Flexible Mathematics Thinking - Foundation to Year 3 by Teacher Superstore

RIC-6092 4.4/108

Flexible mathematics thinking (Foundation to Year 3)

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© Australian Curriculum, Assessment and Reporting Authority 2012. For all Australian Curriculum material except elaborations: This is an extract from the Australian Curriculum. Elaborations: This may be a modified extract from the Australian Curriculum and may include the work of the author(s). ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. In particular, ACARA does not endorse or verify that: • The content descriptions are solely for a particular year and subject; • All the content descriptions for that year and subject have been used; and • The author’s material aligns with the Australian Curriculum content descriptions for the relevant year and subject. You can find the unaltered and most up to date version of this material at http://www.australiancurriculum.edu.au/ This material is reproduced with the permission of ACARA.

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FOREWORD By the time students reach Year 3 they are presented with a NAPLAN test. It is important that before students reach Year 3 they are taught flexible ways to think. This will improve their mathematics and develop lifelong skills that can be used as they progress along the mathematics continuum. The purpose of this book is to look at early childhood strategies which give students the opportunity to use mathematical manipulative materials and graphic organisers before moving onto abstraction. The important issue is action before abstraction.

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Many early childhood strategies can be motivated by storybooks or rhymes that focus on mathematics ideas. This book will endeavour to link storybooks to flexible thinking, word problems and number sentences. Because Year 3s are expected to answer word problem questions in the NAPLAN test, it is important to link the stories to mathematic manipulatives, graphic organisers, pictures, drawings, word problems, number sentences and symbols. Links are made to the Australian Curriculum for Foundation to Year 3. Word problems can be stated in a number of ways. ‘First steps in number’ (number operations) focuses on three important ways in which number sentences can be written to reflect the semantic structure of the word problem. Young students need to recognise these three situations and understand how addition questions can be solved by using subtraction. It is important that students know both the addition and subtraction action, as it leads to writing number sentences and solving word problems. Subtraction, in terms of thinking, is more difficult than addition, and time should be spent on making sure the jump to abstraction is not too fast.

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If we want students to learn different ways of thinking then we need to clearly show the strategies that are available (direct teaching) and then give them time to practise the skill. We need to make sure the opportunities are fresh and that they allow students to make connections and see the relevance of what they are doing. The brain craves for this type of learning, and it is hoped that by using strategies in this book, you will be able to develop flexible thinkers in your classroom who will be able to tackle any mathematical situation that confronts them. Importantly, students also need to be taught strategies that assist them in thinking of solutions mentally. There are a number of important mental strategies that need to be explicitly taught to students so that, with practice, they develop their fluency through mental thinking processes.

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In the early childhood years there is a focus on number and measurement. Students also need to see what geometry, probability and statistics, and algebraic thinking is all about. Number is an important acquisition because the success of measurement, algebra and probability and statistics rely on a student’s understanding of number. It is important to make sure that measurement and data collections do not go beyond the students understanding of number. Therefore, number understanding is most important and many of the activities in this book concentrate on developing flexibility in number.

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Flexible mathematics thinking (Foundation to Year 3)

iii

CONTENTS Foreword ....................................................................iii

Three dice adding game 1 and extension .........41–42

Australian Curriculum links ........................................v

Three dice adding game 2 and extension .........43–44

Ten frame – flexible thinking to the number ten. ...........................................................................2–3

‘Make the Target number’ strategy .........................45

Partitioning grid. .....................................................4–6 Partitioning small numbers – One is a snail ten is a crab ....................................7–10

Target number – subtraction ....................................47 Using playing cards ...................................................48

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Sort the cards.............................................................11 Partitioning addition action number sentences......12

Two dice addition......................................................13 One more, one less....................................................14 Addition grid for solving word problems ................15

Subtraction action ...............................................16–17 Word problems – Chickens .................................18–21 Word problems – Rooster’s off to see the world.....22 Mental thinking strategies .......................................23 Count on ....................................................................24 One more, one less – addition and subtraction ......25

Subitising game ...................................................49–50 Two card shuffle ........................................................51

Two card shuffle and number sentences – 10 .........52 Two card shuffle and number sentences – 12 .........53 Three card shuffle – 17 .............................................54

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Target number – addition.........................................46

Three card additions .................................................55 In the middle – subtraction game ............................56 Writing number sentences .......................................57 Addition and subtraction problem types ..........58–59 Multiplication ......................................................60–62 Multiple sort ..............................................................63

Combinations of 10 ...................................................27 Using doubles ............................................................28 Using doubles to solve near doubles .......................29 More near doubles ....................................................30 Adding 9 ....................................................................31

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Adding 7, 8, 9 ............................................................32 Adding 11 ..................................................................33 Relating addition to subtraction ........................34–35

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Multiplying 2s, 3s, 4s, 5s and 10s ..............................64 Multiplying 2s and 10s ..............................................65 Multiplication ............................................................66 Money problems .......................................................67 What number am I? ............................................68–69 Open-ended money tasks ...................................70–71 Division ......................................................................72

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Make 5 .......................................................................26

Division grids .......................................................73–74 Division – When the doorbell rang ....................75–76 Balancing number sentences ..............................77–78

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Two six-sided dice adding game and extension ......................................................36–37 Two ten-sided dice adding game and extension ......................................................38–39

‘Three bear family’ mathematical manipulative........................................................79–82 Appendix 1–2.......................................................83–84

Three dice bingo .......................................................40

Flexible mathematics thinking (Foundation to Year 3)

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AUSTRALIAN CURRICULUM LINKS Description Foundation Subitise small collections of objects (ACMNA003)

Represent practical situations to model addition and sharing (ACMNA004)

Using subitising as the basis for ordering and comparing collections of numbers Using a range of practical strategies for adding small groups of numbers, such as visual displays or concrete materials

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Year 1 Represent and solve simple addition and subtraction problems using a range of strategies including counting on, partitioning and rearranging parts (ACMNA015) Year 2 Explore the connection between addition and subtraction (ACMNA029)

Developing a range of mental strategies for addition and subtraction problems

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Elaboration

Becoming fluent with partitioning numbers to understand the connection between addition and subtraction Using counting on to identify the missing element in an additive problem

Solve simple addition and subtraction problems using a range of efficient mental and written strategies (ACMNA030)

Becoming fluent with a range of mental strategies for addition and subtraction problems, such as commutativity for addition, building to 10, doubles, 10 facts and adding 10 Modelling and representing simple additive situations using materials such as 10 frames, 20 frames and empty number lines

Recognise and represent division as grouping into equal sets and solve simple problems using these representations (ACMNA032)

Dividing the class or a collection of objects into equal-sized groups Identifying the difference between dividing a set of objects into three equal groups and dividing the same set of objects into groups of three

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Describe patterns with numbers and identify missing Investigating features of number patterns resulting elements (ACMNA035) from adding twos, fives or 10s

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Solve problems by using number sentences for addition or subtraction (ACMNA036)

Representing a word problem as a number sentence writing a word problem to represent a number sentence

Year 3 Recall multiplication facts of two, three, five and ten Establishing multiplication facts using number and related division facts (ACMNA056) sequences Describe, continue, and create number patterns resulting from performing addition or subtraction (ACMNA060)

Identifying and writing the rules for number patterns Describing a rule for a number pattern, then creating the pattern

The above content descriptions have been reproduced with permission from ACARA. © Australian Curriculum: Assessment and Reporting Authority 2012

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Flexible mathematics thinking (Foundation to Year 3)

TEN FRAME When developing initial understanding of numbers up to 10, the ten frame, used with counters, should be introduced to show students that small numbers can be shown in a variety of ways. ‘Show me’ strategy • Show me 3 in the ten frame. Can you show me 3 in another way? • Show me 5. Can you show 5 in another way?

