Have a go… KS1 KS2

Concrete, Pictorial and Abstract Methods My name is Paul Rowlandson, Assistant Principal at Trinity Academy…

Importance of CPA In his research on the cognitive development of children (1966), Jerome Bruner proposed three modes of representation: Enactive representation (action-based) Iconic representation (image-based) Symbolic representation (language-based)

Importance of CPA ‘Bruner's constructivist theory suggests it is effective when faced with new material to follow a progression from enactive to iconic to symbolic representation; this holds true even for adult learners. Bruner's work also suggests that a learner even of a very young age is capable of learning any material so long as the instruction is organized appropriately.’ McLeod, S. A. (2008)

Importance of CPA 3 + 4 = 7 Concrete – The ‘doing’ stage Brings concepts to life by allowing children the experience of handling physical objects Pictorial – The ‘seeing’ stage Encouraging children to visualise concrete experiences and provides a link to the abstract Drawing a representation of their experience Abstract – The ‘symbolic’ stage Using mathematical symbols to model problems 3 + 4 = 7

Multiple Representations Discussion What concrete and pictorial resources do you use in maths?

Using CPA methods Today we are going to focus on: Place value Addition and Subtraction Multiplication and Division

Number bonds

Number bonds Children need to see that numbers can be made in a variety of ways. Using one hand… Show me three Show me another way, and another… Using two hands… Show me five Show me another way, and another How is your way different to a friend?

Number bonds How can we build and show number bonds? How can we use these resources for other number bonds?

Number bonds Ten frames- making 5 in different ways

Number bonds Ten frames- making 5 in different ways

Number bonds Ten frames- making 5 in different ways

Number bonds Ten frames- making 5 in different ways

Number bonds How many ways can you make 6? What number facts can you show placing the counters in different ways?

Number bonds Ten frames

Number bonds Ten frames

Number bonds Ten frames

Number bonds Ten frames

Number bonds 6 6 How many ways can you make the pan scales balanced? Workings 6 6

Number bonds Part - whole model ? ?

Number bonds Part - whole model 3 4 7 5 7 1 7 4 7 7 6

Number bonds Activity How many ways can you make 10 using the part-whole model? Did you use a strategy? 10 10

Number bonds Bar model 10 10 10 6 4 3 7 10 10 10 5 5 2 8 10 10 10 4 6 1 9

Place value columns

Place value columns Use Styrofoam cups to see the value of each digit.

Place value columns Introducing tens and ones Count straws up to ten. When we reach ten we can put these together to make 1 ten. How many do you have? Is this more or less than your partner?

Place value columns Take one lot of 10 away. What is your new number? Which column changed? Build the number 24 Take 6 away. What difficulties did you face? What did you have to do? How did this affect the columns? I have ten bundles of ten. I call this one hundred.

Place value columns Build 243 using place value counters. Add 8 to the ones column. Model Workings H T 0 10 1 100 10 1 100 2 4 3 2 5 1 1 What has happened to the columns? Key vocabulary: exchange Can we exchange any counters?

Place value columns Build 314 using place value counters. Subtract 4 from the ones column. Model Workings H T 0 10 1 100 10 1 100 3 1 4 3 1 0 What has happened to the columns? Key vocabulary: exchange Can we exchange any counters?

Place value columns Build 406 using place value counters. Subtract 10 from the tens column. Model Workings H T 0 100 10 1 100 10 1 100 10 1 4 0 6 3 9 6 What has happened to the columns? Key vocabulary: exchange Can we exchange any counters?

Place value columns We can begin to make rules by using concrete objects. Testing a rule Can you think of a number that changes four columns when you add 8 to it?

Place value Working walls Use the working wall as a reminder of the work you have already done. Encourages children to use equipment or visualise problem.

29 Place value How many ways can you now represent this number? Use other pictorial images, drawings of concrete objects that we have not used.

29 Place value How many ways can you show 29? Twenty nine T O 10 + 19 20 + 9 29 10 + 10 + 9

Addition

Addition- Part whole model Solve… 4 + 3 = Model Calculations  ? 3 4

Addition Solve… 8 + 1 = Model Calculations 8 + 1 = 9 1 + 8 = 9 8 + = 9 8 + = 9 + 1 = 9 8 1

Addition – parts and wholes Solve… 8 + 1 = Model Calculations ? 8 1 8 + 1 = 9 1 + 8 = 9 8 + = 9 + 1 = 9 8 1 ?

