**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 next 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.

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 line down in the table:

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.