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• Show me 2. Now show me 3. How many counters do you have now? (addition concept) • Show me 3 and show me 4. How many counters do you have now? (addition concept)

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• Show me 2 and show me 4. How many counters do you have now? Can you shift one from the 4 to the 2? What do you have now? (linking the ten frame to doubling a number; mental strategy) • Show me 5 and show me 3. How many counters do you have now? Can you shift one from the 5 to the 3? What do you have now? (linking the ten frame to doubling a number; mental strategy)

Extend to use a balance and show what has so far been shown with the ten frames. Put 2 blocks on one side and 2 blocks on the other side of the balance. • On one side of the balance put 2 blocks, then 3 blocks. On the other side put 3 blocks then two blocks. What do you notice? What can you say?

• On one side put 5 blocks, then put 3 blocks. On the other side put 4 blocks and then another 4 blocks. What is happening? What can you say? •

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If you shift one counter from the 7 to the 5; you end up with 6 and 6 – making 12 (doubling). See mental mathematics strategies on page 23.

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Ten frame

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Flexible mathematics thinking (Foundation to Year 3)

PARTITIONING GRID – PART-PART-WHOLE This strategy teaches students to be flexible when thinking about any number. It gives the students the opportunity to link counters to abstract thinking and develops the initial addition concept with numbers up to 20. For this strategy, you wil need the partitioning grid (found on page 5). The teacher chooses a number; e.g. 12. Students write the number 12 in the box at the top. Students show 12 using a combination of counters in each partition of the grid. Students then complete the table below by putting in the two parts and stating what the partition is equal to. They then reform the counters showing a different combination and then record this combination in the table below. This is repeated until all combinations for that number are found. It is important to have students say what they have done i.e. 8 counters and 4 counters makes 12 counters.

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This strategy can be repeated with any number. You can differentiate within the classroom by giving some students the number 8, some the number 12 and so on. The strategy can be extended to have the students record the numbers in a number sentence using the initial language of addition.

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If you want students to use both the ‘show me’ and ‘part-part-whole’ strategy, they need to be taught what is expected of them and they need practice using the strategies over and over again. In the following two activities you will see a change in the partitioning grids to cater for students as they develop toward the abstraction of addition.

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Flexible mathematics thinking (Foundation to Year 3)

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Partitioning grid – part + part = whole Number:

part

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and

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Total:

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Show combinations with counters.

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Flexible mathematics thinking (Foundation to Year 3)

Partitioning grid – part + part = whole Number:

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Show combinations with counters.

part

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Total:

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PARTITIONING SMALL NUMBERS Read the book One is a Snail Ten is a Crab by April Pulley Sayre to the whole class for numbers 1–10. Re-read the story and ask students as you go through the story: • What other things or combinations can represent three? • What other things or combinations can represent four?

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• What other things or combinations can represent five? • What other things or combinations can represent six?

• What other things or combinations can represent seven? • What other things or combinations can represent eight? • What other things or combinations can represent ten?

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• What other things or combination can represent nine?

Some answers that students may suggest during the second reading of the story.

One person and a snail; tripod; triangle.

Two people; tripod and a snail, table; chair; car; square.

Person and a triangle. pentagon; two people and a snail.

Hexagon; two triangles; three people; a car and a trailer.

A square and a triangle; a tripod and two people.

Two cars; two chairs; an insect and a person; octagon.

Octagon and a person; an insect and tripod.

Insect and a table.

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Another book that you could read is Maths Fables by Greg Tang.

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• On the whiteboard list the different suggestions that students make. Ask them to choose a number, draw the objects and then use numbers to show the mathematics in the drawing. The following graphic organisers have been used to stimulate students to think about being flexible with numbers.

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One is a Snail Ten is a Crab is also a very good book to read for multiplication.

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Flexible mathematics thinking (Foundation to Year 3)

Partioning small numbers

6 feet

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Drawing

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8 feet

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Make your own

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feet

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How many feet? . te

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Flexible mathematics thinking (Foundation to Year 3)

Partioning small numbers

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How many feet are in these picture sentences?

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Sort the cards

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What animals would you choose to make your own number of feet in your story?

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Flexible mathematics thinking (Foundation to Year 3)

Partitioning addition action number sentences 12 9

= 12

Put numbers in the squares.

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r e 8o = 12 t s B r e 6 oo p = 12 u k S

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makes © R. I . C.Publ i cat i ons •f or r evi ew pur pose sonl y• and makes and

o c . Extension che e r o t r Make up a grid for the number 15 and write down some number sentences using s su r e p the words and and makes.

Flexible mathematics thinking (Foundation to Year 3)

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Two dice addition You will need 2 six-sided dice and some counters for this adding activity. Throw the dice, add the two numbers and put a counter on the number in the grid below. The winner is the first person to fill the card with counters.

Addition

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Extension:

Add these numbers together.

4 8 12

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3+4+2 4+5+3

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1+2+3

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Flexible mathematics thinking (Foundation to Year 3)

One more, one less Number

One more or one less

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one less

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one less

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one less

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New number

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one more

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Addition grid for solving word problems Richard found 3 sweets on the treasure hunt. Tricia found 5 sweets. Graham found 1 sweet. How many sweets did the children find? You need to add the parts to find the whole.

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Put counters in the grid to replicate what the problem is asking you.

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At the party 15 sweets were hidden in the garden. The three children found a total of 9 sweets. How many sweets are yet to be found? You have one part and the whole; you need to find the other part.

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Flexible mathematics thinking (Foundation to Year 3)

THE SUBTRACTION ACTION Just as we have been doing with addition, students need many opportunities to experience what subtraction is all about. Subtraction for young learners is much harder than addition and we need to spend time on developing the subtraction concept before moving onto abstracted forms using numbers. Once students have had the opportunity to manipulate counters using the subtraction action concept, then you can move them onto tabular formats and then onto abstracted forms. It is important once students understand subtraction that you give them the opportunity to solve word problems in both addition and subtraction situations

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The subtraction action strategy allows students to manipulate counters to reflect the action associated with a subtraction situation. This strategy was developed by David A. Sousa in his book Brain – compatible activities for mathematics Years K–1, 2010.

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1. What do you get when you subtract 3 from 8? 2. What is 9 take away 2? 3. 10 – 4 =

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Use the subtraction action strategy on page 17 to answer these subtraction problems and number sentences.

