# Seven Squares - Group-worthy Task

## 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**

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

### L shapes

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

**Two squares**

- 25 black dots
- 100 black dots
- n black dots

**Square of Squares**

- side length 25
- side length 100
- side length n

**Dots and More Dots**

- side length 25
- side length 100
- side length n

### Rectangle of Dots

- side length 25
- side length 100
- side length n

## Getting Started

How did Phoebe group the matchsticks that she drew?

How did the others, Alice and Luke, group their matchsticks?

In the follow-up activities draw them out for yourself and notice how YOUR drawings develop. Always begin with simple cases and try to PREDICT what will happen.

Look for patterns.

How can you describe the lines? Horizontally? Vertically?

Try to understand why the patterns develop in the ways that they do.

## 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?

### Possible approach

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 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.

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 mathematical, these prompts might be useful: