Fair and square

Students will need square tiles to build the patterns, or you can get them to draw the tile patterns on square grid paper. Although “building on” methods using equipment can solve the problems quite effectively, encourage students to look for rules that may help. For example, with pattern 1, students may notice that three yellow squares are added to the pattern each time a black square is added. This may be shown in a table:

Students can check the predictions they made using tables or equations by continuing the pattern to the required length. In theory, future predictions could be found by graphing the relationships involved. For example, the graph for pattern 2 could be extended to find the tenth term:

In practice, however, finding future values by extending a linear graph can be troublesome because one small slip in the line will lead to incorrect values. However, this unreliability is a useful teaching point.

Some students may find direct rules linking the values of the variables in the relationship. For example, for pattern 3, this might be described as “The number of blue squares, plus one, gives the number of pink squares.” This is a powerful method since it allows students to predict elements in a pattern without building the pattern or continuing a table.

For example, in pattern 2, the number of orange squares is twice the number of red squares plus two. Therefore, for ten red squares, the number of orange squares is given by (10 x 2) + 2 = 22.

1.

31 yellow squares

2.

22 orange squares

3.

20 blue squares

4.

a. 2 green, 9 white

b. 3 green, 12 white

c. 8 green

5.

a. 6 orange

b. 3 blue, 9 orange

c. 30 orange