4. John had 12 lollies and gave 5 to his sister, Janet. How many lollies does he have left?

6. 15 –

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5. 12 – 5 =

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The subtraction action

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Word problem or number sentence:

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How many were taken away?

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How many left?

Flexible mathematics thinking (Foundation to Year 3)

CHICKENS This Kid Picks rhyme is about a mother hen and her brood of chickens.

Chickens

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Two little chickens pecking by the door, 2+ Along came another two, 2= And that made four.

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It is an opportunity for students to:

• interact with different magnet pictures and act out the story in pictures, words and symbols • count in twos

• add with small numbers using partition strategies and tables using counters • subtract using partition strategies and tables using counters • solve number word problems stated in different ways, including starting unknown problems and changing unknown problems using counters before moving onto abstraction • further develop the addition and subtraction word problems with the book Rooster’s off to see the world by Eric Carlés. This book develops both ideas of addition and subtraction. Using the book as a resource allows the development of addition and subtraction word problems which relate to the story. 18

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Chickens In the morning the hen house had 4 chickens in it, and in the afternoon it had 12 chickens in it. How many chickens came into the hen house between the morning and afternoon? (Part – part = whole.) In this problem we know the whole and one of the parts. We need to find the other part.

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There were 10 chickens in the yard; some were white and some were brown. If there were 6 white chickens, how many brown chickens were there?

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In the hen house there were 20 mother hens and chickens altogether. If there were 4 mother hens, how many chickens were there?

. te o hens c . chickens che e r o t r s super The chicken farmer had 20 mother hens on his farm. He gave some to his friend and now he has 12. How many did he give to his friend?

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Flexible mathematics thinking (Foundation to Year 3)

Chickens continued In the chicken’s yard there were 15 chickens. The farmer separated them into two yards. If 7 chickens went into one yard, how many chickens were moved to the other yard?

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r o e t s Bo r e p18 chickens; some were in o In the chicken’s yard there were the hen house and u some were in the yard. If there were 10 chickens pecking in thek yard, how many S chickens were in the hen house?

The farmer had 16 chickens on his farm; he moved some chickens to another farmer’s hen house. He has 6 chickens left on his farm. How many did he move?

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The farmer had 11 chickens. He bought some more and now he has 18 chickens. How many chickens did he buy?

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Stories

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Real things

A mother hen woke up and found she had only four chickens left. How many did she start with and what might have happened to the rest?

Symbols

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Pictures

Think board

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Flexible mathematics thinking (Foundation to Year 3)

Rooster’s off to see the world 1. There were 3 animals off to see the world. Some frogs joined them. Now there were 6 animals off to see the world. How many frogs came along for the trip?

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r o e t s Bo r e p ok u S 2. There were 10 animals off to see the world. Some fish joined them.

Now there were 15 animals off to see the world. How many fish came along for the trip?

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=3 + 5 = 15

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3. 15 animals arrived at the end of the trip. Some animals left, so only 6 animals remained. How many animals left?

Flexible mathematics thinking (Foundation to Year 3)

+2=6

= 10

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MENTAL THINKING STRATEGIES – KEY FOUNDATION TO YEAR 3 MENTAL STRATEGIES • When adding, start with biggest number first • Count on; only 1, 2, 3 • Partitioning: breaking up any number • Make 5

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• Combinations of ten: using the ten frame

• Doubles: 1 + 1 through to 10 + 10 (using two ten frames) • Using near doubles

• Adding 8: 10 take 2 • Adding 11: 10 + 1

• Skip counting: 2s, 3s, 5s, 10s, ……………

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• Adding 9: 10 take 1

It is important to give young students practice at seeing number sentences using small numbers in different ways. Introducing students to the commutative property; e.g. 8 + 4 = 4 + 8 is an important way to get them to think flexibly. Developing activities that link to subtraction is another skill which gives them the opportunity to think differently.

© R. I . C.Publ i cat i ons 1. If 2 + 3 = 5 then 3 = 5 – 2 and 2 = 5 – 3. This thinking must be shown with counters before students arer required tow calculate answers. youo have them time to •f or evi e pu r poOnce ses ngiven l y• practice similar types then it is possible to withdraw some numbers and focus on preInclude examples such as:

2. If 4 + 3 = 7 then

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= 7 – 4 and

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algebraic thinking, as shown in example 2.

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Flexible mathematics thinking (Foundation to Year 3)

Count on strategy Always start with the biggest number and count on the smallest number. Number

Number

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How many?

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6 9

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Flexible mathematics thinking (Foundation to Year 3)

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One more, one less – addition and subtraction Number

One more or one less

3+2

one more

5+3

one less

New number

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4+5

one less

14 – 2

one more

8+6

one less

9–4

one less

12 – 5

one less

5+ . t6

one more

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4+7

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4–3

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5+5

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7+9

one more

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Flexible mathematics thinking (Foundation to Year 3)

Make 5 When adding numbers, think of how to make 5 to make the addition easier. For example; 9 + 6. Look at the 9 and partition it into 5 + 4. Look at the six and break it into 5 + 1. Now you have 9 + 6 = 5 + 4 + 5 + 1. So 5 + 5 + 4 + 1 = 15. Addition

r o e t s Bo r e p ok u S Make 5 strategy

8+5

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6+7

6+9 9+7

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Extension

What is the missing number?

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11 + 6

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+2=3

7–

–5=2

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Combinations of ten Choose a number in set A and join it up with a number in set B to make 10. Set A

Set B

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1 9 R. © I . C.Publ i cat i ons 7l •f or6r evi ew pur poseson y•

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Extension

. te

m . u

o c . c e r 5 6he 7r 8 9 o t s super

How many different ways can you make a total of 12 using 3 or 4 of these numbers?

12 3 4

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Flexible mathematics thinking (Foundation to Year 3)

Using doubles to solve addition To add 3 and 5, you shift one from the 5 making it 4, and give it to the 3, which makes it 4. So now you have a double: 4 + 4 = 8. This idea is clearly shown using the ten frame and counters. Now try these additions by converting the numbers to doubles then adding them. Addition

Convert to doubles

1+3

ew i ev Pr

Teac he r

7+5

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

Total

4+6

9 + 11 6+8

w ww

8 + 10

10 + 12 . 11 + 13

m . u

7+9

o c . che e r o t r s super

Flexible mathematics thinking (Foundation to Year 3)

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Near doubles This strategy builds on a student’s understanding of doubling. When addition problems are stated so that the numbers are close, you can double both numbers, then either add one or take away one. Example: 7 + 8 = 7 + 7 + 1 = 15 or 8 + 8 – 1 = 15. Addition

7+8=

Near doubling strategy

Answer

r o e t 7+ 7 + 1 or 8+ s B8o– 1 r e

Teac he r

8+9= 5+4=

11 + 12 =

ew i ev Pr

p u 6 + 7 =S

w ww

8+7= 11 + 10 .=

9 + 10 =

12 + 13 =

m . u

3+4=

o c . che e r o t r s super

9+8= 6+7= R.I.C. Publications® – www.ricpublications.com.au

Flexible mathematics thinking (Foundation to Year 3)

