Seven squares - group-worthy task

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

Age

11 to 14

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

Three students were asked to draw this matchstick pattern:

This is how Phoebe drew it:

Can you describe what Phoebe did?

How many 'downs' and how many inverted C's are there?

How many matchsticks altogether?

Now picture what Phoebe would do if there had been $25$ squares.

How many 'downs' and how many inverted C's would there be?

How many matchsticks altogether?

If there had been $100$ squares? How many matchsticks altogether?

A million and one squares? How many matchsticks?

This is how Alice drew it:

Can you describe what Alice did?

How many 'alongs' and how many 'downs' are there?

How many matchsticks altogether?

Now picture what Alice would do if there had been $25$ squares.

How many 'alongs' and how many 'downs' would there be?

How many matchsticks altogether?

If there had been $100$ squares? How many matchsticks altogether?

A million and one squares? How many matchsticks?

This is how Luke drew it:

Can you describe what Luke did?

How many squares and how many inverted C's are there?

How many matchsticks altogether?

Now picture what Luke would do if there had been $25$ squares.

How many squares and how many inverted C's would there be?

How many matchsticks altogether?

If there had been $100$ squares? How many matchsticks altogether?

A million and one squares? How many matchsticks?

Now choose a couple of the sequences below.

Try to picture how to make the next, and the next, and the next...

Use this to help you find the number of squares, or lines, or perimeter, or dots needed for the $25^{th}$, $100^{th}$ and $n^{th}$ pattern.

Can you describe your reasoning?

Growing rectangles

This rectangle has height 2 and width 3.

Work out the perimeter, the number of dots, and the number of lines needed to draw a rectangle with:

height 2 and width 25
height 2 and width 100
height 2 and width n

L shapes

This L shape has height 4 and width 4.

Work out the perimeter, the number of squares, and the number of lines needed to draw an L shape with:

height 25 and width 25
height 100 and width 100
height n and width n

Two squares

This pattern with two squares has four black dots.

Work out the number of white dots and the number of lines needed to draw a pattern with:

25 black dots
100 black dots
n black dots

Square of Squares

This pattern has side length 5.

Work out the number of edge squares and the number of lines needed to draw the pattern with:

side length 25
side length 100
side length n

Dots and More Dots

This pattern has side length 3.

Work out the number of dots and the number of lines needed to draw the pattern with:

side length 25
side length 100
side length n

Rectangle of Dots

This pattern is made from two joined squares with side length 3.

Work out the number of lines and the number of dots needed to draw the pattern of joined squares with:

side length 25
side length 100
side length n

Student Solutions

Elliott from Wilson's School described what Phoebe, Alice and Luke did:

Phoebe began with one vertical match stick and then added seven inverted C shapes of match sticks to make seven squares. Overall, there were seven inverted Cs and one downs, making 22 all together.

If there were 25 squares, there would be 1 downs and 25 inverted Cs. There would be 76 match sticks in total.

If there were 100 squares, there would be 301 match sticks and if there were 1000001 squares, there would be 3000004 match sticks.

You can work this out by multiplying by 3 and adding 1.

Alice made the seven squares by placing the horizontal match sticks down first, then placing the vertical matches to join them up.

There were 14 'alongs' and 8 'downs', totalling 22 match sticks.

If there were 100 squares, there would be 301 matches and if there were one million and one squares, there would be three million and four match sticks.

You can find this by doubling the number of squares and adding the number of squares plus one.

Luke made seven squares by making the first square, and then adding inverted Cs. There was one square and six inverted Cs in total.

If there were 25 squares, Luke would make one square and then add 24 inverted Cs, using 76 match sticks altogether.

If there were 100 squares, there would be 301, and if there were one million and one squares, there would be three million and 4 match sticks.

You can work this out by subtracting one from the number of squares, multiplying by three and adding four.

Leonie and Pippa from the Mount School in York described what Phoebe and Alice did and then pictured what would happen when the number of squares increased:

Phoebe started with 1 vertical matchstick. She added 3 more matchsticks at a time to make a square.

Alice started with laying out all the top horizontal row of matchsticks, then she added the bottom horizontal row of matchsticks. She then added the vertical matchsticks.

For 7 squares, there are 8 downs and 14 alongs. There are 22 matchsticks in total.

For 25 squares, there are 26 downs and 50 alongs. There are 76 matchsticks in total.

For 100 squares, there are 301 matchsticks in total.

For 1 million and 1 squares, there are 3000004, matchsticks in total.

Sophie and Rachael, also from the Mount School in York, included a formula:

Where N is the number of squares.

Alice laid out all the top matchsticks then all the bottom ones, then all the middle ones.

Alongs = 2N

Downs = N+1

Altogether = 3N+1

Jamie, from Highfields School in Derbyshire, sent us this clear solution to the L shapes extension problem.

Hannah from Fullbrook had a go at a couple of the extension questions:

Growing Rectangles:

For a rectangle with a height of two and a width of 25, the number of dots required would be $(2+1) (25+1) = 3 \times26= 78$.

Therefore, 78 dots are required.

The number of lines required would be $76 + (2\times25) + 1 = 76 + 50 + 1=127$

Therefore, 127 lines are required. (The number 76 is known from the Seven Squares problem.)

For a rectangle with a height of two and a width of 100, the number of dots required would be $(2+1)\times(100+1) = 3 \times101 = 303$.

