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Year 4 Block A - Counting, partitioning and calculating - Unit 2

Objectives Children's learning outcomes are emphasised	Assessment for learning
Report solutions to puzzles and problems, giving explanations and reasoning orally and in writing, using diagrams and symbols I can explain how I solve problems, using diagrams and symbols to help me	What information did you use to solve this problem? Why? Tell me why you chose this way to record your solution to the problem. Could you have done it differently? Make up a word problem that could be solved using each calculation: 6 × 5, 30 ÷ 3, 30 7, 26 + 19 Sort these problems into those you would do mentally and those you would do with pencil and paper. Explain your decisions.
Use decimal notation for tenths and hundredths and partition decimals; relate the notation to money and measurement; position one-place and two-place decimals on a number line I can use decimals when I work with money and measurement	Can you tell me what the digit 7 represents in each of these amounts: £2.70, 7.35 m, £0.37, 7.07 m? Which is larger: 239p or £2.93? Why? Put these in order: £0.56, 125p, £3.60, 250p, 7p, £5, 205p. Which is the smallest? How do you know? Which is the largest? How do you know? What amount of money comes next: £1.76, £1.86, £1.96,...?
Add or subtract mentally pairs of two-digit whole numbers (e.g. 47 + 58, 91 - 35) I can add and subtract mentally pairs of two-digit numbers and find a difference by counting on	What strategies would you use to work out the answers to these calculations: 47 + 58, 91 - 35? Could you use a different method? How could you check that your answer is correct? The difference between a pair of two-digit numbers is 13. What could the pair of numbers be? How would you calculate the answer to 93 - 86? Why would you choose that strategy?
Refine and use efficient written methods to add and subtract two-and three-digit whole numbers and £.p I can add and subtract three-digit numbers using a written method	Which of these are correct/incorrect? What has this person done wrong? How could you help them to correct it? How does partitioning help to solve 436 + 247? What tips would you give to someone to help them with column addition/subtraction?
Recognise and continue number sequences formed by counting on or back in steps of constant size I can count on and back in sevens	Count on in sevens from zero. Now count back to zero. This time, count on eight sevens from zero. Show me seven hops of eight from zero on the number line. Now show me eight hops of seven. What do you notice?
Derive and recall multiplication facts up to 10 × 10, the corresponding division facts and multiples of numbers to 10 up to the tenth multiple I know my tables to 10 × 10 I can use the multiplication facts I know to work out division facts	The product is 40. What two numbers could have been multiplied together? How many multiplication and division facts can you make, using what you know about 24 (or 20, 30)? How did you work out the division facts?
Multiply and divide numbers to 1000 by 10 and then 100 (whole-number answers), understanding the effect; relate to scaling up or down I can multiply and divide numbers by 10 or 100 and describe what happens to the digits	What number is ten times bigger than 500? Explain the calculation you would use to change 25 to 2500. How many tens are there in 200? How many hundreds in 2000? If 4 × 6 = 24, what is 40 × 6 and 400 × 6? How could you quickly work out the answers to these calculations: 3 × 80, 120 ÷ 4? The product of two numbers is 2000. What could the two numbers be?
Develop and use written methods to record, support and explain multiplication and division of two-digit numbers by a one-digit number, including division with remainders (e.g. 15 × 9, 98 ÷ 6) I can multiply and divide a two-digit number by a one-digit number	How would partitioning help you to calculate 27 × 6? How does knowing that 10 × 6 = 60 help you to calculate the answer to 72 ÷ 6? Do all divisions have remainders? Make up some division questions that have no remainder. How did you do this? Why don't they have a remainder? Make up some division questions that have a remainder of 1. How did you do it?
Use knowledge of rounding, number operations and inverses to estimate and check calculations I can estimate and check the result of a calculation	Roughly, what answer do you expect to get? How did you arrive at that estimate? Do you expect your answer to be greater or less than your estimate? Why?
Respond appropriately to others in the light of alternative viewpoints I can explain how I solved a problem and can decide whether someone else solved it in a better way	Explain what information you used to solve the problem. What stages did you go through to complete it? What calculations did you do? Did you draw any diagrams? Why? Did anyone solve the problem in a different way? Which do you think was the best way to solve the problem? Why?

Learning overview

Children develop understanding of decimal notation for tenths and hundredths in the context of money and length. They understand that the decimal point is used to separate whole amounts and parts of the whole. They respond to questions such as: What does the digit 6 represent in £1.65? and recognise that because there are ten lots of 10p in £1, then 60p is six tenths of £1. They count on and back in equal steps to develop a sequence. They use patterns and relationships between numbers to predict the next term in a sequence such as £1.37, £1.47, £1.57, and they describe the pattern or rule. They order money and measurements involving decimals. For example, they locate 1.2 m, 2.1 m, 1.5 m and 2.5 m on a line numbered from 0 to 3 metres and marked in tenths.

Children continue to add or subtract mentally pairs of two-digit whole numbers. They use a 100-square to derive pairs of numbers that sum to 100. When presented with calculations such as 93 - 86 (e.g. I have 93p and Sam has 86p. How much more money do I have?) they recognise that the numbers are close together and can find the difference by counting up. They suggest other calculations where counting up would be an appropriate strategy, e.g. 403 386.