Addition- Regrouping to make 10 Solve… 7 + 4 Model Calculations 7 + 4 = You try: 5 + 8 = 6 + 5 = 11

Addition- Regrouping to make 10 Solve… 7 + 4 Model Calculations 7 + 4 = +3 +1 11 10 3 1 11 7

Addition- Regrouping to make 10 Solve… 7 + 4 Model Calculations 7 + 4 = 11 ? 11 10 7

Addition- Column method Solve… 15 + 34 = Model Calculations 15 +34 ___ T O 4 9 Key vocabulary: exchange Can we exchange any ones for a ten?

Addition- Column method Solve… 15 + 34 = Model Calculations 15 +34 ___ T O 4 9 Key vocabulary: exchange Can we exchange any ones for a ten?

Addition- Column method Hafsa is counting cars. She counts 15 blue and 34 red. How many cars does Hafsa count? Model Calculations 15 + 34 = = 15 + 34 = 34 + 15 - 15 = 34 - 34 = 15 ? 15 34

Addition- Column method Solve… 35 + 17 = Model Calculations 35 +17 ___ T O   10   10   10 1 1 1 1 1   10 1 5 2 1   10 Key vocabulary: exchange Can we exchange any counters?

Addition- Column method Patryk earns £17. He adds this to the £35 he has already saved. How much money does Patryk have now? Model Calculations 35 + 17 = 17 + 35 = = 35 + 17 = 17 + 35 - 35 = 17 - 17 = 35 ? 35 17

Addition- Column method Solve… 243 + 368 = Model Calculations 243 +368 ____ 10 1 100 100 100 100 10 1   10 10 1 6 1 1 1 1 100 10 1 You try: 327 + 175 = 402 + 249 = Key vocabulary: exchange Can we exchange any counters?

Addition – column method 23.7 + 21.5 = = 40.5 + 19.6 Remember Build both numbers in the correct columns Exchange counters if needed Complete abstract method alongside Draw bar model Write all calculations bar model shows

Subtraction

Subtraction- Part whole model Solve… 10 – 6 = Model Calculations  10 6 ?

Subtraction- find the difference Solve… 12 – 11 = Model Calculations 12 – 11 = 1

Subtraction - make 10 Solve… 14 – 5 = Model Calculations 14 – 5 =

Subtraction - make 10 Solve… 14 – 5 = Model Calculations 14 – 5 =

Subtraction - make 10 Solve… 14 – 5 = Model Calculations 14 – 5 = 9

Subtraction- column method Solve… 49 – 27 = Model Calculations 49 -27 _ T O     2 2

Subtraction- Column method Solve… 35 - 17 = Model Calculations 35 -17 ___ T O 1   10   10   10 1 2 1 1 1 1 1 1 1 1 1 1 1 8 Key vocabulary: exchange Can we exchange any counters?

Subtraction- column method Solve… 234 – 88= Model Calculations 234 - 88 ____ 10 1 100 1 1 2 1 100 100 10 10 10 1 1 1 1 10 10 10 10 1 1 1 1 14 6 10 10 10 10 1 1 1 1 10 10 1 1 Key vocabulary: exchange Can we exchange any counters? You try: 345 – 167= 402 – 58=

Multiplication and division Importance of multiplication and division facts How can we help children learn these facts? Make sure children understand the link between multiplication and division by using the inverse. Ensure that children practice their times tables every day but in slightly different formats. Teach it in a variety of ways – missing numbers, concrete with numicon and cubes, solve problems.

Multiplication Muffins come in boxes of 4. Peter buys 6 boxes of muffins. How many muffins does Peter buy all altogether? Model Calculations ? 6 x 4= 24 4 4 4 4 4 4

Multiplication 23 × 6 _ 13 8 Solve… 6 x 23 = Model Calculations 100 10 1   100 10 1 1 10 100 1 10 1 10 23 × 6 _ 1 10 1 1 10 13 8 1 10 1 10 You try: 5 x 16 = 4 x 34 = 1 1 1 10

What’s the same? What’s different? Tom has 5 packets of sweets. Each packet contains 42 sweets. How many sweets does Tom have altogether? Tom has 42 packets of sweets. Each packet contains 5 sweets. How many sweets does Tom have altogether?

Division

Division – 2 ways Partitive (sharing) I have 42 sweets and I want to give them to 3 people. Each person will receive an equal amount.

Division (sharing) Jane has 30 cakes. She wants to share them equally between five boxes. How many should go in each box? Model Calculations 30 30 ÷ 5 = 6 6 6 6 6 6 ? ? ? ? ? Number of cakes in each box = 6 ? In this version, we are splitting 30 into 5 equal groups.

