and then measure at least five objects or distances within the room. If partners don’t agree on the estimation, both estimations should be recorded. One partner should then do the actual measuring, and the other partner should record the measurement.

Emphasise the need for all pairs to record both their estimation and their measurement. Estimations and measurements should be recorded in metres or centimetres or both. The abbreviated forms of m and cm can be introduced at this point, or at a later stage.

Allow time for students to compare their estimations and measurements. You could have them write about three things they found out from today’s session.


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Steppin’ Out

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Activity 10 Measurement

With your partner, estimate the length of the object or the distance. If you and your partner don’t agree, record both estimations.Now measure the length or distance. Record the measurement.

Maths on the Go Book 2 © Rob Vingerhoets/Macmillan Education Australia. This page may be photocopied for non-commercial classroom use.

Object or distance Estimation (in metres and/or centimetres)

Measurement (in metres and/or centimetres)

Height of the door

Length of my left foot

Length of maths notebook

Length of my pencil

Length of the bookshelf

Length of a paper clip

Length of the window sill

Student Name _______________________________

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Objectives

To have students make estimates of the number of pendulum swings, then predict, based on prior experience, what may happen when conditions for the pendulum swing alter.

To have students hypothesise what is likely to happen next, based on data and recognition of patterns.

Organisation

For Years 0 to 2, start with students seated on the floor. Years 3 to 6 students can be seated at their tables/desks.

Introduction

I have used this activity with classes from Years 0 to 6. It’s a comfortable mix of science (physics) and mathematics. With younger classes, it can be used for simple counting, and making and checking of predictions. With older students, this activity leads naturally to interpretation of data to form hypotheses, and identification of patterns to make predictions.

This activity fits in very well with measurement units of work as well as data/pattern work.

Procedure

For Year 4, but easily worked up or down for higher or lower grades

Before the activity begins, you need to spend 2 minutes making the pendulum. Simply take some Blu-Tack, or similar material, and wrap it around the final 4 or 5 centimetres of a 60 cm length of string. Shape the Blu-Tack into a ball. This ball should enclose enough of the string so that when you hold the end of the string, the ball hangs happily and securely at the bottom. Test your newly formed pendulum by allowing it to swing from side to side.

Activity 11 Like a Pendulum Swings

FOCUS Making predictions, identifying trends and patterns, hypothesising



EQUIPMENT A piece of string approximately 60 cm long

Small ball made of Blu-Tack or similar material

Metre ruler


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I tend to open this activity by telling students that today they are going to be making some estimations and some predictions.

Tell students that you are going to let this device (does anyone know its name?) swing from side to side for exactly 30 seconds. Hold the end of the string between the index finger and thumb of one hand and the ball in the same way with the other hand. Have the string extended to its full length so that the resultant line is stretched horizontally in front of you and at about eye level.

Inform the class that a swing up and back equals one.

Now ask students to record their estimate of how many swings of the pendulum (up and back) they believe will occur in 30 seconds. I often have Years 1, 2 and 3 students watch 30 seconds pass on the classroom clock to give them a concept of the duration of 30 seconds.

With all estimations locked in, either appoint someone with a stopwatch or have a student use your own stopwatch to time 30 seconds exactly. Using the classroom clock will suffice, although it won’t be as accurate as a stopwatch. Ensuring the string is as horizontal as you can get it, give the ‘Ready, set, go!’ countdown and let the ball end swing freely on the ‘go!’ command. The 30 second timing also starts with the ‘go!’ command.

I normally appoint two official counters who count the swings out aloud for the rest of the class. I also make sure I count the number of swings quietly myself. They, and you, should stop counting as soon as the 30 second duration expires. Although they already know it, announce the number of swings. Place the number on the board.

Experiment 1 = 19 swings

(I have a 58 cm length of string (from end of string to start of ball) and it almost always swings back and forth 19 times in 30 seconds. Occasionally I will get an 18 or a 20.)

Now place the pendulum length of string along a metre ruler and ask one of the students to come out and find the halfway point as accurately as they can.

Show students how long the pendulum is now — half the length of the first one. (With my pendulum, this is a case of 58 cm ÷ 2 = 29 cm.) Ask students to record their estimate of how many full swings of the pendulum will occur in 30 seconds this time. You will get a vast range of estimations with this one, even with the upper grades. Some will believe there will now be fewer swings and halve their previous estimation, or halve the recorded result; others will believe there should now be more swings and double or otherwise increase the recorded result.

Again, hold the string horizontally extended to its full length and let the ball swing on the ‘go!’ signal. Count the swings. Experience has taught me to wrap excess string from the halving of the length around my wrist, as the ball can sometimes entangle itself if the string is left dangling down.


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Inform students of the number of swings, and have them compare the result with their estimation. Record the data on the board.



Allow students some time to discuss the results. What is happening? Why?

Tell students that you are going to halve the length again. Have a volunteer come out to the metre ruler and string and find half of its length. (With my pendulum, this is a case of 29 cm ÷ 2 = 14.5 cm.)

Having once again reinforced the fact that the length of the string has been halved, and directing students’ attention to the data on the board, ask them to record their estimate of how many swings this time in 30 seconds. Wrap up excess string and let it swing! Count the swings, announce the number and allow some animated comparisons, speculation and discussions. Record the data.




Circulate among students while they are discussing, and encourage their hypotheses and theories, for example ‘The string gets shorter and the ball moves faster and you get more swings.’

Halve the length of the string again. (With my pendulum, this is a case of 14.5 cm ÷ 2 = 7.25 cm.) Have students look at the data and record their estimate. Remove excess string, make sure the fingers that aren’t holding the end of the string are out of the way, check with the timer and counters, and away it goes again.

Record the result on the board.





Announce that this will be the final experiment and, yes, the string length will be halved yet again. Direct students’ attention to the data. What do they notice? Who wants to make a prediction for the number of swings in 30 seconds this time? Ask them to explain their prediction. Years 5 and 6 (and regularly Year 4s) will identify what appears to be a pattern: 19 to 26 is +7, 26 to 36 is +10, 36 to 50 is +14, increasing by 3, then by 4, so the next experiment should see an increase of 5, which means +19. Someone may predict 69 swings of the pendulum. And guess what — there usually are! Give 5 minutes (rocket writing) to students to write a brief report on the pendulum experiment.


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Objectives

To have students discover ways to calculate the area of triangles by applying knowledge of shapes and prior knowledge for finding the area of squares and rectangles.

To have students feel good about themselves, having made these discoveries, and so raise their confidence and enhance their attitude towards mathematics.

Organisation

You will need an area of flooring or wall that is tiled, bricked or otherwise made up of squares or rectangles. Students work in pairs.

Introduction

While there may be occasions where there is some merit in instructing students on how to do something and then letting them try it out, I have no doubt that learning gained by discovery is far more valuable. This activity allows you to create a situation in which students can work out how to find the area of triangles. Why tell them something they can very happily find out for themselves?

Procedure

Take some masking tape and using the tiles, bricks or whatever as a guide tape a sufficient number of triangles so that there is one for each pair of students. As no pair will be able to get around to all the triangles, repeating shapes and sizes is fine. They might look like this:

Use self-stick notes to label each triangle a, b, c, d, etc.

Activity 12 Triangular AreasFOCUS Working out area of right-angled and equilateral triangles



EQUIPMENT Masking tape

Calculators — one per pair of students

Tape measures or rulers — one per pair of students

Self-stick notes

a

a

b

c

d

e f

gh

i

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Start the lesson with students seated at their tables or desks. Tell students they will be working in pairs. Let them know that you have taped a number of triangles in and/or outside the classroom, and that their mission will be to try to work out the area for some of those triangles.

Now is an appropriate time to review the concept of area. Don’t presume students will know it. For many upper primary school students the concept of area is very abstract — many know how to work it out, but not so many actually understand it.

Organise students into pairs. Tell students that the triangles all have a letter identifying them. Assign at least three triangles to each pair, and tell each pair which triangle they are to work on first. Make sure each pair starts on a different triangle: this will not only help give a structure and necessary order to the lesson, but also gives each pair the opportunity to make their discoveries independently. One student from each pair should record on paper which triangles they will be working on.

Each pair should ensure they have a notebook or similar for recording their answers, a calculator, a tape measure/ruler and the letters of the triangles they will be working on.

Your main concern now is to ensure you circulate among the pairs and offer advice or hints (not answers or solutions) as necessary. You may also — even after the review of the concept of area — be required to prevent pairs from simply finding the perimeter of their particular triangle. Frustration often leads to invention, so don’t be too hasty in offering help or advice to pairs — even if they appear to be struggling and without direction.

Hints that are useful if any pair are obviously and hopelessly inert include: ‘Remember that work we did on flipping, sliding and turning shapes?’; ‘Too bad we’re not trying to find the areas of rectangles or squares’; ‘It’s a bit like a puzzle — where could I put another triangle like that one so they fitted side by side?’

It is because they are free of any prescribed formula that students will often actually ‘stumble’ on the official formula for the area of a triangle: A = 1−2 base x perpendicular height.

Your students will report strategies such as the following:

– ‘We saw that if you flipped a right angle triangle, you get a rectangle (or square). Working out the area of a rectangle is easy, and then you just have to halve it.’ (It is appropriate that students use a calculator to do this as the process here is far more important than the exact final answer.)

– ‘With the equilateral triangle, we thought that if you added a right-angle triangle in the top left empty space and another right-angle triangle in the top right empty space you would make a rectangle. The rectangle would be twice as big as the triangle, so you just have to find the area of the rectangle and then halve it.’

You could and should certainly show the official formula to students (A = 1−2 base x perpendicular height) — after they have experienced the delight of working this out for themselves.

Ask students to do some report writing — complete with illustrations — as a conclusion to this activity. You will invariably get some very good writing.

Variations

Does the formula work for isosceles or scalene triangles? Yes or no? Prove it.


34

Maths on the Go Book 2 SPACE

Activity 13 Tell Me Ten Things About . . .

Objectives

To have students think, in depth, about a given 2D shape.

To find out what students already know about the shape specified, and to ascertain what geometrical knowledge and understandings they have, individually and collectively.

Organisation

A whole-class activity. Students will need a clear view of the board.

Introduction

This is a very effective warm-up or mini-lesson type activity. It focuses students very quickly on the topic. I have used this activity with 5-year-olds and with Year 6 students: all you need do is vary the specific shape you wish students to focus on.

