Home > Patterns > Misunderstandings > Number sequences > Find a rule

Find a rule

A teacher asked students to continue a growing pattern by two terms.

Three collections of dots: a single dot, a 2 by 2 square array, a 3 by 3 square array.

The given pattern.

Ashah started to repeat the first three terms. While she would have made a correct repeating pattern, she had not drawn a growing pattern.

Five collections of dots: a single dot, a 2 by 2 square array, a 3 by 3 square array, a single dot, a 2 by 2 square array.

Ashah’s pattern.

Ben counted 9 dots in the third term and then made pictures with 10 and 11 dots. His rule did not include the first two terms, so it is not correct. He also ignored the shape of the given terms.

Five collections of dots: a single dot, a 2 by 2 square array, a 3 by 3 square array, a single dot, a 2 by 2 square array, an approximately circular cluster of 10 dots, an approximately circular cluster of 11 dots.

Ben’s pattern.

Cora added one row at a time. Her rule took some account of the shape of the terms, but again did not include the first two terms.

Five collections of dots: a single dot, a 2 by 2 square array, a 3 by 3 square array, a 3 by 4 array, a 3 by 5 array.

Cora’s pattern.

Students who drew a correct continuation argued that "there is one more row and one more column" or "they are squares getting bigger by one". Their rule applied to all the terms.

Five collections of dots: a single dot, a 2 by 2 square array, a 3 by 3 square array, a 4 by 4 square array, a 5 by 5 square array.

A correct solution.

This growing pattern gives the sequence of square numbers: 1, 4, 9, 16, 25…

Find a rule

Curriculum links