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Ma1 Using and applying mathematics

Make 50

Illustration of Ben applying different ways to make an answer of 50.

Teacher's notes

  • Responds to 'Find different ways to make an answer of 50'.
  • Works independently choosing to use addition and subtraction.
  • Uses '+', '-' and '=' signs correctly to record solutions.
  • Knows the number that is one more/less, ten more/less than 50 and uses this to create 49 + 1 = 50, 60 - 10 = 50 and 40 + 10 = 50.
  • Uses his understanding of place value to create 10 + 10 + 10 + 10 + 10 = 50.
  • Knows pairs of numbers that add to ten and uses this with place value to create 44 + 6 = 50 and 43 + 7 = 50.
  • Is starting to check work but does not spot that 64 - 4 = 50 is incorrect.
  • Sometimes reverses the number symbol '5'.
  • Transposes some two-digit numbers, e.g. writes 10 as 01 but reads it as 10 when calculating.

Next steps

  • Ensure that the digits of two-digit numbers are not transposed.
  • Find and record all solutions to a number problem, e.g. using number cards 10, 20, 30 and 40, find all totals that can be made by adding numbers on pairs of cards.
  • Talk about his methods, the strategies he uses and how he knows he has found all possible totals.

Sharing story

Teacher
Let’s have hedgehog come to the party as well then, so now, how many people are going to be at Percy’s party?
Ben
Three.
Teacher
Three. Do you think this time you could share the cakes out so that they all have the same amount of cakes each and show me how you do it?
Teacher
Right, so how many have you given each animal this time?
Ben
Three.
Teacher
Three cakes, did you use all the cakes then?
Ben
No.
Teacher
No, how many cakes have you got left over?
Ben
Three.
Teacher
Do you think you could share those cakes out for me as well?
Teacher
Good boy, well done. How many cakes have they got each now?
Ben
Four.
Teacher
Let’s sit owl here. Okay. Sitting up. Do you think you could share those cakes out now for me?
Teacher
Oh, that was quick. Well done! How many cakes has each animal got now then?
Ben
Three.
Teacher
Three cakes.
Teacher
What do you think would happen if you and I were at the party as well?
Ben
We would share the cakes out.
Teacher
Do you want to have a go and see if you can share the cakes out for all of us?
Teacher
How many cakes have we all got now?
Ben
Two.
Teacher
There were twelve people at his party including him. How many cakes do you think they’d have all had each then?
Ben
One.
Teacher
One cake. They would, wouldn’t they. Well done.

Teacher discusses the 'sharing story' activity

Teacher
I was very impressed with Ben with the number story this morning. I was expecting him to share out the cakes for Percy the park keeper one by one, but while he was doing it he was actually almost sharing them out in groups, so when we had five of us or six of us at the party he was actually putting two cakes at a time and in his head he was managing to work out that two multiplied by six was twelve and he was actually a bigger step forward than I expected him to be.

[End of clip]

Download formats:
sf_asf_ma_ben_sharingstory.mov (15.5 MB) sf_asf_ma_ben_sharingstory.wmv (14.5 MB)

Teacher's notes

  • Sees relevance of mathematics by solving division problems set in context of story.
  • Shows how to share 12 cakes between two, three, four then six characters.
  • Interprets solutions to problems by responding to questions, e.g. 'How many cakes each do they have now?'
  • Shows some mental calculation strategies by sharing out 12 cakes among six characters in groups of two rather than one by one.
  • Knows that 12 cakes divided among 12 characters gives each character one cake.

Next steps

  • Without prompting, explain what he finds out after each sharing of cakes.
  • Record the different 'sharing' problems and their solutions.
  • Start to use the division sign to record results, e.g. 12 ÷ 2 = 6.
  • Make different arrays with 12 counters and record results.
  • Explain and compare his results with a partner.

Fifty pence

Interviewer
How many five pence coins did you use?
Ben
Nine.
Interviewer
Right, let's have a check, both of you. So get your five pence coins…
Ben
I checked again.
Interviewer
Right, Ben could you count them out on the table for me? So, five…?
Ben
Five, ten, fifteen, twenty, twenty-five, thirty, thirty-five, forty, forty-five, fifty.
Interviewer
So how many five pence coins, Ben, were there there?
Ben
Ten.
Interviewer
Ten five pence coins, okay. Right, Ben, could you show me one other way of making fifty pence?
Ben
Ten, ten, ten, ten, ten.
Interviewer
How many ten pence coins did you have there?
Ben
Five.
Interviewer
Well done.

