Number and place value
National Curriculum Objective, Y2, Number and place value • count in steps of 2, 3, and 5 from 0, and in tens from any number, forward and backward
Counting Challenge 1:
Sophie is counting in threes from one.
Sophie is not correct. Starting from 1 and counting in threes, the counts will always be one more than a multiple of 3. The numbers 30, 33 and 36 are all multiples of 3, so they won’t be included in the count.
Assessment:
I think my count will include 30, 33 and 36.
Challenge 2:
Challenge 1 Answer:
During the task, notice if pupils work systematically, listing the counts in order. If they don’t, systematic working is something they probably need to focus on.
Is Sophie correct? Explain your thinking to a friend.
Once they have proved Sophie is incorrect, assess their confidence in explaining their reasoning to a friend and in writing an explanation about the mistake she has made. Doing this independently and accurately would indicate depth of mastery.
Now write your explanation on paper. You can use words and numbers.
Ask pupils to predict the numbers that will be counted if Sophie starts at 2. They should then list them. Can they see a pattern when comparing with the count from 1? What do they notice?
Ben is counting back in twos and fives from 60 to zero. I think that only 7 numbers will come in both counts.
Note: Ensure that pupils know the meaning of multiple. Knowing the correct vocabulary will make explanations simpler and more succinct. Some pupils might be able to make a verbal or written generalisation such as: when counting from one in threes the numbers will be a multiple of three add one which could look like this: C (the count in threes) = M (multiple of three) + 1.
Is Ben correct? Explain your thinking to a friend. Now write your explanation on paper. You can use words and numbers.
Place value Challenge 3:
Write all the two-digit numbers less than 50 that can be made using each of these digits once.
So if pupils want to find the 5th number in the count, they find the 5th multiple of 3 and add 1. The number statement would look like this: 5th = (5 × 3) + 1 = 16.
Challenge 2 Answer: Ben is correct. When counting backwards in twos from 60 the numbers will always be even. When counting in fives the ones digits alternate between zero and five. Those ending with zero are the only even ones. There are only six multiples of ten when counting in 5s from 60 to zero and these will match when counting in twos. Both counts start at 60 and end at zero, so 7 numbers will be the same in total.
Assessment:
How do you know you have them all? Prove it to your friend. Now write your explanation on paper. You can use words and numbers.
Ask pupils what they notice about the numbers when counting in twos and then fives. Expect them to know that counting forwards and backwards in twos from and to zero will give even numbers and when counting in fives the ones digits will alternate between zero and five. Encourage pupils to make lists to show this and to find the common multiples. You could introduce this vocabulary. What do they notice about the numbers that come in both counts? Can they tell you that these are multiples of ten as well? This shows depth of understanding.
Can you make the number which is closest to 50? Prove it.
Challenge 4:
I made one two-digit number that is more than 56 and one that is less than 56 using each of these digits once: 2, 3, 9, 6
What numbers could Sophie have made? Could she have made any others? Find all the possibilities. Make a list of them. 2
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National Curriculum Objectives, Y2, Number and place value • recognise the place value of each digit in a two-digit number (tens, ones) • use place value and number facts to solve problems
Challenge 3 Answer: The tens digit needs to be 1 or 4. Any of the other digits would make the number greater than 50. If we begin with a tens number of 1, the two-digit numbers we can make are 14, 15, 17 and 18. If we position the 4 in the tens place, the numbers are 41, 45, 47 and 48. The closest number to 50 is 51.
Assessment: During the task, notice if pupils can identify what the tens digits must be. Do they work systematically, listing the numbers as in the answer section? If they don’t, systematic working is something they probably need to focus on. Assess how they can prove their results to a friend and in writing. If they can do both independently, they are working with a greater depth of mastery. If they can’t, make one of these areas something to focus on with the class. Were they able to think outside the constraints of the first part of the task for the closest number to 50? Encourage them to do this. There were no rules, they simply had to make the number closest to 50, which in this case is 51.
Note: When pupils learn about place value they need to develop understanding of three important aspects: positional, multiplicative and additive. Pupils will only be working at greater depth if they can explain these. The positional aspect of place value is where the digit sits in the number, for example in 54, the digit 5 is positioned in the tens and the digit 4 in the ones. This is important for success in the written methods pupils will learn in KS2. The multiplicative aspect tells us that the digit must be multiplied by its position to give its true value, for example in 54, the digit 5 is multiplied by 10 to give its true value of 50 and the digit 4 is multiplied by one to give its true value of 4. This aspect is really important for mental calculation. The additive aspect indicates that we need to add the values together to get the whole number, for example 50 add 4 is 54.
Challenge 4 Answer: There are several possibilities: 92, 36 93, 26 62, 39 63, 29 69, 23 69, 32 96, 32 96, 23
Assessment: Watch pupils work through the task. Are they showing perseverance and resilience, a determination to find all possibilities? Can they find them? As they work, are they showing a systematic approach, for example, beginning with numbers with 9 in the tens position for the greatest number, then 6?
