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# Ma2 Number

## Sorting numbers

Harjeev
Sixty-four.
Teacher
Sixty-four.
Harjeev
It goes in the twos.
Teacher
Why does it go in the twos?
Harjeev
It goes in the twos because it goes two, four. So when it goes two, four…
Teacher
So when you are looking at these numbers do you look at the first number or do you look more at the second number?
Harjeev
I look more at the second number just in case it's an odd or even number.
Teacher
Okay.
Harjeev
Multiple of five… Inaudible
Teacher
Where does it go?
Harjeev
Thirty-seven. So it’s… Don’t goes in the twos and it don’t go in the fives. Thirty-seven.
Teacher
What… what numbers are we expecting to find to put in here then? What numbers would you want to pick up to put in that box?
Harjeev
I think, maybe … twelve.
Teacher
Twelve would go in this box?
Harjeev
I think so, because, it goes five, ten, fifteen. So it is a five… I mean it’s four, five… no fifteen, I mean, because… because it has got a five in it and it’s got a… and I think… No. I don’t know really what goes in here.
Teacher
Right Harjeev, Rees has got three numbers in that box.
Harjeev
Oh.
Teacher
What do you think he’s got? Dara’s got two numbers in her box.
Harjeev
Fifty-two? No. Fifty-…
Harjeev
Five, ten, fifteen, twenty, twenty-five, thirty. Yeah thirty.
Harjeev
Two, four, six, eight, ten, twelve, fourteen, sixteen, eighteen, twenty, twenty-two, twenty-four, twenty-six, twenty-eight, thirty. Yeah, I think it’s thirty.
Harjeev
It’s lots whats ends with zero.
Teacher
Oh, really.
Harjeev
Now I get it.

sf_asf_ma_harjeev_sorting.mov (16.9 MB) sf_asf_ma_harjeev_sorting.wmv (15.1 MB)

### Teacher's notes

• Sorts numbers between 1 and 100 into a Carroll diagram.
• Uses two criteria: multiples of 2 and multiples of 5.
• Knows the units digit is significant when deciding if a number is a multiple of 2, an even number.
• Reasons about a number that is a multiple of both 5 and 2.
• Counts in fives to thirty and recognises 30 as a multiple of 2.
• Makes a generalisation about multiples of both 5 and 2, 'It's lots [of numbers] that end with zero.'
• Puts a three-digit number in each region.

### Next steps

Use multiples of other numbers as criteria, such as multiples of 2 and multiples of 3, and reason about the numbers that are multiples of both 2 and 3.

## Fractions

Interviewer
How many are you trying to split it into?
Harjeev
I'm trying to put it in six groups.
Interviewer
Why are you doing six groups?
Harjeev
Because it's one and a six and it means that you need to do it in six groups.
Interviewer
Go on then, carry on.
Harjeev
Just getting a bit confused.

sf_asf_ma_harjeev_fractions.wmv (8.7 MB) sf_asf_ma_harjeev_fractions.mov (9.4 MB)

### Teacher's notes

• Chooses a lined whiteboard to show what he knows about fractions.
• Decides to represent each 'whole' as a rectangle.
• Understands how to write fractions and which part of the fraction determines the number of parts.
• Refers to putting his diagram into 'six groups'.
• Perseveres to create a rectangle divided into six parts, although they are not equal parts.

### Next steps

• Talk about fraction names: one-half, one-quarter, one-sixth.
• Talk about one-sixth as one of six equal parts or three-quarters as three of four equal parts.
• Consider how to represent his fractions more accurately.

## More fractions

### Teacher's notes

• Folds a rectangle and an isosceles triangle into two equal. regions and labels each as 1/2
• Folds squares into equal regions to demonstrate quarters
• Decides to continue folding and labels each region as 1/16
• Refers to his diagrams to respond to questions about equivalent fractions, e.g. to find how many quarters there are in one-half of a shape and how many sixteenths in one-half.

### Next steps

• Represent and name other fractions, e.g. given a rectangle divided into 12 equal regions represent fractions such as 1/4, 3/4, 1/3, 1/6, 3/6  and say their names.
• Find other fractions that are equivalent to one-half.

## Multiplication 1

Teacher
What is a different wording for four times six? What other wording can we use instead of times?
Harjeev
Four multiplied by six. Four groups of six.
Harjeev
One, two, three, four, five, six. One, two, three, four, five, six. Six times six… Inaudible.
Teacher
What did you get?
Harjeev
I got thirty-six.
Teacher
I got thirty-six as well. Well done.
Teacher
I think for Harjeev I would next give him some counters, say twenty counters, and say to him, ‘How many different arrays can you make using those different counters?’ and see what answers he come up with and how he would record it.

sf_asf_ma_harjeev_multiply41.wmv (5.5 MB) sf_asf_ma_harjeev_multiply41.mov (6.8 MB)

### Teacher's notes

• Knows 4 × 6 represents four groups of six and six groups of four.
• Reads 4 × 6 as 'four times six', 'four multiplied by six' and 'four groups of six'.
• Represents multiplication as arrays.
• Knows 6 × 6 = 36 but checks the array has six rows of six crosses, and checks the total by counting in ones.

### Next steps

• Represent numbers such as 20 using as many different arrays as possible.
• Count large numbers of objects in groups rather than in ones.

## Multiplication 2

Harjeev
Now I'm just doing a… figuring out another one, two, three, four, five, six, seven, eight, nine, ten, eleven…
Child 1
I'm ready. What should I do now?
Child 2
Oh, I made my pyramid but – two minutes – two missing, just. Only eighteen. Where did you get that from?

