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Year 3 Block E - Securing number facts, relationships and calculating Unit 2

Objectives Children's learning outcomes are emphasised	Assessment for learning
Solve one-step and two-step problems involving numbers, money or measures, including time, choosing and carrying out appropriate calculations I can recognise when a word problem involves multiplication or division	Look at this problem. There are 20 legs. How many zebras is this? What calculation did you do? What was it about the problem that made you decide to use this operation? Make up your own word problem that would lead you to working out the calculation 32 ÷ 4. How do you recognise that this problem involves division?
Read and write proper fractions (e.g. ³/₇ , ⁹/₁₀ ), interpreting the denominator as the parts of a whole and the numerator as the number of parts; identify and estimate fractions of shapes; use diagrams to compare fractions and establish equivalents I know that the number on the bottom of a fraction tells me how many pieces the whole is divided into	What fraction of this shape is shaded? How do you know? Is there another way that you can describe the fraction? Approximately what fraction of this shape is shaded? Explain how you decided on your answer.
Derive and recall multiplication facts for the 2, 3, 4, 5, 6 and 10 times-tables and the corresponding division facts; recognise multiples of 2, 5 or 10 up to 1000 I know the 2, 3, 4, 5, 6 and 10 times-tables.	How many sides do six triangles have? What multiplication fact do you need to work out to answer this problem? What is the answer? How can you use the fact 7 × 3 = 21 to find the answer to 7 × 6? What tips would you give to someone who cannot remember the 4 times-table? Complete this division fact in as many ways as you can: 20 ÷ = What multiplication facts did you use to help you do this?
Multiply one-digit and two-digit numbers by 10 or 100, and describe the effect I can multiply a number by 10 or 100	What is the value of the 5 in the number 15? Multiply 15 by 10. What is the value of the 5 in your answer? What operation would change 37 into 370? What operation would change 4 into 400? How did you decide on your answers?
Use practical and informal written methods to multiply and divide two-digit numbers (e.g. 13 × 3, 50 ÷ 4); round remainders up or down, depending on the context I can multiply a multiple of 10 by a one-digit number	What calculation do you need to do to work this out? Rulers are 30 cm long. If you place six of them end to end, how long a line will they make? Explain how you solved this problem. Did you write anything down? How can you use 4 × 6 = 24 to work out 4 × 60? How many threes make 36? How do you know?
Understand that division is the inverse of multiplication and vice versa; use this to derive and record related multiplication and division number sentences I can give the multiplication fact that is linked to a division fact	Write down the two multiplication facts and two division facts that are linked to this array. Mary keys 27 ÷ 3 into a calculator to get the answer 9. What operation should she type in to turn the 3 back into 27?
Find unit fractions of numbers and quantities (e.g. ¹/₂ , ¹/₃ , ¹/₄ and ¹/₆ of 12 litres) I can find a fraction of a number by using division	Look at this problem. Barry has saved 60p. He decided to spend ¹/₃ of it. How much does he spend? What calculation did you do to find your answer? 50 ÷ 5 = 10. Now complete: of 50 = 10 Explain how to find ¹/₄ of a number. Is there another way to do it?
Develop and use specific vocabulary in different contexts I can pick out words that tell me that I should subtract one number from another	In a group, sort this set of word problems into those that involve addition, those that involve subtraction, those that involve multiplication and those that involve division. What words were clues? Do you all agree?

Learning overview

Children derive and recall the 3, 4 and 6 multiplication facts and the related division facts. They use these facts to respond to questions like:

How many sides do six triangles have?
There are 20 legs. How many zebras is this?

Children recognise when word problems involve multiplication or division. For example, groups of children work together reading aloud a set of one-step word problems. They discuss and agree which problem involves which operation, placing them into sets. They look at those that involve multiplication and those that involve division, and discuss and explain to each other what clues in each problem helped them to identify the operation.

Children understand that multiplication and division are inverse operations. They use this to state the multiplication calculation that is linked to a division calculation, or vice versa. They explore what happens when you multiply then divide by the same number. In this way they develop their understanding that division reverses multiplication and vice versa. This helps them to solve problems such as:

I think of a number, double it and add 5. The answer is 35. What was my number?

