# The prompt Mathematical inquiry processes: Identify and analyse structure; generate other examples; extend a pattern; graph results. Conceptual field of inquiry: Sum and product of integers and decimals.

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 .

### Inquiry pathways

The prompt has the potential to initiate different inquiry pathways:

• 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.

# Classroom inquiry

Pal Rayit, a teacher of mathematics at St Joseph's Catholic High School Slough, Berkshire (UK), describes how she started the lesson with 60 x 1.2 on the board:

"I just wrote it down on the spur of the moment after visiting the Inquiry Maths website. Even though I hadn't thought about the numbers, the pupils automatically linked 0.2 to 20%. When I questioned them about what 1 stood for, they immediately came up with 100%. So what percentage is 1.2? 'It's 120%, miss!' I was gobsmacked. What really impressed me most was the eagerness of the pupils in trying to explain."

One student then noticed that 60 x 1.2 equals 60 + 12. Pal concludes the report on her lesson by saying: "I couldn't believe how easy it was to teach the topic. The prompt automatically led to the multiplier being used. I intend to use the Inquiry Maths website a lot more. Keep up the good work! You have introduced a new way of teaching which is actually working and more useful then a whole exercise on a topic." # Pattern and Structure

In a year 9 class, students posed questions and made observations about the prompt:

• Is it right?

• The numbers are the same on both sides.

• How do you multiply by a decimal?

• It works because both sides equal 72.

• 0.2 of 60 is 12.

• Why do you have to multiply because adding is easier?

• Is there another equation like this with different numbers.

At first, the class chose to 'find more examples' when offered a choice of the regulatory cards. The students were enthusiastic to find another equation of the same type. As they sought that equation, their fluency in calculating with decimals improved quickly. The breakthrough came when one student found 30 x 1.5 = 30 + 15. The teacher took advantage of the new equation to structure the inquiry by writing on the board:

30 x 1.5 = 30 + 15

__ x 1.4 = __ + 14

__ x 1.3 = __ + 13

60 x 1.2 = 60 + 12

Students immediately speculated that the missing numbers were 40 and 50 by following the pattern. They were disappointed to find that neither worked. By checking different numbers with a calculator, they came to:

30 x 1.5 = 30 + 15

35 x 1.4 = 35 + 14

__ x 1.3 = __ + 13

60 x 1.2 = 60 + 12

However, the remaining case proved more difficult.

At this point in the inquiry, the teacher introduced a structural approach. He referred back to the prompt and assumed the 'starting number' (60 in the case of the prompt) is to be found.

__ x 1.2 = __ + 12

Multiplying by one would not alter the starting number: __ x 1 = __ + 0. Therefore, multiplying the starting number by the number of tenths in the multiplicand has to achieve the same result as adding on the amount on the right-hand side of the equation - that is, multiplying by 0.2 is equivalent to adding 12. This means that 12 is two tenths of the starting number, six is one tenth, and 60 is the starting number. Using the same method for next case:

__ x 1.1 = __ + 11

A tenth of the starting number is 11 and, hence, the starting number must be 110.

This explanation proved challenging. A small group who understood was able to extend the table up to 1.9. Others who commented on the seemingly 'random' pattern set about producing graphs by plotting the multiplicand against the starting number. This new representation showed the students that there was a pattern to the results after all.