[End of clip]

Download formats:
sf_asf_ma_ben_fiftypence.mov (7.9 MB) sf_asf_ma_ben_fiftypence.wmv (7.3 MB)

Teacher's notes

  • Uses coins and recognises values up to 50p.
  • Makes amounts totalling 50p using coins of the same value and of mixed values.
  • Predicts that nine 5p coins will equal 50p and checks.
  • Counts on in 5p and 10p coins up to 50p.
  • Checks some amounts by re-counting.

Next steps

  • Make a record of some amounts he finds.
  • Check totals, explaining his mental strategies to a partner.
  • Investigate whether 50p can made with 1 coin, 2 coins, 3 coins, … 10 coins and record findings.
  • Find as many ways as possible of making 50p with silver coins only.
  • Talk about how he knows he has found all possible ways.

3-D shapes

Hannah
They has eight edges.
Teacher
It has eight edges.
Ben
Is it the… triangle… um.
Other
Just do the square.
Ben
Square-based pyramid.
Other
Yes.
Teacher
Is he right? Okay, can you count then, let’s have a look. Well done.
Teacher
Ben, could you think of another clue we could give about the cylinder?
Ben
It’s had a face at the top and a face at the bottom.
Teacher
It has a face at the top and a face at the bottom.
Ben
And it has three faces.
Teacher
Okay! What shape are these two that you just told me about?
Ben
Circles.
Teacher
Circles, well done.
Teacher
Can they open their eyes?
Ben
Yes they can.
Teacher
Open your eyes. Right, Ben, could you give us your first clue?
Ben
It’s round at the top and round at the bottom.
Teacher
It’s round at the top and round at the bottom.
Ben
Hannah.
Hannah
Cylinder.
Teacher
Don’t tell us if we are right yet. Let’s see if Ben can give us another clue? Hannah thinks it is a cylinder. Ben, can you give us another clue?
Ben
It’s got three faces.
Other
A cylinder
Ben
A flat cylinder!

Teacher discusses the 3-D shapes activity

Teacher
While Ben was doing the 3-D shape activity he had all the language of the 3-D shape names so he could tell you cuboid, cube and I was very impressed with how he recognised the square-based pyramid, very good. I wasn’t expecting to have that one because we’ve only just covered that but he did, did know that one, that was great. But when we came to the sorting he’s confident with the sorting by one criteria but now we need to move on to sorting by two criteria…
Other
Yes.
Teacher
… with him, and just to make sure his shape language with edge and faces is secure as well.
[End of clip]

Download formats:
sf_asf_ma_ben_dshape.mov (16.8 MB) sf_asf_ma_ben_dshape.wmv (15.9 MB)

Teacher's notes

  • Recognises and names common 3-D shapes including square-based pyramid.
  • Visualises that a hidden 3-D shape with eight edges is a square-based pyramid.
  • Describes two cylinders using a range of vocabulary, e.g. 'has a face at the top', 'is round at the bottom' 'has three faces'.
  • Describes one cylinder about 3 cm in height as 'flat'.
  • Is starting to count faces and edges.

Next steps

  • Order 3-D shapes by their number of edges or faces.
  • Write clue cards about 3-D shapes to which a partner matches the shapes.
  • Write as many facts as possible about one 3-D shape.
  • Compare two 3-D shapes to find ways in which they are the same and how they differ.

Programmable toy

Handwritten note of a teacher's observation on Ben following instructions on working with a Beebot

Teacher's notes

  • Programmes toy to reach destination on square grid.
  • Uses single commands, e.g. 'forward' then 'go'.
  • Interprets symbols to make robot move towards target.
  • Uses vocabulary such as 'forward', 'turn', 'down' to explain moves.
  • With the help of arrow symbols, is starting to distinguish between left and right turns.

Next steps

  • Programme toy to use multiple commands, e.g. pressing 'forward, forward, turn left' then 'go'.
  • Programme toy to reach destination using a given number of moves or as few moves as possible.
  • Record the toy's journey in words, symbols and pictures.
  • Instruct another child on how to programme toy to reach a destination.
  • Start to use 'clockwise' and 'anticlockwise' to describe direction of turn.

