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# Ma1, Using and applying mathematics

## Sorting and ordering numbers

### Teacher's notes

• Assembles a new classroom resource, which has a 10 by 10 arrangement of clear pockets to hold cards numbered from 1 to 100.
• Chooses to find numbers 1 to 10 and position them in turn across the top row.
• Adapts his approach to include positioning numbers in columns.
• Uses the units digit to decide the column and the tens digit to decide the row.
• Orders all of the numbers from 1 to 100.

### Next steps

• Talk about what would happen if:
• the numbers started with zero
• there were only five columns in the grid.
• Pose his own questions, predict where numbers will be placed and test the prediction.

## Archaeological finds

Teacher
What would we write on the pot? What labels would we write? Daniel?
Daniel
Teeth.
Teacher
Teeth. Fantastic!
Daniel
As well as…
Teacher
What other things do…
Daniel
Bones.
Teacher
Bones. Brilliant!
Other
Metal.
Teacher
Metal. Leah?
Leah
Glass.
Teacher
Glass. We might find glass.
Teacher
So we have our pots to sort and group our finds, but how are we going to remember how many plastic artefacts we find, how many metal artefacts we find, how many bone artefacts we find? Daniel?
Daniel
We could record it on the board.
Teacher
Could you write it on the board for us, Daniel, to show us how you’d organise that information?
Evangeline
Or a tally chart.
Teacher
A tally chart. Is that what you think Daniel is doing, Evangeline?
Teacher
Who would like to count how many tally marks we have in wood? Daniel, I’m going to choose you.
Daniel
One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen,seventeen, eighteen, nineteen, twenty, twenty-one, twentytwo, twenty-three, twenty-four, twenty-five.
Teacher
Twenty-five. Who had the closest guess?
Others
Roshan
Teacher
Daniel?
Daniel
I did because I went er… thirty-one.
Teacher
Roshan said…
All
Twenty-six.
Teacher
Roshan said twenty-six. Now how many did you count, Daniel?
Other
Twenty-five.
Daniel
Twenty…
Other
Five.
Daniel
Twenty-five.
Teacher
Twenty five. So is thirty-one closer to twenty-five or is twenty-six?
Daniel
Twenty-six.
Teacher
Twenty-six. Well done.

In the classroom

Teacher
Daniel, how would you count… how would you count the tallies?
Daniel
There’s twenty-five of er… wood because I counted in fives and it got to twenty-five.
Teacher
Can you show us how you counted in fives, Daniel?
Daniel
Five, ten, fifteen, twenty, twenty-five.
Teacher
Good counting, Daniel. Well done.
Daniel
And next we’ve got… ten… five, ten, fifteen, twenty, twenty-five… twenty-six.
Teacher
How many artefacts would be in the not bone case? Work it out with your partner. Working out.
Daniel
Inaudible… forty-five, fifty, fifty-one… fifty-two, fifty-three, fifty-four, fifty-five, fifty… -six.
Other
Fifty-seven.
Daniel
Fifty-seven.

[End of clip]

sf_asf_ma_daniel_archfind.mov (19.4 MB) sf_asf_ma_daniel_archfind.wmv (17.8 MB)

### Teacher's notes

• Suggests labels for pots to sort and group finds from a sample of garden soil: teeth, bones, metal…
• Divides the whiteboard and labels regions for children to keep a tally as they place finds into labelled pots.
• Estimates that there are thirty-one tally marks for wood.
• Counts the twenty-five tally marks for wood, ignoring the mark that is partially erased, and recognises his estimate was close but not the closest.
• Uses the teacher's grouping of tallies on the whiteboard, counts in fives and confirms the number of wooden artefacts found.
• Counts in fives and one more to find the total of metal artefacts.
• Uses the tally chart and counts in fives to find how many artefacts would be in the 'not bone' case.

### Next steps

• Suggest a criterion for sorting the finds into two roughly equal groups to display in the two available museum cases, e.g. metal/not metal.
• Use a block graph or pictogram where one symbol represents two to record the information about finds from garden soil.
• Compare the graphs of finds from the garden soil and finds from the beach sand.
• Pose and answer questions, e.g. 'What types of find were found only in the sand from the beach?', 'Were some types found only in the garden soil?'.
• Begin to group tallies in fives to make counting more efficient as he collects other data.