Procedure

Draw a shape on the board and write the numbers 1 to 10 directly underneath. For example, for Year 3, using a trapezium as the ‘target shape’:

1.

2.

3. etc.

Make sure there is enough space for at least one full sentence to be written alongside each number.

Advise students that their challenge, as a group, is to come up with at least ten things that they know to be true about the shape on the board. At all stages avoid naming the shape, as this may deprive one of the students of providing this fact. Discourage students from calling out the name or anything else they might know about the shape.

Inform students that since this task is a team effort, no single person can provide more than one fact about the shape they are looking at. I tell students directly that this rule stops them from relying on any one, two or three students in the class who they believe will provide most of the answers.

FOCUS Shape recognition, attributes of shapes, spatial terminology



EQUIPMENT None

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With Years 0 to 3, I now have students turn around to someone next to them and share or discuss what they were thinking of contributing as a fact about the shape. This helps students who have a fact to offer to clarify their thoughts, and gives them an opportunity to rehearse the terminology. It also makes students who may have had no clear contribution to make more aware of what is required, and alerts them to one or more possibilities.

Ask for a volunteer to start you off with the first fact, and record the response next to number 1 on the board. Here’s an example of how the activity might proceed for the Year 3 class using the trapezium example:

1. It has four sides.

2. It is a trapezium.

3. The bottom side is longer than the top side.

4. It has four corners.

5. It has four angles.

This can be a good point at which to review what students have already contributed and allow them to again talk with someone next to them. If they are starting to slow down with their contributions, this discussion time often stimulates additional responses, if only through two or three individuals who have already contributed now being able to share other ideas with a classmate.

6. It has two parallel lines (top and bottom).

7. The left and right sides are not parallel.

8. It is a polygon.

9. There are no right angles.

10. If you flip it, it makes a hexagon.

You may also get facts about symmetry (make students identify, with your guidance, whether it is horizontal or vertical symmetry), specific types of angles (two acute and two obtuse), lines drawn to make a rectangle and two triangles, and so on.

From Year 2 up, I am reasonably tough with students in terms of my expectations for clear and precise directions or explanations. If a students says there are two acute angles, I respond by saying, ‘Great! Now tell me where they are.’ It may take a little time, but the rest of the class quickly comprehends that you require precise and clear terms such as ‘top and bottom sides’ and ‘bottom right-hand corner’. I believe this makes students more knowledgeable, more precise in their instructions, and brings in some useful mathematical terminology.

Variations

For Years 0 to 1, shapes such as the square, rectangle and triangle (equilateral) work well. For Years 2 and 3, use shapes such as the trapezium, rhombus, parallelogram and hexagon. For Years 4 to 6, a pentagon, right-angle triangle, scalene triangle and semi-circle will work well.

With Years 2 and up, you can also try the activity using 3D shapes such as a cube, rectangular prism, triangular prism, square pyramid, cone and sphere. Hold these as solids in your hand while students provide the facts.

With Years 4 to 6, I generally have them tell me 12 things about . . . They consistently achieve at least 12 facts, even if some of them are ‘easy’ facts.

This activity also works well for measurement concepts and terms. For example, you could put the word ‘perimeter’ up on the board and list the numbers 1 to 6 underneath. Then challenge students to give you six facts that are true about perimeter.


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Activity 14 Left, Right, Compass and Clock — Find the Teddy Bear!

Objective

To give students practice in using appropriate spatial terminology when following and giving directions.

Organisation

A whole-class activity. For lower primary classes, have students sit on the floor or use a relatively spacious area in the classroom for the warm-up. Students can then return to their seats. Middle and upper primary students can start off seated at the tables/desks.

Introduction

We hear and use spatially specific terms every day. (‘What have you got behind your back?’; ‘It’s underneath the table’; ‘Sit down in front!’) In this activity, students initially follow specific directions, and then progress to providing clear and unambiguous directions to others in order to bring them to the treasure. On the way, they are likely to learn some specific terms and vocabulary, or perhaps hear them in a new context.

Procedure

For Years 0 to 2

To review the basic directional/positional terminology, play a simple game of ‘Simon Says’ or similar. For example, ‘Simon says put your right hand on your head, Simon Says put your left hand on your right foot’, and so on.

Tell students that you are going to choose someone to go outside while you hide an object somewhere in the room. After it has been hidden, the person waiting outside will come back in and you will give them accurate directions to follow so that he or she finds the hidden object. I like to hide the object down low, up high, next to or inside something so that this type of terminology can be used in the directions.

A sample game (for a Year 1 or 2 student)

Once the object has been hidden and the searcher has been called back in, stand them near the classroom door, facing the opposite side of the room. Instructions might go as follows (and can go on for as long as you like): ‘Walk forwards for five steps. Stop.

FOCUS Spatial and directional terminology



EQUIPMENT Small plastic teddy bear, or any small object that is easy for you to hide and that students will enjoy finding

3�

Put your left arm out and turn to your left. Walk forwards for seven steps. Put your right arm out and turn to your right. Walk forwards for eight steps. Walk backwards for three steps. Stop. Turn to your right. Open the cupboard. Find the box below the big red book and next to the jar. Look underneath the box. The teddy bear! ’

Some Year 2 students may well be ready for quarter, half and three-quarter turns.

After three or four students have had a turn, tell students that you are going to wait outside while one of them hides the treasure and will follow their directions to find it. They will make mistakes, but the excellent thing about the activity is they realise instantly when they should have given you ‘left’ rather than ‘right’. It is cause for much celebration and praise when the student gets you to the object.

Now a new student can be chosen to give directions while another student goes outside.

Procedure

For Years 3 to 6

Draw a large circle on the board. Tell students it is a clock face missing its numbers, and you will need them to help you fill it in.

Tell students to give you directions using degrees in a circle to fill in the numbers. Students usually respond by saying, for example, ‘Go to 90°, Mr. V. That’s 3 o’clock.’

When a student gives a correct direction, fill in the number on the clock, and write the associated number of degrees outside the circle.

You may need to provide prompts such as ‘If 90° is 3 o’clock, I wonder how many degrees 4 o’clock (or 2 o’clock) would be?’ Some students will be able to take you directly to each hour being equal to 30°. Eventually your blank circle/clock face will have all 12 hours on the inside and the associated number of degrees on the outside, increasing by gradients of 30° and starting and ending at 0°/360°.

Have students add the turns, with or without guidance: quarter, half, three-quarter and full. I record these next to the 90°, 180°, 270° and 360° labels.

Now choose a student to wait outside while you hide an object somewhere in the classroom.

Have the searcher come back in and face the wall opposite the clock diagram. Tell the searcher to imagine the clock on the board directly in front of them, and say that no matter which way they are facing, they need to imagine that the clock is directly in front of them. Students who are able to do this (and this is most) have no real difficulty with this activity.

Instructions might go as follows: ‘Walk forward. Stop. Turn 270° (left turn). Walk forward. Stop. Make a quarter turn (90° or right turn). Walk forward. Stop. Make a 180° turn (turn around, half turn). Walk forward. Stop. Do a 360° turn (just for the fun of seeing it!)’ and so on.

I like to give a cryptic clue for the final location, for example: ‘The teddy bear likes his food’ (if he’s hidden inside the class lunch basket).

With Year 6 students, include partial turns; for example ‘Turn 60°’, or ‘Walk forward five steps at 30°’.

The next time, have a student hide the object and give you directions; then have a student give another student directions.


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Activity 15 We’re Going on a Shape Hunt

Objective

To have students recognise 2D and 3D shapes within and outside the classroom.

Organisation

A whole-class activity, with students working in pairs. If you can, find someone who can assist you in supervising the class outdoors during the shape hunt. If this isn’t possible, it may be wise to conduct this activity within the confines of the classroom or the corridor area or similar area immediately outside the classroom.

Introduction

Far too often our students think mathematics is something confined to the immediate environment of the classroom. If it can be done safely and effectively, take as many opportunities as you can to show students that maths exists everywhere. Do this by physically taking them from the classroom and into the outside world, even if that is simply the corridor or playground.

For the lower primary years, good entry points into this lesson include reading a story to students that features shapes in the environment; showing students pictures of circles, squares and triangles (or actual cubes or rectangular prisms etc.) and asking students where they have seen these shapes in their homes or on the way to school; and telling the story ‘We’re Going on a Bear Hunt’, then telling students that today we won’t be hunting bears but instead will be hunting shapes.

For the middle and upper primary years, good entry points into this lesson again include reading a story to students that features shapes in the environment. Ask the class to give you specific examples of some essential roles that shapes play in our daily lives. Expect responses related to architecture/buildings/structures (for example, bridges), transport, packaging, utensils, furniture, art (design, clothing, tiling) and many other aspects of their daily lives. Or have students imagine what their lives would look like and be like if there were absolutely no rectangles or no rectangular prisms. What would be different?

FOCUS Recognition of 2D and 3D shapes



EQUIPMENT One worksheet (pp. 40 and 41) for each pair

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Procedure

For Years 0 to 2

Give students the ‘We’re Going on a Shape Hunt’ worksheet and have them identify for you as many of the ten shapes as possible. There is a mix of 2D and 3D shapes and I believe this is a good thing, even for the very young classes. Three-dimensional shapes are everywhere — no point keeping them a secret until a later year.

Inform students that their task will be to find each shape in the environment around them. You may wish to give some examples, for example a brick is a rectangle; a die (as used in board games) is a cube.

Tell them that in the ‘Where did you find it?’ column you want them to write or draw where they saw this shape. This is very important, especially when they share their discoveries at the end of the lesson. Often a cylinder or some other shape will turn up in a place that no-one else has recorded.

Obviously Year 0 and even Year 1 students will need assistance from you in filling in the ‘Where did you find it?’ column. You can alleviate the pressure that may be placed on you by using some good communicators from Years 4, 5 or 6 to work with one or two pairs of students.

Now students know what to do, it’s time to go on a shape hunt!

Ensure students have time to share with each other where they found the shapes. As this happens, make sure shapes are named accurately, and verify the identified shape was in fact (for example) a trapezium.

You may wish to finish by asking questions such as, ‘Why are cricket balls (basketballs, soccer balls) spheres and not cubes?’

