## Mathematics: Using percentages to increase quantities

### Mathematics: Using percentages to increase quantities

At Boundary Oak School, we work towards embedding a love for learning in all of our pupils, in all that we do, something we feel vital in aiding a child’s engagement and progress. In Mathematics, Year 7 have been working on inverse relationships with Mr Polansky.

This card matching activity helped them to identify inverse relationships when calculating increases/decreases using fractions, percentages and decimals. For example, did you know that to undo (or reverse) an increase of one fifth you need a decrease of one sixth; to undo an increase of one sixth you need a decrease of one seventh and so on?

Most pupils will have met the basic concepts before. Many, however, will have been introduced to the ideas in a basic, standard format- “this is how you calculate a percentage of a quantity” – rather than in a conceptual one. Typically, for example, it is believed that an increase of 50% followed by a decrease in 50% takes us back to the original value. Mr Polansky confronted such misconceptions with his pupils for them to be able to build deeper conceptual links between percentages, decimals and fractions and how to reverse their changes.

By discovering and testing relationships themselves and through dialogue and teamwork they were able to spot patterns and relationships:

For example, 3/4 of an amount can be returned to the original amount by x 4/3. 5/6 of an amount can be returned to the original amount by x 6/5.

Multiplying by 1.2 can be reversed by x1/1.2. The reverse of x 3.4 is x1/3.4 and so on. Increasing by 1/8 can be reversed by decreasing by 1/9, etc.

The aim of the lesson was to enable pupils to rapidly answer the following questions:

If a price increases by 10% . . .
How can you write that in words?
How can you write that as a decimal multiplication?
How can you write that as a fraction multiplication?
How much will the price need to go down to get back to the original price?
How can you write that in words?
How can you write that as a decimal multiplication?
How can you write that as a fraction multiplication?

Whilst this lesson promoted fluency in the three most common formats for representing numbers and change (fractions, percentages , decimals) it also raised some very interesting, deeper thinking questions:

• Do we need all three (fractions, percentages , decimals)?
• Which is most useful in today’s world?
• Would you change your answer if we had a base 12 or base 60 number system, instead of a base 10 system?
• Which of the three would the Babylonians, Sumerians or Romans have chosen? Why?