### Teacher's notes

• Chooses to list addition facts he knows for different numbers.
• Works in an organised way finding pairs of numbers that add to 1, 2, 3…
• Records addition sentences using symbols '+' and '='.
• Understands addition can be done in any order.
• Checks he has all addition pairs for each number.
• Looks for pattern in results.
• Notices that the number 1 has two facts, number 2 has three facts…
• Predicts that for the number 9 there will be ten facts and for 10 eleven facts.

### Next steps

• Reorganise his calculations as a way of justifying his prediction, e.g.
0 + 3 = 3
1 + 2 = 3
2 + 1 = 3
3 + 0 = 3
• Test the prediction by listing the addition facts for 9 and 10.
• Find different pairs of numbers to make these subtraction sentences correct:

```☐ - ☐ = 1, ☐ - ☐ = 2
```

• Talk about why there are an infinite number of ways to complete the subtraction sentences.

## Coins problem

### Teacher's notes

• Shows how to pay for various items using as few coins as possible.
• Works in an organised way selecting the largest coin values first.
• Places coins next to items.
• Records by tracing around the coins, accidentally moving a 1p coin into the wrong group.
• Records coins that total 72 pence but not the fewest possible.

### Next steps

• Review and check his work.
• Investigate a related situation:
• find all the ways to pay exactly 10 pence and talk about how you know you have them all
• find ways to pay each amount from 1p to 50p using as few coins as possible and talk about amounts that can be made with just one coin, with just two coins.

## Repeating patterns

### Teacher's notes

• Identifies the repeat in patterns made by the teacher.
• Continues those patterns, mostly accurately.
• Explains how he knows what comes next.

### Next steps

• Check patterns to identify a missing shape in one of them.
• Predict how many cubes there will be in the next shape, the fifth shape…and give his reasons.
• Make the shapes and check predictions.
• Create a similar sequence for a partner to continue.

## What the teacher knows about Daniel's attainment in Ma1, Using and applying mathematics

Daniel identifies his own starting points for solving a range of problems that relate to numbers and data particularly. He uses apparatus such as coins and cubes to represent the situations he investigates and problems he solves. He is beginning to approach some problems systematically, for example starting with the largest coin denominations when finding ways to pay different amounts using as few coins as possible. He applies the handling-data skills developed in one survey when undertaking another. For example, having used a tally to record votes for favourite crisp flavours he suggested a tally for recording different types of archaeological find.

Daniel discusses his work using everyday and mathematical language. For example, he describes how to use known doubles facts to find half of an even number. He talks about sorting, counting, tallying, listing and drawing a table or graph when handling data. Daniel is beginning to represent his work using symbols and diagrams. He uses symbols to record addition and subtraction. He uses diagrams, tables and graphs to record and retrieve information.

Daniel tests simple statements about numbers to see if they are true or false. When testing ‘There are five odd numbers between 10 and 20' he listed 11, 13, 15, 17 and 19 on his whiteboard. In this instance he explained, 'There are five. I worked them out. Ten is even. Then there's an odd number, then even, then odd… The odd numbers are all numbers that aren't multiples of two.' Daniel predicts what comes next in a simple number or shape pattern. For example, he continued an ascending sequence of multiples of five and then created a descending sequence by reversing the order.

## Summarising Daniel's attainment in Ma1, Using and applying mathematics

In each assessment focus Daniel's teacher decides that his attainment is best described as level 2. Reading the complete level descriptions for levels 1 and 2 confirms that the level 2 description is the best fit overall.

Although Daniel meets all of the assessment criteria for level 2, much of his attainment relates to using and applying number and handling data. He has yet to demonstrate his attainment as fully in the contexts of shape and space or measures. His teacher refines her judgement to secure level 2.

To make further progress within level 2 Daniel should solve problems in a wider range of contexts, particularly contexts relating to shape and space. For example, using construction materials, he might identify which collection of linking shapes could be used to create a given 3-D shape. Investigating 2-D shapes, he might find different ways to fold a square into quarters and investigate the different shapes that can be made by fitting all four quarters of one square together edge to edge. He should also solve measurement problems such as comparing lengths that cannot be placed together for direct comparison.