Number – Number and place value Above and below zero Challenge 5:
I think that zero is in the middle of our number system, and that numbers below zero and numbers above zero go in the same order.
Is Adam correct? Explain your reasoning on paper. What else can you tell Adam about numbers that are below zero? Give Adam some examples.
Challenge 6:
Samira thinks that the difference between negative 7 and positive 12 is 5. Is she correct? Explain your reasoning and illustrate with several examples, including some from real life.
Whole numbers Challenge 7:
Use the digits 3, 5, 1, 9 to make up all the possible four-digit numbers that you can. How do you know you have them all? Explain your reasoning on paper. Choose one of your numbers and explain how it is made. Use the words ‘positional’, ‘multiplicative’ and ‘additive’ in your explanation.
Challenge 8:
5 7 0 8 I think the greatest four-digit whole number that I can make with the digits is 8750.
Is Samira correct? Explain your reasoning. List all the possible numbers in order from greatest to least, to prove your reasoning.
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National Curriculum objective, Year 4, Number and place value • count backwards through zero to include negative numbers
Challenge 5 Answer Adam is correct.
Assessment Before the task, assess how much the pupils know about negative numbers. Can they tell you these go in order from negative one to infinity, just as positive numbers go from positive one to infinity? Negative numbers mirror positive numbers. Can they tell you why we say ‘to infinity’? They may be aware that no one has found the largest number; there are always more. Expect your pupils to write about this within their reasoning for why Adam is correct. They are asked to give Adam more information. One of the things he has not mentioned is that we call numbers below zero ‘negative numbers’ (not ‘minus numbers’, which is what some people call them). Ensure this is included in this extra information. Can they tell you where negative numbers are found in real life? Below zero temperatures is an obvious one; so look out for pupils who can tell you, for example, about land below sea level, lifts that travel below ground, overdrafts when a person has no money in the bank.
Note You could ask your pupils to research negative numbers on the internet and make an information poster to share with the class.
Challenge 6 Answer Samira is incorrect. The difference between negative 7 and positive 12 is 19.
Assessment Assess if the pupils can explain why Samira is incorrect. They should be able to tell you that Samira has found the difference as if both numbers were positive. If they had been positive, she would have been correct. However negative 7 is much further from 12 than positive 7 is, so the difference is going to be more. Expect them to be able to draw a number line and position negative 7 and positive 12 on it. Expect them to be able to count on from negative 7 to zero and then on to 12, giving a difference of 19. They are asked to give several examples of finding the difference between positive and negative integers. Encourage them to draw number lines to show exactly how these differences can be found. For example, they draw arrows from the negative integer to zero, and write the number added to get to zero and then from zero to the positive integer and again write the number added. They write the addition statement to show how the difference is calculated. Observe the contexts that the pupils choose. If they can’t think of one, encourage them to consider temperature and to draw thermometers to show differences between negative and positive differences. They could make up scenarios, for example: ‘At 6 a.m. the temperature was –5°C, and by noon it had risen 15 degrees to 10°C.
Note It would be helpful to plan in opportunities to count in steps of different sizes crossing zero, for example, counting in twos from 20 to negative 20, or counting in tens from 100 to negative 100. This reinforces the fact that negative integers mirror positive ones.
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National Curriculum objective, Year 4, Number and place value • recognise the place value of each digit in a four-digit number (thousands, hundreds, tens and ones)
Challenge 7 Answer 3519, 3591, 3915, 3951, 3195, 3159, 5319, 5391, 5913, 5931, 5193, 5139, 9531, 9513, 9315, 9351, 9135, 9153, 1935, 1953, 1395, 1359, 1539, 1593
Assessment During the task, notice if the pupils work systematically, listing the numbers as in the answer section or in a similar systematic way. If they don’t, systematic working is something you probably need to focus on with the whole class. Are they showing perseverance and resilience, a determination to find all possibilities? Can they find them? Assess how they explain that they have all the numbers in writing, for example, beginning with the same digits in the thousands and hundreds positions and finding the possible other numbers, then repeating this by keeping the thousands digit the same and changing the hundreds. Once they have written all the possibilities for a particular thousand digit, they repeat again with a new thousands digit. Assess how they explain the place value of the number they chose. In their explanation they should include positional, multiplicative and additive vocabulary. For example, if they chose 3519 they should be able to say that: the 3 is a digit; it is positioned in the thousands position (positional) and multiplied by 1000 to give its value of 3000 (multiplicative). The 5 is a digit; it is positioned in the hundreds position and multiplied by 100 to give its value of 500. They repeat this for the digits 1 and 9. Ensure they include the fact that 9 is multiplied by 1 to give its value of 9. When they have all the values, they add them together to make the whole number, for example: 3000 + 500 + 10 + 9 = 3519 (additive).
