Fewest squares

A thinking mathematically targeted teaching opportunity focussed on reasoning to create and explore squares and square numbers on a grid

Syllabus

Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2023

• MAO-WM-01
• MA2-2DS-03

• MAO-WM-01
• MA3-GM-02

You will need:

Watch

Watch Fewest square part 1 video (1:44).

Explore ways to fill the grid with the fewest possible squares.

Transcript of Fewest squares part 1

[White text on a navy-blue background reads ‘Fewest squares From youcubed’. Small white text at the bottom reads ‘NSW Mathematics Strategy Professional Learning team (NSWMS PL team). In the bottom right corner, the NSW Government red ‘waratah’ logo.]

Speaker

Hello, and welcome back, mathematicians. Today, I've got an exciting challenge for you. One that comes from youcubed, and it's called fewest squares.

[A blue text on white header reads ‘You will need…’ Three bullet points below (as read by speaker). Below, in a still colour image, a wooden ruler alongside a sheet of blank white paper and a collection of coloured marker pens. On the right, a grid square has a blue text header that reads ‘Fewest Squares Grid’.]

Speaker

For this challenge today, you're going to need the fewest squares grid paper template or a blank piece of paper and a ruler to draw your own grid on, and a few different coloured markers or pencils.

[White text on a blue background reads ‘Let’s play!’]

[The ‘Fewest Squares Grid’ from earlier. Various areas of the grid are highlighted in larger pink, purple and blue squares.]

Speaker

So, my challenge today is to work out what's the fewest number of squares I can fit inside of this 11 by 13 grid without any overlapping on top or underneath each other, without any extending outside of the grid, and without leaving any blank spaces inside of the grid as well.

[Various areas of the grid are highlighted in different coloured squares as they are mentioned.]

Speaker

When I first explored this task, I didn't have a set strategy in mind. I decided to have a play and to see how many square regions I could fit into the grid. Here's one way that I did it. I started with a 4 by 4 square and placed another one underneath. Then I drew a 3 by 3 square and repeated this thinking on the other side. When looking at the middle of my grid, I noticed that I could draw a 5 by 5 square region and then use my understanding of squares to finish it off. So, that was one way and here's another.

[A second grid square filled with larger coloured squares of differing sizes.]

[The two grid squares side by side. The left square is labelled ‘Attempt 1’ and the right square ‘Attempt 2’.]

Speaker

There are lots of different ways to fill the grid with squares. So, mathematicians, now it's over to you to have a play and to fill the grid with squares in two different ways.

[White text on a blue background reads ‘Over to you mathematicians!’

[The NSW Government waratah logo turns briefly in the middle of various circles coloured blue, red, white and black. A copyright symbol and small blue text below it reads ‘State of New South Wales (Department of Education), 2021.’]

[End of transcript]

Discuss

• Draw an 11 x 13 grid on paper or use the 11 x 13 grid provided.

• What is the fewest number of squares you can draw inside your 11 x 13 grid?

• Make sure you don't have any overlapping squares or leave any blank spaces.

Watch

Watch Fewest square part 2 video (4:10).

Investigate possible solutions to fill the grid with the fewest squares.

Transcript of Fewest squares part 2

[Two grid squares side by side. The left square is labelled ‘Attempt 1’ and the right square ‘Attempt 2’. Coloured squares of various sizes are placed inside the grid square so that all the small squares are covered.]

Speaker

Welcome back, mathematicians. As you were drawing squares, I was wondering how I could be more strategic with my thinking and wondered how I could end up with fewer squares compared to my first two attempts, if I was to use what I know about square numbers, for instance.

[A large blank grid square.]

Speaker

As a mathematician, I know that a square number is the result of multiplying a number by itself. So, I have to make sure that when I draw out my square numbers, I have the same amount of rows as I do columns, so that the width and the length are the same or equivalent in value.

[Blue squares increase in size from left to right. Pink squares outline each different stage to highlight the increase in size of the next blue square.]

[An array of circles has 4 pink circles in 3 rows. The array is rotated clockwise.]

Speaker

This reminds me of an array. When looking at this 3 by 4 array, for example, the number of rows and the number of columns change when I rotate my array. This also changes the way that the shape of the array is represented, as when I rotate the three by four array clockwise, I notice the orientation of the rectangle changes. But looking at a square array, I can see that it stays the same.

[A second array of circles has 4 pink circles in 4 rows.]

Speaker

So, when I think about square numbers, they should look like this or this.

[A 4 by 4 array of pink circles alongside a 5 by 5 array.]

[White text on blue reads ‘Let’s investigate!’]

Speaker

So, let's investigate.

[A blank grid square. Coloured squares of various sizes are placed onto it as mentioned.]

Speaker

To start with, I know that ten squared is equivalent in value to 100, but I'm worried that if I drew a 10 by 10 square here, then that would leave me with 3, 3 by 3 squares, one 2 by 2 square, and 12 individual, 1 by 1 squares. Using this strategy means that I'll have 17 square regions altogether, which is even more than my first attempt. This makes me think that my square number needs to be larger or possibly smaller than ten squared. And this got me thinking about what square numbers actually nest inside of 11 x 13 and what combinations of square numbers can I use to make a total of 143? And how can I represent that?

[A blank grid square. Coloured squares of various sizes are placed onto it as mentioned.]

Speaker

I'm thinking that a strategy that I could use is to start with a smaller square number and use it repeatedly to cover the grid. So, I'm thinking that I could use my knowledge of 4 squared and see how many I can fit inside of the grid. Using this strategy, I can fit 6 4 squared square regions, which leaves me with 4 3 squared square regions, and yep, you guessed it, 11 individual one by one squares. Doing it this way, however, means that I'll end up with 21 square regions, which I know is far too many.