More near doubles Addition

Near doubling strategy

Answer

11 + 10 = 11 + 11 – 1 or 10 + 10 + 1 11 + 12 = 13 + 12 =

Teac he r

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

ew i ev Pr

15 + 16 =

18 + 17 = 16 + 17 = 18 + 19 =

w ww

14 + 15 =

14 + 13 =. t 25 + 26 = 34 + 35 =

m . u

19 + 20 =

o c . che e r o t r s super

24 + 25 = 30 + 29 = 39 + 40 = 30

Flexible mathematics thinking (Foundation to Year 3)

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Adding 9 When adding 9 to any number, you think add 10 and take 1. For example; in 8 + 9, the 9 becomes 10; so 8 + 10 = 18. Now we need to take off the one we added to the 9, so the answer is 17. Addition

Adding 9 strategy

Answer

8+9=

8 + 10 – 1 =

r o e t s Bo r e p ok u 7 + 9 =S

ew i ev Pr

Teac he r

4+9=

11 + 9 = 14 + 9 =

5 + 9 = © R. I . C.Publ i cat i ons

8 +• 9f = orr evi ew pur posesonl y•

w ww

6+9= 10 + 9. = t

13 + 9 = 16 + 9 =

m . u

12 + 9 =

o c . che e r o t r s super

15 + 9 = 18 + 9 = 17 + 9 = R.I.C. Publications® – www.ricpublications.com.au

Flexible mathematics thinking (Foundation to Year 3)

Adding 7, 8, 9 When adding 7, 8, 9 to any number, you think 10 take 1, 2 or 3. Addition

Adding 9 strategy

Answer

7+4=

10 + 1 =

You have shifted 3 from the 4 to the 7 to make 10.

5+7=

ew i ev Pr

Teac he r

8+5=

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

11 + 9 = 4+8= 7+9=

w ww

6+9=

11 + 7 =. t 13 + 9 = 8+9=

m . u

7+6=

o c . che e r o t r s super

8+6= 7 + 12 = 8 + 11 = 32

Flexible mathematics thinking (Foundation to Year 3)

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Adding 11 When adding 11 to any number, you think add 10 add 1. For example; in 8 + 11, the 11 becomes 10; so 8 + 10 = 18. Now we need to add the one we took from the 11, so the answer is 19. Addition

Adding 11 strategy

Answer

8 + 11 =

8 + 10 + 1 =

ew i ev Pr

Teac he r

r o e t s Bo r 4 + 11 = pe ok u 7 + 11 =S

11 + 11 = 14 + 11 =

5 + 11 =© R. I . C.Publ i cat i ons

8 +• 11f =rr o evi ew pur posesonl y•

w ww

6 + 11 = 10 + 11 . t=

13 + 11 = 16 + 11 =

m . u

12 + 11 =

o c . che e r o t r s super

15 + 11 = 18 + 11 = 17 + 11 = R.I.C. Publications® – www.ricpublications.com.au

Flexible mathematics thinking (Foundation to Year 3)

Relating addition to subtraction If you know one fact you can work out related facts for subtraction. Knowing 2 + 3 = 5 you can see that 3 + 2 = 5, therefore 5 – 3 = 2 and 5 – 2 = 3. Addition fact

Related + fact

Related – fact

2+3=5

3+2=5

5–3=2

5–2=3

2+4=6

Teac he r

ew i ev Pr

4+3=7

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

3+5=8 5+2=7 6+2=8

w ww

6 + 5 = 11

7 + 2 = 9. t

7 + 3 = 10 7 + 4 = 11

m . u

6 + 4 = 10

o c . che e r o t r s super

7 + 5 = 12 7 + 6 = 13 8 + 2 = 10 34

Flexible mathematics thinking (Foundation to Year 3)

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Relating addition to subtraction (continued) If you know one fact you can work out related facts for subtraction. Knowing 8 + 3 = 11 you can see that 3 + 8 = 11, therefore 11 – 3 = 8 and 11 – 8 = 3. Addition fact

Related + fact

Related – fact

8 + 3 = 11 3 + 8 = 11 11 – 3 = 8 11 – 8 = 3

r o e t s Bo r e p ok u 9 + 3 = 12 S 8 + 5 = 13 9 + 4 = 13 8 + 6 = 14

ew i ev Pr

Teac he r

8 + 4 = 12

w ww

9 + 6 = 15 9 + 7 =. t16

m . u

8 + 7 = 15

o c . 8 + 9 = 17 c e her r o t s super 14 + 5 = 11 12 + 6 = 18 13 + 6 = 19 14 + 6 = 20 R.I.C. Publications® – www.ricpublications.com.au

Flexible mathematics thinking (Foundation to Year 3)

Two six-sided dice adding game Use two different coloured six-sided dice to play this game. Throw the dice, record the numbers in the table and add the numbers shown on the dice. Repeat this ten times. Dice 2

Total

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

ew i ev Pr

Teac he r

Dice 1

w ww

. te

m . u

o c . che e r o t r s super

Flexible mathematics thinking (Foundation to Year 3)

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Two dice adding game (extension) Extension: Throw one ten-sided dice and one six-sided dice and add the numbers shown. Record the additions by writing a number sentence after each throw. Dice 2 r o e t s Bo r e p o u =k S+

Total

ew i ev Pr

Teac he r

Dice 1

w ww

. te

R.I.C. Publications® – www.ricpublications.com.au

m . u

c =.

e her r o st super

= Flexible mathematics thinking (Foundation to Year 3)

Two ten-sided dice adding game Use two different coloured ten-sided dice to play this game. Throw the dice, record the numbers in the table and add the numbers shown on the dice. Repeat ten times. Dice 2

Total

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

ew i ev Pr

Teac he r

Dice 1

w ww

. te

m . u

o c . che e r o t r s super

Flexible mathematics thinking (Foundation to Year 3)

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Two ten-sided dice adding game (extension) Extension: Throw two ten-sided dice and add the numbers shown. Record the additions by writing a number sentence after each throw. Dice 1

Dice 2

Total

= r o e t s B r e oo p u k S+ = +

ew i ev Pr

Teac he r

w ww

. te

m . u

o c . c+he e = r o t r s super

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Flexible mathematics thinking (Foundation to Year 3)

Three dice bingo In this game you will need three six-sided dice and each player will need a set of coloured counters (a different colour for each player). Throw the three dice, add them together and put a counter on the number in the grid that matches the total on the dice. The first person to get four in a row horizontally, vertically or diagonally wins. Play again. Keep score of who wins.

w ww

11 15 40

m . u

Teac he r

ew i ev Pr

r o e t s Bo r e p ok u 5 S 4

. te

o c . che e r o t r s super

Flexible mathematics thinking (Foundation to Year 3)

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Three dice adding game – 1 Use three six-sided dice to play this game. Throw the dice, record the numbers in the table and add the numbers shown on the dice. Repeat ten times. Dice 2

Dice 3

Total

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

ew i ev Pr

Teac he r

Dice 1

w ww

. te

m . u

o c . che e r o t r s super

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Flexible mathematics thinking (Foundation to Year 3)

