Patterns of Change

Two quantities that change together are organized into a table, and by watching what happens to one when the other goes up by 1, children find the expression behind them. Arrangements of beads, matchsticks, and tables with people seated around them are laid out, and the question of what the nth arrangement holds extends as far as a two-step rule table.

Seeing that the count rises by 3 each time and answering with × 3 alone, forgetting to add the first piece, stands out. An answer short by a fixed amount is this pattern. Children also look only at the gap between the first and second rows of the table and assume the same step continues from there on.

The view organized in Investigating Patterns of Change in grade 4, for a relationship that grows by the same amount each step, is broadened here to arrangements of figures: rows of beads, matchsticks. Deriving an expression from a table this way returns unchanged in grade 6, in Proportion and Inverse Proportion.