Division Solve… 42 ÷ 3 = Model Calculations 10 10 10 10 1 1 42 ÷ 3 =

Division 42 ÷ 3 = Solve… 42 ÷ 3 = Model Calculations 10 1 1 1 1 1 1 1

Division 42 ÷ 3 = 14 Solve… 42 ÷ 3 = Model Calculations 65 ÷ 5 = 10 You try: 65 ÷ 5 = 10 10

Division – 2 ways Quotative (grouping) I have 42 sweets and I want to give them out 3 at a time. How many people will receive 3 sweets?

Division (grouping) Jane has 30 cakes. She wants to pack them into boxes with 5 cakes in each box. How many boxes will she need? Model Calculations 30 30 ÷ 5 = 6 5 5 5 5 5 5 Number of boxes needed = 6 ? In this version, we are counting how many fives go into thirty.

Division Solve… 615 ÷ 5 = Model Calculations 5 615 H T O

Division Solve… 615 ÷ 5 = Model Calculations 5 615 H T O 1

Division Solve… 615 ÷ 5 = Model Calculations 5 615 H T O 1 1

Division Solve… 615 ÷ 5 = Model Calculations 5 615 H T O 1 2 1

Division Solve… 615 ÷ 5 = Model Calculations 5 615 H T O 12 1 1

12 3 Division 5 615 Solve… 615 ÷ 5 = Model Calculations H T O 5 615 H T O 12 3 1 1 You try: 936 ÷ 4 =

Multiple representations ‘Used well, manipulatives can enable pupils to inquire themselves- becoming independent learners and thinkers. They can also provide a common language with which to communicate cognitive models for abstract ideas.’ Drury, H. (2015) “If we do not use concrete manipulations, then we can not understand mathematics. If we only use concrete manipulations, then we are not doing mathematics.” Gu (2015)

Fractions and Decimals

Introducing fractions How many ways can you show 1 4 ? 0.25 2 8 25%

Comparing fractions How can you use the strips of paper to compare fractions? Calculations 1 2 > 1 3 2 4 = 4 8 1 5 < 2 5

Finding fractions of amounts Find 3 5 of 20 Calculations 20 ÷ 5 = 3 x 4 = 4 12

Finding fractions of amounts Solve….. 5 6 of 18 = Calculations 18 ÷ 6 = 3 x 5 = 3 15

Finding fractions of amounts On your white boards use the counters to work out: 1 3 of 15 2 3 of 15 2 3 of 21 2 7 of 21 What are the links between the questions? What question could you ask next?

Finding fractions of amounts Design a tower which follows one of the following rules: Each colour represents 1 3 One in every four is blue There are 3 4 of one colour

Adding Fractions + = 1 5 3 5 Model Calculations 1 5 3 + 4 5 =

Adding fractions Solve… Model Calculations + = You try: 2 + = 6 2 + = 3 5 + 1 = 5 9 + 2 =

Adding Fractions + = 3 4 1 4 Model Calculations 3 4 1 + 4 = = 1

Adding fractions Solve… Model Calculations You try: 3 5 1 2 5 3 5 1 2 + 1 = 2 5 + Model Calculations You try: 3 5 + 1 = 2 5 + 3 5 - 1 = 4 2 8 + = 6 -

Adding fractions 3 6 + 5 6 Solve: 2 5 + 4 5 Model Calculations = Model Calculations 2 5 + 4 5 = You try: 3 6 + 5 6 1 1 5 2 1

Subtracting fractions Solve… 5 7 - 2 = Model Calculations You try: 5 7 - 2 = 3 7 7 9 - 3 =

Subtracting fractions Solve… 5 6 - 2 = Model Calculations You try: 3 6 5 6 - 2 = 7 9 - 3 =

Converting fractions Convert 8 6 into a mixed number fraction Model Calculations You try: 8 6 2 6 = 1 7 5 1 2 6 9 3

Fractions and Decimals Convert 2 10 into a decimal Model Calculations 2 ÷ 10 = 1 0.1 1 0.1 0.2

Fractions and Decimals Convert 1 4 into a decimal Model Calculations 1 ÷ 4 = 1 0.1 0.01   0.1   0.1 0.01 1 0.25

Fractions and Decimals Prove… 3 10 is 0.3 Model Calculations 3 ÷ 10 = 1 0.1 1   0.1 0.3

Fractions and Decimals Prove… 1 5 is 0.2 Model Calculations 1 ÷ 5 = 1 0.1   0.1 1 0.2

Multiple representations ‘Used well, manipulatives can enable pupils to inquire themselves- becoming independent learners and thinkers. They can also provide a common language with which to communicate cognitive models for abstract ideas.’ Drury, H. (2015) “If we do not use concrete manipulations, then we can not understand mathematics. If we only use concrete manipulations, then we are not doing mathematics.” Gu (2015)