Three dice adding game – 1 (extension) Extension: Throw three six-sided dice and add the numbers shown. Record the additions by writing a number sentence after each throw. Dice 1

Dice 2

Dice 3

Total

r o t +eB s r e oo p u k S + + +

+ . te

ew i ev Pr

Teac he r

w ww 42

m . u

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

Flexible mathematics thinking (Foundation to Year 3)

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Three dice adding game – 2 Use three ten-sided dice to play this game. Throw the dice, record the numbers in the table and add the numbers shown on the dice. Repeat ten times. Dice 2

Dice 3

Total

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

ew i ev Pr

Teac he r

Dice 1

w ww

. te

m . u

o c . che e r o t r s super

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Flexible mathematics thinking (Foundation to Year 3)

Three dice adding game – 2 (extension) Extension: Throw three ten-sided dice and add the numbers shown. Record the additions by writing a number sentence after each throw. Dice 1

Dice 2

Dice 3

Total

+eB r o t s r e oo p u k S + + +

+ . te

ew i ev Pr

Teac he r

w ww

+ +

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

m . u

Flexible mathematics thinking (Foundation to Year 3)

+ +

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‘MAKE THE TARGET NUMBER’ STRATEGY The use of this strategy is meant as a mental thinking activity, however, the use of a calculator has been found to be very helpful in allowing students to guess, check and improve their skills. Students should be given the opportunity to be flexible when working with numbers. This flexibility gives them the necessary skills to solve problem situations in their head or by manipulating numbers based on non-standard partitioning.

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

When introducing this strategy it has been found that working in pairs and then using the pairs – pair strategy supports the uptake of knowledge needed to be successful at the ‘make the target number’ strategy.

Teac he r

ew i ev Pr

Using the strategy with an overhead projector or an electronic whiteboard has been found to be an important teaching strategy for students who have difficulties. Overhead materials, such as the number set with the 4 operations and equal signs, give students the opportunity to show a solution to the target number. How does the strategy work? It is similar to ‘Today’s number is …’. For all target number activities students should be given the opportunity to manipulate materials. The same rule applies for both the addition and subtraction target number problems. ‘Make the target number’ – addition: instruct students to collect 10 craft sticks/counters and solve the problem. Students manipulate the 10 craft sticks/counters to make the combinations of ten. They write the numbers used each time; e.g. 7 and 3 make 10. Each target number has an extension for students to work on.

‘Make the target number’ – subtract: instruct students to collect 15 craft sticks/counters and solve the problem. Students manipulate the 15 craft sticks/counters to make a difference of two. They write the numbers used each time; e.g. 8 take 6 makes 2. Each target number has an extension for students to work on.

. te

m . u

w ww

When solving the target number problem some students can achieve this without using counters. These students will need to be given more challenging situations such as the extension activities or activities from Mental thinking: Using the target number strategy by Richard Korbosky, published by R.I.C. Publications® 2012.

o c . che e r o t r s super

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Flexible mathematics thinking (Foundation to Year 3)

Target number – addition How many ways can you make the number 10 by adding any two numbers in the grid? 9

ew i ev Pr

Teac he r

2 r o e t s Bo r e p o u k Find the combinations of 10 and write them below. S

w ww

Extension:

. te

m . u

o c . che e r o t r s super

What 3 numbers can you put together to make 10?

Flexible mathematics thinking (Foundation to Year 3)

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Target number – subtraction How many ways can you make 2 by subtracting any two numbers in the grid? 2

r o e t s B8o r e 0 7 p ok u S List the numbers that when subtracted will make 2: 6

10 1

ew i ev Pr

Teac he r

w ww

. te

m . u

o c Use the same grid to work out combinations with a difference of 3, 4, 5 and 6. . che e r o t r s super Extension:

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Flexible mathematics thinking (Foundation to Year 3)

USING PLAYING CARDS TO PROMOTE FLEXIBLE THINKING Playing cards are a useful tool because they represent the numbers 2–10. They are readily available and can be used to represent two or three digit numbers and two or three cards can be added or subtracted. The idea of same, less and more can be established using cards, particularly 2–10. Mathematics activities that can be developed with playing cards include: • subitising • place value

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

• recognising single-digit, two-digit and three-digit numbers • subtracting; including the idea of difference

ew i ev Pr

Teac he r

• adding; including adding 2 two-digit numbers

• partitioning; making numbers using two cards, extending idea to use three cards • problem solving and reasoning

To use the activities on the following pages, get the students to take out the picture cards and the aces, leaving the numbers 2–10. In some activities you use the whole pack of 2–10 cards and in other cases only one of the four sets of 2–10 are used.

It is important to use materials such as counters, craft sticks and multi-based arithmetic blocks (MABs) to support students who need extra help in completing these activities. It is also important to give students the opportunity write the mathematics in a graphic organiser. To support this idea, graphic organisers are included on the following pages.

w ww

. te

m . u

o c . che e r o t r s super

Flexible mathematics thinking (Foundation to Year 3)

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SUBITISING GAME Use all the playing card numbered from 2–10. Play this game in pairs. 1. One player shuffles the card deck containing the 2–10 cards and deals out an equal number of cards to both players. 2. With cards facedown on the table, each player removes the top card and places it face up on the table in front of them. 3. The player whose card has the higher value keeps both cards. The winner of the round must explain why they have the greater of the two cards.

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

4. If both cards have the same value then both players shout ‘wooh’ and put three cards facedown on the table. They then turn the next top card from their pack over and compare numbers. The player with the higher value takes all of the cards.

Teac he r

ew i ev Pr

5. Play continues until someone wins all of the cards or until time is up. If time is called, the winner is the player with the most cards. 6. During the game, the teacher can call ‘switch’, which now means the player whose card has the lower value keeps both cards.

w ww

. te

m . u

o c . che e r o t r s super

To make a pack of cards from page 50, copy four sets onto card and play with the cards 1–10.

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Flexible mathematics thinking (Foundation to Year 3)

Number cards 1–10 1

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m . u

w ww

ew i ev Pr

Teac he r

r o e t s B r e oo 4 p u k S

o c . che e r o t r s super 10

Flexible mathematics thinking (Foundation to Year 3)

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Two card shuffle Use only one set of 2–10 cards. Put the cards face up. Using only two cards at a time, how many different combinations of 10 and 12 can you make? Record attempts in the tables below. Card 2

Total

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

10 10

ew i ev Pr

Teac he r

Card 1

Card 2

Total

Extension

m . u

w ww

. te

o c . che e r o t r s super

Using three cards, how many combinations can you make to get a total of 14? You should be able to find more than 5 combinations. Create a table and record the numbers that you use. You cannot repeat a number in a combination; however, you can use the same number in a different combination.