Note Place value underpins all aspects of number work, so it is important that it is mastered fully. It is worth spending time on this and applying it to activities involving measurement. In the past, place value wasn’t generally taught in enough depth.
Challenge 8 Answer Samira is correct. Possible numbers in order from greatest to least: 8750, 8705, 8570, 8507, 8075, 8057, 7850, 7805, 7580, 7508, 7085, 7058, 5870, 5807, 5780, 5708, 5087, 5078, 875, 857, 785, 758, 587, 578
Assessment Assess the pupils’ reasoning about why Samira is correct. They should be able to explain that because 8 is the greatest digit it needs to be positioned in the position of greatest value, which is the thousands; 7 is the next greatest digit and needs to be positioned in the position with the next greatest value, which is the hundreds; 5 needs to be positioned in the tens; and zero, as the place holder, should be positioned in the position with the least value. These digits need to be multiplied by the value of their position and then added, to give 8000 + 700 + 50 = 8750. Do the pupils understand that in terms of place value zero is a place holder, not a value? During the second part of the task, notice if the pupils work systematically, listing the numbers as in the answer section or in a similar systematic way. If they don’t, work with them to show how this can be done. Are they showing perseverance and resilience, a determination to find all possibilities? Can they find them? Assess how they explain in writing that they have all the numbers (see the Assessment for Challenge 7). Can they explain that because zero is a place holder in a number, six of the possibilities will be three-digit numbers?
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Number – Number and place value Tenths and hundredths Challenge 9:
Explain how the number in the place value grid is made. Write about the positional and multiplicative aspects of place value for each digit, and the additive aspect to make the whole number.
Challenge 10:
1000
100
10
1
.
1 10
1 100
6
8
1
3
.
5
4
Samira thinks that this number is two thousand, three hundred and forty eight, and six-tenths and seven-hundredths.
1000
100
10
1
.
1 10
1 100
2
3
4
5
.
6
7
Is she correct? Explain your reasoning. Include two more efficient ways she could have read the number.
Ordering numbers Challenge 11:
My teacher asked me to order these numbers from greatest to least value.
7198 7845 7199 7834 I think it’s impossible; they all begin with 7000.
Explain why Adam is incorrect. Show Adam how to order the numbers from least value to greatest.
Challenge 12:
I’ve put these numbers in order from greatest to least.
3581.79 3581.48 358.13 3581.91 What do you think Samira has done to get this order? Explain what she should have done. 10
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National Curriculum objective, Year 4, Number and place value • recognise the place value of each digit in a four-digit number Notes and guidance: begin to extend their knowledge of the number system, to include tenths and hundredths as decimal numbers and fractions that they have met so far.
Challenge 9 Answer No definitive answer.
Assessment Assess pupils’ explanations of how the number in the grid has been made. Do the pupils refer to the positional, multiplicative and additive aspects of place value? This is something they need to know in order to develop a deeper understanding of place value. They should be able to say that: the 6 is a digit, it is positioned in the thousands position (positional) and is multiplied by 1000 to give its value of 6000 (multiplicative). The 8 is a digit; it is positioned in the hundreds position and multiplied by 100 to give its value of 800. They repeat this for the digits 1 and 3. They should be able to say that the 5 is in the tenths position and is multiplied by one-tenth to give five-tenths and that 4 is in the hundredths and multiplied by one-hundredth to give four-hundredths. When multiplying the 5 by tenths, they should use a repeated addition method and know that will give five tenths. Likewise for four hundredths. Can pupils tell you that
5 10
is equivalent to equivalent to point five four.
