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### Number and algebra

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# Seven Squares

### Why do this problem?

### Possible approach

*The Article Go Forth and Generalise may be of interest.*

Have the "seven squares" image preprepared on the board so that students cannot see how you drew it. "I have drawn seven matchstick squares on the board, and I would like you to make a rough copy of it - no need to use a ruler."
### Key questions

### Possible support

### Possible extension

*"Some students succeeded in building the patterns and working numerically, but were not yet ready to work algebraically, while other students progressed to finding, and even simplifying, formulae for the patterns. All students experienced success and there was appropriate challenge in this problem for everyone."*## Related Collections

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Age 11 to 14

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

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.

Have the "seven squares" image preprepared on the board so that students cannot see how you drew it. "I have drawn seven matchstick squares on the board, and I would like you to make a rough copy of it - no need to use a ruler."

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?"

"How would 25 squares be drawn using this method?"

"How many matchsticks would be needed altogether?"

"What if there were 100 squares?"

"Or a million squares?"

"Or $x$ squares?"

The answers to these questions could be recorded on the board, so that the results and the algebraic expressions emerging from each method can be compared at the end.

For example, for Phoebe's method from the problem you could initially write $$1+ 7 \times 3$$ leading to $$1 + 25 \times 3$$ $$1 + 100 \times 3$$ and so on, eventually finishing with $$1 + 3x$$

*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.

Can you see a pattern in the image? How might you draw it?

Can you tell how someone drew the pattern from the way they write the calculation?

How does your formula relate to the structure of the pattern?

Students could spend time exploring the first three patterns before moving on to the harder cases.

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

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

A teacher's comments after using this activity:

"It gave rise to much discussion about how to describe the patterns. It led naturally to building algebraic expressions and seeing them as easily understandable ways to record the patterns. It provided motivation for checking that the different algebraic expressions (used to describe the different ways in which a pattern can be built) are in fact equivalent."

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?