Foot lengths

Illustration of a graph drawn by Ben to represent number of students having different feet sizes.

Handwritten note of a teacher's observation on Ben's ability to use a ruler, how to measure everyone's foot and make a graph on the data.

Teacher's notes

  • Applies recent work on measuring in centimetres to collect data on foot length.
  • Chooses which data to collect.
  • Designs table to record names of children and length of their feet.
  • Chooses ruler and uses it correctly to measure feet to nearest centimetre.
  • Collects data from other children and records data independently.
  • Records length of feet using 'cm' notation, e.g. 23 cm.
  • Tries to design a block graph independently.
  • Labels graph 'graph (on) feet size'.
  • Responds to questions such as, ‘How many children have feet that are 24 centimetres long?' by counting blocks.

Next steps

  • Check that all of the children in his list are represented on the graph.
  • Use the convention of leaving a space between columns.
  • Number the vertical axis accurately in a way that helps to read the number represented rather than counting individual 'blocks'.
  • Label the axes to explain what each represents.
  • Talk or write about what he has found out, e.g. 'The shortest foot was 17 cm long.'.
  • Think of questions to ask about his graph then try them out on a partner.

Shape pattern

Interviewer
Okay boys, now in front of you on the table I've put a pattern, a shape pattern. Can anyone tell me how I've made the pattern? What have I done?
Joshua
Well, you've repeated it.
Interviewer
What have I repeated, Joshua?
Joshua
This.
Interviewer
Okay, so what have I done?
Joshua
Made a pattern.
Interviewer
How? Can you tell me how I've made that pattern?
Joshua
You put that there and that there and that there and that there.
Interviewer
If I put this here then, would that be right next, Ben?
Ben
No.
Interviewer
Why would that not be right?
Ben
Because there's not another triangle.
Interviewer
Okay, because there isn't another triangle on there. Do you want to put the green triangle back then? Okay. Ben, do you think you can carry on the pattern for me, then? Well done, excellent. How many different shapes have you used in your pattern, Ben?
Ben
Count them?
Joshua
Just count one of them. No, just one pattern from one colour.
Ben
One, two, three, four, five of the diamonds.
Joshua
That's five are different.
Ben
And one, two, three, four, five of the red ones.
Interviewer
Excellent. Okay, what if I asked you to make me another repeating pattern using two different shapes? Can you both do that for me? Can you make a pattern each, though? Well done. Ben, can you tell me about your pattern?
Ben
Well, it's got six of these round yellow things.
Interviewer
Do you know what that shape is called, either of you?
Ben and Joshua
Hexagon.
Interviewer
Hexagon. How do you know it's a hexagon?
Joshua
Because it's got six sides.
Interviewer
Because it has six sides. Well done.
Ben
And it's got six of each.
Interviewer
It has six of each. And what else in your pattern, Ben?
Ben
Triangles.
[End of clip]

Download formats:
sf_asf_ma_ben_shapepattern.mov (19.3 MB) sf_asf_ma_ben_shapepattern.wmv (17.2 MB)

Teacher's notes

  • Given the rule that patterns can use only two types of shape, continues the teacher's pattern.
  • Describes shapes in pattern as 'diamonds' (rhombi) and 'red things' (trapezia).
  • Responds to teacher's instruction to make his own repeating pattern with two shapes.
  • Chooses yellow hexagon, green triangle as repeating unit.
  • Positions 11 shapes that show five repeats of unit and one extra hexagon.
  • Describes pattern as having 'six of these roundy yellow things' (meaning hexagons) then names hexagon correctly after prompt from teacher.
  • When partner says that hexagon has six sides, implies that it has six corners also.
  • Names triangles correctly.

Next steps

  • Create a shape pattern and describe to a partner how to copy it.
  • Make a pictorial record of patterns created with dividing lines to show each repeat and details of shapes used.
  • Make repeating patterns that involve counting sides or corners, e.g. any shape with 3 sides, 4 sides, 5 sides and 6 sides as the repeating unit.

