The single equation in the prompt is intriguing because it can lead to conjectures that turn out to be false. However, the prompt's ambiguity means it is suitable for students who are experienced in inquiry and show high levels of perseverance. For classes who are new to inquiry and require more structure, a prompt with two equations is more appropriate.

If the teacher uses the prompt with one equation, then students will need to find at least one other example before they can identify a pattern and generalise. Perhaps the most obvious generalisation from the prompt is (10a + b)2 = (10a)(10b) + b2. However, students quickly dismiss that with a counter-example - such as, 752 ≠ 70 x 50 + 25. The teacher guides the class if students get stuck by sharing another example or by encouraging an approach based on mathematical structure (see the section on 'Exploration' below).

Question, notice, and wonder

In the initial phase of the inquiry, students' responses have included:

• 4 and 5 are on both sides and 25 is 52.

• The equation is true because 45 x 45 = (40 x 40) + (40 x 5) + (5 x 40) + (5 x 5) = (40 x 40) + (10 x 40) + (5 x 5) = 50 x 40 + 25 = 40 x 50 + 25.

• Does the rule always work? Would 352 = 30 x 50 + 25?

• Do squares of other two-digit numbers follow the pattern?

• Does it work for the square of three-digit numbers?

• What would 453  equal?

April 2025