This made me think that using the same squared number repeatedly is not the best strategy, and that I actually have more squares than when I started with my 10 by 10 grid. So, I think my 10 by 10 strategy was more on the right track.

[A blank grid square. Coloured squares of various sizes are placed onto it as mentioned.]

Speaker

This time I'm going to start by creating an 11 by 11 square here. Now, that just leaves me with this rectangle. But I can visualise and I can see that I actually have a 2 by 2 square or 2 squared square here. I can see that too. I can see that I have a few of them that are the same going down. I'm going to use colour to help me mark those out, so as to be more clear and precise in my thinking.

Now, I'm just left with this rectangle down here, and I know that a square is a special kind of rectangle. And as a mathematician, I can actually see that there's 2 squares inside of that rectangle. So, I'm going to keep using different colours to help mark those out.

OK, and that's it. Using colour intentionally, I can see that I have 8 squares. But this then got me thinking, and I'm really curious to find out, is this as few as possible squares? I wonder if you can find less, and I wonder what the largest square number would be that you would start with. Over to you, mathematicians, to find the fewest squares.

[The NSW Government waratah logo turns briefly in the middle of various circles coloured blue, red, white and black. A copyright symbol and small blue text below it reads ‘State of New South Wales (Department of Education), 2021.’]

[End of transcript]

Discuss

Test out some different starting places, or start with different square numbers to help you find the fewest squares within the 11x13 grid.

Watch

Watch Fewest square part 3 video (02:00).

Investigate similarities and differences between possible solutions.

Transcript of Fewest squares part 3

[3 colourful grids sit in a row on the top half of the screen. Another two grids in a row are on the bottom half. Each grid is made up of colourful square tiles of different sizes. In the lower right-hand corner is the red waratah of the NSW Government logo.

Small font text on the top left corner reads: NSW Department of Education.]

Speaker

So, mathematicians, here are a few of our attempts from today.

[The first grid from the left of both the top and bottom rows are outlined with a blue line].

Speaker

Let's explore these two and discuss what's the same and what's different about them.

[A title on a white background reads: What’s the same? What’s different?

Below the title are the 2 grids from the previous slide. The grid on the left is made up of 3 squares of different colours across all sides of the grid, with some squares containing a few smaller squares within. The grid on the right is made up of a large blue square on the left, and a column of smaller squares of a variety of colours next to it.

Underneath the grids are 2 blue textboxes. One is titled: Similarities. Bullet points below read:

· Both grids show square regions

· Both grids are equivalent to 143 square centimetres or 11 thirteens

· Both grids show colour to show the number of square regions

· Some square regions are repeated.

The other textbox is titled: Differences. Bullet points below read:

• The squares are different sizes
• There are less squares in the one on the right than the one on the left
• The grid on the left has double the amount of square regions as the one on the right.]

Speaker

On the left is the image of my first attempt, and on the right is the last investigation I completed with you before.

Some similarities are that both of the grids show square regions, both grids are equivalent to 143 square centimetres, or eleven thirteens. Both grids use colour to show the number of square regions, and some square regions are repeated.

For example, on the grid on the left, I noticed that there are five three by three square regions. And on the one on the right, I notice that there are five two by two square regions. Some differences are the squares are different sizes, there are less squares in the one on the right than the one on the left, and the grid on the left has double the amount of square regions as the one on the right.

[Text over a navy background: What’s (some of) the mathematics?]

Speaker

What's some of the mathematics?

[A title on a white background reads: What’s some of the mathematics? A text below reads: We can change or adjust the sizes of our squares in order to find the fewest squares possible. A bullet point below says:

• Changing the size of the square regions can change how many squares we can fit in our rectangle grid.

Underneath this is the same 2 grids from the previous slide.]

Speaker

Throughout the investigation, we saw that we can change or adjust the sizes of our squares in order to find the fewest squares possible. This was quite clear when we started with the bigger square region and ended up with less. We saw that changing the size of the square regions can also change how many squares we can fit in our rectangular grid.

[A title on a white background reads: What’s some of the mathematics? A text below reads: To add onto this idea, we can use this task to see that mathematicians can even look at grids with flexibility. So, even in an 11 by 13 grid, which we could rename as eleven thirteens, we can see smaller grids or smaller square numbers hiding inside. Look, inside of eleven thirteens, I can see one big square of eleven elevens and then four squares showing two twos and two squares of one ones!

Underneath the text are 3 grids in a row. The grids on the left and right are the two previous grids. The grid in the middle is made up of one large blue square, a column of smaller squares on the right, and a row of even smaller squares on the bottom.]

Speaker

To add onto this idea, we can use this task to see that mathematicians can even look at grids with flexibility. So, even in an 11 by 13 grid, which we could rename as eleven thirteens, we can see smaller grids or smaller square numbers hiding inside.

[The grid on the right is outlined with a blue line.]

Speaker

Look, inside of eleven thirteens, I can see one big square of eleven elevens and then four squares showing two twos and two squares of one ones.

[Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]

[End of transcript]

Instruction

Watch the video and compare your 11 x 13 grid to the ones created in the video.

• What was the fewest number of squares you could fit into the 11 x 13 grid?

• How were the grids you made similar or different to the ones in the videos?

Discuss/Reflect

• Is 8 the fewest number of squares we can use to fill the 11 x 13 grid?

• What would the dimensions (size) of a grid need to be so we could fill it with exactly 7 squares?