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In this learning overview are suggested assessment opportunities linked to the assessment focuses within the Assessing Pupils’ Progress (APP) guidelines. As you plan your teaching for this unit, draw on these suggestions and on alternative methods to help you to gather evidence of attainment, or to identify barriers to progress, that will inform your planning to meet the needs of particular groups of children. When you make a periodic assessment of children’s learning, this accumulating evidence will help you to determine the level at which they are working.
To gather evidence related to the three Ma1 assessment focuses (problem solving, reasoning and communicating), it is important to give children space and time to develop their own approaches and strategies throughout the mathematics curriculum, as well as through the application of skills across the curriculum.
In this unit the illustrated assessment focuses are:
Children recall multiplication facts to 10 × 10 and the related division facts. They use these to multiply and divide multiples of 10 and 100, such as 30 × 7, 4200 ÷ 6. They use patterns to extend the facts that they know and they look at the relationships between the number of zeros that form the final digits
| 4 × 80 = 320 | 320 ÷ 80 = 4 |
| 4 × 800 = 3200 | 3200 ÷ 800 = 4 |
| 40 × 8 = 320 | 320 ÷ 8 = 40 |
| 40 × 80 = 3200 | 3200 ÷ 80 = 40 |
| 40 × 800 = 32 000 | 32 000 ÷ 800 = 40 |
| 400 × 8 = 3200 | 3200 ÷ 8 = 400 |
| 400 × 80 = 32 000 | 32 000 ÷ 80 = 400 |
They use this new knowledge to extend the mental methods that they use for multiplication and division. They appreciate that multiplication can be done in any order and make sensible choices about how to multiply three numbers such as 4 × 7 × 5. They multiply two-digit by one-digit numbers mentally by using partitioning, calculating 26 × 7 by working out 20 × 7 and 6 × 7 then putting the answers together to get 182. They use factors where appropriate to help them to multiply numbers efficiently, for example calculating 35 × 6 by working out 35 × 2 × 3.
Children understand that 6 ÷ 3 gives a different answer from 3 ÷ 6. They divide two-digit by one-digit numbers mentally also by using partitioning, finding 51 ÷ 3 by splitting 51 into 30 and 21, dividing each part by 3 and then putting the answers back together to get 17. They use factors where appropriate; for example, they work out 90 ÷ 6 by dividing 90 by 3 and then dividing the answer by 2. Children explore patterns in linked division calculations; for example, they use a calculator to find the answers to the calculations 4000 ÷ 32, 2000 ÷ 16 and 1000 ÷ 8. They explain the patterns they notice, and suggest other linked calculations that will have the same answer as each other.
Children use mental methods (with jottings where appropriate) to solve problems involving multiplication and division, such as:
Children develop and refine written methods for multiplication. They move from expanded layouts (such as the grid method) towards a compact layout for HTU × U and TU × TU calculations. They suggest what they expect the approximate answer to be before starting a calculation and use this to check that their answer sounds sensible. For example, 56 × 27 is approximately 60 × 30 = 1800.Is 81 a multiple of 3? How do you know?
Find a number that has exactly six factors.
Find a number that is a common multiple of 6 and 8.
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Children collaborate in a group to solve problems and puzzles such as:
Use the digits 2, 3, 5 and 7 and the × symbol once each to create a multiplication calculation, for example 572 × 3 or 35 × 72. How many different products can you make? What is the largest product? What is the smallest product?
Children record their working systematically. They use reasoning to explain how they know that they found every possible solution.
Assessment focus: Ma2, Written and calculator methods
Look for the written methods that children choose to use as they solve problems, and at how and when they use a calculator. Look for children who use and can explain the steps in, for example, the grid method of multiplication quickly and accurately, and who begin to recognise how the same partial calculations can be recorded in a more compact form. Look for evidence of children recording the calculations they perform using a calculator, so that they can review and check them. Look for children who check calculator work by repeating a calculation or entering it in a different way.
Children use practical equipment and diagrams to extend their understanding of fractions. They recognise equivalence between fractions. For example, they fold a strip of 20 squares into quarters and colour 3/4 of the strip to establish that 3/4 is the same as 15 out of 20 or 15/20. They find other fractions that are equivalent to 3/4, recording their results and identifying patterns and relationships in the set of equivalent fractions. They use these patterns to predict other fractions that are equivalent to 3/4 and test their predictions. (The ITP 'Fractions' can also be used to establish equivalent fractions.)
Children express a smaller number as a fraction of a larger one. For example, they compare a base-ten 'ten' stick to a 'hundred' flat, appreciate that it would take 10 'tens' to make 1 'hundred' so 10 is 1/10 of 100. They compare a strip containing 3 squares with a strip containing 15 squares to establish that 3 is 1/5 of 15. They use their knowledge of the relationships between measures to answer questions such as:
Children find fractions of amounts using division and multiplication. For example, to find 3/10 of 20 they first find 1/10 by dividing 20 by 10, then multiply the answer by 3 to find 3/10. They use diagrams to confirm their calculations.What fraction of £1 is 50p, 75p, 30p, ...?
What fraction of 1kg is 500g, 400g, ...?
What fraction of a day is 1 hour, 12 hours, 8 hours, ...?
