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Reasoning and explaining
Year 6 Autumn 5
Explain methods and reasoning orally and in writing
Previous learning
Core for Year 6
Extension
Understand, read and begin to write these words:
Understand, read and write these words:
Understand, read and write these words:
problem, puzzle, solution, method, justify, explain how you
know, give your reasons, …
sequence, pattern, relationship, rule, general statement, …
problem, puzzle, solution, method, justify, explain how you
know, give your reasons, … general statement, …
sequence, pattern, relationship, rule, formula, substitute, …
problem, puzzle, solution, method, justify, explain how you
know, give your reasons, … general statement, …
sequence, pattern, relationship, rule, formula, substitute, …
Explain reasoning and methods to solve problems such as:
Explain reasoning and methods to solve problems such as:
Explain reasoning and methods to solve problems such as:
• Find:
two consecutive numbers with a product of 182;
three consecutive numbers with a total of 333.
• Each ✸ represents one of the digits 1 to 6.
Replace each ✸ to make a correct product.
• Debbie has a pack of cards numbered from 1 to 20.
She picks four different number cards.
• Choose any four numbers from the grid. Add them up.
Find as many ways as possible of making 1000.
✸✸ ×✸ =✸✸✸
?
• 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.
• 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.
A+A=B
A × A = DF
A + C = DE
C + C = DB
C × C = BD
A × C = EF
• One whole number divided by another gives 1.1818181…
What are the two numbers? Use your calculator.
• Each ✸ represents a missing digit.
Use your calculator to solve this.
✸ 2 ✸ × ✸ ✸ = 11 316
• Complete this addition table.
+
?
?
Exactly three of the four numbers are multiples of 5.
Exactly three of the four numbers are even numbers.
All four of the numbers add up to less than 40.
Write what the numbers could be.
• 30 children are going on a trip. It costs £5 including lunch.
Children who take their own packed lunch pay only £3.
The 30 children pay a total of £110.
How many children are taking their own packed lunch?
• Here are five number cards.
A
A
A
B
B
A and B stand for two different whole numbers.
The sum of all the numbers on all five cards is 30.
What could be the values of A and B?
99
120
27
• Complete this multiplication table.
?
100
146
150
• I think of a number.
I divide it by 4, then subtract 5. I get 11.
What is my number?
© 1 | Year 6 | Autumn TS5 | Reasoning and explaining
• I think of a number, square it and add 4..
The answer is 40.
What is my number?
• I think of a number. I add 4. I multiply the result by 3.
Then I take away 9. My final answer is 90.
What number did I start with?
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Previous learning
Core for Year 6
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:
• 7003 – 6994
6994 + 6 = 7000, add 3 more is 7003,
so the answer is 6 + 3 = 9.
• 42 × 15
• 17.5% of £30 000
• 24 × 4
Double 24 is 48, and double 48 is 96.
• 400 × 80 is equivalent to 4000 × 8 = 32 000.
• 87 ÷ 2
Half of 80 is 40, and half of 7 is 3.5, so it’s 43.5.
42 × 10 = 420
42 × 5 = 210
42 × 15 = 630
half of 42 × 10
the sum of 420 and 210
• 15 × 12 is 15 × 4 × 3 = 60 × 3 = 180.
•
1
20
of 400
One tenth of 400 is 40, so one twentieth of 400 is 20.
• 387 ÷ 9
10%
5%
2.5%
17.5%
= £3000
= £1500
= £ 750
= £5250
387 ÷ 3 = 129
129 ÷ 3 = 43
• 49 × 30
50 × 30 = 1500, subtract 30 is 1470.
Make general statements about odd and even numbers, including their products
Previous learning
Core for Year 6
Extension
Make general statements about odd and even numbers
including their sums and differences, e.g.
Make general statements about odd and even numbers
including their products.
Consider whether general statements about odd and even
numbers are true.
• The sum of:
Make or draw rectangles to show sums or products, and
which can be broken into two equal parts and are even.
Respond to questions such as:
two odd numbers is even;
two even numbers is even;
one odd and one even
number is odd.
odd + odd = even
even + even = even
even + odd = odd
• The difference between:
two odd numbers is even;
two even numbers is even;
one odd and one even number is odd.
• The sum of:
three odd numbers is odd;
three even numbers is even;
two odd and one even numbers is even;
two even and one odd numbers is odd.
• The product of:
two odd numbers is odd;
two even numbers is even;
one odd and one even number is even.
• The sum of three odd numbers is odd:
odd + odd + odd = even + odd = odd
The sum of two odd and one even number is even:
odd + odd + even = even + even = even
The sum of one odd and two even numbers is odd:
odd + even + even = odd + even = odd
The sum of three even numbers is even:
even + even + even = even
Respond to questions such as:
• Harry says: ‘When you add three odd numbers the answer
is always even.’ Is he correct? Explain how you know.
• Jack has a bag of seven counters numbered 1 to 7.
Anil has a bag of twenty counters numbered 1 to 20.
Each secretly chooses a counter from their own bag.
For each statement, put a tick (9) if it is true.
Put a cross (8) if it is not true.
–
–
–
–
Jack is more likely than Anil to choose a '5'.
They are equally likely to choose a number less than 3.
Jack is more likely than Anil to choose an odd number.
Anil is less likely than Jack to choose a '10'.
• Jay says:
‘Divide an even number by 2.
The answer must be an odd number.’
Give an example to show that Jay is wrong.
For example: 20 ÷ 2 = 10.
20 is even but when you divide it by 2 you get 10, which is
even, not odd.
• Hannah says: ‘I added three odd numbers and my answer
was 50.’ Explain why Hannah cannot be correct.