For Years 3 to 6

As for the lower grades, hand out the appropriate ‘Shape Search’ worksheet (a slight variation on the title to reflect their greater maturity!) and have students identify the shapes for you.

Tell students that their task is to find each shape in the environment. Remind them that they must record exactly where each shape was found.

Tell students to look everywhere — not just on the floor immediately in front of them!

As with the younger classes, ensure students have time to share with each other where they found the shapes, and check as this is done that shapes have been identified correctly.

Again, you may wish to finish by asking questions such as, ‘Why are almost all pipes cylinders and not rectangular prisms?’


40 Maths on the Go Book 2 © Rob Vingerhoets/Macmillan Education Australia. This page may be photocopied for non-commercial classroom use.

We’re Going on a Shape Hunt

Activity 15 Space

Shape Where did you find it?

Student Name _______________________________Years 0 to 2

41

Student Name _______________________________


Shape Search

Activity 15 Space

Shape Where did you find it?

Years 3 to 6

42

Objective

To review, in a game format, the knowledge, concepts and space-specific vocabulary students have acquired during their work on space.

Organisation

Divide the class into five tables-based teams (teams of five or six work best). Appoint one person in the team as the recorder. This person will be responsible for keeping his or her team’s score as well as recording which team members have answered questions. Name the teams Team 1, Team 2, etc.

Introduction

The quiz can be used as an alternative to a test that might normally be given at the end of a unit of work, or it can be given at any stage of the year purely as a review activity. A well-delivered quiz should not be a daunting thing; it should be challenging, but it should be fun. Set the tone or atmosphere for the quiz by de-emphasising the competitiveness and keeping the quiz dynamic and fast-moving.

Procedure

Take students through the quiz rules:

1. No student, in any team, is permitted to answer more than three questions, including ‘steals’. (This prevents some team members from leaving all the answers to one or two students who they believe are knowledgeable in this area.)

2. If any team answers a question incorrectly, the next team is eligible to take their question, answer it and earn an extra point if their answer is correct; for example, Team 3 answers incorrectly, so Team 4 may now ‘steal’ their question and potentially earn an extra point.

3. Any team that ‘steals’ an incorrectly answered question also receives their own questions (hence the opportunity to earn an extra point).

4. Each correct answer is worth one point, including ‘steals’.

5. While they should be encouraged to not be impulsive and to consider team strategies, the first student in a team with their hand in the air is asked the question.

6. After a predetermined number of rounds or set time limit, each scorer totals their team’s points and announces the total to you.

7. The quizmaster’s decision is final and absolutely no correspondence will be entered into!

Activity 16 The Big Space QuizFOCUS 2D and 3D shapes, angles, directions, spatial terminology



EQUIPMENT Quiz sheet (p. 43)


43


The Big Space Quiz

Shape/measurement questions

How many sides does a rectangle have? [4]

How many faces does a cube have? [6]

If one side of a pentagon is 8 cm long, what is its perimeter? [40 cm]

What is the perimeter of an equilateral triangle with one side 12 cm long? [36 cm]

How many right angles are there in a trapezium? [0]

How many degrees are there in a circle? [360°]

How many sides are parallel in a pentagon? [0]

Shape knowledge

Draw a shape on the board. Each member of the team must volunteer at least one fact about the shape. Five facts = 1 point; four facts = 0.5 points; three facts = 0!

semi-circle rectangle trapezium right-angled triangle hexagon

Real-life space

Ask teams to find the following in the classroom:

parallel lines an acute angle a cylinder

a semi-circle a triangular prism something symmetrical

Guess my shape

I have three sides. All of my three sides are equal. All of my three angles are equal. My shape can tessellate. I am only symmetrical vertically. [equilateral triangle]

I have four sides. My bottom and top sides are parallel. I am only symmetrical vertically. My shape can tessellate. I have two angles that are obtuse and two angles that are acute. [trapezium]

21 I have six faces. I have twelve edges. I am 3D. I have eight vertices. All my faces have the same area. [cube]

Physical shapes

Have a volunteer from each team come out to the board and draw:

22 an isosceles triangle

23 a trapezium

24 a cube

25 a shape with more than four sides that is symmetrical both vertically and horizontally

44

Maths on the Go Book 2 NUMBER

Objective

To explore, reinforce and revise the concepts of place value, odd and even numbers, before and after, number facts and physical characteristics of numbers.

Organisation

A whole-class activity. You will need a sufficiently large area so that 10 to 15 students can stand in a line. The other students will need to be able to see the standing students clearly.

Introduction

Working in so many different classes and in many different schools, there are a few essentials I always take with me and one of those essentials is a now reasonably tattered set of numbers — 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. They are invaluable and what follows are just some examples of the ways I have used the number cards from Year 0 all the way through to Year 6.

Procedure

For Years 0 to 2

Distribute your ten number cards in random order and ask the number recipients to stand at the front of the classroom (or in whatever suitable space is available).

Ask students to please get into order from the smallest number through to the largest. Then ask students to count up their numbers (‘zero, one, two,’ etc.) to 9 and then do a countdown to check the order is correct.

On some occasions, I do not give out all the numbers (for example, I keep hold of 2, 5 and 8) and then have students identify which numbers are missing

Activity 17 The Human Number Line

FOCUS Place value, odd and even numbers, cardinal and ordinal numbers



EQUIPMENT Set of numbers 0 to 9, commercially or self-made — should be at least 15 x 20 cm, on thick card, with clear numbers.

For Years 3 to 6 — some regular or medium-sized self-stick notes

4�

After placing the numbers in order, ask numbers 1, 3, 5, 7, and 9 to take one step forward. Ask students why they think you’ve done this. Introduce or revise the notion of odd and even numbers. Have the even numbers step forward and recognise themselves while the odd step back.

At about this stage, you may want to have students standing with numbers give their number to a seated child. This can be made into an exercise in itself, for example, ‘If you have a number that is even, sounds like door and only has straight lines in it, please give it to Giovanni’, and so on.

Ask students to join with a partner so that when they add their two numbers together, they add up to 10. You will be left with two students looking a little forlorn — poor unwanted 0 and 5. Reassure them that they have just been unlucky as 0 and 5 don’t add up to 10. Guide students to see the pattern in the number facts by pairing up the numbers in order: 9 + 1 = 10; 8 + 2 = 10; 7 + 3 = 10; 6 + 4 = 10; 5 + ? — we don’t have another 5. Now ask 9 and 1 to swap places, and note that 1 + 9 still equals 10; 2 + 8 = 10, and so on.

Ask these groups of numbers to stand together: 1, 4 and 7; 2, 5 and 9; 0, 3, 6 and 8. How have I made these three groups? Allow discussion and speculation from all students. Not always, but frequently, someone will be able to tell you that 1, 4 and 7 all have straight lines; 2, 5 and 9 have curved and straight lines; and 0, 3, 6 and 8 are all curved lines.

Randomly put two numbers together. For example, 2 and 5 stand together; 6 and 4 stand together; 9 and 1; 7 and 0; 3 and 8. Tell students that this time, instead of adding the numbers together you are going to place them side by side so that they make a two-digit number. Ask 2 and 8 what number they make. ‘28’. Now ask the partners to stand side-by-side so that they make the smallest number they can. Now have them make the largest (they simply swap positions).

With Year 2 students, ask 0 to stand with you and then ask students to get into groups of three. Ask students to make, in their group of three, the largest number they can (for example, 8, 4 and 5 make 854); the smallest number they can; the smallest even number they can; the largest odd number they can. Now send in zero to join with a group, and make the smallest number, the largest number, the largest odd number and so on.

For Years 3 to 6

Give the number 5 card to a student and ask her or him to stand at the front of the room. Now give someone the number 4. Invariably they will stand next to or at least close to each other.

Now write a number such as 4.2 on a self-stick note, give the note to a randomly chosen student and ask them to join the line wherever they think they fit in. Fractions and decimals are very abstract concepts to primary students, and it is fascinating to watch students realise that 4.2 is bigger than 4, so it must be between 4 and 5 — ‘and I thought there couldn’t be anything between 4 and 5!’

With 4.2 standing somewhere between 4 and 5, write 4 1−2 on a self-stick note. Hand it to a randomly chosen student, and ask them to join the line. With Year 3 and 4 students, asking them to think of money can be a helpful hint. With upper primary students, I suggest they think about metres and centimetres.


4�

Write more numbers on self-stick notes (for example, 4.9, 4.25, 4.1, 5.2, 3.9, 4 1−4, 4.55, 4.6, 5.05) and hand them to students one at a time. When you get to about a dozen students, the front of the room will be quite busy. You may want to stop here and have each student, individually and in turn, read out his or her number. Then have students read their number out again as dollars and cents; as metres and centimetres; as kilograms and grams.

In a second round with new students, give the number 0 to someone and the number 1 to another person. This allows you to include percentages. Use self-stick notes with 0.7, 10%, 3−4, 0.5, 0.05, 90%, 0.3, 1−3, 75%, 0.8, 40%, and have students (one at a time) place themselves in the appropriate position along the line.

Variations

With Year 0 students, use the numbers to reinforce concepts of ‘before’ and ‘after’. You can precede this by asking students to nominate something they do before breakfast; something they do after breakfast; what the day after today is; what the day before today was; something we do before lunch; something we do after lunch?

Now sit on the floor with students in a large circle and place the number 4 in front of yourself. Scatter the other numbers directly in front of where you are seated. Tell students that if they know the number that comes before the number 4, they can come out and pick up that number and sit next to you. The activity proceeds this way: ‘If you know the number after 4 . . . If you know the number before 3 . . .’. Do this until all numbers 0 to 9 are in line seated on the floor. To involve the other students, you can say, for example, ‘If you know the number before 6, come out and get it from the person who is holding it now’, and so on.


4�


Objectives

To introduce young students to digital numbers through making and copying them; and to have them discover that from digital number 8 you can make all of the other digital numbers.

To have students use a calculator and get to know some of its functions.

Organisation

This activity works well with students working individually or with a partner.

Introduction

‘The Magic 8’ activity can often help young students who have a problem with reversal of numbers — particularly twos and fives, threes and sevens.

Procedure

Ask all students in the room to tell you something about the number 8. It can be how it looks, how you can make it (4 + 4 = 8, etc.), where they have heard or seen the number (‘It’s on the clock’) or something personal about the number (‘My brother is 8 years old’).