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Flexible mathematics thinking (Foundation to Year 3)

Two card shuffle and number sentences – 10 Use only one set of 2–10 cards. Put the cards face up. Using only two cards at a time, how many different combinations of 10 can you make? Record attempts in the table below and write a number sentence. Card 1

Card 2

Total

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

ew i ev Pr

Teac he r

w ww

. te

m . u

10 o

c . che e r o st super + r = 3 5

Flexible mathematics thinking (Foundation to Year 3)

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Two card shuffle and number sentences – 12 Use only one set of 2–10 cards. Put the cards face up. Using only two cards at a time, how many different combinations of 12 can you make? Record attempts in the table below and write a number sentence. Card 2

Total

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

ew i ev Pr

Teac he r

Card 1

w ww

. te

m . u

12 o c . che e r o st= super +r 3

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Flexible mathematics thinking (Foundation to Year 3)

Three card shuffle Use only one set of 2–10 cards. Put the cards face up. Using three cards, how many combinations can you make to get a total of 17? You should be able to find more than 5 combinations. You cannot repeat a number in a combination; however, you can use the same number in a different combination. Record attempts in the table below. Card 2

Card 3

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

Total

17 17

ew i ev Pr

Teac he r

Card 1

17 17 17

w ww

. te

Extension

m . u

17 17 17

o c . che 17 e r o r st super

Using three cards, how many combinations can you make to get a total of 21? You should be able to find more than 5 combinations. Create a table and record the numbers that you use. You cannot repeat a number in a combination; however, you can use the same number in a different combination.

Flexible mathematics thinking (Foundation to Year 3)

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Three card additions Use all of the set of 2–10 cards (four sets in total). Shuffle them and put the cards face down. Choose three cards and add them together. Record attempts in the table below and write the number sentence below. Card 2

Card 3

Total

r o e t s Bo = r + pe + ok u S +

ew i ev Pr

Teac he r

Card 1

+ R. = © I . C.Pu+ bl i cat i ons •f orr evi ew pur posesonl y• +

w ww

m . u

. te+ + =o c . che e r o r st super +

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Flexible mathematics thinking (Foundation to Year 3)

IN THE MIDDLE Focus of the activity Students play to explore the use of probability and prediction.

Materials required A full pack of playing cards (minus the jokers).

How to play the game

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

• Have groups of four or five students. • One player is nominated the dealer.

• The highest value cards in the pack are the aces; the lowest are the 2s.

Teac he r

• The dealer deals two cards to each player, face up.

ew i ev Pr

• Players take turns to ask the dealer for another card or to tell the dealer they don’t want another card, in which case they score one point.

• The object of the game is to receive a third card from the dealer that falls between the two cards previously received. • If the third card is between the two numbers, the player receives 3 points; otherwise the player receives zero points. • When everyone has had a turn, a new dealer takes over.

Player 1

w ww

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• Points are recorded during the game and totalled at the end.

o c . che e r o t r s super

Flexible mathematics thinking (Foundation to Year 3)

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Number sentences 1. If 3 + 5 = 8, then 5 = 8 – 3 and 3 = 8 – 5. Show this with counters. 2. If 4 + 9 = 13, then 3. If 6 + 11 = 17, then

= 13 – 4 and = 17 –

= 13 – 9. and

= 17 –

r o e t s Bo r e p ok u 15 S

4. If 9 +3 =12 and 8 + 4 = 12, then 9 + 3 = 8 + 4. Show this with counters.

= 15

ew i ev Pr

Teac he r

5. Complete the table.

= 15 = 15

w ww

m . u

= 15 = 15

. t e o 6. What other number sentences that balance could be written for the number c . che 15? e r o r st super 7. Write your own number sentences that balance. (a) (b) (c) R.I.C. Publications® – www.ricpublications.com.au

Flexible mathematics thinking (Foundation to Year 3)

ADDITION AND SUBTRACTION PROBLEMS TYPES Students should be taught a variety of strategies so that they can make a decision to use them when presented with different mathematical situations.

Part

Part Whole

Result is unknown:

r o e t s Bo r e p ok u S One part is unknown:

Teac he r +

A basket contains 34 apples; some red and some green. If there are 16 green apples, how many red apples are there? +

Eden had 17 marbles and George had 21 marbles. They put them together into a bag. How many marbles did they have?

ew i ev Pr

Hugo has 7 swap cards. His friend Eve gives him another 8. How many cards does he have now?

Greer had 119 shells. She gave some to Jacinta. Greer now has 73 shells. How many shells did she give to Jacinta?

– = © R. I . C.Publ i ca t i on s A double-decker bus can hold 50 passengers. If s there were 20 • Ryan had 15 swap cards butr he gave 8 wp • f o r e v i e u r p o s e o n l y passengers upstairs and 5 passengers to Robert. How many swap cards does =

downstairs, how many more passengers can ﬁt onto the bus?

Ryan have now? –

w ww

–

There are 32 students in the class. If 18 are buying their lunch today, how many students are not buying their lunch? –

. te

m . u

–

Richard has 148 places for stamps in his stamp book. If he has 136 stamps, how many more does he need to ﬁll his book?

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

I had $10.00 in my money box. I took out $3.50 and went to the shops. How much money is now in the money box?

LONDON

Flexible mathematics thinking (Foundation to Year 3)

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ADDITION AND SUBTRACTION PROBLEMS TYPES CONTINUED Start is unknown: Sean ate 7 jelly beans; now he has 8 left. How many did he start with? –

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

Bev had some stickers. She gave 27 to Sean. Bev now has 18 stickers. How many stickers did Bev start with? –

ew i ev Pr

Teac he r

At the theatre, there were some people already seated when another 10 walked in. Now there are 35. How many people were in the theatre to start with?

Use this grid to solve these problems.

w ww +

= 28

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m . u

There are 28 children in the class. Some are playing sport today; some are not. How many are not playing sport?

o c . che e r o t r s super

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Flexible mathematics thinking (Foundation to Year 3)

MULTIPLICATION Multiplication is repeated addition. It is important for young learners to understand the language of repeated addition and the language of multiplication. 2 + 2 + 2 + 2 + 2 becomes 5 lots of 2, 5 sets of two or 5 groups of 2. These can be written as 5 x 2. These multiplication ideas can be shown in a multiplication word wall. The important idea is to show that 5 x 3 = 3 x 5. This commutative property should be taught and understood by students in addition and then understood in multiplication.

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

Multiplication Word Wall

groups of

lots of

w ww

. te

5 lots of 3 or 5 times 3

m . u

Array

times

ew i ev Pr

Teac he r

5 sets of 3 counters = 5 x 3

3 lots of 5 or 3 x 5

o c . che e r o t r s super

Here are some problems. Discover the different strategies students use to solve them. 1. One octopus has eight legs. How many legs do 5 octopi have?

2. A muffin tray has three holes in a row and there are six rows. How many muffins can be made? 3. In the classroom students sit in groups of four. If there are 7 groups in the class, how many students are in the class? 4. A tricycle has 3 wheels. How many wheels are there on 8 tricycles?

Flexible mathematics thinking (Foundation to Year 3)

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MULTIPLICATION Multiplication is also represented as an area concept.

