The students can use both the tables and the graphs to predict the values of further terms in each pattern.
The table below shows Emma’s problem extended to the tenth and the fifteenth person.

The graph of the first four terms is a straight line (hence, the relationship is linear). This line can be extended to predict the values for the tenth and fifteenth terms.

The students can use formulas on a spreadsheet to find further values of a sequence. The spreadsheet table for Emma’s sequence shows that when the number of people increases by one, the corresponding increase in the number of sticks is five. Instead of writing “2” in cell A3, the students could input the formula (rule) =A2+1 and fill it down for as many cells as desired. Similarly, they could input the formula =B2+5 in cell B3 and fill it down to give the matching number of sticks.
These rules can be adapted for the spreadsheets for question 2. For example, for 2a, the formula for the number of sticks in cell B3 is =B2+6 and for 2b, the formula in cell B3 is =B2+5.

Answers to Activity

1. a. 51 sticks
b. 76 sticks
2. a. The computer spreadsheet should look like this:

The graph could look like this:

The next member of this pattern is 32 sticks and the tenth member is 62 sticks.
b. The computer spreadsheet should look like this:

The graph could look like this:

The next member of this pattern is 27 cubes, and the tenth member is 52 cubes.