Alternatively, start by showing students the numbers on a digital clock or watch and asking them to compare them with the numbers on the 1 to 100 chart, a number line, analog clock or any other source of non-digital numbers in your room. How are they similar? How are they different? Besides clocks or watches, where else might they have seen digital numbers? If no one suggests calculators, introduce the idea yourself and show students a calculator.

After the introduction to digital numbers generally and/or digital 8 specifically, ask students (working individually or in pairs) to pick up their calculators and turn them on using the ON/C button.

Draw students’ attention to the fact that the 0 number displayed on the calculator is not continuous — it is broken up into smaller parts. Ask them how many smaller parts or lines make up the digital number 0. Students will see that it is 6 — not 4, but 6. Have them make digital 0 with six ice-block sticks, MAB ones, or whatever material you are using. Then have them copy the digital number zero onto a piece of paper.

Activity 18 The Magic 8FOCUS Number



EQUIPMENT Enough ice-block sticks, MAB ones, toothpicks or similar so that there are 14 to 15 (very important) for each student or pair of students

One calculator each or at least one per pair of students

Pencils and paper

4�

Now have students press the number 1 button and ask what they see. How many pieces make up the 1? Ask students to make it and copy it onto their paper.

Have students press the ON/C or CE or AC button to clear the number 1 and bring you back to digital 0. Now press the number 2 button. How many ice-block sticks will you need to make your number 2? Have students make it and copy it onto the paper.

Just about now students are likely to be running out of ice-block sticks. Tell them that, very unfortunately, you don’t have any more sticks and that if only there was one of the digital numbers from 0 to 9 that you could make all of the other digital numbers out of, they wouldn’t need any more sticks.

Presuming that no-one has named digital 8 as the magic number, ask students to take all the sticks they used to make 0, 1 and 2 and place them together, in preparation for making another number — why not number 8? Have students press ON/C or CE or AC and then press the 8 button. Have them make number 8 exactly the way it’s shown on their calculators and then copy it onto their paper.

Check all students have made the 8 correctly, then ask them to clear the 8 on the calculator and press the 3 button. Ask them to look at the number 3 on their calculator, then look at the number 8 they made with the sticks in front of them. Ask them to take away two sticks from their number 8 so they have digital number 3, then to copy the digital 3 onto their paper.

Have students put their digital 8 back together again, clear the calculator, and press the 9 button. Ask them to take one stick away from the digital 8 to make digital 9. Check and then have them copy the digital 9s.

Around this stage, one or more students will usually tell you that they think you can make all the numbers from digital 8. Ask them to prove the theory by putting their number 8 back together and using it to make digital 4; then digital 5, 6 and 7. Be on hand as some individuals or pairs may need assistance. Remind students to copy each number onto their paper.

To wind up the activity, ask what students have discovered today or why digital number 8 is such a magic number. I have found that you get some very good writing and illustrating from students if you ask them to write and draw about ‘magic 8’.

Variations

Students thoroughly enjoy challenges such as the following: turn digital 0 into a digital 9 (suggest students use their calculators to help them); turn digital 4 into digital 7 in one move; turn digital 5 into digital 2 in two moves; make two digital 8s with your sticks and make the digital number 47; 33; 25; 52.


4�


Objective

To build number sense through giving number values to known pattern block shapes. This develops basic addition, multiplication and equation creating skills as well as introducing algebraic thinking to students.

Organisation

A whole-class activity. You can start this activity with students on the floor, but have paper, pencils and pattern blocks ready at their tables.

Introduction

The following is one of my favourite ways of using pattern blocks for teaching number sense and revising and reinforcing shape knowledge and concepts. The appeal and versatility of number blocks extends right through primary school to Years 5 and 6.

Procedure

Show students a yellow hexagon pattern block. Ask them to tell you something they know about this shape. Do the same for the trapezium, rhombus and triangle. Remind students to use the correct terminology for each shape, rather than saying, for example, ‘the yellow one’. Ask the students to compare the shapes to each other, and make observations such as the following: the trapezium is half the size of the hexagon or ‘two trapeziums make a hexagon’; three rhombuses fit on a hexagon; six triangles fit on a hexagon; two triangles fit on a rhombus; two rhombuses and two triangles equal a hexagon; a trapezium and a rhombus and a triangle make a hexagon.

All these observations should be demonstrated on the board by the student who suggested them so that all students can see the evidence that three triangles fit on a trapezium, etc.

It may now be timely to say something like, ‘I wonder what would happen if we said that this hexagon is worth 6?’ Ask students if this hexagon was worth 6, what would the trapezium, rhombus or triangle be

Activity 19 Pattern Block Numbers

FOCUS Addition, shape, fractions, logic

AGE LEVEL Years 1 to 2 (6- to 7-year-olds) Variations for Years 3 and 4/ Years 5 and 6


EQUIPMENT Enough pattern blocks (hexagons, trapeziums, triangles and rhombuses) so that each pair has at least five hexagons and ten each of the other shapes

Paper and pencils

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worth? If necessary (and it is likely it will be with Year 1s), show the hexagon, place the trapezium on top of it and then repeat the question. Confirm that the trapezium must be worth 3, as it is half the size of the hexagon (6). (Be prepared for some students to assume that since the hexagon is 6 and has 6 sides, the trapezium must be 4 because it has 4 sides.)

Now show the rhombus. Place the rhombus on top of the hexagon. If the hexagon is 6, then the rhombus must be . . . ? If necessary, place all three rhombuses on top of the hexagon. Students should then conclude that the rhombus must have a value of 2 since 2 + 2 + 2 = 6 (the hexagon) or 3 x 2 = 6.

Finally, have students work out what the triangle must be worth. Reiterate that we know the hexagon is 6, the trapezium is 3 and the rhombus is 2. By placing a single triangle on a rhombus, perhaps 2 triangles on a trapezium and 3 triangles on a hexagon, if necessary, have students inform you that a triangle must have a value of 1.

Ask students to return to their tables and get ready to make a shape with the pattern blocks.

Tell the class that their challenge is to build a shape for you that is worth exactly 12, remembering that the hexagon = 6, etc. The shape can be made vertically (stacking the pattern blocks) or horizontally (laying out the shapes with edges touching). They should then record how they made their shape — just the numbers will suffice. For example, 6 (hexagon) + 3 + 3 (trapeziums) = 12; 6 (hexagon) + 3 (trapezium) + 1 + 1 + 1 (triangles) = 12; or 3 + 3 (trapeziums) + 2 + 2 + 2 (rhombuses) = 12. Wander among the tables ensuring that all pairs have understood what is required.

When students have done this, tell them that this time you would like them to make a shape that is worth 12 again, but it must look different and use different pattern block shapes than the last one.

Now ask students to make you a shape that is worth more than 15 but less than 20. I write it up on the board as > 15 and < 20 and repeat the instructions. Once again, ensure they record what their shape is worth so they can prove to you it is between 15 and 20. Writing basic equations is going to be challenging for some students at Year 1 level, but persevere as they will get there.

Here are some further challenges:

– if hexagon = 6: make and record a shape that is worth between 20 and 25

– if hexagon = 12: work out the value of the other pattern blocks; make and record a shape that is worth 24; make and record a shape that is > 30 and < 36

– if hexagon = 24: make and record a different shape that is worth 24

– if hexagon = 3: work out the value of the other pattern blocks (if students are able to get two or three of the four shapes, they will be doing okay); make and record a shape that is worth 6; make and record a shape that is > 6 and < 10.

Have students draw and label their favourite of the shapes they made and write a sentence about two things they found out from the lesson.


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Variations

Have Year 3 or 4 students seated at their tables and working in pairs, with a range of hexagon, trapezium, rhombus and triangle pattern blocks at their disposal. Try these challenges:

– if hexagon = 3: prove to me what the trapezium, rhombus and triangle must then be worth; make and record a shape worth 13, 10 1−2, 7 1−2, 20; make a shape using more than three pieces that equals 4; make and record a shape that is > 18 and < 21

– if hexagon = 1: prove to me what the trapezium, rhombus and triangle must then be worth; make and record a shape worth 3, 2 1−2, 1 1−3; make a shape using more than ten pieces that equals 4; make and record a shape that is > 4 and < 6; make and record at least four different ways to equal 2.

Try these challenges with pattern blocks for Years 5 and 6:

– if two hexagons = 1: prove to me what the trapezium must then be worth; make and record a shape worth 1 3−4, 2 1−2, 3 1−4 ; make a shape using more than four pieces that equals 2

– if three hexagons = 1: prove to me what the trapezium, rhombus and triangle must then be worth; make and record a shape worth 1 1−2, 2 1−2, 1 1−3, 2 4−9; make a shape using 14 pieces that equals 2; make and record a shape that is > 1 1−2 and < 1 2−3 ; make and record at least three different ways to equal 1 1−3 ; make and record a tessellating shape that uses a minimum of three shapes and is worth between 5 and 7.


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Objectives

To consolidate/revise place value concepts in a game format. In middle to upper grades, to use the same format to revise/reinforce subtraction, multiplication and division skills.

To revise odd and even numbers.

Organisation

This game is best played individually. All students will need is a pencil or pen and paper or their maths notebook.

Introduction

This is a great little activity that is very versatile. It requires little equipment or preparation and you can use it from Years 0 to 6, and for topics from place value to division.

Procedure

For Year 3

Tell students that in the container you are holding are the numbers 0 through to 9 on separate cards.

Draw the following on the board, complete with place value headings:

tens of thousands hundreds tens ones thousands

Tell students that they are going to help take the numbers (0 to 9) out of the container, one at a time. As each number comes out, they have to place it in one of the place value boxes, with the aim of making the largest number they possibly can. For example, if a 9 were drawn out of the container, what would be the best place value box to place that in? Ones? Hundreds? Ten of thousands? Thousands? What do students think, and why? Invariably, the majority of the class will be able to tell you that the 9 should go in the tens of thousands, as the 9 is the largest number you have and tens of thousands is the largest place value.