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

3 lots of 4 equal twelve

ew i ev Pr

Teac he r

Give students the opportunity to use square tiles to represent different rectangles. Copy these rectangles onto grid paper. It is important that the commutative idea is shown by the grid representations.

w ww

. te

m . u

o c . che e r o t r s super

5 lots of 2 equal 10

2 lots of 5 equal 10 R.I.C. Publications® – www.ricpublications.com.au

Flexible mathematics thinking (Foundation to Year 3)

r o e t s Bo r e p ok u x =8 S

ew i ev Pr

Teac he r

Look at the following diagrams

w ww

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m . u

o c . che4 x = 20 e r o t r s super

x 62

Flexible mathematics thinking (Foundation to Year 3)

= 24 www.ricpublications.com.au – R.I.C. Publications®

Multiple sort Look at these numbers. Read the labels in the columns below and decide where each number should go. Some numbers can go into more than one column and some numbers cannot go into any column.

6, 32, 27, 20, 12, 40, 2, 11, 16, 60, 8, 45, 30, 25, 9, 15,

Multiples of 10

ew i ev Pr

Teac he r

r o e t s Bo r e 18, 35, 7, 24, 13, 28, 36, 10 p o u k S Multiples of 2 Multiples of 3 Multiples of 4 Multiples of 5

w ww

List the numbers which will go into two or more multiples.

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m . u

What numbers do not fit into any column?

o c . che e r o t r s super

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Flexible mathematics thinking (Foundation to Year 3)

Multiplying 2s, 3s, 4s, 5s and 10s Look at the numbers. Put a red counter on all the even numbers. Now count along the numbers in fours and put a blue counter on all these numbers. Which numbers have both red and blue counters? What can you say about these numbers?

3 4 5 6 7 8 9 r o e t 13 14 r 15 16 17 s Bo18 19 e p 25 26 27 28 ok 29 23 u 24 S 33 34 35 36 37 38 39 43 44 45 46 47 48 49 53 54 55 56 57 58 59

Teac he r

2 12 22 32 42 52

10 20 30 40 50 60

ew i ev Pr

1 11 21 31 41 51

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Extension:

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What is the first number covered by two counters when you count in 3s and 4s?

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What is the first number covered by two counters when you count in 2s, 3s, and

o c . cheby two counters when your What is the first number covered count in 2s, 4s, 5s e o r st super and 10s? 4s?

What is the first number covered by two counters when you count in 2s, 3s, 4s, 5s and 10s?

Flexible mathematics thinking (Foundation to Year 3)

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Multiplying by 2s and 10s Look at the multiplication problems in column 1 and draw a line to the correct answer in column 2. Answer

3x2

5x2

7x2

5 x 10

6x2

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Multiplication

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2 x 10 10 . te o c . e 4 xc 10 8 her r o st super 7 x 10

9x2

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Flexible mathematics thinking (Foundation to Year 3)

Multiplication problems Problem 1 There were two boys and one dog. How many legs altogether? Drawing

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Problem 2

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Number sentence

In the paddock there were three sheep, four goats and five geese. How many legs altogether? Drawing

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Number sentence

. te o c At the house there were two cars, three bicycles and two tricycles in the front . che e r yard. How many wheels where there altogether? t o r s super Problem 3

Drawing

Number sentence

Flexible mathematics thinking (Foundation to Year 3)

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Money problems Problem 1 One boy had 20c and his friend had 35c. How much money did they have altogether? Drawing

Teac he r Problem 2

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

On the table there was 2 five-cent coins, a 10-cent coin and 3 twenty-cent coins. How much money was on the table?

Drawing

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Number sentence

. te o c There were 5 five-cent coins, 3 ten-cent coins and 2 twenty-cent coins in the . che e r money jar. How much moneyr was in the jar? t o s super Problem 3

Drawing

Number sentence

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Flexible mathematics thinking (Foundation to Year 3)

What number am I? Read the clues to work out what the number is. Number clues

What number am I?

I am larger than 10 but smaller than 18. I am an even number between 15 and 17.

r o e t s Bo r e p ok I am not an even number. I am less than 20 u but more than 17. S What number am I? I am an even number. I am less than 20 but more than 12. I am not double 7 or double 9.

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What number am I?

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I am an odd number. I am greater than 5 + 8 but less than 8 + 8.

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20.

o c . c e r I am a double number. I amh greater than e o t r s su r 10 but less than 20. I am greater than 6 +p 6 e but less than 8 + 8. What number am I?

What number am I? I am an even number. I am greater than 16 but less than 10 x 2. What number am I? 68

Flexible mathematics thinking (Foundation to Year 3)

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What number am I? (continued) Read the clues to work out what the number is. Number clues

What number am I?

I am an odd number. I am greater than 20 but less than 30. I am a multiple of 5. What number am I? I am an odd number between 20 and 30. When you add my two digits together you get 11.

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If you double 12 and add the difference between 5 and 3, you get my number. What number am I? I am an odd number. I am between 30 and 40. When you add my two digits together you get 10.

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r o e t s Bo r e p ok u What number am I? S I am an even number between 20 and 30.

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What number am I? I am an even number. I am between 30 and 40. I am greater than double 16. 2, 3 and 4 will divide evenly into me.

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What number am I? I am an odd number between 20 and 40. The difference between my tens and ones is 5. I am a multiple of 3.

o c . che e r o t r s super What number am I? I am two odd numbers. I am between 40 What number am I? I am an odd number. I am between 30 and 50. I am 5 less than double 25.

and 50. When you divide me by 5 you have 1 left over. What numbers can I be? I am an even number. 2, 3, 4, 5, and 10 will go evenly into me. What number am I? R.I.C. Publications® – www.ricpublications.com.au

Flexible mathematics thinking (Foundation to Year 3)

Open-ended money tasks Aim To experience a variety of ways to make up $1 using 5c, 10c, 20c and 50c coins.

100.00

7 - x 4 5 8 9 % 6 2 3 + =

Materials You need money and calculators.

r o e t s Bo10c, 20c and 50c r e Make $1.00 in as many different ways as you can using 5c, p ok coins. (Show your drawings.) u S Extension

Make 75 cents.

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Open task

Make $2: include a $1 coin.

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. te o c Make $3: include a $1 coin. . che e r o t r s super Make $5: include a $2 coin.

Flexible mathematics thinking (Foundation to Year 3)

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Open-ended money tasks (continued) Work out how much a letter would cost. AUSTRALIA

60c

ESTERN W

23•JULY•2012

SYDNE

Steffie Chang 19 Honeycherry Lane Tilba Tilba NSW 2546

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r o e t s Bo r e p ok u Sit would cost to send: Work out how much • 2 letters

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• 4 letters

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• 5 letters

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• 3 letters

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Extension: Change the price of the stamp, write a similar task and ask another student to solve your problem. R.I.C. Publications® – www.ricpublications.com.au

Flexible mathematics thinking (Foundation to Year 3)

DIVISION Division is a difficult mathematics idea for students. It is important to give them many opportunities to share discrete objects into groups, sets and lots of. Division is closely linked to fractions. When you share a whole object or a packet of sweets between 3 people then each person gets one third of what you are sharing. What students find difficult is that after many sharing activities where the result is a whole number, in the same situation the sharing can be named as a fraction quantity. Division sub-set set notion – division grid

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Division is also seen as a sub-set of a set. i.e. share 20 sweets between 3 children. A handy graphic organiser to use is a division grid. Division grids can be created to show a variety of one digit divisors. The division grid gives students the opportunity to use manipulatives to show answers to division problems, therefore giving them the opportunity to develop the action before the abstraction.