They record their working efficiently using symbols, for example:
20 ÷ 10 = 2 1/10 of 20 = 2
2 × 3 = 6 3/10 of 20 = 6
Assessment focus: Ma2, Fractions, decimals, percentages and ratio
Look for children who recognise equivalence between one fraction and another, and between fractions and decimals. For example, look for children recognising that 1/2 = 50/100 = 0.5, 1/5 = 2/10 = 0.2, or 3/4 = 75/100 = 0.75. Look for children who express one number as a fraction of another, for example the score in a quiz out of the total number of points available. Look for children who find a unit fraction of a number and those children who begin to calculate with fractions that are several parts of the whole, for example, children who find 1/5 then 3/5 of 30.
Children solve word problems that involve fractions, choosing to use a calculator where the calculations involved merit its use. They solve problems such as:
I pour 2/5 of a litre of juice into a jug. How many millilitres is this?
I have cycled 7/10 of a distance of 50km. How far do I still have to go?
I have saved £194.40. I plan to spend 5/12 of this on a bicycle. How much will I have left?
Children check that each answer sounds reasonable in its original context.
Assessment focus: Ma1, Reasoning
Look for evidence of children recognising patterns in their results and using these to predict and check other possible results. For example, look for children who recognise patterns in equivalent fractions such as 3/5 = 6/10 = 12/20 and 3/5 = 9/15 = 18/30. Look for evidence of children beginning to use the pattern to suggest other equivalent fractions. Similarly, as they investigate multiplication, look for children who recognise patterns in sets of calculations they are given, such as 24 × 5 = 120, 12 × 10 = 120 and 6 × 20 = 120, and who suggest other related multiplication calculations.
| Objectives Children's learning outcomes are emphasised | Assessment for learning |
|---|---|
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How many calculations are needed to solve this problem? |
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How will you solve this problem? Will you use a mental, written or calculator method? Why did you choose this method? |
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Explain what this diagram tells you. Use this fraction wall to find a fraction equivalent to 3/4 . |
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Tell me some fractions that are equivalent to one quarter. How do you know? Are there others? What about two thirds? What is the missing number?
How do you know? |
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How many times bigger is 2400 than 6? How do you know? What is 1800 ÷ 20? Explain how you know. |
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5 is a factor of 85. What is its factor pair? |
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Explain how you would work out 25 × 5, 1000 ÷ 5, 23 × 7. |
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What numbers did you multiply to find this part of your answer? Which parts of your answer will change if I change this digit? Roughly, what do you expect the answer to be for this calculation? What tips would you give to someone to help them to multiply one two-digit number by another? |
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What operation must you do to find one seventh of a number? One sixth of a number is 3. What is the number? What calculation would you key into a calculator to find 1/13 of 403. |
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What two steps are involved in finding 3/4 of a number? I divide a number by 5, then multiply the answer by 4. What fraction of the number have I found? |
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These cards describe the steps in adding up 2.86m, 97cm and 5.08m. Arrange the cards in order. Explain to the class why you solved the problem in that way. |
| Activities | PDF 1MB |
| Activity 53 - Square it up | |
| Activity 55 - Money bags | |
| Activity 57 - Presents | |
| Activity 58 - Spot the shapes 2 | |
| Activity 60 - Three digits | |
| Activity 61 - Make five numbers | |
| Activity 62 - Maze | |
| Activity 63 - Jack's book | |
| Activity 64 - Flash Harry | |
| Activity 65 - Age old problems | |
| Activity 66 - Zids and Zods | |
| Activity 67 - Franco's fast food | |
| Activity 69 - Coins on the table | |
| Activity 70 - A bit fishy | |
| Activity 74 - Anyone for tennis |
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Objectives for Springboard intervention unit |
Springboard unit |
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Recognise 1/2, 1/4, 1/10, 1/5 and use them to find fractions of shapes and numbers |
Springboard 5 Unit 4 (PDF 283KB) |
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Calculate a difference mentally by counting up from the smaller to the larger number |
Springboard 5 Unit 7 (PDF 305KB) |
| Springboard 5 Unit 7 supplementary (PDF 77KB) | |
| Know the three- and four-times tables Begin to know the six-times tables |
Springboard 5 Unit 9 (PDF 269KB) |
| Springboard 5 Unit 9 supplementary (PDF 110KB) | |
| Develop and refine written methods for multiplication (two- or three-digit x single-digit) Approximate the answer first |
Springboard 5 Unit 10 (PDF 269KB) |
| Springboard 5 Unit 10 supplementary (PDF 63KB) |
| Diagnostic focus | Resource |
| Is not confident when recalling multiplication facts | 1 Y4 ×/÷ DfES 1150-2005 (PDF 104KB) |
| Does not apply partitioning and recombining when multiplying and confuses the value of two digit numbers | 4 Y4 ×/÷ DfES 1153-2005 (PDF 104KB) |
| Assumes the commutative law holds for division also | 5 Y4 ×/÷ DfES 1154-2005 (PDF 85KB) |
| Writes a remainder that is larger than the divisor | 6a Y4 ×/÷ DfES 1155-2005 (PDF 76KB) |
| Discards the remainder; does not understand its significance | 6b Y4 ×/÷ DfES 1156-2005 (PDF 93KB) |
| Does not recognise when the remainder is significant when rounding up or down | 6c Y4 ×/÷ DfES 1157-2005 (PDF 65KB) |
| Continues to subtract 2s without using knowledge of times tables | 7 Y4 ×/÷ DfES 1158-2005 (PDF 89KB) |
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