[Harjeev is lining up his counters throughout and counting under his breath]

Harjeev
Now I'm just going to… twenty times one. One, two, three, four, five, six, seven, eight, nine…

[Harjeev makes two lines of ten]

Harjeev
Now I'm just doing ten times two.

[Harjeev makes five lines of four]

Interviewer
What are you doing now, Harjeev? What are you doing now?
Harjeev
Five times four.
Interviewer
Okay.

sf_asf_ma_harjeev_multiply42.wmv (19.2 MB) sf_asf_ma_harjeev_multiply42.mov (20.1 MB)

### Teacher's notes

• Uses counters to demonstrate multiplication facts for 20.
• Works systematically and describes his arrangements as '20 times 1', '10 times 2' and '5 times 4'.
• Represents the first and last examples as arrays.

### Next steps

• Arrange the counters for 10 × 2 as an array.
• Discuss how each array shows two multiplication facts, e.g. one array shows 5 × 4 and 4 × 5.

## Understanding division

### Teacher's notes

• Derives division facts from known multiplication facts.
• Represents division as sharing.
• Tries to represent division as repeated subtraction but records incorrectly.

### Next steps

• Discuss how 'sharing among 5' and 'how many fives make…' are related, e.g. every time there is a five, then five people can have one each.
• Discuss why 20 - 5 – 5 – 5 – 5 = 4 is not correct, but how 20 – 5 – 5 – 5 – 5 = 0 shows that 20 ÷ 5 = 4

## Word problems

Clarke, Peter. (2006) Apex Maths Word Problems CD-ROM 2, Cambridge University Press. Used with kind permission.

### Teacher's notes

• Solves problems involving measures.
• Underlines key information.
• Interprets difference problems as ‘how many more’, e.g. 16 + ? = 30
• Explains how he uses recall of addition and multiplication facts to check results:

50 + 20 = 70 so 50 + 22 = 72

16 + 10 = 26 so 16 + 14 = 30

6 × 2 = 12

### Next steps

• Use and record multiplication where appropriate.
• Discuss how subtraction can be used to represent difference problems as well as 'take away'.

### Teacher's notes

• Draws lines to show the digits he adds.
• Knows which digits represent hundreds, tens and ones.
• Demonstrates his method in a different way to show how he knows he is correct.

### Next steps

Begin to add three-digit numbers where examples involve bridging the tens or the hundreds.

## What the teacher knows about Harjeev’s attainment in Ma2, Number

Harjeev reads and records numbers with up to four digits, for example some of the totals he makes when he adds two three-digit numbers. He partitions numbers into hundreds, tens and ones demonstrating his understanding of place value. Given a number he is beginning to know which is the nearest ten. Harjeev recognises multiples of 2, 5 and 10 and works out rules for other sequences that increase or decrease in steps of equal size.

Harjeev understands unit fractions as one of an equal number of parts. Given paper squares he folds them to create halves, quarters and sixteenths. He is beginning to understand and represent fractions that are several equal parts of the whole. He knows that two-quarters equal one-half. Harjeev finds one-half of numbers including three-digit even numbers. He is beginning to use decimal notation to represent sums of money.

Harjeev knows that in addition and multiplication order does not matter. For example, he knows that 4 × 6 and 6 × 4 give the same answer and can be represented by the same array. He understands subtraction as the inverse of addition. He is beginning to derive division facts from known multiplication facts. For example, given 30 ÷ 2 he knows 2 × 15 = 30 and uses this to solve the division. He represents division as ‘sharing’. He also understands that 30 ÷ 2 can be solved as 'how many twos in thirty'. He is beginning to represent division as repeated subtraction although he does not yet record this accurately.

He adds and subtracts two-digit numbers mentally. Given 36 + 17, for example, he describes his method, 'first I added the ten and then the ones' and shows 36 + 10 = 46, 46 + 7 = 53. When subtracting mentally he often uses the strategy of counting on from the smaller number. For subtraction he often counts on in ones rather than in larger steps. Harjeev knows multiplication facts for 2, 5, and 10. He is beginning to count in fours, threes and sixes and to learn these multiplication facts. He uses mental strategies to complete calculations such as 6 × 25 and 'half of 150'.

He solves problems involving addition, subtraction and multiplication in the context of money and measures. He is not yet consistent in completing the second calculation in two-step problems such as finding the total cost and then calculating change. For multiplication problems he sometimes uses a strategy of repeated addition. He understands when to round up in simple division problems that involve a remainder such as how many packs of five balloons to buy if eighteen are needed for the party.

To add three-digit numbers, Harjeev is beginning to use a partitioning method and to apply this to examples that involve bridging the tens. He does not yet have a reliable method for subtracting three-digit numbers.

## Summarising Harjeev's attainment in Ma2, Number

Harjeev's attainment is best described as level 3 in almost all of the assessment focuses for Number. It is only in written methods that level 2 is still the best description. Reading the complete level descriptions for levels 2 and 3 confirms the level 3 judgement for Number.

In Numbers and the number system, Mental methods and Written methods there are assessment criteria that Harjeev does not yet meet or meet fully. His progress in Fractions is recent and he has yet to demonstrate his attainment consistently. He has yet to solve a range of numerical problems involving larger numbers. Consequently his teacher refines her judgement to low in level 3.

To make further progress within level 3, Harjeev needs to work with a wider range of fractions to consolidate his understanding of fractions that are equivalent to one-half, for example. He should begin to use decimals in contexts such as measuring accurately in centimetres and recording half-centimetre lengths. He also needs to solve a wider range of numerical problems including those that involve larger numbers or two operations such as addition followed by subtraction. He needs to develop his methods for adding and subtracting three-digit numbers.