Generalizing Patterns: The Difference of Two Squares

Mathematical goals

This lesson unit is intended to help you assess how well students working with square numbers are able to:

  • Choose an appropriate, systematic way to collect and organize data, examining the data for patterns.
  • Describe and explain findings clearly and effectively.
  • Generalize using numerical, geometrical, graphical and/or algebraic structure.
  • Explain why certain results are possible/impossible, moving towards a proof.


This lesson unit is structured in the following way:

  • Before the lesson, students work individually on an assessment task designed to reveal their current understanding and difficulties. You review their responses and create questions for them to consider when improving their work.
  • At the start of the lesson, students reflect on their individual responses and use the questions posed to think of ways to improve their work. They then work collaboratively in small groups to produce, in the form of a poster, a better solution to the task than they did individually.
  • In a whole-class discussion students compare and evaluate the different strategies they have used.
  • Working in the same small groups, students analyze sample responses to the task.
  • In a whole-class discussion, students review the methods they have seen.
  • At the end of the lesson, or in a follow-up lesson, students reflect individually on their work.

Materials required

  • Each student will need a copy of the assessment task: The Difference of Two Squares, some plain paper, and a copy of the How Did You Work? review questionnaire.
  • Each small group of students will need a large sheet of paper for making a poster, some felt- tipped pens and copies of the Sample Responses to Discuss.
  • There is a projector resource to support whole-class discussions.

Time needed

20 minutes before the lesson, an 80-minute lesson (or split into two shorter lessons), and 15 minutes in a follow-up lesson (or for homework). Timings are approximate and will depend on the needs of the class.