Pal Rayit devised the prompt from an observation made by one of her year 10 students. She had started a lesson on percentage change by writing 60 x 1.2 on the board with the aim of introducing the concept of a multiplier. The student noticed that the product equalled the sum of 60 and 12.

Students of all ages in secondary school find the prompt intriguing. They wonder whether it is true and whether there are more examples of the same type. The inquiry might start with a period of exploration as students test different starting numbers for the same multiplier. Is it true that 50 x 1.1 = 50 x 11? Or perhaps 70 x 1.1 = 70 + 11. Students soon realise that the starting number for 1.1 must be higher than 70.

At some point, the teacher encourages students to use deductive reasoning. Using the multiplier 1.1 gives:

n x 1.1 = n + 11

n + 0.1n = n + 11

0.1n = 11 (one tenth of the starting number is 11)

n = 110 (divide both sides by 0.1 or multiply by 10)

The teacher might focus on the use of the reciprocal for the final step. For example, to solve 0.3n = 13 requires the student to divide both sides by 0.3 (3/10) or multiply by 10/3 .

Lines of inquiry

The prompt has the potential to initiate different lines of inquiry:

• Search for examples of a similar type and graph the results (see below).

• Consider two decimal places, for example 50 x 1.25 = 50 + 12.5.

• Explore what happens when the decimal number is less than one (90 x 0.9 = 90 - 9) or greater than two (16.67 x 2.5 = 16.67 + 25).

• Explore what happens when the decimal number approaches one.

• Find more examples through deductive reasoning, rather than through pattern spotting or exploration.

• Change the operations in the prompt, for example 72 ÷ 1.2 = 72 - 12.

• Replace the decimals by fractions, for example 72 ÷ 6/5 = 72 x 5/6.