Reasoning and explaining
Year 5 Spring 6
Solve mathematical puzzles
Previous learning
Core for Year 5
Extension
Understand and read these words:
Understand, read and begin to write these words:
Understand, read and write these words:
problem, puzzle, solution, method, explain how you know,
give your reasons, …
pattern, relationship, rule, …
problem, puzzle, solution, method, justify, explain how you
know, give your reasons, …
pattern, relationship, rule, general statement, …
problem, puzzle, solution, method, justify, explain how you
know, give your reasons, …
pattern, relationship, rule, formula, general statement, …
Solve mathematical puzzles and problems, e.g.
Solve mathematical puzzles and problems, e.g.
Solve mathematical puzzles and problems, e.g.
• The 5 key of Li’s calculator is broken so she cannot use it.
How should she do each calculation?
• The 8 key of Jo’s calculator is broken so he cannot use it.
How should he do each calculation?
• Each ✸ represents one of the digits 1 to 6.
Replace each ✸ to make a correct product.
42 + 25
42 + 52
64 – 15
64 – 52
• This is a number triangle with some numbers missing..
Put all the numbers 20, 30, 50 and 60 in the circles so that
the numbers along each edge add up to 90.
42 × 8
84 × 4
28 × 4
42 ÷ 8
84 ÷ 4
✸✸ ×✸ =✸✸✸
28 ÷ 4
• Write a number in each circle so that the number in each
square box equals the sum of the two numbers on either
side of it.
• Investigate how many three-digit numbers there are with a
digit sum of 25.
• Each letter from A to G is a code for one of these digits:
1, 3, 4, 5, 6, 8, 9.
Crack the code.
• Use each of the digits 1, 2, 3 and 4 once to make a
difference of 11.
• Use each of the digits 3, 5, 4 and 6 once to make a total
that is a multiple of 5.
• Joe has three 20p and two 15p stamps.
What values can he make using one or more of the
stamps?
• Use 2, 4 and 5, and the signs +, × and =.
How many different answers can you make between 40
and 200?
© 1 | Year 5 | Spring TS6 | Reasoning and explaining
• Complete this multiplication table.
x
3
6
18
32
A+A=B
A × A = DF
A + C = DE
C + C = DB
C × C = BD
A × C = EF
Calculator investigations
30
40
• Find as many different ways as possible to complete this
calculation.
• One whole number divided by another gives 1.1818181…
What are the two numbers?
• Each ✸ represents a missing digit.
Use your calculator to solve this.
✸ 2 ✸ × ✸ ✸ = 11 316
Calculator investigation
• Each ✸ represents a missing digit.
Use your calculator to solve this.
✸ ✸ × 6✸ = 6272
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Explain methods and reasoning orally
Previous learning
Core for Year 5
Extension
Describe and explain methods, reasoning and solutions to
puzzles and problems, orally and in writing, with diagrams.
Describe and explain methods, reasoning and solutions to
puzzles and problems, orally and in writing, with diagrams.
Describe and explain methods, reasoning and solutions to
puzzles and problems, orally and in writing, with diagrams.
For example, explain orally or write that:
For example, explain orally or write that:
For example, explain orally or write that:
• 24 × 4
Double 24 is 48, and double 48 is 96.
• 400 × 80
This is the same as 4000 × 8 = 32 000.
• 49 × 30
50 × 30 = 1500, subtract 30 is 1470.
• 87 ÷ 2
Half of 80 is 40, and half of 7 is 3.5, so it’s 43.5.
•
•
1
of 424
8
Half of 424 is 212, half of 212 is 106, and half of 106 is 53,
so one eighth of 424 is 53.
1
of 400
20
One tenth of 400 is 40, and half of 40 is 20, so one
twentieth of 400 is 20.
Make general statements and justify these
Previous learning
Core for Year 5
Extension
Make general statements about a given number, e.g.
Make and justify general statements, e.g.
Describe a general relationship in words, e.g.
• Choose a three-digit number (e.g. 240) and say that:
• Choose a three-digit number (e.g. 256) and say that:
• Explain how to find the number of months in a given
number of years.
– It’s between 200 and 250 but nearer 250.
– It’s between 200 and 300 but nearer 300 because it’s
more than 250..
– It’s 200 rounded to the nearest 100.
– It’s an even number because its last digit is 6.
• Explain how to find the change from 50p for a given
number of chews at 4p each.
