Division 5
Patterns, hypotheses & algebra
Objectives

Use mental, written and calculator methods to divide numbers

Look for patterns, make and test hypotheses

Use algebra as a language to explain why something works
For this unit you will need:
calculators, 100 square (see resources)
Watch out for pupils who:

when they spot a pattern, do not test out their theory with further
examples;

are not estimating so don’t spot obvious arithmetical mistakes;

are not confident in using algebra to generalise.
HSNP © Hamilton 2014
Shining Term 3
Division 5
Session 1
Objectives: Use mental, written and calculator methods to divide
numbers; Look for patterns, make and test hypotheses
Teacher input with whole class
 Ask pupils to work out the following and describe what they notice:
(11 – 2) ÷ 9
(111 – 3) ÷ 9
(1111 – 4) ÷ 9 and so on.
 Take feedback and agree the interesting answers of 1, 12, 123…
Paired pupil work
 Ask pupils to work out answers to the following:
(22 – 4) ÷ 9
(222 – 6) ÷ 9
(2222 – 8) ÷ 9
 They discuss what happens.
 Ask them to see if they can predict which digits in the following will give
a similar pattern:
(33 - ?) ÷ 9
(333 - ?) ÷ 9
(3333- ?) ÷ 9
 Can they come up with a sequence of calculations with a similar
pattern?
Teacher input with whole class
 Ask pairs of pupils to feedback what they found out.
HSNP © Hamilton 2014
Shining Term 3
Division 5
Session 2
Objectives: Use mental, written and calculator methods to divide
numbers; Look for patterns, make and test hypotheses
Teacher input with whole class
 Write the following rules on the board:
If a number is odd, triple it and add 1.
If a number is even halve it.
 Write 25 on the broad and ask the first pupil in the class to use this
rule. With the answer (76) on the board. The next pupil uses the rule.
Write the answer on the board (38). Keep going round the class until
you get an answer of 1:
25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26,
13, 40, 20, 10, 5, 16, 8, 4, 2, 1.
Paired pupil work
 Ask pupils to work in pairs to choose their own two-digit number and
write a similar chain, until it ends at 1.
 They repeat with other two-digit numbers. What do they notice?
Teacher input with whole class
 Display a 100 square (see resources) and ask pupils up to the board to
ring the first 10 prime numbers. Ask them to share how they came up
with them.
HSNP © Hamilton 2014
Shining Term 3
Division 5
Session 3
Objective: Use mental, written and calculator methods to divide numbers;
Use algebra as a language to explain why something works
Teacher input with whole class
 Write the digits 2, 4 and 7 on the board and ask pupils to make as
many three-digit numbers using these as they can (247, 274, 427, 472,
724, and 742). They then find the total of these six three-digit
numbers.
 Next they then find the total of the three single-digit numbers, and
divide the total of the six three-digit numbers by the total of the three
single-digit numbers. Write down the answer (222).
Paired pupil work
 Ask pupils to choose their own three digits and repeat above process.
 Repeat with a different pair of three-digit numbers. What happens?
Does it work if the digits are the same? If they choose zero as one of
them? (Obviously, this will generate some two-digit numbers.)
Teacher input with whole class
 Choose a pair of digits, e.g. 3 and 6, and write the two possible
numbers with these digits 36 and 63. Ask pupils to find the total and
divide by the sum of the digits, i.e. 9. What do they get? (11) Ask them
to try other pairs of digits and see if the same happens.
 Explain that if we call these two digits a and b, then we can make the
numbers (10a + b) and (10b + a). Ask pupils to find the total, i.e. 11a +
11b. Show that we can write this as 11(a + b). Say that we now divide
by the sum of the digits, which is a + b, and so this will always give us
11!
Paired pupil work
 Ask pupils to work in pairs to see if they can use this algebraic
approach to explain why the answer is always 222 when they use
three digits. (The sum of three six three-digit numbers is 222 (a + b + c)
if a, b and c are the three digits.)
HSNP © Hamilton 2014
Shining Term 3