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# Ma1 Using and applying mathematics

### Teacher's notes

• Identifies what she has to do.
• Models the problem with paper circles and counters.
• Starts by placing three counters on each paper circle.
• Moves one counter at a time and checks results.
• Writes numbers in the given diagram to record her arrangement of counters.
• Finds arrangements where lines total 8 but forgets there must be a different number on each plate.
• Places a different number on each plate and creates lines that add to 9.

### Next steps

• Position cards with numbers 1 to 5 on the 'cross roads' so that each line totals 10.
• Reposition the cards to make totals of 9 and then 8.
• Apply knowledge to solving a 'magic square' puzzle by placing each number from one to nine on a 3 x 3 grid so that each row, column and diagonal totals 15.

## Doubles

### Teacher's notes

• In teacher-led activity, identifies 1, 2, 4, 8 as a doubling pattern.
• Explains that she doubles 1 to make 2, 2 to make 4, 4 to make 8.
• When asked what comes next, says 'double eight'.
• Uses fingers to calculate double 8 and give answer of 16.
• In follow-up work, records doubles of 2 to 10 in order using words and numbers.

### Next steps

• Record a halving pattern starting with 20 initially; compare results with doubling pattern.
• Record other doubles she knows or can work out.

## Make 12

### Teacher's notes

• Identifies what she needs to do.
• Understands and applies the 'rules' consistently after two incorrect attempts.
• Without help, records addition number sentences using '+' and '='.
• Records results in a random way but avoids duplication.
• Finds seven of the ten correct solutions but omits three solutions that include zero.
• Checks work.

### Next steps

• Find the three missing solutions that include zero.
• Rearrange each of the ten solutions so that numbers in each addition are in order, e.g. 1 + 2 + 9; with support, look for patterns in the results.

## Odd and even

### Teacher's notes

• Sorts given numbers using criterion odd/not odd.
• Organises work by crossing out each number as it is entered on to sorting diagrams.
• Goes on to choose and enter additional numbers.
• Reasons that 'not odd' numbers are 'even' and are the 'answers' when she recalls multiplication facts for 2.

### Next steps

Use Venn and Carroll diagrams to sort using two criteria, e.g. odd numbers/numbers greater than 50; describe numbers in each region.

## 3-D shapes

Teacher
My 3-D shape has two… square faces. Hannah, what do you think?
Hannah
Cuboid.
Teacher
Hannah, could you tell me anything else about a cuboid?
Hannah
It has six vertices.
Teacher
Can you check Hannah that it has got six vertices?
Hannah
Okay.
Teacher
See if you are right.
Hannah
One, two, three, four… five, six… seven, eight.
Teacher
So how many vertices did it have?
Hannah
Eight.
Teacher
Hannah, can you tell me what label you’ve made there?
Hannah
Green and three faces.
Teacher
Green and three faces. Where are you going to put that?
Hannah
Here.
Teacher
Well done. What label could we make for this group here?
Hannah
Yellow and… and eight sides… edges.
Teacher
Eight edges. Okay. Shall we just check?
Teacher
So yellow and eight edges. Could you show me where those eight edges are for me?
Hannah
One, two, three, four, five, six, seven, eight.
Teacher
Good girl.

sf_asf_ma_hannah_3dshape.mov (7.5 MB) sf_asf_ma_hannah_3dshape.wmv (7.0 MB)

### Teacher's notes

• Visualises that a hidden 3-D shape with two square faces is a 'cuboid' and has 'six vertices'.
• After shape is revealed, checks number by pointing to and counting each vertex, then revises answer to 'eight'.
• Follows up preceding activity by sorting 3-D shapes using own criteria and makes labels to describe each category.
• Makes label 'green and 3 faces' and places it with set of green cylinders.
• When teacher holds up yellow square-based pyramid and asks, 'What label could we make for this one here?', replies, 'yellow and eight sides' then refines 'sides' to 'edges'.
• At teacher's request, points to and counts number of edges individually as confirmation.

### Next steps

• Use Venn or Carroll diagrams to sort and classify 3-D shapes using two criteria such as 'yellow' and 'cylinders'.
• Make fact sheets for 3-D shapes, e.g. drawings accompanied by properties/features such as number of edges, faces and vertices, shapes of faces, length of edges, etc.
• Visualise a range of shapes focusing on counting the faces, vertices and edges.

## Programmable toy

Teacher
What sort of words do you think we are going to use when we move Bee-Bot around? Hannah?
Hannah
Forwards.
Teacher
Turn. Okay. Emily?
Emily
Clockwise.
Teacher
Clockwise. Hannah, another one?
Hannah
Anticlockwise.
Teacher
Anticlockwise. Which way, show me with your hands, which way would he turn if he went clockwise?
Emily
Oh.
Teacher
Which way is he going to go?
Emily
That way.
Teacher
That way. And which way is he going to go if he goes anticlockwise? Well done. What other words might we need to use?
Emily
Left and right.
Teacher
Left and right. Show me left… Well done. And show me right… Brilliant. And do we think we are going to get Bee- Bot from his home up to the flower?
Both
Yes.
Teacher
I’d like you to see if you can get him back from the flower to his home.
Hannah
Did you go left twi… yeah you did go left twice.
Emily
No. It was facing that way… and it…inaudible
Hannah
Left twice.
Emily
Yeah… was left twice.
Hannah
Forward.
Emily
Yes. Forwards twice.
Emily
Turn… turn left, then… then go forwards once.
Teacher
Well done. Excellent, girls. That’s brilliant. Can you count on your list there and see how many different instructions you had to give him?
Emily
Forwards. Oh yeah, that’s turn.
Hannah
That’s turn…inaudible
Emily
Turn left and then one space forwards…
Teacher
Okay, how many instructions did you have to give?
Both
Eleven.