Activity 20 Pick It OutFOCUS Place value, operations



EQUIPMENT Numbers 0 to 9 written on separate cards

Opaque container to place number cards in

One worksheet (p. 55) per student

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Go through 2 or 3 numbers to establish that most students understand the concept of the game. To ensure that all students are following, ask them to tell you the best possible result. ‘If I am trying to make the largest number that I can, what is the best number to have in the tens of thousands? The thousands? Hundreds?’ As students tell you the answer, record it underneath the place value boxes:

tens of thousands hundreds tens ones thousands

9 8 7 6 5

Ask one of the students to read out the best possible result: ninety-eight thousand, seven hundred and sixty-five. Having students read out large numbers at every relevant opportunity is a great idea. Primary students feel very empowered by being able to read out large numbers — it’s a substantial confidence builder.

Give each student a copy of the worksheet.

Ask for a volunteer to draw out the first number. Once the first number is drawn, the onus is on each student to choose the most appropriate place value box within which to place the number. Make sure students understand that once a number is written in a box, it stays in that box — no erasing allowed! Another trick students will try is waiting until the next number is drawn before making a decision on the first one. Tell students they must make each place value decision before the next number is drawn.

Have a volunteer draw out the second number. Remind students that they are trying to make the largest possible number. They have a guide as to what the best possible result would be, but they will need to do their own thinking if a 3 or 0 or 2 is drawn.

Continue playing until the five numbers have been drawn. Ask that each person now turn and read their number to the person next to them. Emphasise that it should be read not as ‘eight, six, four, five, two’ (for example) but as ‘eighty-six thousand, four hundred and fifty-two’.

Now inform students that they are playing for points. The largest possible answer receives 3 points, the second-largest 2 points, and third-largest 1 point. Doing this keeps more students in the game and also means that they will be comparing and ordering their large numbers.

Call for someone to read out the final number, as drawn using the cards. Then ask if anyone would like to try for the ‘big 3 points’. Write up the candidate number that is read aloud by the volunteer, for example 84 625. The student should of course be congratulated heartily for reading out their number so well. Ask if anyone has a number larger than 84 625. If someone does, have them read out their number, say 86 452. Write the second number above the first. In this case, you could point out that both numbers have an 8, 6, 5, 4 and 2 in them, but that where the numbers are placed makes a difference. Both students have an 8 in the tens of thousands, but the second student’s 6 is worth six thousand while the first student’s 6 is only worth six hundred.


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Having established the 3, 2 and 1 point answers, you are now ready to move onto the second game. In about 45 minutes you should be able to play:

– largest possible number

– largest possible even number

– smallest possible number

– smallest possible odd number.

Variations

If the worksheet on p. 55 is not at an appropriate level for your class, have students draw up place value boxes just as you have them on the board, with the place value names written above the boxes.

For Year 0, play the games to tens and ones with the aim being to make the largest possible number. Many students will actually devise the strategy that if a number bigger than 5 is drawn, put it in the tens; if it’s less than 5, put it in the ones. It will take a number of games for students to be able to appreciate this concept, but they thoroughly enjoy and gain from the activity.

For Years 1 and 2, play to hundreds and then to thousands. Aim first to achieve the largest number, and then to achieve the smallest. Try odd and even with Year 2s if you think they are ready for it.

For Year 4, play to hundreds of thousands initially, then take the game into the millions.

For Years 5 and 6, having played the game into the millions, you can remove the place value headings, include a decimal point card in the container (or place a decimal point on the reverse side of the 0 card), and tell students they need to make the largest possible number to tenths, hundredths or thousandths. This way, they not only need to have the decimal card emerge, but they need to know which place value box to place it in to ensure their answer will make sense.

For Years 4, 5 and 6, you can also use this game to revise and/or reinforce the operations. First remove the numbers 0, 1, and 2. The object is to use the numbers drawn to make the largest or smallest total, product or quotient. In each case, it is very valuable to have students first work out the best possible result. This tunes in all students and helps with place value concepts as they apply to subtraction, multiplication and division. For example, is 987 – 654 going to bring you the largest answer? Will 98 x 76 produce a larger product than 97 x 86? Does 987 ÷ 6 give the largest possible quotient?


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Pick It Out

Activity 20 NumberStudent Name _______________________________


Game 1: Largest possible number tens of

thousands hundreds tens ones

thousands

Game 2: Largest possible number tens of


thousands

Game 3: Largest possible even number tens of


thousands

Game 4: Smallest possible number tens of


thousands

Game 5: Smallest possible odd number tens of


thousands

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Objective

To have students make connections between numbers using multiplication, division, addition and subtraction, and to employ associative and commutative properties of operations to make additional connections.

Organisation


Introduction

I have seen a number of variations of this activity, and what is presented here is akin to a definitive version. Students respond very well to the challenge inherent in this version as it allows them to contribute at their own level and is relatively open-ended in its format.

Procedure

For Years 5 and 6

Start by writing four or five numbers on the board, for example 6, 4, 18, 24, 3. Challenge students to find, as a class, at least 12 connections between the numbers. A connection exists when you can do something to two of the numbers on the board and it produces a third number that is also on the board. For example, 4 x 6 = 24 is a connection because it uses three numbers, and all the numbers are on the board. 3 x 4 = 12 is not a connection; it is an equation, but there is no 12 on the board.

Tell them that no student can present more than one possibility.

If students have not suggested it by now, point out that if you know something like 4 x 6 = 24, then there are three other things you also know. It’s a bit like buy one, get three for free!

Having come up with 4 x 6 = 24; 6 x 4 = 24; 24 ÷ 6 = 4; 24 ÷ 4 = 6 and done the same for 3 x 6 = 18, students may stall for a little while until they look harder and realise that 24 – 6 = 18, so 18 + 6 must equal 24; and 18 – (3 x 4) = 6 so 18 – (4 x 3) will also equal 6, and it follows that 3 x 4 + 6 must equal 18. You may even have a student tell you that 24 ÷ 3 + 6 + 4 = 18 and thereby make a connection using all five numbers.

Activity 21 Connect the Numbers

FOCUS Operations



EQUIPMENT One worksheet (page 58, 59, or 60, depending on the year level) per student

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Students should now be primed and ready for the activity. Give each student a copy of the appropriate worksheet.

Explain to students that every time they are able to find a connection between any three (or more) numbers on the worksheet, that earns them a point. I generally set every student in the room a minimum of 20 to 25 points. I then go to individual students and inform them that I have different expectations of them. I do this quietly and quickly, simply saying, ‘Anna-Maria, 30 points for you’; ‘Kevin, I think you should get 35 points’, ‘Inga — it’s 40 points for you.’

I approach students who I know will struggle with the set target individually, ensure that they understand the task, and then get them started by doing the first one or two with them.

You can choose to target certain skills or knowledge and, as an incentive, offer extra points for these. For example:

– If students can square a number and find the answer among the other numbers, they get 2 extra points.

– If students can use 3 numbers to make a connection to another number, they get 3 extra points (and for 4 numbers you get 4 extra points, etc.).

– If students can form a fraction from two numbers and use it on another number and the answer to that is also there, you get 5 extra points (for example, 2−3 of 18 = 12).

– If students can use brackets in a mathematically appropriate way, they get 1 extra point.

– If students can make a connection that includes a number marked with a hash (#), they get 2 extra points.

Variations

With Year 2 students, ask them to find connections using addition and subtraction. Each connection should be recorded on the appropriate worksheet. They receive a point for each connection, and their aim is to get to 15 points/connections. Give an extra point for any multiplication/division connections students find.

With Years 3 and 4 students, ask them to look for addition, subtraction, multiplication and division connections, with an extra point being given for any four-number connections. Each connection should be recorded on the appropriate worksheet. Students should aim to find 20 connections.


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Connect the Numbers 1

Activity 21 Number

Try to find 15 connections between these numbers. A connection uses at least three numbers, for example 10 + 5 = 20.All the numbers in the connection must be on this worksheet.

Student Name _______________________________


3 12 5 30 16 9

20 4 20 100 6 2

200 18 10 7 50 8

15 28 14 21 40 25

Year 2

Write your connections here:

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15.

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Activity 21 Number

Find connections between these numbers. A connection uses at least three numbers, for example 3 x 10 = 30 is a connection. You can only use the numbers on this worksheet to make your connections.

Student Name _______________________________Years 3 and 4

3 12 5 30 16 9

20 4 20 100 6 72

200 18 10 7 50 8

15 28 14 48 40 25

1 000 54 60 64 32


Write your connections here:

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

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Activity 21 Number

Find connections between these numbers. A connection uses at least three numbers, for example 6 x 12 = 72 is a connection. You can only use the numbers on this worksheet. If you use a number marked with a # you earn 2 bonus points.

Student Name _______________________________


35 12 500 30 16 9020 4 206# 100 6 72 200 18 10 77# 50 8815 28 14 48 40 25 1 000 54 60 64 32 596 71# 100 000 20 1 000 000# 3 7 624# 8 720 7 200 101#

Years 5 and 6

Record your connections here: 1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25.

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Objectives

To reinforce division and multiplication connections between given numbers and to apply knowledge of zero as a place value holder in finding extended connections.

To revise and give practice in dividing by multiples of 10, 100, etc.

Organisation

This game works well with students sitting in groups of four, five or six at their tables.

Introduction

Students respond to the challenge of this activity and the fact that it is an enjoyable way to review and actively employ prior knowledge about multiplication/division and number facts associated with 10.

Procedure

For Years 4 and 5

Write three factors on the board, for example 4, 5 and 20. Above these numbers, write 16 zeros. For example, 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

4 5 20

Tell the whole class that you are going to give them 3 minutes to come up with as many multiplication or division connections as possible. If they wish to use any of the 16 available zeros, they should specify exactly how many before they read out their number sentence. Ask students to raise their hands, rather than calling out. Call on someone to give you an example of a number sentence that uses the three numbers/factors and one or more of the zeros provided, for example ‘I am using two zeros and my number sentence is 5 x 40 = 200.’ Cross out two of the zeros and record the connection. Tell students they now have 14 zeros and 2 minutes and 40 seconds remaining. Ask for another number sentence. Record connections and cross out used zeros until the 3 minutes has expired.

Activity 22 Grab a Zero FOCUS Operations, properties of operations, multiplying by tens



EQUIPMENT Pen and paper for each student


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Now place students in groups, and inform Group 1 that you are going to place three new numbers on the board as well as 16 zeros, just for their group.

Appoint or have the group select one student who will record the number sentences/equations/connections the group produces within the 3 minutes. (This time can and should be varied in accordance with the general abilities of the class.) Another student should be allocated the announcer role. At the end of the 3 minutes, they will call out to you the number of zeros used and the equations created.

