Reasoning: Explain how you know Learning objective To give an explanation using words and diagrams and using correct mathematical vocabulary.
Links to year 5 problem solving and reasoning pupil target sheet Reasoning I can explain my reasoning using clear sentence structures, calculations and diagrams. I can make conjectures (statements) about mathematics and further develop the ability to convince others by proving and justifying my answers using further examples to back up reasoning.
Teaching notes • Ask the class a straightforward mathematical question which has a yes or no answer. The questions should require some reasoning and not just be factual knowledge. • Model how to explain the answer orally and in writing, emphasising the correct use of mathematical vocabulary. • Give pupils a range of simple questions to enable them to practise writing out their explanations using the correct vocabulary. Example: Java says that there are 126 hours in three-quarters of a week. Is he correct? Explain how you know. • Ask: What does the question want to know? (Are there 126 hours in three-quarters of a week?) – In order to know whether our answer is going to be a sensible one, first we need to estimate by working out how many hours are in a whole week. 24 × 7 = 168 so we know our answer has to be less than 168. When estimating, pupils should also realise that their answer will be more than a half (84) as we want three-quarters. So a sensible estimate could be 100. • Ask: How do we calculate three-quarters of 168? – Model how to calculate three-quarters of 168 in a variety of ways. For example: Half of 168 = 84, half again = 42 then add them together 84 + 42 = 126 or 168 ÷ 4 = 42 then multiply by 3, so 42 × 3 = 126 • Ask: So do we know that Java is correct? – Model how to explain the answer. ‘Java is correct. I know this because there are 168 hours in one week. Three-quarters of 168 is 126. So there are 126 hours in one week.’
For pupils – Steps to success 1. Identify what the question is asking you to do. 2. Identify a strategy to solve the problem. 3. Work out an estimation to know if the answer is a sensible one. 4. Explain clearly using a range of mathematical vocabulary and diagrams.
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Reasoning: Explain how you know Bank of ‘Explain how you know’ questions 1. Here are the library opening times. If Sidney arrives at the library at 12.50 on a Sunday, how long is it until the library closes? Monday Tuesday Wednesday Thursday Friday Saturday Sunday
Opens 09.30 08.45 10.00 09.45 08.50 10.10 10.00
Closes 18.00 18.45 17.00 17.35 18.30 18.00 16.30
2. Fazam writes a large number: 857 641. He says that the value of the digit 7 is 70 000. Is he correct? Yes or no. Explain how you know. 3. Two friends are adding up their computer game scores. Harry’s scores are 27 037 for game 1; 38 208 for game 2 and 83 369 for game 3. Chris’s scores are 15 469 for game 1; 18 402 for game 2 and 88 325 for game 3. They work out that Harry scores 26 418 points more than Chris. Do you agree with them? Yes or no. Explain how you know. 4. Jessie says that three of these calculations have an answer that is between 0.2 and 2.1. Is she correct? Explain how you know. a) 28.32 − 26.82
b) 21.75 − 12.88
c) 42.29 − 40.39
d) 0.83 + 1.27
e) 14.24 − 13.83
f) 31.40 − 28.99
5. Harley, Bailey and Summer have 403 keyrings between them. Summer has 28 fewer than Bailey. Harley has 53 more than Bailey. How many keyrings do they have each? Explain how you know. 6. The product of two numbers is 14 greater than the sum of the two numbers. Both numbers are even and are multiples of 12. One number is 50% greater than the other number. What are the two numbers? Explain how you know.
See the pupil activities marked with the this strategy.
icon on pages 34 to 67 for topic-based practice of
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Reasoning: Would you rather? Learning objective To use reasoning to make a decision between two mathematical situations and explain the outcome clearly.
Links to year 5 problem solving and reasoning pupil target sheet Reasoning I can explain my reasoning using clear sentence structures, calculations and diagrams. I can make conjectures (statements) about mathematics and further develop the ability to convince others by proving and justifying my answers using further examples to back up reasoning.
Teaching notes • Pose a question where pupils compare two situations and justify one over the other using mathematical reasoning. • Choose a real-life context that pupils are familiar with. Also use opportunities that require pupils to compare and convert different measures. For example: 3500mL or 2L. • Highlight the fact that unless stated otherwise, pupils always assume that they must try to find the larger amount. • Model how to reason effectively why they would prefer one over the other. Example: Would you rather go on holiday for 338 hours or two weeks? • Ask: What is the question asking us? (Which is the greater number in hours?) – What is the first thing we need to do before we can work out the greater amount? (We need to convert one measurement into the other so that they are measured in the same unit. So we need to convert two weeks into hours.) – Model how to convert two weeks into hours: 24 × 14 = 336 • Ask: So if two weeks converted into hours is 336 hours, which is the larger amount? (338 hours) For pupils – Steps to success 1. Understand that you are comparing two things and trying to find out which is better. 2. Identify what you need to do to make a comparison and make any necessary conversions. 3. Use the answers you have found to make a reasoned mathematical argument as to why you would rather have one over the other.