What the teacher knows about Ben’s attainment in Ma1, Using and applying mathematics

Until the start of the summer term, Ben was largely dependent on approaches used by the teacher or classmates to solve problems. He is now starting to show more independence. For example, he chooses to use addition and subtraction to make answers equal to 50 using a range of one-digit and two-digit numbers. He also finds and displays different sets of coins that total 50p. He recognises that counting up to 100 objects in ones 'will take too long' and chooses to group and count them in tens. He chooses apparatus such as rulers and weighing scales. He also chooses number lines and 100-squares to help with number work. He is starting to apply knowledge gained in one mathematical context to other contexts.

One way he shows this is by transferring knowledge of the sharing aspect of division to solve problems that feature in a story. When given a project on handling data independently, he identifies what he has to do by bringing together several skills previously modelled by the teacher. He starts by choosing to compare foot lengths of classmates, designs a table to record the data, uses a ruler to measure in centimetres, and records measurements. With some support, he then transfers data from the table to a block graph. He is starting to programme a floor 'robot' and identifies routes to reach target destinations. When encouraged by his teacher, he checks work, for example by re-counting some amounts that total 50p but, as yet, he is not always accurate when checking answers to his written work.

Ben responds to questions about mathematics using a range of language but offers descriptions or explanations of his work, especially numbers and calculations, less often. Ben mostly records his work practically and is less likely to use written recording. He contributes to group discussions, for example to give clues about a hidden 3-D shape and to predict that a classmate has hidden a square-based pyramid. Ben discusses work on shape and space with more success than other aspects of mathematics. For example, when describing a cylinder, he says, 'It's round at the top; it's round at the bottom; it has three faces'. He uses a range of vocabulary related to position and movement, for example to describe the movement of a floor ‘robot'. He does not yet pose questions to ask about his work. He answers simple questions, for example, but cannot ask his teacher a question about his graph of foot lengths. Even with support, Ben finds it difficult to explain why an answer is correct. In independent work, he uses the '+', '-' and '=' signs correctly and abbreviations such as 'cm'. He is starting to create simple tables and graphs to collect and represent data as he shows in his investigations of foot lengths.

He shows emerging reasoning skills, for example when considering the best way to count up to 100 objects. He recognises that counting in ones 'could take too long' and he ‘will forget' so he suggests counting in tens. When given the start of a repeating pattern of shapes, he continues the pattern. He also creates simple shape patterns that meet the teacher's criterion, such as a repeating pattern with two shapes. He chooses his own criterion for sorting, such as choosing to find objects that are longer or shorter than 20 centimetres.

Summarising Ben's attainment in Ma1, Using and applying mathematics

Ben meets most of the assessment focuses at level 2 in Ma1. In Problem solving and Communicating, he meets all assessment focuses. In Problem solving he selects the mathematics and apparatus he needs to use. In Communicating, Ben discusses what he does in shape, space and measures in a more knowledgeable way than in number. He is beginning to choose ways to present his work using symbols and diagrams.

After reading the complete level description for level 2, his teacher decides that Ben's attainment in Ma1, Using and applying mathematics, is best described as level 2.

To decide whether his attainment is low, secure or high at level 2 in Ma1, his teacher considers four criteria how much of the level he has covered, how consistently he demonstrates the assessment focuses, how independently he works and the range of contexts where he engages with mathematics. Ben's teacher judges that, overall, he is not yet working securely at level 2 in Ma1. She makes this judgement for several reasons. First, because much of his progress is recent, he has not yet demonstrated that he can use and apply mathematics consistently. Second, he has not demonstrated his attainment in a wide enough range of contexts, especially in number work. Third, in Reasoning, there are elements of the assessment focus that he has not yet covered, i.e. explaining why an answer is correct, predicting what comes next in simple number patterns and giving reasons for the prediction.

After this consideration, Ben's teacher refines the judgement to low level 2.

To become secure at level 2 in Ma1, Ben should communicate using a wider range of mathematical vocabulary, especially in number work, so that he describes his work and methods in more detail and more accurately. He should develop written methods to show how he works out answers to calculations, especially those involving addition and subtraction that are too difficult to calculate mentally. Similarly, he should write number sentences involving all four operations to represent what he does practically. He should begin to appreciate the need to record. Ben should further develop his reasoning skills. For example, he should visualise familiar 2-D and 3-D shapes. As well as predicting what comes next in simple shape and spatial patterns, he should experience a range of number sequences, explain what comes next and why. He should also have opportunities to explain why answers are correct, for example by testing simple statements.