– It’s a three-digit number.
– It’s not a multiple of 5 because it doesn’t end in 5 or 0.
• Explain how to calculate the area of a rectangle.
– The digits add up to 6.
– It’s not a multiple of 9 because the digits don’t add
up to 9.
– It’s even.
– It’s a multiple of 2, and of 3, and of 4, and of 5, and of 10.
– It’s a square number because 16 × 16 = 256.
Give an example to match given criteria, e.g.
Give an example to match given criteria, e.g.
Express a relationship in a formula using letters as symbols, e.g.
• Write down:
• Write down a square number between 60 and 90 with a
digit sum of 10.
• Use symbols to write a formula for:
– a number between 30 and 40 with a digit sum of 8;
– an even multiple of 3 greater than 30.
• Here is a sorting diagram for numbers.
Write a number less than 100 in each space.
even
© 2 | Year 5 | Spring TS6 | Reasoning and explaining
not even
the number of months m in y years;
Tom’s age T, which is 3 years’ less than Kate’s age K;
the cost of c chews at 4p each;
the number of socks s worn by c children;
a square number
the number of days d in w weeks;
not a square number
the cost D of a DVD which is £3 more than the cost C of a CD.
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Previous learning
Core for Year 5
Extension
Give examples to match a true statement, e.g.
Give examples to match a true statement, e.g.
Solve problems involving formulae, e.g.
• Any odd number is one more than an even number.
• A multiple of 6 is always twice a multiple of 3.
• Write a formula for the nth term of this sequence:
For example: 23 is an odd number and 23 is 1 more than
22, which is an even number.
For example: 24 = 2 × 12, and 12 is a multiple of 3;
60 = 2 × 30, and 30 is a multiple of 3.
• Any even number can be written as the sum of two odd
numbers.
• A number is not a multiple of 9 if its digits do not add up to
a multiple of 9.
For example: 16 is an even number, and 16 = 13 + 3,
which are both odd numbers.
For example: 58 is not a multiple of 9, since 5 + 8 = 13,
and 1 + 3 = 4, which is not a multiple of 9.
• Multiples of 4 are always even.
For example: three fours are 12, which is even.
• A multiple of 6 is both a multiple of 2 and a multiple of 3.
For example: 48 = 6 × 8 or 3 × 16 or 2 × 24
3, 6, 9, 12, 15, …
• The perimeter of a rectangle is 2 × (l + w), where l is the
length and w is the width of the rectangle.
What is the perimeter if l = 8 cm and w = 5 cm?
• n represents the number of magazines that Dan reads
each week. Which of these represents the total number of
magazines that Dan reads in 6 weeks?
A 6+n
B 6×n
C n÷6
D 6÷n
Give an example to show that a statement is not true, e.g.
Give an example to show that a statement is not true, e.g.
Give an example to show that a statement is sometimes true.
• John says:
‘Every multiple of 5 ends in 5.’
Is he right or wrong? Explain how you know.
• Holly says:
‘The product of two consecutive numbers is odd.’
Is she correct? Explain how you know.
• Is this statement always, sometimes or never true?
John is wrong. For example, 20 is four 5s, so it is a
multiple of 5, but it does not end in 5.
• Anna says:
‘Multiply any number by 3.
The answer must be an odd number.’
Holly is wrong. For example, 5 and 6 are consecutive
numbers, and 5 × 6 = 240, which is even.
• Jay says:
‘Divide an even number by 2.
The answer must be an odd number.’
Give an example to show that Anna is wrong.
Give an example to show that Jay is wrong.
For example: multiply 10 by 3.
10 × 3 = 3 × 10 = 30, and 30 is even, not odd.
For example: 20 ÷ 2 = 10.
20 is even but when you divide it by 2 you get 10, which is
even, not odd.
‘Multiples of 5 are even.’
Give examples to show how you know.
The statement is sometimes true. For example,
10 is an even multiple of 5 but 15 is an odd multiple of 5.
• Is this statement always, sometimes or never true?
‘In a triangle, each angle must be 90 degrees or less,
because the three angles add up to 180 degrees.’
Give examples to show how you know.
© 3 | Year 5 | Spring TS6 | Reasoning and explaining
The statement is sometimes true. For example, an
equilateral triangle has angles of three angles of 60° but an
isosceles triangle could have angles of 140°, 20° and 20°.
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999