Children use doubling or halving to establish new facts using known facts. They appreciate that one way to multiply by 4 is to double and double again. They know that they can double the answer to the multiplication fact 7 × 3 = 21 to find the answer to 7 × 6. They double the answer to 10 × 7 = 70 to work out 20 × 7. They use the fact that halving is the inverse of doubling to check results. They apply these skills to scale measurements, for example, to make a half-size model.

Children explain how to multiply a number by 10 or 100. They extend this to multiply one-digit numbers by multiples of 10, recording their methods informally, for example:

3 × 50 = 3 × 5 × 10
= 15 × 10
= 150

Children work out calculations that divide exactly and those that give rise to remainders. They discuss the images in the ITP 'Remainders'.

A screen shot from the ITP 'Remainders'

They use their experience to predict, for example, a number that will have a remainder when divided by 5 or a number that won't have a remainder when divided by 10. They investigate general statements such as:

When you divide a number that ends in 3 by 10, it will give a remainder of 3.
When you divide an odd number by an even number there will be a remainder.

Children share their findings in a class discussion and respond to the findings of others.
Children understand division as sharing. They solve problems such as:

42 crayons are divided equally between six pots. How many crayons are there in each pot?
Three children want to buy their grandmother a present costing £1.50. They each give the same amount. How much does each child give?
An 80 cm length of ribbon is cut into four equal pieces. How long is each piece?

Children link fractions to division. For example, they find ¹/₃ of 18 objects by sharing them into three equal groups. They appreciate that ¹/₃ of 18 is equivalent to 18 ÷ 3.

Children read, write and understand unit fractions such as ¹/₃, ¹/₁₀ and ¹/₅. They realise that the denominator shows the number of parts that the whole is divided into, and they know that each part must be of equal size. They find unit fractions of amounts by sharing out collections of objects. They draw diagrams to show unit fractions of shapes, for example investigating the problem:

In how many different ways can you colour half of a 2 by 2 square?

Children compare unit fractions. For example, they find different unit fractions of strips made from 12 squares. They colour half of one strip, a third of another and a quarter of another, then use these to decide which is the biggest and which is the smallest of the fractions. They use other visual images such as a fraction wall or the ITP 'Fraction' to consolidate their understanding of the relative sizes of unit fractions.

A screen shot from the ITP 'Fraction'

Children estimate a fraction of a shape or object, for example to suggest what fraction of a jar is full of marbles or to cut an apple approximately into thirds. They draw an arrow on a 0 to 10 line without divisions, and estimate the number that the arrow is pointing to. They check their guesses by revealing the divisions.

A time line numbered 0 through 10 with an arrow pointing at 7 and a 2nd timeline with 0 and 10 at either end and an arrow pointing at where number 7 should be

They choose a number on a number line to 100 and estimate where half of that number lies.

Resource links to existing published material

Mathematical challenges for able pupils Key Stages 1 and 2

Activities	PDF 923KB
Activity 30 - Suzie the snake

Intervention programmes

Objectives for Springboard intervention unit	Springboard unit
Know by heart doubles of numbers to 10; doubles of multiples of ten up to 50 Identify near doubles using doubles already known Halve even numbers from 20 to 2 Measure and compare lengths using a standard measure	Springboard 3 Unit 4 sessions 1 and 2 (PDF 181KB)

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

Diagnostic focus	Resource
When sharing, can sometimes make equal groups but has no strategies to deal with any left over	4 YR×/÷ DfES 1140-2005 (PDF 77KB)
Does not use partitioning to find double twelve or double thirty-five	4b Y2×/÷ DfES 1146-2005 (PDF 68KB)
Is not systematic when sharing into equal groups using a 'one for you' approach; does not use the language of division to describe the process	6 Y2×/÷ DfES 1148-2005 (PDF 96KB)
Is not confident in recalling multiplication facts	1 Y4×/÷ DfES 1150-2005 (PDF 104KB)
Describes the operation of multiplying by ten as 'adding a nought'	3 Y4×/÷ DfES 1152-2005 (PDF 68KB)
Writes a remainder that is larger than the divisor	6a Y4×/÷ DfES 1155-2005 (PDF 76KB)
Discards the remainder; does not understand the significance	6b Y4×/÷ DfES 1156-2005 (PDF 93KB)

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