Maths Mastery with Greater Depth - Year 5 Maths Mastery with Greater Depth

Number - Number and place value

National Curriculum objective, Year 5, Number and place value • read, write, order and compare numbers to at least 1 000 000 and determine the value of each digit

Numbers up to 1 000 000 Challenge 1:

Challenge 1 Answer Possible passcodes are 12856, 12865, 18256, and 18265

Abdul has forgotten the passcode to unlock his mobile. He knows it is made up of the digits below and that:

Observe how pupils approach this challenge. Do they work systematically? Do they have a method? Did they notice straight away that the digit in the ten thousands place must be 1? They should be able to explain their thinking to a partner. As an extension, pupils could use the same digits and create different hints for a classmate to guess the passcode.

–

Each number is used once

–

It is between 10 000 and 20 000

–

The tens place and the units place are either a 5 or 6

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2

1

6

Challenge 2 Answer

5

What are his possible passcode combinations?

Challenge 2:

Use the cards pictured here to make: a)

5

the smallest number you can using 4 cards

7

1

4

3

3 even numbers using all the cards

d) a multiple of 5 using all the cards

2

9

8

1023 (NB: pupils may write 0123, in which case, it should be discussed that we do not write leading zeros.)

b)

9876543

c)

any three numbers with 0, 2, 4, 6, or 8 in the ones place.

d)

any number with 0 or 5 in the ones place.

Assessment

0

Encourage pupils to discuss place value by deciding where to put each digit, in order to make the smallest and largest numbers. They should be able to name each place value and explain why they have chosen to place each digit in that spot. Assess whether pupils are able to say and write the numbers in words. Pupils could be asked to repeat the challenge of finding the largest and smallest number if they are also given a decimal point.

Note

Counting in 10s, 100s and 1000s Challenge 3:

a)

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b) the largest number you can using 7 cards c)

Assessment

Pupils could compare answers to part c) and d) with a partner or with the class to start to get a sense of the vast combinations of numbers that are possible when given 10 digits. Did any pupils pick the same numbers?

Pupils could be challenged to use divisibility rules they may know in order write numbers that are multiples of 3, 4, etc.

Hebe is counting up in tens.

No matter what number I start with, when I count up in tens it takes me 10 steps to reach one hundred more than my starting number.

Is Hebe correct? Explain why or why not.

Challenge 4:

Fill in the blanks for the sequences below.

123,

,

, 153,

b) 570,

,

,

a)

c) d) 2

,

12 796, ,

, , 530,

, 13 096, , 709 743,

, , 689 743, 3

National Curriculum objective, Year 5, Number and place value

Number - Number and place value

• count forwards or backwards in steps of powers of 10 for any given number up to 1 000 000

Negative numbers

Challenge 3 Answer Yes, Hebe is correct.

100 divided by 10 is 10, so it will always take 10 steps of 10 to count up to 100 more from any starting number.

Challenge 5:

Starting at 11, count up in 3’s until you reach your first number which is bigger than

Assessment

30. Now count backwards in 7’s from your last number until you reach the first number

Ensure that pupils are able to explain their answer. Challenge them to discuss division rather than simply citing examples. Pupils could be asked how many steps it will take to count in tens between different numbers, e.g. 458 and 508. You could also ask them to find the number of steps using different-sized steps, such as 100. Challenge pupils to write a rule explaining the number of steps for any number, e.g. N (Number of steps) = M (Total Increase or Decrease) ÷ S (Size of step)

which is smaller than -10.

Challenge 6:

123, 133, 143 153, 163, 173

b)

570, 560, 550, 540, 530, 520

c)

12 796, 12 896, 12 996, 13 096, 13 196, 13 296

d)

729 743, 719 743, 709 743, 699 743, 689 743, 679 743

40 30

The thermometer shows a warm summer’s day

Challenge 4 Answer a)

°C

Karla says,

20 10 0 10 20 30 40

Assessment Observe the methods that pupils use to solve these problems. Are they working out addition and subtractions or are they able to simply look at the relevant place value? Discuss the merit in different methods.