50 100
so the total fraction is
54 ? 100
1 10
added five times
Ensure that they can say that
54 100
is
5 + 4 = 6813 54 or 6813.54 (additive). Assess the The values are then added together: 6000 + 800 + 10 + 3 + 10 100 100 pupils’ clarity of understanding in all of these to ensure they have mastered with greater depth.
Note Tenths and hundredths are mentioned in the fractions part of the national curriculum. In fact they are a very important part of number and place value. The notes and guidance confirms this. It would be a good idea to teach these when working on place value.
Challenge 10 Answer Samira is correct. More efficient ways to read the number are 2348 and
67 , 100
and 2348.67.
Assessment Assess the pupils’ explanations of how Samira is correct. Expect them to explain that the whole numbers are 2000, 300, 40 and 8, and that when added together they make 2348. They should be able to tell you that Samira is correct in 6 and 7 – but because 6 is equivalent to 60 , it would have been more efficient to say saying that the fractional part is 10 100 10 100 67 . Assess their confidence at reading this as a number with two decimal places: 2348.67. 2348 and 100 The place value grid in the challenge is a really clear way to help your pupils make these links. To have achieved mastery with greater depth, they should have been able to do all of the above.
Note As a practical application of this concept, link to measurement of metres and centimetres and pounds and pence. You could ask the pupils to measure things around school that are over a metre in length, and to represent their lengths in whole metres and fractions and decimals of a metre. You could also ask them to count amounts of money and represent these as pence, and then pounds and pence using the fewest coins.
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National Curriculum objective, Year 4, Number and place value • order numbers beyond 1000
Challenge 11 Answer Adam is incorrect. The correct order is 7198, 7199, 7834, 7845.
Assessment Assess the detail, clarity and accuracy of pupils’ explanations as to how Adam should have ordered the numbers. They should be able to explain that ordering by the first digit of a number only works if these digits are different. If numbers have the same first digit, the second digits are considered. Adam should have looked at the digits in the hundreds position. He would have seen 1 and 8. 800 is greater than 100, so 7845 and 7834 are greater than 7198 and 7199. In order to find the greater of 7845 and 7834, Adam needed to look at the digits in the tens position. He would have seen 4 and 3. 40 is greater than 30, so the greater number is 7845 followed by 7834. To find the greater of the other two numbers, the tens digit is the same so Adam needs to look at the ones. He could then see that because 9 is greater than 8, 7199 is greater than 7198. Look out for pupils who can explain in this much detail. They were asked to order from least to greatest. Do they know what this means? Did they order the numbers in this way: 7198, 7199, 7834 and 7845?
Note When the pupils practise ordering numbers, give opportunities for them to do this within the context of measurement, for example ordering lengths, masses and money. This is a real-life application of number; it helps to link mathematical concepts.
Challenge 12 Answer Samira has only considered the digits in the hundredths position when she ordered.
Assessment Assess if the pupils can identify that Samira has only ordered the last digit of the numbers. She may have seen that the first four digits are the same, and gone straight to the hundredths. Can the pupils give a plausible explanation such as this? If so, they are correct. You might want to ask them if there is another possibility. Can your pupils confidently read all the numbers? Can they 7 and 9 – and explain how to put the two together to make 79 ? identify the decimals as fractions – for example 10 100 100 7 into the equivalent hundredths? They Does their explanation show that they understand that they need to turn the 10
should be able to tell you that they multiply the numerator and denominator by 10 to give to tell you all this information in response to your questions.
70 . 100
Expect them to be able
Assess the pupils’ understanding of ordering numbers. Can they explain that they look at the most significant digits first and order those? And that if some or all are the same, they need to consider the next significant digit and so on? They should be able to write that because the thousands, hundreds, tens and ones digits are the same, Samira needs to look at the next digit, which is the tenth, in each number, and order those. The order from greatest to least that she should have made is 3581.91, 3581.79, 3581.48, 3581.13.
Note When ordering while working on number and place value, it is important to include tenths and hundredths. They are part of place value, and shouldn’t be kept to work on fractions and decimals. If you don’t already include tenths and hundredths, it would be helpful to do so the next time you work on this area; it helps builds connections and deepen understanding.
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