Seven Squares

Why do this problem?

This problem challenges students to describe patterns clearly - verbally, numerically and algebraically. It does not assume prior knowledge of algebra and could be a good way to introduce, practise or assess algebraic fluency.

Similar-looking questions are often asked, expecting an approach that uses number sequences for finding a formulae for the $n^{th}$ term. This problem deliberately bypasses all that, instead focusing on the structure of the pattern so that the algebraic expressions emerge naturally from that structure.

Possible approach

While the students are sketching, look out for students creating the image in different ways, such as Phoebe's, Alice's and Luke's methods in the problem.

Select at least three students who have used different methods, and invite them to draw the image on the board (perhaps using colours to emphasise the order in which it was drawn). 

"Without counting individual matches can you say how many matchsticks there are in the drawing?"

Alternatively, you could show the class the videos provided in the problem showing three different methods.

Next, hand out this worksheet. There are six different patterns with the simpler ones at the start. Invite students to work in pairs:

"With your partner, choose two or three of the six patterns and have a go at the questions. Make sure you can explain clearly how you worked out your answers, focusing on the order in which you would draw the diagram, like we did for the Seven Squares problem."

While students are working, circulate and listen to the conversations, identifying students who have really elegant ways of seeing the general case in the initial picture.

"I'm going to give you ten minutes to prepare a poster presenting one of the problems you worked on and explaining how you arrived at your solution."

Students could choose which problem to work on, and you could guide particular students towards problems where you have noticed them reasoning clearly.

Once they have produced their poster, there are a number of different ways that sharing and feedback could be organised:

• Half the class stand by their posters and the other half of the class visit them, read, ask for clarification on anything that is unclear, and suggest improvements as 'critical friends'. After five minutes, swap over.
• All the posters are laid out. Students visit each other's posters and write any comments, questions or feedback on post-it notes.
• Selected students could present the content of their poster on the board, with the rest of the class feeding back.

Key questions

Possible support

Encourage students to draw a few examples of each pattern and notice how their drawings develop.

Possible extension

Here are a couple of suitable follow-up problems that use the structure of a situation to lead to algebraic generalisations: