Challenge Level

This problem looks at generic patterns, and challenges students to describe them 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.

This problem lends itself to collaborative working, both for students who are inexperienced at working in a group and students who are used to working in this way.

Many NRICH tasks have been designed with group work in mind. Here we have gathered together a collection of short articles that outline the merits of collaborative work, together with examples of teachers' classroom practice.

This is an ideal problem for students to tackle in groups of four. Allocating these clear roles (Word, pdf) can help the group to work in a purposeful way - success on this task should be measured by how effectively members of the group work together as well as by the solutions they reach.

Introduce the four group roles to the class. It may be appropriate, if this is the first time the class have worked in this way, to allocate particular roles to particular students. If the class work in roles over a series of lessons, it is desirable to make sure everyone experiences each role over time.

For suggestions of team-building maths tasks for use with classes unfamiliar with group work, take a look at this article and the accompanying resources.

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.

Once everyone has sketched the image - "Can anyone describe the order in which they drew the lines?" "Without counting individual matches can you say how many matchsticks there are in your drawing?"

Collect at least three different methods, selecting students who you know have something new to offer. For each method, draw it on the board (perhaps using colours to emphasise the order in which it was drawn) and pose the following questions:

"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 animations provided in the problem showing three different methods.

Select some of these tasks (Word, pdf) and hand them out, along with this instruction sheet (Word, pdf). You might want all groups to work on the same task(s), or you may
want different groups to attempt different tasks. There are six different tasks, with the easier ones first.

Explain that by the end of the sessions they will be expected to report back to the rest of the class, showing how they saw the patterns growing, and how this helped them to work out the hundredth pattern and how they arrived at an algebraic expression. Exploring the full potential of these tasks is likely to take more than one lesson, allowing time in each lesson for students to feed back ideas
and share their thoughts and questions.

While groups are working, label each table with a number or letter on a post-it note, and divide the board up with the groups as headings. Listen in on what groups are saying, and use the board to jot down comments and feedback to the students about the way they are working together.

You may choose to focus on the way the students are co-operating:

Group A - Good to see you sharing different ways of seeing how the pattern grows

Group B - Facilitator - is everyone in your group contributing?

Group C - I like the way you are keeping a record of people's ideas and results.

Group B - Facilitator - is everyone in your group contributing?

Group C - I like the way you are keeping a record of people's ideas and results.

Alternatively, your focus for feedback might be mathematical:

Group A - I like the way you are trying to use letters to represent the pattern you have described in words.

Group B - Have you tried checking that your rule works with some simple examples?

Make sure that while groups are working they are reminded of the need to be ready to present their findings at the end, and that all are aware of how long they have left.

We assume that each group will record their diagrams, reasoning and generalisations on a large flipchart sheet in preparation for reporting back. There are many ways that groups can report back. Here are just a few suggestions:

- Every group is given a couple of minutes to report back to the whole class. Students can seek clarification and ask questions. After each presentation, students are invited to offer positive feedback. Finally, students can suggest how the group could have improved their work on the task.
- Everyone's posters are put on display at the front of the room, but only a couple of groups are selected to report back to the whole class. Feedback and suggestions can be given in the same way as above. Additionally, students from the groups which don't present can be invited to share at the end anything they did differently.
- Two people from each group move to join an adjacent group. The two "hosts" explain their findings to the two "visitors". The "visitors" act as critical friends, requiring clear mathematical explanations and justifications. The "visitors" then comment on anything they did differently in their own group.

If your focus is effective group work, this list of skills may be helpful (Word, pdf). Ask learners to identify which skills they demonstrated, and which skills they need to develop further.

If your focus is mathematical, these prompts might be useful:

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

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

How does your formula relate to the structure of the original problem?

Here are three suitable follow-up problems:

By working in groups with clearly assigned roles we are encouraging students to take responsibility for ensuring that everyone understands before the group moves on.

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."

"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."