Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

Squaresearch

Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?

Seven Squares

Stage: 3 and 4 Challenge Level:

Why do this problem?

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.

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."

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 produce an A3 display showing 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 sheets, 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

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?

Possible extension

Here are a couple of suitable follow-up problems:

Painted Cube

Steel Cables

Possible support

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.

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."

Mathematical reasoning & proof. Manipulating algebraic expressions/formulae. Circles. Creating expressions/formulae. Working systematically. Visualising. Real world. Powers & roots. Games. Sequences.

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