sf_asf_ma_hannah_toy.mov (18.4 MB) sf_asf_ma_hannah_toy.wmv (17.2 MB)

### Teacher's notes

• Communicates with partner throughout.
• When teacher asks both children to suggest words related to movement, contributes 'forwards', 'turn' and 'anticlockwise'.
• Demonstrates with arm movements clockwise, anticlockwise, left and right.
• With partner, programmes toy to move along grid of squares from home to flower target then back home.
• Programmes toy to make return journey in fewer moves than outward journey.
• Recalls return route with partner using language of programming and records it.
• When asked by teacher, counts with partner and records number of instructions needed for return journey.

### Next steps

• Get a classmate to try out written instructions, review and adapt if necessary.
• Instruct another child on how to programme toy to reach a destination.

## Fruit survey

### Teacher's notes

• Chooses to find favourite fruit of classmates from choice of four fruits.
• Tests hypothesis that, 'The favourite fruit of children in our class is an apple.'
• Works independently of teacher with partner.
• Creates tally on whiteboard to record choices.
• Transfers data from tally to simple column graph and related table using a familiar computer program.
• With teacher-prompt, concludes that apples and nectarines are 'both the favourite fruits'.
• Uses terms 'tally', 'table', and 'graph' when describing what she has done.

### Next steps

• Put titles on graphs, tables, etc.
• Interpret then create graphs in which the frequency axis is labelled in twos.
• Test a hypothesis that requires a comparison, e.g. 'More children like apples than like bananas.'

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

When given simple problems to solve, Hannah draws on a range of strategies. For example, in 'Cross roads', she identifies what she has to do and decides on the strategy of moving only one counter at a time to adjust the total of counters in each row and column. She checks results after each move until she achieves totals of ten. She demonstrates similar skills in 'Make 12' when she has to try to find ten different solutions to an addition problem. When programming a toy robot, she identifies with a partner what they have to do to move the toy from its 'home' on a grid to a target destination. She then shows flexibility by programming the toy to return 'home' in fewer moves.

In 'Fruit survey', again working with a partner, she identifies what needs to be done to test a simple hypothesis. Together, they choose what strategies to use, i.e. what data to collect, and how to collect and represent the data. She is starting to make connections between different mathematical situations, for example when recognising that even numbers are the 'answers' when she recalls multiplication facts for 2.

With questions to prompt her, Hannah explains her work using appropriate mathematical vocabulary. For example, when discussing results of her fruit survey with her teacher, she communicates her findings by referring to the tally, graph and table she constructs. Hannah is starting to recognise the need to record findings, for example using a tally to collect data. When programming the toy robot, she recalls several terms associated with movement prior to programming. She uses relevant vocabulary while programming and records the instructions used to move the toy, such as ‘turn right, 2 spaces forward’. She writes addition sentences using symbols '+' and '=' as she shows when recording solutions to the 'Make 12' problem. Hannah often works collaboratively with partners, two examples of which are programming the toy and conducting the fruit survey. She listens to explanations given by others and contributes to group and class discussions.

Hannah shows reasoning skills in different aspects of mathematics. In the 'Doubles' activity, she recognises that '1, 2, 4, 8' is the start of a doubling pattern, explains that 'double 1 is 2, double 2 is 4' and so on and reasons that the next number will be 'double 8'. When she programmes the toy, she reasons with a partner about how to get the toy to move to a specified destination then back ‘home’ using fewer moves. She is starting to test simple hypotheses. For example, with a partner she finds out if the statement, 'The favourite fruit of children in our class is an apple.' is true. She chooses how to collect, represent and interpret the data.

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

Hannah meets almost all of the assessment criteria for level 2 in Using and applying mathematics. Her teacher judges Hannah’s attainment in each assessment focus is best described as level 2. After reading the complete level descriptions for levels 2 and 3, she confirms her level 2 judgement for Ma1, Using and applying mathematics.

To decide whether to describe Hannah's attainment as secure level 2 or high level 2 her teacher takes other factors into consideration. She knows that Hannah has covered most of level 2 in Ma1 but goes on to reflect on how consistently, how independently and in what range of contexts. Her teacher knows that much of Hannah's attainment in Ma1 is recent or undertaken with the support of a partner. She also recognises that, in independent work, Hannah’s recent problem solving involving number has focused mainly on addition with smaller numbers.

Additionally, in reasoning, Hannah is only starting to explain why answers are correct. Her teacher concludes that Hannah has not yet demonstrated her attainment consistently or in enough contexts including individual work to be described as high at level 2. Nor has she shown enough problem-solving skills involving numbers to 100 or subtraction. Her teacher refines the judgement to secure at level 2 but recognises that she is on the border between secure and high.

To make further progress within level 2, Hannah should continue to work collaboratively but also work individually using and applying her mathematics. She should solve a wider range of puzzles and problems involving numbers up to 100 and each of the operations. She should record her methods. Hannah should explain why answers are correct in a range of contexts including number, for example by using inverse operations to check answers.