Each number sentence that is true will earn 1 point. No points for incorrect equations. One bonus point if exactly 16 zeros are used.

Give students the following ‘handy hints’ before they start: students should try to use as many of the zeros as they can; it’s a good idea for the recorder to copy the zeros and numbers as they are presented on the board; group members should relate their equations to their recorder in an orderly way; the group must ensure the number of zeros used does not exceed 16 — after the zeros are used up, any equations using zeros will be disallowed; students should remember their ‘turnaround’ or commutative property facts (if 4 x 6 = 24, then you know that 24 ÷ 6 = 4); and every group member should contribute to the group’s effort.

While Group 1 has their turn, students in the other groups should practise by attempting to write as many equations, collectively or individually, as possible.

Announce when Group 1 has 2 minutes, 1 minute, 30 seconds and then 10 seconds remaining. At the conclusion of their time, the announcer should call out the zeros used and the resultant number sentences. Cross out the zeros as they are announced and record the number sentences underneath or alongside the zeros and factors. Total the number of accurate equations. Give a bonus point if all zeros were used.

Ask Group 2 to get themselves organised with a recorder and announcer, briefly review your ‘handy hints’ and then place 16 zeros and three more factors on the board.

The following groups of factors and zeros work well at the Year 4 and 5 levels:

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 27 9 36 4 9 7 6 42 8 32 4

Variations

For Year 6 students, use 20 zeros and comparatively more challenging factors such as 54, 3, 18; 4, 72, 18; 12, 7, 84; 16, 80, 5. I usually specify that they can only make division connections.

With Year 3 students, place 12 zeros and three factors on the board and ask them to find as many multiplication facts as they can in the allocated time, say 4 minutes. Any division number facts can be awarded a bonus point.

You can readily vary the number of factors and time allocations to suit the ability levels of your students.


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Objective

To give students confidence in moving between different forms for numbers smaller than 1 by having them add and/or subtract fractions, decimals and percentages to give a total of 1.

Organisation


Introduction

Fractions often fill primary students with dread, but students consistently respond well to ‘Match-up to 1’. The wide range of fractions, decimals and percentages included allows students to contribute at their own level, and I have seen some excellent work from students of all abilities with this activity. You can make this activity more or less challenging through the 26 rational numbers you choose.

Procedure

Give each student a copy of the worksheet.

Tell students that their mission is to find pairs — or more — of cards that when added together will equal exactly 1. The same number cannot be used twice in an equation so, for example, b + b (0.5 + 0.5) is not permitted. It is also acceptable to go past 1, then subtract to bring the total back to 1. (Be prepared for some students to provide efforts that are 7, 8, 10 or more cards long.)

Students must record each attempt as a number sentence. Ask them to include the letter that appears on each card as well, for example:

d + r

3−4 + 0.25 = 1

Tell students that if they use two cards to make an equation that equals 1, they earn 2 points; if they use three cards, they earn 3 points, etc. The aim is to earn 15 points. (You may wish to tell some students that they need to earn a greater or smaller amount of points.)

Remind students that there is a big difference between 0.05 and 0.5.

Encourage equations that are a mix of fractions, decimals and/or percentages. I occasionally give a bonus point to particularly well-thought-out equations.

Activity 23 Match-Up to 1 FOCUS Adding and/or subtracting simple

fractions, decimals and percentages



EQUIPMENT One worksheet (p. 64) per student


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Match-Up to 1

Activity 23 Number

Add two or more cards so that they equal 1 exactly. You can also go past 1 and then subtract to bring yourself back to 1.Record each attempt as a number sentence. You cannot use the same number twice in a number sentence.

Student Name _______________________________

Record your number sentences here

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.


a b c d e

k l m n o

v w x y z

q r s t u

f g h i j

p

710

10100

510

14

12

25

48

210

35

45100

0.5 25% 34

30%

0.75 90%

0.05 75% 0.4 10%

0.8 0.25 60%

0.1 50% 0.6

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Objectives

To have students compare and order fractions and express these fractions in terms of decimals or percentages in order to make comparisons.

To recognise equivalences of fractions if or when they occur in the context of this activity.

To have students use calculators to convert fractions to decimals or percentages in order to make comparisons.

Organisation

Students work in pairs. Have them seated at their tables/desks, preferably sitting opposite one another.

Introduction

This activity is an example of how a basic little game can lead to some worthwhile revision of fractions and some advanced maths thinking.

Procedure

One of each pair should be the dealer and the other the scorer.

The dealer deals two cards to the scorer and two cards to her- or himself.

Each partner then needs to place one card above the other so one represents the numerator and the other card the denominator. Aces equal 1, and the other cards take their face value. The objective is to make the fraction that is as close to one as possible.

When each partner has made a final decision as to how to place their two cards to make the fraction closest to 1, they compare fractions. The person with the fraction closest to 1 earns 1 point.

Often determining who is closest to one is relatively simple. For example, anyone fortunate enough to receive a six and a six ( 6−

6 ) has exactly 1 and will be hard to beat. If one partner receives a seven and an eight ( 7−8 ) and the other an ace and 5 (1−5), there will be little difficulty in acknowledging that 7−8 is significantly closer to 1. But what if one partner is dealt a five and a seven ( 5−7) while the other partner receives a two and a three ( 2−3 )? To work this out —and so that there is an accepted procedure to resolve any disputed calls — I have the pairs do one or more of the following:

Activity 24 Closest to 1

FOCUS Comparing and ordering fractions



EQUIPMENT Pack of playing cards with all picture cards removed

Pen and paper for each student

Calculator for each pair

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– work it out by finding the lowest common denominator, then comparing fractions

– if possible, work it out by mentally converting the fraction to a decimal or percentage, then comparing

– work it out by using the calculator to convert the fraction to a decimal and/or a percentage, then comparing.

In the instance of 2−3 and 5−7 , the pair would need to establish that 21 was the lowest common denominator, so that 2−3 is 14−21 and 5−7 is 15−21. The closest to one is 5−7, by a mere 1−21, and that partner earns a point.

Mentally converting 2−3 to a percentage or decimal is achievable (0.66 or 66.66%) as it is a well-known and often used fraction, but the same is not true of 5−7 — so bring out the calculator!

5−7 expressed as a percentage is 71.428 etc. or 71%, rounded down. This is closer to 1 than 67%, so 5−7 is closer to 1 than 2−3.

There will be many close calls, but if the partners become proficient at using some or all of the three options above to determine closeness to 1, there will be little cause for angst. Although the calculator is a viable and valuable asset in this activity, ensure you roam around the pairs and not only help settle any disputes (if necessary) but also observe how well (or otherwise) students are able to work with fractions.

Encourage mental computations such as when comparing 1−3 and 2−5 where a student may say that 1−3 is 33% and 2−5 is the same as 4−10 and that’s 40%, so 2−5 is closest to 1. Encourage and positively reinforce students who recognise equivalences, such as when one partner has 2−4 and the other 5−10.

Students may deliberately choose a mixed number. For example, in the case of five and seven, is 5−7 the only option? Is 7−5 actually closer to 1? It’s 1 and 2−5 as a mixed number, which in turn is 1 and 4−10, which is 1.4. Good try, but 0.71 is closer to 1 than 1.4.

You can vary this activity to suit your students, but I generally have them play to see who is first to reach 15 points.


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Objective

To have students identify attributes in order to then sort and classify objects or people. From this sorting, students should then be able to make and justify simple statements based on the data.

Organisation

This activity requires as much available space as possible, as it will result in groups of students standing with their arms linked. When choosing an area or space to conduct the activity, try to ensure that students seated on the floor can clearly see what is happening when other students come up to link arms.

Introduction

This activity will get students standing, moving, observing and thinking. It is a dynamic activity that involves students physically, and for this reason it is an excellent activity to start the day off with. It is not a one-off, and can be used as part of your classroom routine on a regular basis.

Procedure

I initiate this activity by looking at students and choosing an attribute by which I can distinguish different groups of them. It could be, for example, hair colour or eye colour (obviously in some classes these may not work); mode of transportation (students who arrive by bus, are driven to school, walk); names (first names starting with letters in the first half of the alphabet, etc.); number of letters in first names; siblings (those with a brother at the school; a sister; both; neither); birthdays (those with birthdays in January, February, etc.). The important thing is to not announce the particular attribute you intend to employ.

If hair colour were the attribute chosen, you might observe that three of your students have black hair. Call them by name and direct them to stand in a specified area with their arms linked. You then ask the seven students with dark brown hair to stand nearby but distinct from the other group. Three students have blonde hair. Ask them to form their own arm-linked group near to, but not touching, the other groups. Two have red hair. Have them stand near but not mingling with another group. Three more have light brown hair. Have them stand as a distinct group.

Activity 25 Thinking Linking 1

FOCUS Sorting/classifying people by attributes



EQUIPMENT None

Maths on the Go Book 2 CHANCE AND DATA

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During this sorting process there will be much speculation and discussion by students as to what it is exactly that you are doing. As they perceive that you are sorting by some undisclosed criteria or attribute, they will start trying to work out why they personally are together as a group and then how or why other groups have been formed.

At about this time I generally say to students. ‘You can see I have sorted you into different groups. You have already been talking about it but now drop arms and talk again with people in your group about how I have sorted you and all the other students.’

Only rarely have I found it necessary to provide a hint or clue. There is usually an individual or group of students who will have identified the attribute you have applied to sorting the class. If a hint is required, don’t make it too obvious.

Now draw two students to one side and inform them that they will now be sorting students using any attribute of their choice. They should be given 2 or 3 minutes of discussion and selection time. Sometimes students will need your assistance or guidance with this, but more often than not they will come up with a very appropriate and feasible way to sort their peers.

I choose not to know the attribute the pair has decided on. In this way I too can be sorted (as regularly occurs) and be no wiser as to how this has happened than the students. On a regular basis, one of the students will have identified the attribute before I have. The student who initially and correctly identifies the attribute being applied can then choose a partner, decide on a new way to sort the whole class, and then place students into linked groups accordingly. Once again, it is up to those being sorted to try and identify how this is being done.

Once students have been sorted like sheep on three or four occasions, call a halt. Tell those who have not yet had a turn at sorting that you will be doing this activity again soon, and that you have noted who had an opportunity today (use your class list of names, with the date and the activity) and that different students will get their chance next time. This tends to appease the masses.