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Reasoning: Would you rather? Bank of ‘Would you rather?’ questions 1. Would you rather have 72 934 points in a video game or 172.934?
3. Would you rather have 42 coins shared between 6 of you or 18 coins shared between 3 of you?
5. Would you rather have 6250mL of orange juice or 6.5L of orange juice?
7. Would you rather have 1 of 2L or 3 of 600mL? 4 4
9. Would you rather have 4 of a bar of 5 chocolate or 0.80 of a bar of chocolate?
See the pupil activities marked with the this strategy.
2. If you collected shells, would you rather be given 16 buckets containing 1051 shells in each bucket or 5 buckets containing 3439 shells in each bucket?
4. Would you rather have 2 of the pizza or 4.5 of 5 10 the pizza?
6. If you wanted to walk the shortest distance, would you rather walk 10 001m or 10km?
8. Would you rather have 60% of £3927 or 1 of 2 £4840?
10. Would you rather buy a 3kg jar of jam costing £6 or six 500g jars costing £1 each?
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Addition and subtraction To complete addition and subtraction calculations, you may want to use the column method. Example: Calculate 12 537 + 3619 1 2 5 3 7
Calculate 82 649 − 22 942 7 11 1 8 2 6 4 9
3 6 1 9
− 2 2 9 4 2
1 6 1 5 6
5 9 7 0 7
+
1
1
Getting started 1
What is the same and what is different about these two calculations? Convince me. 19 730 − 16 927 and 18 745 − 15 942
2
Harry says that 48 327 subtract 7122 is 41 205 but Josh has worked it out as being 41 305. Who do you agree with? Explain your reasoning.
3
How many different numbers can you insert in the boxes to make the calculation correct? + 2715 = 7
4
5
4
Which calculation is the odd one out? 18 243 + 8298
44 449 − 17 898
19 482 + 7059
43 518 – 16 977
Ted says that the answer to the problem below is 15 313. Is he correct? Explain your reasoning. Four friends are playing a video game. The total score for all four players was 95 752. Player 1 scored 19 513 points. Player 2 scored 24 918 points. Player 3 scored 36 008 points. How many points did player 4 score?
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Challenge 1 6
Are the following statements always, sometimes or never true? a) The sum of three consecutive numbers is a multiple of 3. b) The sum of the same six even numbers is divisible by 4.
7
Connor says two of the following calculations have an answer that is between 0.5 and 0.8. Do you agree? Explain how you know. b) 25.48 − 24.78
a) 16.67 − 16.07
c) 46.03 − 45.63 d) 77.55 − 76.95
e) 106.42 − 105.52
Challenge 2 8
Join any four numbers. Joins can go up, down or sideways, but not diagonally. Find their total. For example, the score shown is 9279 + 6214 + 4978 + 7504 = 27 975. 1102
2616
4813
9279
8019
4115
6120
6214
5337
2090
7504
4978
1879
9793
8780
1999
a) If the answer is 32 197 what is the question? b) Rosalie says that the lowest possible score is 13 099. Is she correct? Explain how you know.
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Multiplication Remember that when you multiply, you are making a number bigger. It is quicker to use your knowledge of multiplication facts than it is to add numbers over and over. Example: Casey opened a bottle of bubbles and blew three bubbles all at once. On her second try she blew three bubbles. For her final go she blew another three bubbles. How many bubbles did she blow altogether? 3 × 3 = 9 is quicker than writing out 9 lots of 1.
Getting started 1
Work out which statement is the odd one out. 4 × 35
70 × 2
22 × 8
17.5 × 8
5.6 × 25
2
Work out which statements are true and which are false. a) 305 × 12 = 3660 b) 270 × 4 = 1100 c) 268 × 31 = 8318 d) 473 × 20 = 9460 e) 1543 × 7 = 10 801
3
Assuming you want the most, which would you rather have? a) 50 × £2.48 b) 980 × 12p c) 20 × £6.50
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Challenge 1 4
If these are the answers what are the questions? Can you think of two questions for each? Make your questions as difficult as you can. a) 545
b) 948
c) 2994
d) 5070
e) 8664
5
If you collected buttons, would you rather be given three bags containing 82 buttons in each bag or four bags containing 61 buttons in each bag?
Challenge 2 6
Use the digits 2, 4, 6 and 9 and the × symbol once to create a multiplication calculation, for example 496 × 2 or 24 × 96. a) What is the largest product you can make? Explain how you know. b) What is the smallest product you can make? Explain how you know.
7
On planet Zog, sea monsters have been found, deep in the Squazi Ocean. The green sea monsters have three eyes and the blue sea monsters have seven eyes. If there are 52 eyes altogether in the ocean, how many of each monster are there in the ocean? List as many combinations as you can.
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