Therefore, 303 dots are required.

The number of lines required would be

$301 + (2\times100) + 1 = 301 + 200 + 1=502$.

Therefore, 502 lines are required.

For a rectangle with a height of 2 and a width of n, the number of dots required would be $3(n+1)$, and the number of lines required would be $5n + 2$ (because $(3n+1)+(2n+1)=5n +2$)

L shapes:

If it is of height 25, width 25.

Perimeter: 100

Number of squares: 49

Number of lines: 148

If it is of height 100, width 100

Perimeter: 400

Number of squares: 199

Number of lines: 598

If it is of height n , width n.

Perimeter: 4n

Number of squares: 2n-1

Number of lines: 6n -2

Laura from Ramapo submitted a clearly explained solution:

Pattern with a height of 2 and a width of 25:

Perimeter (add total number on each side) $= 2 + 2 + 25 + 25 = 54$

Number of dots: 26 dots on each line, total of 3 lines $26 \times 3 = 78$

Total number of lines: $54$(perimeter) $+ 25$ (middle line) $+ (2 \times24) =127$. $2 \times 24$ represents the vertical lines within the rectangle.

Pattern with a height of 2 and a width of 100:

Perimeter (add total number on each side) = $2 + 2 + 100 + 100 = 204$

Number of dots: 101 dots on each line, total of 3 lines $101 \times 3 = 303$

Total number of lines: $204$ (perimeter) $+ 100$ (middle line) $+ (2 \times 99) = 500$

$2 \times 99$ represents the vertical lines within the rectangle.

Elijah, who is home-educated, sent us a comprehensive answer to the entire problem. Bravo!

GROWING RECTANGLES

height $2$, width $n$

$P = 2(n+2)$

dots $= 3(n+1)$ [there is one more dot than line on each side]

lines $= 3n$ [the vertical lines] $+ 2(n+1)$ [the horizontal lines] $= 5n + 2$

height $m$, width $n$

$P = 2(m+n)$

dots $= (m+1)(n+1)$

lines $= (m+1)n + m(n+1) = 2mn + m + n$

L SHAPES

height $n$, width $n$

$P = 4n$

squares $= 2n-1$

lines $= 4n + 2n-2 = 6n-2$

height $m$, width $n$

$P = 2(m+n)$

squares $= m + n - 1$

lines $= 2m + 2n + m + n - 2 = 3m + 3n - 2$

TWO SQUARES

4 black

white = 27

lines = 48

25 black

white = 1224

lines = $25\times24\times4 = 2400$

100 black

white = 10000 + 10000 - 101 = 19899

lines = $100\times99\times4 = 39600$

$n$ black

white = $n^2 + n^2 - (n+1) = 2n^2 - n - 1 $

lines = $4n(n-1)$

SQUARE OF SQUARES

side length 5

edge squares = 16

lines = 48

side length 25

edge squares = 96

lines = $25\times4 + 24\times4 + 23\times4 = 288$

side length 100

edge squares = 396

lines = $100\times4 + 99\times4 + 98\times4 = 1188$

side length $n$

edge squares = $4n - 4 = 4(n-1)$

lines = $4n + 4(n-1) + 4(n-2) = 12n - 12 = 12(n-1)$

height $m$, width $n$

edge squares = $2m + 2n - 4 = 2(m+n-2)$

lines = $2m + 2n + 2(m-1) + 2(n-1) + 2(m-2) + 2(n-2) = 6m + 6n - 12 = 6(m+n-2)$

DOTS AND MORE DOTS

side length 3

dots = 25

lines = 24

side length 25

dots = $625 + 676 = 1301$

lines = $25\times26\times2 = 1300$

side length 100

dots = $10000 + 10201 = 20201$

lines = $100\times101\times2 = 20200$

side length $n$

dots = $n^2 + (n+1)^2 = n^2 + n^2 + n + n + 1 = 2n^2 + 2n + 1$

lines = $2n(n+1) = 2n^2 + 2n$

height $m$, width $n$

dots $= mn + (m+1)(n+1) = mn + m(n+1) + (n+1) = mn + mn + m + n + 1 = 2mn + m + n + 1$

lines $= 2mn + m + n$

RECTANGLE OF DOTS

For 2 squares:

side length 3

lines = $7\times3 = 21$

dots = $4\times7 = 28$

side length 25

lines = $7\times25 = 175 $

dots = $26\times51 = 1326$

side length 100

lines = 700

dots = $101\times201 = 20301$

side length $n$

lines $= 7n$

dots $= (n+1)(2n+1)$

Generalising to $p$ squares of side length $n$

lines $= (3p+1)n$ [got $3p+1$ from Tom's matchsticks]

dots $= n + 1 + pn(n+1)$ [start with n+1 dots, then for every square added, need to add $n(n+1)$ dots] $= (n+1)(pn+1)$

Teachers' Resources

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.

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.

Possible approach

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.

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.

Key questions

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?

Possible extension

Here are three suitable follow-up problems:

Coordinate Patterns

Painted Cube

Christmas Chocolates

Possible support

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."

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For younger learners

Seven squares - group-worthy task

Problem

L shapes

Rectangle of Dots

Getting Started

Student Solutions

Teachers' Resources

Why do this problem?

Possible approach

Key questions

Possible extension

Possible support