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A useful book to read is When the Doorbell Rang by Pat Hutchins. It is also important to give students situations where there are some left over. For many years teachers have said that when students are presented with leftovers in Year 5, they do not know what to do with them. By giving younger students situations where they have to deal with leftovers, and linking this with counters or other mathematics manipulatives, we can begin to rectify that problem. Use the appropriate division grid such as the one on page 73 to solve these problems.

© R. I . C.Publ i cat i ons 3. There are 28 students at school today. We need groups of 4. How many groups will be •f orr evi ew pur posesonl y• in the classroom? 1. Share 18 lollies equally between three students.

2. Group the 24 counters into equal groups of 6. How many groups do you have?

4. Three straws make a triangle. How many triangles can we make with 15 straws?

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5. I see 16 wheels. How many cars can I see?

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6. There are 5 children at the party. There are 22 lollies in the bag. How many lollies does each child get?

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Flexible mathematics thinking (Foundation to Year 3)

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Division grid – dividing quantities in half You have 12 toys in a box and wish to share them with a friend. How many do you each get?

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Direct students to take out 12 counters. Direct them to now share the toys, one at a time, into each part of the grid so that each section gets an equal amount.

Ask these questions: How many do you each get? What is half of 12?

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In this example we have started with the number 12. For younger students you could start with a one digit number, while for older students you could use bigger numbers (or even change the material to money).

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Flexible mathematics thinking (Foundation to Year 3)

Division grid – dividing quantities in thirds

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The teacher wants to break your class into three equal teams for the sports afternoon. How many students will be in each team? (You will need to find out how many students are in your class.)

Division grid – dividing quantities in fourths Share 20 chocolates equally between yourself and 3 friends.

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Flexible mathematics thinking (Foundation to Year 3)

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When the doorbell rang What happens when there are 12 biscuits? Number of children

How many biscuits will each child get?

Division number sentence

2 4

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

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. te How many biscuits will Division number sentence o each child get? c . che e r o t r s super

What happens when there are 18 biscuits? Number of children 2

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4 6 12

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Flexible mathematics thinking (Foundation to Year 3)

When the doorbell rang (continued) What happens when there are 20 biscuits? Number of children

How many biscuits will each child get?

Division number sentence

2 4

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. How many biscuits will te Division number sentence o each child get? c . che e r o t r s super

What happens when there are 24 biscuits? Number of children 2

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4 6 12

Flexible mathematics thinking (Foundation to Year 3)

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NUMBER EQUALISER BALANCE The number balance helps students to compare numbers, practise addition of small numbers, multiplication and division, and discover number properties of partitioning and commutivity. The number balance is not a balance scale but a mathematical balance which links early number concepts to algebra. Number skills highlighted by the use of the number balance include:

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(a) Greater that or less than

Is 4 greater or less than 6? Show this on the number balance.

Is 5 greater or less than 3 + 4? Show this on the number balance.

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(b) Partitioning

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How many ways can you make ten using the number balance? Show different possibilities.

Put 5 on one side and 8 on the other side. What do you have to put on the number balance to make it level?

=4+5

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(e) Multiplication 4x2= 2x3= (f) Division

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5+3=3+

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How many 4s in 12? Put 12 on one side and find out how many 4s are needed make the mathematics manipulative balance. The activity on page 78 allows students to experiment with the balance to make up their own number sentences.

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Flexible mathematics thinking (Foundation to Year 3)

Number equaliser balance Use the balance scales or number balance to equalise a number sentence.

and

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and

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r o e t s Boand r e and = p ok u S

Change ‘and’ to mathematical addition symbols and create new number sentences.

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+ . t e

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Flexible mathematics thinking (Foundation to Year 3)

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‘THREE BEAR FAMILY’ The ‘Three bear family’ counters are a mathematical manipulative focusing on concepts of mass, sorting, classifying, comparison, counting and number, while linking to algebra. The following activities require a balance scale so that bears can be put on both sides so students develop ideas about their masses. Students compare the bears by seeing how many small bears are needed to balance one large bear. Students investigate how many small bears are needed to balance one medium (middle) bear. The data is recorded in a table. The focus of this activity is multiplication of 2s and 3s. With this data students can compare and use the information to answer questions such as:

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Are 2 large bears equal to 6 small bears?

Are 2 large bears equal to 2 medium bears and 2 small bears?

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Are 2 large bears equal to 1 medium bear and 4 small bears?

Students can check their answers by putting the bears on both sides of the scale to see if they balance. This information can be recorded in number sentences such as: 2 big bears =

small bears or

2 big bears =

medium bears and

2 big bears =

medium bears and

small bears

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This activity is getting students to weigh the bears, compare their sizes, record the information in a table and then use it to develop number sentences. Students employ algebraic thinking and develop multiplication ideas which underpin the idea of rates. Students can be further tested by looking at the different relationships that can be developed using 4, 5 or 6 large bears.

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Flexible mathematics thinking (Foundation to Year 3)

‘Three bear family’ mathematics Using the ‘Three bear family’ mathematics manipulative to link measurement to algebra. On one side of the balance place 1 big bear. Now balance it with the correct number of small bears. Record this in the table. Big bear/s

Small bear/s

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•f orr evi ew pur posesonl y•

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Repeat this activity by adding the medium bear. Medium bear/s

. 1e t 2 3 4 5 6 7 8

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Write about what you have discovered.

Small bear/s

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Flexible mathematics thinking (Foundation to Year 3)

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‘Three bear family’ mathematics Look at the table below. Big bear/s

Medium bear/s

Small bear/s

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1 3 2 3 6 3 9 r o e t s B r e o p ok 12 4 u 6 5 S 15 6 9 18 7 21 8 12 24 9 © R. I . C.Publ i cat i ons 27 15 •10 f orr evi ew pu r posesonl y30 • 11 33 12 18 36

What do you notice? Big bear

1 big bear 1 big bear 2 big bears

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Small bear o c = c 1 medium bear and .1 small bear e her or r o st super Medium bear

0 medium bears

and

3 small bears

1 medium bear

and

4 small bears

and

2 small bears

and

6 small bears

or 2 big bears

2 medium bears or

2 big bears

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0 medium bears

Flexible mathematics thinking (Foundation to Year 3)

‘Three bear family’ mathematics Answer these number sentences.

small bears

1. 2 big bears = or

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

2 big bears =

and

2. 3 big bears =

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medium bears

small bears

3 big bears =

medium bears and

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small bears

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3 big bears =

small bears . te medium bears and o or c . che e r o t r s s r u e p medium bear and small bears 3 big bears = 3 big bears =

Extension What number sentences can you create using 4, 5 or 6 big bears?

Flexible mathematics thinking (Foundation to Year 3)

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

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Ten frame – Appendix 1

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Flexible mathematics thinking (Foundation to Year 3)

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Addition grid for solving word problems – Appendix 2

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Flexible mathematics thinking (Foundation to Year 3)