www.inquirymaths.org © 2012 Andrew Blair

Lesson Notes As multiplication is commutative, they argued, this showed that the two sides must be equal.

40% of 70 = 70% of 40 Initial reactions to this prompt are mixed. Even if some students doubt its veracity, there is always one student in a key stage 3 all ability class who can calculate the percentages by deducing them from 10% of the number. Comments in the orientation phase include attempts to show both sides equal 28, while questions have centred on whether it is possible to create more complicated examples, including the use of decimals and percentages greater than 100%.

My aim at this point was to try to reconcile the two approaches, and use the algebra to explain the pattern spotted by the first group. I wrote this: A B AB B A BA x = = x = 10 10 100 10 10 100

After a period of thinking, students attempted to explain the links between the two algebraic statements. The discussion ended with some students attempting to convince their peers that the second one was limited to particular examples, and the first was superior as it worked for all cases. ..................................................................

Students enjoy creating more examples. The teacher might aim to consolidate students’ understanding of a calculator method to work out percentages.

The prompt can be made more complex by presenting it alongside a second statement.

An interesting discussion arose in one class when a group noticed that: 40% of 70 = 28 = 4 x 7 70% of 40 = 28 = 7 x 4 They created similar examples using multiples of ten. (Later in the inquiry, another group showed how they could adopt this approach for their examples. So, for instance, 34% of 72 = 3.4 x 7.2) At the same time, another group attempted to show algebraically why the two sides of the equation had to be equal. This is what they wrote on the board:

The second part of the statement here is not true. Students have raised the question, ‘Can it ever be true for two increases if the number and percentage are different?’ and gone on to conclude that it cannot. Another inquiry has started when students changed one part of the statement to include the word ‘decreased’. Both inquiries have involved students using a multiplier to calculate percentage increases and decreases.

A AB B BA xB= = xA= 100 100 100 100

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www.inquirymaths.org © 2012 Andrew Blair

.................................................................. A second additional statement is added to the prompt below:

½ of 20% of 60 = 2/3 of 10% of 90

Key stage 3 classes have found the second statement difficult to develop. If students wish to explore it, I have offered them a worksheet to match ‘percentage pairs’. This offer is normally in response to their selection of the regulatory card ‘Ask the teacher for something to do.’ In the prompt above, the percentages have been changed by a factor that is a reciprocal of the factor used to change the number. This is shown in the diagrams below:

.................................................................. Two alternatives to the original prompt that can be used with older classes are as follows:

20% of 10% of 40 = 40% of 20% of 10

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