Year 6 Teaching Sequence 5 – Reasoning and explaining (three days)
Prerequisites:
 Recognise multiples of 2-10 up to at least the 10th multiple (see oral and mental starter bank 5)
 Find factors of two-digit numbers (see Year 5 Summer teaching sequence 3)
 Make general statements about patterns and relationships (see Year 5 Summer teaching sequence 6 and oral and
mental starter bank 5)
Overview of progression:
Children revise finding factors and prime numbers. They are challenged to find numbers with large and small numbers of
factors as well as discussing those with an odd numbers of factors (square numbers). They investigate the products of pairs
of even and odd numbers, making generalisations. Children continue sequences and are encouraged to describe each term,
leading towards a generalisation for each. ‘n’ is introduced as the number of a term, and this will be built on in Spring
sequence 6.
Note that it is important that children focus on describing and then making a generalisation for each term in a sequence
rather than only looking for the difference between consecutive terms. Otherwise in order to find the 100th, or later the
nth term, they would have to work out every term up to this point!
Note that it is not easy to write generalisations, but paired discussion should help, i.e. talking through generalisations
before recording them.
Watch out for children who are not confident with the vocabulary in this sequence, such as prime number, factor, multiple,
term, sequence. Also watch for children who seem determined to stick with ‘trial and error’ rather than pausing and actively
looking for patterns.
© Original teaching sequence copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users.
Y6 Maths TS5 – Aut – 3days
Objectives:
 Explain methods and reasoning orally
 Make general statements about odd and even numbers including their products
 Recognise and extend number sequences
 Revise finding factors of two-digit numbers
Whole class
Launch the ITP Number Grid and click to
show the prime numbers:
Group activities
Paired/indiv practice
Group of 4-5 children
Draw the following Venn diagram on
the flipchart:
Factors
2
What do you notice about these numbers?
Why are there no prime numbers in the 4th,
6th, 8th or 10th columns? Work with a
partner to find how many factors 48 has.
Take feedback. How did you work out all the
factors? Draw out dividing by 2, 3, 4, 6, and
7. Why did you not need to go past 7? Also
explain how we can write the pairs of
factors in a list to make it easier for
ourselves.
48 × 1
3
5
Challenge chn to find the
number between 20 and 40
with the most factors, the
number with the least
factors, discuss and record
what is special about this
number, and also find
numbers with an odd number
of factors, discussing and
recording what is special
about these numbers.
Easier: Chn investigate
numbers between 10 and 30.
Harder: Chn investigate
numbers between 20 and 50.
Resources
 ITP Number grid
 1-20 number cards
Where would we put 6 on this
diagram? It belongs in two sets!
Discuss how we put it in the overlap
between the two. Can you think of a
number that has 2, 3 and 5 as factors?
Where will that number go? Where
does 7 go? Write these outside the
© Original teaching sequence copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users.
Y6 Maths TS5 – Aut – 3days
24 × 2
12 × 4
6×8
3 × 16
i.e. halving and doubling to arrive at the
same product. (Starting with the number
itself and 1 won't always give all of the
factors, but it can help chn to spot more.)
Agree that 48 has ten factors. Challenge
chn to work in pairs to find a number with
more than ten factors, (e.g. 84). Take
feedback. Do all big numbers have more
factors than smaller ones?
Ask chn to work in pairs to choose a pair of
odd numbers to multiply together, a pair of
even numbers to multiply together, and a
pair, one odd and one even to multiply
together. Take feedback. What do you
notice? Why do you think the product of
two even numbers is even? Discuss how even
numbers have 2 as a factor, and so
multiplying any number by a number with
two as a factor will give another number
with two as a factor, i.e. an even number.