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Number and place value
National Curriculum Objectives, Y2, Number and place value • identify, represent and estimate numbers using different representations, including the number line • use place value and number facts to solve problems
Identifying, representing and estimating numbers Challenge 5:
Challenge 5 Answer: You need a tub of counters and three other pupils to work with. Each pupil needs to take a handful of counters and put them on the table. Take turns to estimate how many counters you each have. Discuss the best way to count them and then count them. Who had the closest estimate?
Assessment:
Now find out how many counters you have altogether. You need to represent this
During the task, assess the accuracy of pupils’ estimates. Are these random guesses or well thought out? Can they decide on sensible group sizes to count the counters rather than counting them one at a time?
amount in as many different ways as you can.
Do the pupils count rather than estimate? It is tempting to count if there is enough time. If they do, they have not mastered the art of estimation in this context. Encourage them to say what they think very quickly by covering the counters, showing them quickly and then covering again. Ask pupils to write what they think.
Draw your representations.
Challenge 6:
Draw these three number lines on paper. 0 – 100
25 – 75
Were they able to represent the total in different ways, for example, one coloured cube to represent tens and another colour for ones, showing the number on a ruler or metre stick or as minutes on a clock?
10 – 90
Mark 50 on all three number lines. Prove they are positioned in the correct place.
Compare and order numbers Challenge 7:
To complete this task you need to take a pair of numbers and compare them. Write two number statements using the symbols > and <
Note: Pupils often find estimating difficult – it doesn’t come naturally to them and they seem to always want to find the correct answer. We need to encourage estimating skills. As a whole class it would be helpful to ‘train’ the pupils to estimate. You could hold a bundle of straws or show a spread of counters on a visualiser. Ask the pupils to write down how many they think there are. Then, as a class, start counting them in groups of 10. When you are half way through, ask the pupils to consider their estimates and change them if they want to. Repeat this when you are about three-quarters of the way through. If you do this often enough, estimating should become more natural.
Challenge 6 Answer:
Here is an example: 24 < 83, 83 > 24
0
50
100
How many different ways can you pair these numbers?
25
50
75
10
50
90
24
56
37
83
How do you know that you have them all?
Challenge 8:
There is no specific answer. Encourage pupils to write down their estimates. They should aim to count the counters in sensible amounts, for example, make groups of 10. Provide equipment such as money, base 10 equipment, clock faces, thermometers, rulers, metre sticks and different-coloured cubes. Allow them to choose from these and other equipment they can think of to use.
There is something wrong with these number statements. Find two ways to make each of them correct. Here is an example: 36 = 60
36 + 24 = 60, 36 = 60 – 24
1.
35 = 25
2.
56 = 75
3.
80 = 25
4.
14 = 36
5.
78 = 12
6.
29 = 54
Pupils should be able to tell you that 50 is the middle number between each of those on the number lines. They should be able to prove this.
Assessment: Can pupils recognise that 50 is the middle number between 0 and 100, 25 and 75, 10 and 90? Observe how they found this out. Did they use rulers? If not, what strategy did they use? Observe how they drew their number lines. Did they put divisions on the lines to help them or did they estimate? If they worked out that 50 should be half way, then divisions would not be necessary. Pupils do need to talk to you about this and prove it!
What do you notice about the numbers you have added and subtracted? Explain the strategies you used to find the missing numbers to a friend. What do you notice?
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National Curriculum Objective, Y2, Number and place value • compare and order numbers from 0 up to 100; use <, > and = signs
Challenge 7 Answer: 24 < 56, 56 > 24 24 < 37, 37 > 24 24 < 83, 83 > 24 56 > 37, 37 < 56 56 < 83, 83 > 56 37 < 83, 83 > 37
Assessment: During the task, assess whether pupils are systematic in their approach. The answers show a systematic way of finding all pairs. If pupils were random in their approach, work with them on becoming more systematic. Were they showing resilience and persistence in finding all possibilities? Assess their ability to prove verbally. Some pupils may be able to write an explanation of what they did. Encourage this as appropriate.
Challenge 8 Answer: 1.
35 = 25
35 – 10 = 25, 35 = 25 + 10
2.
56 = 75
56 + 19 = 75, 56 = 75 – 19
3.
80 = 25
80 – 55 = 25, 80 = 25 + 55
4.
14 = 36
14 + 22 = 36, 14 = 36 – 22
5.
78 = 12
78 – 66 = 12, 78 = 12 + 66
6.
29 = 54
29 + 25 = 54, 29 = 54 – 25
Pupils should notice that the same number has been added and subtracted to make each statement correct.
Assessment: How quickly did pupils realise that the same number had to be added and subtracted each time? Could they explain why this is? Assess the strategies. Ideally they would have used a mental strategy to find the first missing number. These could include counting on, sequencing or using known facts.
Note: Before giving pupils this task, it would be helpful to provide base 10 equipment so that the pupils can make the numbers and then add to the smaller number or subtract from the higher number to make the other. This should help them to see that the same number needs adding or subtracting to make the two numbers equal.
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