Is Karla correct? Explain your answer.

Note

Rounding

Pupils may find part d) particularly challenging because the starting number is missing. Encourage them to think about the methods they used in the first three parts and see how they can apply their methods to this problem.

Challenge 7:

Maja has completed the following table showing numbers rounded to the nearest 10, nearest 10 000 and nearest 100 000. Mark her work and correct any errors.

Challenge 8:

Number:

Nearest 10

Nearest 10 000

Nearest 100 000

459832

459820

450000

400000

865994

865990

865000

900000

523986

523980

500000

500000

743023

743020

740000

743000

For this challenge you will need a die. Roll the die 6 times and record each roll. Use the digits rolled to create the largest possible number. Round this number to the nearest 10, nearest 100, nearest 1000, nearest 10 000 and nearest 100 000.

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5

National Curriculum objective, Year 5, Number and place value • interpret negative numbers in context, count forwards and backwards with positive and negative whole numbers, including through zero

Challenge 5 Answer

National Curriculum objective, Year 5, Number and place value • round any number up to 1 000 000 to the nearest 10, 100, 1000, 10 000 and 100 000

Challenge 7 Answer Number:

11, 14, 17, 20, 23, 26, 29, 32

Nearest 10

459832

32, 25, 18, 11, 4, -3, -10, -17

Assessment

865994

523986

743023

Challenge pupils to describe each sequence in words, e.g. start at 11 and add 3; start at 32 and subtract 7.

Pupils often struggle with counting backwards through zero. Encourage them to use a number line and help them to understand that the numbers are smaller the further away they are from zero.

Challenge pupils to write a generalised rule for the sequences, such as: S (sequence) = M (multiples of 3) + 11 for the first sequence and S (sequence) = M (multiples of -7) + 32 for the second sequence.

You could encourage pupils to start using correct mathematical language in regards to sequences, such as term and term-to-term rule.

Challenge 6 Answer Karla is not correct. The thermometer is showing a temperature of -21°c, which is in fact, very cold.

Assessment Pupils may not notice that the temperature shown is a negative temperature since there is not a minus symbol in front of the numbers, hence they must recognise that the number is below 0 °C. Challenge pupils to accurately read the thermometer. Ask questions such as ‘What would the temperature be if it was 5 degrees warmer/colder?’ etc. Can they name a country in which the measurement may have been taken?

Nearest 100 000

459820 ✗

450000 ✗

400000 ✗

459830

460000

500000

865990 ✓

865000 ✗

900000 ✓

870000

Observe the method pupils use to solve this challenge. Do they count on their fingers, use a number line, solve mentally or use an alternative method? Discuss the merits of different methods with them, particularly when counting through zero to negative numbers. Pay particular attention to their answer to the second part of the question; do they realise that -3 is not smaller than -10 and that they need to continue the sequence?

Note

Nearest 10 000

523980 ✗

500000 ✗

523990

520000

743020 ✓

740000 ✓

500000 ✓ 743000 ✗ 700000

Assessment: Check that pupils understand each place value and what it means to round to that place value. Challenge pupils to explain what Maja may have been thinking when she wrote down her incorrect answers.

Note Pupils are sometimes confused by the terms ‘round up’ and ‘round down’ and will decrease the digit by 1 instead of rounding (as shown in the incorrect rounding of 459,832 to the nearest 10). Watch out for this error.

Challenge 8 Answer Answers will vary depending on numbers chosen by pupils.

Assessment Check that pupils understand what it means to round to each place value. A visual representation of rounding on a number line can help pupils struggling with the idea of rounding. Explain that rounding is finding the nearest 10, 100, etc., and avoid discussing the rules of rounding up or down.

Note Pupils are sometimes confused by the terms ‘round up’ and ‘round down’ and will decrease a digit by 1 instead of rounding (e.g. rounding 143 to 130 because they think they should decrease the tens when rounding down). Watch out for this error.

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