So this will also be the case when
multiplying an odd number by an even
number. Write an example, 8 × 5 = 2 × 4 × 5
(using factors of 8) = 2 × 20, an even
number. Repeat with other examples chn
have given. We can show the product of two
numbers as an array. If one number is even,
we could always draw the array as one with
sets, but inside the rectangle. Ask chn
to work in pairs to think of at least
one number to go in part of the
diagram. Take feedback and write
these on. Which kinds of numbers did
you find it most difficult to think of?
Easier: Use numbers cards 1-20 and
work together to place them on the
diagram. Afterwards, ask chn to think
of another numbers to go in each part
of the diagram.
Group of 4-5 children
Together investigate the products of
three numbers: three even numbers,
two even and one odd, one even and
two odd, and three odd numbers. Use
cuboids to show each multiplication,
and then discuss which can be broken
into two equal parts and therefore
have an even number of cubes. For
example, 5 × 4 × 3
This can be split into two cuboids, 2 by
3 by 5, each having an equal number of
Give each pair of chn a copy
of a 1-100 square and ask
them to investigate the
products of two squares in a
vertical, horizontal or
diagonal line, writing general
statements about whether
their products are always
odd, sometimes odd, always
even etc. They should discuss
and then write about why
they think these statements
are true. Give chn calculators
so that they can focus on the
reasoning, rather than taking
so long to work out the
calculations themselves.
Harder: Chn could also
investigate lines of three
© Original teaching sequence copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users.
 Calculators
 1-100 squares
Y6 Maths TS5 – Aut – 3days
an even number of rows, and so the number
in the array is even.
What happens when you multiply two odd
numbers together? Discuss how neither is a
multiple of two, and so the answer is not a
multiple of 2 either. As 2 has only two
factors, 2 and 1, one number must have 2 as
a factor to produce an even number.
Draw the following sequence on the board:
These numbers are called triangular
numbers, you can see why! How many do you
think might be in the next shape? Sketch it
on your whiteboards. And the next? What
do you think the tenth shape might look
like? Discuss this with your maths partner.
Take feedback. The fourth shape has
columns of 4, 3, 2 and 1. The fifth has
columns of 5, 4, 3, 2, and 1 so what will the
tenth have?
Draw the following sequence of square
cubes, therefore 3 × 4 × 5 gives an
even number.
Easier: Choose single-digit numbers to
multiply together, and show this using
multilink as an array, e.g. 6 × 3
Show how this array can be broken
into to equal parts, and therefore
must be even numbers. Repeat,
multiplying even numbers and odd
numbers together.
Group of 4-5 children
Write a list of triangular numbers, 1,
3, 6, 10, 15, 21 and square numbers 1,
4, 9, 16, 25, 36. Ask chn to investigate
the following general statement: every
square number can be made by adding
two consecutive triangular numbers.
Ask them first to choose numbers
from the list that suggest this might
be true. Then challenge them to use
sketching of triangular and square
numbers to find out why this might be
the case.
Draw out how two right-angled
triangles fit together to make a
square e.g. 9 being made of 6 and 3.
squares.
Ask chn to write/draw the
next few terms of a
sequence, and then to
describe the sequence to a
partner, before writing a
description (see resources).
They then write a 3-digit
number that will be in each
sequence, and a 3-digit
number that will not be in
each sequence.
Harder: Challenge chn to
choose two sequences and
describe the nth term in
each.
© Original teaching sequence copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users.
 Squared paper, scissors
 Activity sheet (see
resources)
Y6 Maths TS5 – Aut – 3days
numbers. How many will be in the next
shape? And the next? And the tenth? What
will that shape look like? We call the
number of term (shape) n, so the number of
spots in each term is n2. In the second term
this is 22, the third 32 and so on.
Do the triangular numbers have to be
consecutive? Why?
Easier: Give chn squared paper and ask
them to colour in squares to form
triangular numbers. Ask them to cut
these out and try to make square
numbers. What do you notice? Draw
out the general statement.
© Original teaching sequence copyright Hamilton Trust, who give permission for it to be adapted as wished by individual users.
Y6 Maths TS5 – Aut – 3days