Variations

This activity can be gainfully used to tally, record and analyse data. How many students have a brother and a sister? Are there more people with blue eyes than brown eyes?

The concepts of ‘same’ and ‘different’ can be developed in students through free and directed play using materials such as attribute blocks; a large supply of buttons of various shapes, sizes, textures, colours and number of button holes; ‘people’, fruit, dinosaur or transport sets (and many other similar products and materials). Students will readily see that sometimes applying a single attribute such as colour still leaves many options, and that they may need to go to a second attribute such as size to sort even further. Young students are capable of sorting buttons by multiple attributes such as colour, shape and number of button holes, for example.


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Objective

To have students identify attributes in order to then sort and classify objects or people. From this sorting, students should then be able to make and justify simple statements based on the data.

Organisation

This activity requires as much available space as possible, as it will result in a circle or line of students standing with their arms linked. When choosing an area or space to conduct the activity, try to ensure that students seated on the floor can clearly see what is happening when other students come up to link arms.

Introduction

This activity will get students standing, moving, observing and thinking. It is a dynamic activity that involves students physically, and for this reason it is an excellent activity to start the day off with. It is not a one-off, and can be used as part of your classroom routine on a regular basis.

Procedure

Have students seated on the floor where they can see you clearly. Stand with your right hand on your right hip and your left hand on your left hip so that you have formed a < and a > shape on either side of your body.

Make a statement such as ‘I have two brothers’; ‘I am wearing shoes without laces’; ‘We have a pet dog’; ‘I wear glasses’; ‘We drive a Ford’; ‘I have no sisters’; ‘My favourite colour is blue’; ‘My favourite number is 8’; ‘I like apples more than bananas’ or ‘We go to the beach at summer time’. The statement in itself is relatively unimportant; it just needs to be something readily understood by students in your class and something that they can relate to. Standing with your hands on your hips and making a statement like ‘I have no sisters’ can leave you sounding and looking just a little ‘different’ as students will have no real idea of what is required of them. On a few occasions I have actually had students, with no prompting whatsoever, come out and join me.

I generally follow my original statement with something like, ‘and I’m wondering if anybody is like me. That is — they have no sisters.’ At this point, those students who do not have sisters will walk over to you.

Activity 26 Thinking Linking 2

FOCUS Sorting/classifying people by attributes, collecting simple data



EQUIPMENT None

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Invite them to link arms so that there is a group of you who do not have sisters all standing with your arms interlinked.

Now invite either of the two students at the end of the chain to make a statement. Students who are currently seated listen to the statement, judge whether the statement holds true for them and then either come out and link arms with the person who made the statement or the person at the other end of the chain, or remain seated. This new statement does not have to be related to the one that you made. ‘I have no sisters’ can be followed by ‘I don’t like carrots’, or something equally diverse. Anyone who shares a similar dislike for the carrot should now leave the floor and link arms with a student at either end of the chain.

One of the two people at either end of the growing chain should now be asked to make a statement, and so on. If no-one is drawn by the statement, ask the other person at the end of the chain to try. The statements I’ve heard over the years have ranged from the very funny to the sad to the absolutely weird! Be prepared for just about anything. Students will often follow your lead, so if you’ve begun proceedings with a statement about your family, they’re likely to stick to that theme. Stress to students that this is not necessary and that their statement can be about anything.

Continue with the activity until all students in the class have been able to link arms with someone else.

If this activity is conducted on a regular basis, the quality of the statements simply continues to improve and it can be almost as if they have rehearsed their statement in readiness for the next time you play.

Variations

The data from this activity can easily be used to make Venn diagrams. One circle entitled ‘No sisters’ may contain a number of names. Another circle entitled ‘Dislikes broccoli’ may contain a number of different names. But what of those students who do not have any sisters and who dislike broccoli? Where do we place them? Even with Years 0 and 1, it is possible to have three overlapping circles. With ‘No sisters’ and ‘Dislikes broccoli’, you may have had a third group of students under ‘We have a red car’. It is then possible that you may have in your class a broccoli-disliking student with no sisters whose family has a red car.


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Objective

To have students appreciate that data can be represented in a dynamic and physical form (in this instance, a human column graph) and that from this data conclusions and inferences can be drawn.

Organisation

Have lower primary students seated on the floor to start the activity. You will need a long, straight, open area such as along the front, sides or back of the room. This area should be free of equipment and materials so that students have sufficient space to line up one behind the other, perhaps five or six students deep. For Year 3, you may need to take students out into the corridor or some similar longer and larger area.

Introduction

‘Human Birthday Graph’ is a dynamic and visual activity. Students are actively involved in the lesson — in fact, they are the lesson! Data is so often found stationary in books and magazines and on TV and computer screens. This activity has living, breathing, 3D data — a human histogram. Students enjoy it and like being part of it because it’s about them.

The end result should be that you have all your students lined up in the month they were born in and that the line is in order from January through to December.

Procedure

For Year 0 students, I have found that it is a good idea to take some masking tape and lay it in one long line across the front of the room, allowing sufficient space for perhaps five or six students to line up behind any particular segment of the line. Then mark the line like this:

January February March April May June July August September October November December

For Years 1 to 3, and depending on your students, writing the months on a provided line can actually diminish the challenge inherent in the activity. It will usually suffice to have clearly visible a calendar with the 12 months listed that students can see at a glance.

Activity 27 The Human Birthday Graph

FOCUS Sorting/classifying people by attributes, collecting/interpreting simple data



EQUIPMENT Masking tape and a marker for Years 0 and 1

A calendar showing all the months of the year at a glance

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Having established that students know the twelve months of the year and are relatively comfortable with the order of the months, walk to one end of the imaginary line (or the real one, if you have Year 0) and announce to students that this end of the line is January. Walk to the other end and inform them that this end of the line or area represents December.

The next instructions need to be clear and precise and as unambiguous as possible. I usually say something like the following: ‘With this end being January and the other end being December and all the other months being in between, I want you to stand up quietly and place yourself in the month that you were born in.’ Repeat the instructions if necessary.

Position yourself near the line and assist students only if they are completely at a loss. Even then, provide hints and advice rather than telling them where they should stand. One of the valuable aspects of this activity is that it actively promotes cooperation: it is a whole-class effort to get every student in their birthday months and all the months in order.

Hints to be used if students are totally lost could include:

– If January is that end, and December is this end, what is a month that is in the middle of the year?

– Look at the calendar and see if your month is closer to January or December.

– Look at what month comes before the month you were born in and after the month you were born in. See if you can find someone from those months.

You may find there are students born in May who are placing themselves next to December; students born in June who are unsure where to go because there are no student birthdays in May or July; even students who do not know what month they were born in at all. However, you will have to solve few, if any, of these problems as students themselves are generally excellent at providing the necessary help and advice that sees someone finding their appropriate position along the birthday line. It’s a great cooperative activity.

When students eventually have themselves lined up, check the accuracy of the line by starting at January and saying ‘Hands up January birthday people’. Do this all the way through to December.

Once it has been determined that the line is, in fact, an accurate representation of which students have their birthdays within which months, ask one of the students to come out to where you are standing, so that they have a full view of the human birthday graph. Ask them to tell you one thing they notice about the data assembled in front of them. By placing the onus on the individual student to tell you something, rather than asking a question about the data, you are inviting him or her to take a broader view and comment on something that they personally have noticed. I have consistently been pleased and impressed by the ability of young students to see so much in the data.


Students are likely to tell you things such as that (for example) there are no birthdays in April and September; the most birthdays happen in May; more people are born in the first six months of the year than the last six months; three months have three people born in them; eight people were born in May and June altogether.

If necessary, ask some specific questions of students — but there is great value in allowing and encouraging them to make observations and interpretations about the data.

I normally seat students again by saying things such as ‘If your birthday month has four letters in it, please sit down. If your birthday month is next to September, please sit down. If your month ends in the letter r, please sit down. If your month starts with the letter m, please sit down. If your birthday month ends in a y, please sit down. If your birthday month is the same as Mr V’s, please sit down!’

Variations

For Year 3 (and Year 2 students, if you think they will cope), challenge them to not only line up in their birthday month but in order within their birthday month. The students within May, for example, then need to determine the order of their actual birth dates and line themselves up within May from first birthday through to last birthday in May.

For Year 3 (or older students), I challenge them to find their birthday month and line themselves up in order within their month without saying one word! Signals are fine; even mouthing the month or date is okay, but there is to be no audible communication. This adds a different perspective to the activity and heightens students’ need to work cooperatively and communicate effectively.

Older students can do the ‘Human Birthday Graph’ and use the data to find mean, mode, median and range.

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Objective

To have students make reasoned decisions based on the likelihood of an event occurring.

Organisation

‘Four Corners’ is a dynamic whole-class activity that involves students frequently moving from one side of the room to the other. Given the movement and enjoyment level of ‘Four Corners’, whatever you can get out of the way to facilitate movement within the room is a good thing!

Introduction

It may be worth your while at the outset to establish or elicit the following from your students: that there are 52 cards (minus jokers) in a deck of playing cards; that there are four suits, or types, of cards: hearts, diamonds, spades and clubs; that hearts and diamonds are red cards and spades and clubs are black cards; that in any deck there are picture cards (Jacks, Queens and Kings) and aces as well as number cards from 2 to 10; that there are four each of every number card and picture card — so there is, for example, a ten of hearts, a ten of diamonds, a ten of clubs and a ten of spades. Inform students that for today we will be playing the game without the picture cards.

This activity is about luck, but more pertinently it is about informed decision-making based on data, making predictions based on probability and a good dollop of common sense. Plain old-fashioned luck will be a significant factor in the activity, though, despite the best applications of mathematics.

Procedure

For Year 3

Place a clearly visible club in one corner of your classroom and a spade, a heart and a diamond in the other three corners. Large self-stick notes work well for this, and it takes about 5 minutes using a red and a black marker.

Activity 28 Four Corners

FOCUS Probability



EQUIPMENT Pack of playing cards with all picture cards removed.

A large spade, club, heart and diamond

A large supply of counters

Calculators for Years 4 to 6 (optional)

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Tell students that we are going to play a game called ‘Four Corners’. Point out the picture in each corner of the room. See if they can name the spade, club, heart and diamond.