© 2 | Year 6 | Autumn TS5 | Reasoning and explaining
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Recognise and extend number sequences
Previous learning
Core for Year 6
Extension
Use, read and begin to write these words:
Use, read and write these words:
Use, read and write these words:
multiple, divides exactly by, factor, divisor, divisible by,
test of divisibility, … sequence, term, rule, …
prime number, multiple, divides exactly by, factor, divisor,
divisible by, test of divisibility, … sequence, term, rule, …
prime number, multiple, divides exactly by, factor, divisor,
divisible by, test of divisibility, … sequence, term, rule, …
Recognise and extend number sequences, e.g.
Recognise and extend number sequences, e.g.
Recognise and extend number sequences, e.g.
• Here is a number sequence.
• Fill in the missing numbers in these sequences.
Describe each rule.
• A sequence starts at 500 and 80 is subtracted each time.
0 1 1 2 3 5 8 13 F F
Describe the rule for the sequence.
Write the next two numbers in the sequence.
• Draw the next pattern for each sequence.
Describe each rule.
10, 25, F, F, 80, …
1, 4, F, F, 25, 36, F, …
F, F, –61, –42, –23, …
420
340
...
The sequence continues in the same way. Write the first
two numbers in the sequence which are less than zero.
• Ann makes a pattern of L-shapes with sticks.
• Here is the start of the sequence of triangular numbers.
Shape–number: 1
Number of sticks: 7
• Here is a repeating pattern of shapes.
Each shape is numbered.
Draw the next two patterns in the sequence.
How many dots will there be in pattern 10?
The pattern continues in the same way.
Write the numbers of the next two stars in the pattern.
500
• Here is a sequence made from squares and circles.
2
11
3
15
Ann says : ‘I find the number of sticks for a shape by first
multiplying the shape-number by 4, then adding 3.’
Work out the number of sticks for the shape that has
shape-number 10.
Ann uses 59 sticks to make another L-shape in this
pattern. What is its shape-number?
Complete this sentence.
Shape number 35 will be a circle because ...
Draw the next two patterns in the sequence.
How many squares will there be in pattern 10?
How many circles?
© 3 | Year 6 | Autumn TS5 | Reasoning and explaining
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Revise finding factors of two-digit numbers
Previous learning
Core for Year 6
Extension
Know that factors occur in pairs, e.g.
Know that factors occur in pairs, e.g.
Find the prime factors of any two-digit number, e.g.
Know that:
Know that:
Know that, for example:
• A factor is a number that divides exactly into a bigger
number, e.g.
• Factors occur in pairs so that if 3, say, is a factor of the
number 24, then 24 ÷ 3 = 8 is also a factor of 24.
5 is a factor of 15
because 5 divides exactly 3 times into 15.
• If a number is a multiple of, say, 2, then 2 is a factor of the
number.
6 is not a factor of 15
because 6 does not divide exactly into 15.
15 is not prime, but can be written as the product 3 × 5.
3 and 5 are prime numbers which are factors of 15.
They are called the prime factors of 15.
12 is not prime, but can be written as the product of its
prime factors:
12 = 2 × 2 × 3
• Factors occur in pairs, e.g. since 12 = 3 × 4, both 3 and 4
are factors of 12. The simplest factor pair of any number is
the number itself and 1.
Find factors of two-digit numbers, e.g.
Revise finding factors of two-digit numbers, e.g.
Write a number as the product of its prime factors, e.g.
• Find the factor pairs of, say, 20 by arranging 20 counters in
the shape of different rectangles:
• Find the factor pairs of 36 by arranging 36 counters in the
shape of different rectangles.
• To write 75 as the product of its prime factors, look for the
smallest prime factor of 75, which is 3.
Divide 75 by 3
75 ÷ 3 = 25
20 × 1, 10 × 2, 5 × 4
There are no other pairs of numbers which have a product
of 20.
So the six factors of 20 are 1, 2, 4, 5, 10 and 20.
The factor pairs of 36 are:
1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6
The nine factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.
• Make a ‘factor finder’.
Repeat with the new number (25).
Look for the smallest prime factor of 25, which is 5.
Divide 25 by 5
25 ÷ 5 = 5
Repeat with the new number (5).
Divide 5 by 5
5÷5=1
So 75 = 3 × 5 × 5
3 75
5 25
5 5
1
By colouring factors on a grid, show for example that
factors of 6 are also factors of 12.
You can also use factor trees, e.g. to show that
48 = 2 × 2 × 2 × 2 × 3.
48
4
2
2
© 4 | Year 6 | Autumn TS5 | Reasoning and explaining
4
12
3
2
2
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Previous learning
Core for Year 6
Extension
Solve problems such as:
Investigate problems such as:
• Which number(s) between 20 and 40 have:
• Give an example of a two-digit number with:
the most factors
exactly two factors (e.g. 17)
the least factors
exactly three factors (e.g. 49)
an odd number of factors?
exactly four factors (e.g. 10)
• Which numbers less than 100 have exactly three factors?
(the squares of prime numbers, e.g. the factors of 49 are 1,
7 and 49)
• Write a number in each circle so that the number in each
square is the product of the two numbers on either side of it.
Use factors to multiply when appropriate, e.g.
15 × 6 = 15 × 2 × 3
= 30 × 3
= 90
© 5 | Year 6 | Autumn TS5 | Reasoning and explaining
exactly five factors (e.g. 16)
exactly ten factors (e.g. 48)
more than ten factors (e.g. 84)
• Which numbers have exactly five factors?
(the squares of the squares of prime numbers, e.g. the
factors of 81 are 1, 3, 9, 27 and 81)
Use factors to multiply when appropriate, e.g.
35 × 12 = 35 × 2 × 6
= 70 × 6
= 420
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999