Åk 6–9

3.2 Patterns and Algebraic Expressions
When figures are constructed according to a pattern, we can express this pattern as an algebraic expression.  This means that we can reconstruct any figure with a given pattern by using an algebraic expression.

Example 1

How many matches does the figure n have? Figure 1 has in the example above 4 matches.  Figure 2 has 7 matches.  Figure 3 has 10 matches and so on.  It is more practical to show this in a table form in order to get an overview and then be able to create an algebraic expression for how many matches n has. We can see that the number of matches increases by 3 in every figure. This can even be written in the follow way:

 Figure 1: 3 · 1 + 1 = 4 matches Figure 2: 3 · 2 + 1 = 7 matches Figure 3: 3 · 3 + 1 = 10 matches Figure 4: 3 · 4 + 1 = 13 matches Figure 5: 3 · 5 + 1 = 16 matches . . . Figure n: 3 · n + 1 = ? matches

If we want to calculate the number of matches in figure 14 in the order, we replace the variable n with 14:

Figure 14 has 3 · 14 + 1 = 43 matches

Example 2

How many black squares does figure n have? We begin by recreating the number of black squares in table form. By doing so, we can more easily find a pattern. Now, we need to find a pattern for how many black square each figure increases by. We can see that the whole time each figure is squared:

1 · 1 = 1
2 · 2 = 4
3 · 3 = 9
and so forth

It seems though that it is moved one step to the right.

 Figure 1: (1 - 1)2 = 02 = 0 black squares Figure 2: (2 - 1)2 = 12 = 1 black squares Figure 3: (3 - 1)2 = 22 = 4 black squares Figure 4: (4 - 1)2 = 32 = 9 black squares Figure 5: (5 - 1)2 = 42 = 16 black squares . . . Figure n: (n - 1)2 = ? black squares

Answer: In the figure n, the numbers of black squares is (
n - 1)2.

How many black squares are there in figure 9 in the order?

We replace the variable n with 9:

Figure 9 has (
9 - 1)2 = 82 = 64 black squares    