Take them through the essentials of what constitutes a deck of playing cards (see the Introduction) and tell them that the game of ‘Four Corners’ is all about trying to work out what type or suit of card you, as the teacher, are going to flip over.

Inform students that before you flip or turn over the card that is at the top of the deck, the students need to go to the corner of the suit that they believe will be turned over.

Before students make their decisions, verify with them what the chances or likelihood of a heart, diamond, club or spade being turned over are. They should realise that at this stage all suits have an equal chance. There are 40 cards in the deck. There are 10 of each of the four suits (since picture cards have been removed). All suits have a 10/40 or 1−4 or 25% chance of being turned over.

I regularly use the following display on the board

Diamonds Hearts Spades Clubs

Ace Ace Ace Ace

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

6 6 6 6

7 7 7 7

8 8 8 8

9 9 9 9

10 10 10 10

This tally board helps the students appreciate the vital role data plays in informing decision-making. Tell students that whenever a card is turned over, you will record that card on the tally board. Any turned-over card stays out of the deck.

Now you have two options. Both work well and it really depends on the age of your students. With Years 1, 2 and 3 students I simply have a large pile of counters readily available, enough to supply about 5 counters per student. For older students, I issue each of them with 5 counters before the activity begins.

Now that students know that each suit has an equal chance of being turned over, have them make their decision. Emphasise that students need to make their own decision rather than simply following a friend. They should quietly walk over to the corner of the room of the suit that they believe you will turn over.

It is certainly not always the case (especially with Years 1 and 2 students — baaa syndrome!), but often you will get a reasonably even spread of students across the four suits. Give students one last opportunity to change their minds, then flip the first card over. And it’s a . . .


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A sample game (for Year 3)

The first card you have turned over is a diamond. There’s no getting around the fact that there is going to be a significant amount of cheering and moaning (about 25/75 proportion wise) in the room. It’s understandable — just close your door!

Go to the group of students congregating around the diamond and hand each of them a counter. This is the point at which I announce to students that each time they are fortunate or ‘wise’ enough to choose the matching suit to the card I turn over, they will receive a counter.

For Years 4 to 6 students, I start them with 5 counters and remove one from each student not standing adjacent to the diamond. This pains them greatly — I enjoy it!

Go to the ‘Four Corners’ tally board and record that it was, in fact, the 6 of diamonds that was turned over by marking a cross next to the 6 in the diamonds column.

Invite students to move to another suit if they wish. Emphasise that they do not have to move. When all have chosen a suit, ask four or five individuals whether they have moved or not. If yes, why? If not, why not?

Turn over another card. It’s a . . . diamond (10 of diamonds).Those who remained with the diamonds will be ecstatic. Others won’t believe their bad luck, as they considered that diamonds had a lower chance of being turned over since one of them had already occurred. (This is correct mathematically— although by a slim margin. The odds are 10/39 or 25.6% for hearts, spades and clubs and 9/39 or 23% for diamonds, and this is the maths I have the Years 4 and 5 students working out on their calculators.)

Distribute counters to those standing in the diamond corner. (Some students may now have two.)

Mark a cross next to the 10 in the diamonds column.

Tell students to walk quietly to a new suit or remain where they are if they wish to. Once students have made their decisions, ask three or four students for their reasoning, and be sure to help them in making the mathematics explicit.

And the next card is a . . . club! And so on.

Continue the game until one or more students have received 5 counters.


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Objective

To have students understand that some events have an equal chance of occurring while other events have a greater probability of occurring, and that data can help us identify these events and make predictions accordingly.

Organisation

A whole-class activity. Make sure all students can see the board. Ensure that you have a reasonably open area where the tossed coins can land relatively unhindered.

Introduction

‘Two Coins on a Ruler’ blatantly cashes in on two great interests that students frequently share — money and guessing games. It provides ample evidence that maths doesn’t have to hurt, and that enjoyable and engaging activities can allow students to discover vital mathematical concepts and ideas. The activity is based on the premise that when you flip one coin there are only two options — a head or a tail — and both options have an equal chance of occurring. There is one head out of two options, so there is a 1−2 or 50% chance of a head occurring. The same obviously applies to tails.

What, then, if we introduce a second coin? What can happen now? What is most likely to happen now?

Procedure

Produce one of the 20 cent coins. Remembering not to presume anything (especially of the Year 2s and even some Year 3s), ask students what you call the side of the coin that has a profile (or relief) of the Queen’s head on it. Heads! And what do we call the side of the coin that isn’t heads? It doesn’t have a picture of one but, yes, we call it tails.

Tell students that you are going to flip or toss a coin into the air but that before you do this everyone has to try to predict whether it will land with the head side facing up or the tail side facing up — these are the only two possible results.

Activity 29 Two Coins On A Ruler

FOCUS Probability, data interpretation



EQUIPMENT Two 20 cent coins

Standard 30 cm ruler

One worksheet (p. 80) per student for Years 4 to 6

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For this one-coin part of the activity, I ask that students indicate whether they believe heads or tails will occur by doing the following: if they believe it will be heads, they should place both hands on their head; if they believe it will be tails, they should place both hands on their tail (hips is close enough!). Having students indicate their choice visually enables you to make some general summations about the data; for example, ‘It looks like more than half of you are going for tails.’

Flip the coin using finger and thumb or simply place the coin towards the end of the 30 cm ruler and throw it in the air. When the coin has come to rest, but before you reveal the outcome, tell students to make sure they keep their hands where they currently are.

Say, for example, it’s heads. Award the students with hands on their heads 1 point (or give them a counter). Immediately after doing this, go to the board and record the result; for example, 1. heads.

Continue this sequence (students choose heads or tails, coin is flipped, correct choosers receive a point, result is recorded) for 20 flips of the coin.

Now check if anyone has more than 15 points. Who has between 10 and 15? Who has between 5 and 10? Discuss the strategies, if any, used by anyone who has more than 15 points.

Interesting aspects to discuss include:

– Even after three heads in a row, the likelihood of a tail on the fourth flip is still 50/50 or 50%. Yes, it should even out over time or number of flips, but technically each time you flip it is 50/50.

– After 20 flips you may have an uneven count of something like 13 heads and 7 tails. Students may argue that the results should be 10 outcomes each for heads and tails. Very good point, but explain that the sample size affects the likelihood of an even outcome: the larger the sample, the more likely is an even result. 1000 is often considered to be a minimal sample size. I’m sure you will have a volunteer or two willing to be given time to conduct a 1000 flip experiment!

With Year 2 students, you may wish to leave the activity here. With Years 3 to 6 students, ask what would or could happen if you now introduced a second coin.

Without any pre-discussion of the mathematics involved, issue students with the worksheet. Adopt the same procedures as for one coin. Ask students to choose heads and heads, tails and tails, or heads and tails by circling the appropriate combination on the worksheet. Flip the coins using a ruler with the coins placed side by side towards the end of the ruler and announce the outcome. Award those students who selected the correct outcome with a point (or give them a counter). Record the result on the board (so students can see any trend in the results and can make their next choice based on the data).

Follow this procedure for 20 flips of the coin.

After 10 tosses of the coins, draw students’ attention to the results of each toss. It is highly likely that there will be more heads plus tails combinations recorded than heads/heads or tails/tails. One of the


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students will notice this. Ask that student specifically, or the class generally, why this would be the case. If they are unable to answer, ask students to tell you all the possible combinations that can occur when you flip two 20 cent coins. They are: head/head, tail/tail, head/tail, and tail/head.

It is important that students understand that there are two ways to achieve the outcome of a head and a tail: a head can land and be followed a millisecond later by a tail, or a tail can land and be followed by a head. Head plus tail and tail plus head gives you the same outcome, but there are two ways to achieve this outcome. With heads/heads or tails/tails, there is only one way to achieve the outcome. This means that there is double the chance of getting a head/tail than a head/head or a tail/tail outcome.

Now that students have this new knowledge and understanding, return to the activity for the final 10 tosses.

When 20 tosses have been completed, ask students to be seated and discuss strategies used.

Ask students to predict what the data might look like if there had been 60 tosses of the two coins; 80 tosses of the two coins; 100 tosses of the two coins.

Variations

For Year 6 students, you might like to try this activity using three coins. Have students first list all possible combinations. Balance the three coins on the end of your ruler. Have them predict the outcome of the toss on paper. Record the result and have students refer to the data to inform their predictions.


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Two Coins on a Ruler

Activity 29 Chance and Data

Circle one of the coin combinations before each toss of the coins. After each toss, place a tick alongside the coin combination that actually resulted. A circled combination with a tick next to it means a correct prediction.

Student Name _______________________________

Toss Head and Head Tail and Tail Head and Tail/ Tail and Head

1 H H T T H T/T H

2 H H T T H T/T H

3 H H T T H T/T H

4 H H T T H T/T H

5 H H T T H T/T H

6 H H T T H T/T H

7 H H T T H T/T H

8 H H T T H T/T H

9 H H T T H T/T H

10 H H T T H T/T H

11 H H T T H T/T H

12 H H T T H T/T H

13 H H T T H T/T H

14 H H T T H T/T H

15 H H T T H T/T H

16 H H T T H T/T H

17 H H T T H T/T H

18 H H T T H T/T H

19 H H T T H T/T H

20 H H T T H T/T H


Mental

Computation

Measurement

Chance and Data

Space

Number

Book

2

Rob V i ngerhoets

Book

2

Book

2

Also availableMaths on the Go

Mat

hs on t

he Go

Rob Vingerhoets

More ideas from the best-selling author of Maths on the Go Book 1 !

Book

1ISBN 0732978807 ISBN 9780732978808

About the authorRob Vingerhoets is an experienced primary maths teacher and author. He is a popular and engaging presenter of maths ideas for busy teachers and understands the demands of a full curriculum. His books show how it is possible to squeeze more maths into less time!

The activities in Maths on the Go Book 2:• are easy to organise and implement• require minimum equipment, or none

at all• are easily adapted across a range of

year and age levels • cover the important content strands:

Mental Computation, Measurement, Space, Number, and Chance and Data

• can be readily used as part of any maths program

• are perfect for revision or extension.

Make the most of every minute of the teaching day . . .• 30 minutes before morning recess• 20 minutes between library and lunch• 5 minutes at the end of the day.

5to

45 m

inute

mat

hs ac

tivitie

s for all primary levels

the activities in book2 maths on the go book...

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