For additions and subtractions that cannot easily be done mentally, children develop written methods. They rehearse rounding two- and three-digit numbers to the nearest 10 and 100. They use rounding to estimate a calculation; for example, they recognise that the answer to 367 + 185 is less than 400 + 200, and that 725 - 477 is about 700 - 500. They build on their understanding of place value and partitioning to refine and use efficient methods of recording for addition and subtraction. For example, for 367 + 185 children use an expanded method, then move on to recording this vertically:

367 plus 185 is partitioned horizontally and a total given underneath 367 plus 185 is partitioned in a vertical column

Children discuss how adding the ones first gives the same answer as adding the hundreds first. Over time, they move to consistently adding the ones digits first.
For 326 - 178, they extend their understanding of 'difference' by counting up from 178 to 326, initially using an empty number line and then moving on to vertical recording.

Recording and calculating 'difference' on a number line; numbers 178, 180, 200, 300, 320, 326 run along the number line with the corresponding calculations of plus 2, plus 20, plus 100, plus 20, plus 6 above 326 - 178 is recoded vertically by counting as in the number line in previous image

Children explain orally how their method of calculation works and demonstrate an understanding of the place value that underpins written methods. Children with a firm understanding of the expanded methods move towards refining their recording to make them more efficient.
Children continue to derive and practise recalling multiplication and division facts to 10 × 10. They consolidate multiplying and dividing numbers to 1000 by 10 and 100. They work out how many 10p coins there are in £15 or £150 and can complete equations such as 4000 ÷ square = 400. They apply their knowledge of multiplying by 10 to known multiplication and division facts. For example, they recognise that if they know the answer to 3 × 6 they can calculate 30 × 6 or 3 × 600; equally if they know 21 ÷ 3 they also know 210 ÷ 3. They use this knowledge to develop written methods for multiplying and dividing a two-digit by a one-digit number. When calculating 38 × 7 they approximate first (approximately 40 × 10 = 400), partition into 30 × 7 and 8 × 7 and represent this on a grid.
38 × 7 = (30 × 7) plus (8 × 7) = 210 + 56 = 266


The number with the most digits is always placed in the left-hand column of the grid so that it is easier to add the partial products.	The next step is to move the number being multiplied (38) to an extra row at the top of the grid. Presenting the grid like this helps children to set out and add the partial products 210 and 56.	The next step is to reduce the method of recording to a column format, but showing the working. Point out the links with the grid method on the left.

When dividing 64 by 4 children approximate first. They recognise that the answer must lie between 40 ÷ 4 = 10 and 80 ÷ 4 = 20, and use this approximation to do a calculation such as:

A sum using the process of approximation in division

Remainders after division are recorded similarly.

A sum using the process of approximation in division with a remainder

Children use their knowledge of calculations to solve problems and puzzles. Given an equation with missing digits such as 7 square + square 8 = 1 square square , they find how many different ways they can complete it. They find three consecutive numbers which add up to 39 and then consider what other numbers up to 50 they can make by adding three consecutive numbers. They record the stages in the problem using calculations and/or drawings. They explain what information they selected and why. They evaluate the work of other children and modify their thinking in the light of comments and questions from others.

Resource links to existing published material

Mathematical challenges for able pupils Key Stages 1 and 2

Activities	PDF 923KB
Activity 49 - Footsteps in the snow

Intervention programmes

Objectives for Springboard intervention unit	Springboard unit
Know by heart all addition and subtraction facts for each number to 20 Derive quickly all pairs of multiples of 5 with a total of 100	Springboard 4 Unit 2 (PDF 185KB)
Understand division as grouping or sharing. Read and begin to write the related vocabulary Recognise that division is the inverse of multiplication and that halving is the inverse of doubling Know by heart the facts of the 2-, 5- and 10- times tables	Springboard 4 Unit 5 (PDF 201KB)
Add and subtract a ‘near multiple of 10’ to or from a two-digit number by adding or subtracting 10, 20, 30 and adjusting	Springboard 4 Unit 9 (PDF 165KB)

Supporting children with gaps in their mathematical understanding (Wave 3)

Diagnostic focus	Resource
Is not confident when recalling multiplication facts	1 Y4 ×/÷ DfES 1150-2005 (PDF 104KB)
Does not apply partitioning and recombining when multiplying and confuses the value of 2 digit numbers	4 Y4 ×/÷ DfES 1153-2005 (PDF 104KB)
Interprets division as sharing but not grouping	3 Y6 ×/÷ DfES 1161-2005 (PDF 94KB)
Describes the operation of multiplying by ten as 'adding a nought'	3 Y4 ×/÷ DfES 1152-2005 (PDF 68KB)
Is muddled about the correspondence between multiplication and division facts	2 Y4 ×/÷ DfES 1151-2005 (PDF 93KB)
Does not make sensible decisions about when to use calculations laid out in columns	3 Y4 /- DfES 1130-2005 (PDF 101KB)

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