Inverse Operations 2
Four number operations to
solve problems
Objectives

Understand the relationship between addition and subtraction

Understand the relationship between multiplication and division

Use inverse operations to solve missing number problems
For this unit you will need:
sheets of blank pyramids, optional (see resources), Inverse operations from
http://www.topmarks.co.uk/Flash.aspx?f=inversemachinev3, Function
machine from
http://www.topmarks.co.uk/Flash.aspx?f=FunctionMachinev3
Watch out for pupils who:

do not understand addition as the inverse of subtraction or
multiplication as the opposite of division;

do not understand how to use brackets;

find it difficult to solve equations if the missing number is not after
the equals sign.
HSNP © Hamilton 2014
Keeping up Term 3
Inverse Operations 2
Session 1
Objectives: Understand the relationship between addition and
subtraction; Use inverse operations to solve missing number problems
Teacher input with whole class
 Write the following missing number problems on the board and ask
pupils to work out the missing numbers in each:
75 + □ = 187; □ + 22 = 200; □ – 46 = 38
 Take feedback on how pupils solved these, drawing out using inverse
operations, e.g. subtracting 75 from 187 to find the missing number in
75 + □ = 187, adding 46 + 38 to give the missing number in □ – 46 = 38.
 Draw the following pyramid on the board:
172
81
27
12


Explain that the top number is the sum of the two numbers below the
numbers in the second row down are each the sum of the two
numbers below them and so on. So how can we work out what
number goes next to 81?
Ask pupils to work in pairs to complete the pyramid. Take feedback.
Paired pupil work
 Using the sheets of blank pyramids, if required, (see resources), pupils
work in pairs to put a number between 100 and 200 in the top row, a
number between 50 and 100 in the second row, a number between 20
and 50 in the next row and a number between 10 and 20 in the
bottom row. They work together to see if they can find numbers to
make the whole pyramid work.
 Repeat.
HSNP © Hamilton 2014
Keeping up Term 3
Inverse Operations 2
Session 2
Objective: Understand the relationship between multiplication and
division
Teacher input with whole class
 Ask pupils to work out the missing numbers in: 5 × □ = 65; □ × 14 = 28;
□ ÷ 9 = 11, □ × 10 = 650 and □ ÷ 100 = 3.42. Take feedback on how
they did this. Draw out using inverse operations, e.g. working out how
many 5s are in 65, dividing 650 by 10.
 Sketch the following function machine:
?

×2
720
Ask pupils to agree in pairs the input. How did they work out the
answer? Draw out the relationship between doubling and halving. Rub
out × 2 and replace with ÷ 2, asking pupils to work out the new input.
Paired pupil work
 Ask pupils to draw their own multiplication or division function
machines with mystery inputs (they must know what these are!). They
swap with a partner to see if they can guess the inputs.
Teacher input with whole class
 Use Inverse machines from
http://www.topmarks.co.uk/Flash.aspx?f=inversemachinev3, choosing
‘mix’ from the bottom row. Pupils close their eyes whilst you click on
each number to send them through the machines. They open their
eyes and guess each input in turn, writing the answer on their
whiteboards. Click on each number to send them back through the
machine to check. Repeat.
HSNP © Hamilton 2014
Keeping up Term 3
Inverse Operations 2
Session 3
Objective: Use inverse operations to solve missing number problems
Teacher input with whole class
 Write (2 × □) + 10 = 160; 5 (□ + 2) = 50 on the board and ask pupils to
work out the missing numbers in each.
 Take feedback on how pupils solved these, drawing out using inverse
operations, e.g. subtracting 10 from 160, then dividing by 2.
Paired pupil work
 Ask pupils to work out the following:
(□ × 10) + 20 = 70
(7 × □) – 5 = 44
(□ ÷ 5) + 4 = 10
10 (6 + □) = 90
2 (7 + □) = 18
5 (10 – □) = 30
Teacher input with whole class
 Use Function machine from
http://www.topmarks.co.uk/Flash.aspx?f=FunctionMachinev3,
choosing ‘pairs of rules to fit the function machine’. Explain that this is
a two-step function machine.


Ask pupils to choose an input, click on the question mark to see the
output, then click on ‘check values’ to put this pair of numbers in the
space for pairs which follow the function machine’s rule. Repeat with a
few other numbers of pupils’ choice.
Then ask pairs to work together to suggest other pairs of numbers
(inputs and outputs) which might follow the machine's rule. Enter
these, and then ‘check values’ to see if they obey the function
machine's rules. Repeat until more pupils think they have guessed the
rule. Those who have guessed the rule suggest other pairs to help
others to see the rules. Click on the clues to help, then ‘show function’.
HSNP © Hamilton 2014
Keeping Up Term 3