# Multiplication toss

Stage 1 to 3 – A thinking mathematically context for practise resource focussed on using and developing multiplicative thinking

This task is from Dianne Siemon.

## 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

## Outcomes

- MAO-WM-01
- MA1-FG-01
- MA1-2DS-02

- MAO-WM-01
- MA2-MR-01
- MA2-2DS-02

- MAO-WM-01
- MA3-MR-01
- MA3-2DS-02

## Collect resources

You will need:

different coloured pencils or markers

two spinners

paper clip for spinner.

## Watch

Watch the Multiplication toss part 1 video (5:38).

### Speaker

OK, everybody, welcome back!

We're here today to have a look at the game multiplication toss, which some people also call how close to 100.

To play today I'm using a spinner, and I just made it by printing out a decagon and drawing lines across the opposite angles and labelling it from 0 to 9.

[Screen shows 10 by 10 grid paper, one 0 to 9 spinner made from a decagon, and coloured markers. Presenter points to decagon and traces the lines which make the segments in the spinner.]

And I'm going to use my paper clip that I found in the drawer, and a pen and I can flick it...

[Screen shows presenter placing a large paper clip in the middle of the spinner, and placing the point of a pen at one end of the paper clip to keep it in place in the centre of the spinner. Presenter flicks the paper clip with her finger and spins it.]

And that will give me the numbers that I'm going to use.

And in fact, I could start with 5, and I now also have a 0 which is no good for me because what I know is that 5 times 0 or 0 fives is 0.

[The paper clip spins and lands on the number 5. Presenter spins again and lands on 0.]

So, for my first recording of my game, I can't block out anything because 5 zeros is the same as 5 times 0, which is equivalent to 0.

[Presenter points to grid, then writes 5 zeros equals 5 times 0 equals 0 next to the grid.]

So, fingers crossed my next go is more lucky!

Ah 0 and 2, so this time I could say 0 twos is equivalent to 0 times 2, which is also 0.

[Presenter spins and lands on 0, then spins again and lands on 2. Presenter writes 0 twos equals 0 times 2 equals 0.]

Okay, third time lucky!

Come on, spinner!

Excellent, so this time I got an 8 and ah...I think that's a 5 so I can actually now get to colour in my board here and because I got an 8 and a 5, I can choose to make 8 fives or 5 eights.

[Presenter spins and lands on 8, then spins again and lands on 5. Presenter points to grid.]

So, I'm just going to go with 5 eights because I like them better.

So, I need 8 in my rows, so 1 2 3 4 5 6 7 8 and I need 5 down here so that's 2 3 4 5.

[Presenter counts 8 squares across the top row of the grid, then 5 squares down the first column of the grid.]

So, I get to draw a border all around this area of my game board.

And I'm going to record this as 5 eights.

[Presenter uses a green marker to draw a border around the area, outlining an area of 5 squares down and 8 squares across, which are rows 1 to 5 and columns 1 to 8 of the grid. Presenter writes 5 eights within the green outlined area of the grid.]

And I'm also going to record it over here.

So, 5 eights is equivalent in value to 5 times 8, which is equivalent to 40.

[Presenter continues to write next to the grid, writing 5 eights equals 5 times 8 equals 40.]

Now if I wasn't sure I could use the grid here to help me work out how many squares are encased in my green section.

And because mathematicians like to code and keep a record of their ideas, I might also put a green marker here to say that corresponds to this section on my game board.

[Presenter points to grid indicating the 40 squares outlined in green marker. Presenter puts a green dot next to writing about 5 eights and points to the area of 40 squares, showing how this writing corresponds to the section outlined in green on the game board.]

Alright, let's see. I've had a disastrous start, but I could have a successful finish. I'm going to call that a 3.

And a 0. I got too excited so I could say 0 threes or 3 zeros, but I know they're the same as 0, so 3 zeros is equivalent to 3 times zero, which is 0. OK.

[Presenter spins and lands on 3, then spins again and lands on 0. Presenter writes 3 zeros equals 3 times 0 equals 0.]

Come on, spinner!

Four... Fives, so I could do 4 fives so that would be across here like this. Or I could do 5 fours which would...ok, go like this.

[Presenter spins and lands on 4, then spins again and lands on 5. Presenter points to 4 rows and 5 columns, which are rows 6 to 9 and columns 1 to 5 of the grid. Presenter then points to 5 rows and 4 columns. These are rows 6 to 10 and columns 1 to 4 of the grid.]

And I might actually do that. I'm going to use a different colour mark at this time so I know this is 4 because, actually I can subitise that many.

And that takes me all the way down to here.

[Presenter uses a blue marker to draw a line around rows 6 to 10 and columns 1 to 4 of the grid, which is an area of 4 squares across and 5 squares down.]

I realised I didn't actually have to count those 'cause I know my board is 10 by 10. Five fours.

[Presenter writes 5 fours within the blue outlined area of the grid.]

I am I going to record that, over here. Five fours is equivalent in value to 5 times 4, which is 20.

And I actually know that because that's the same as saying 10 twos and you just rename that as 20.

Like this, you could say that's 10 twos which is the same as 2 tens.

[Presenter marks a blue dot next to the grid and writes 5 fours equals 5 times 4 equals 20 equals 10 twos equals 2 tens.]

We could just keep going, but we won't.

[Presenter spins and lands on 2, then spins again and lands on 4. Presenter uses a pink marker to draw a line around rows 1 to 4 and columns 9 to 10, which is an area of 4 squares down and 2 squares across. Presenter writes 4 twos within the pink outlined area of the grid. Presenter marks a pink dot next to the grid and writes 4 twos equals 4 times 2 equals 8.

Next, presenter spins and lands on 0, then spins again and lands on 6. Presenter writes next to the grid 0 sixes equals 0 times 6 equals 0.

Next, presenter spins and lands on 6, then spins again and lands on 4. Presenter uses a yellow marker to draw a line around columns 5 to 10 and rows 6 to 9, which is an area of 6 squares across and 4 squares down. Presenter writes 6 fours within the yellow outlined area of the grid. Presenter marks a yellow dot next to the grid and writes 6 fours equals 6 times 4 equals 24.]

I could write that 0 sevens or 7 zeros. Zero times 7 which equals 0.

[Presenter spins and lands on 0, then spins again and lands on 7. Presenter writes next to the grid 0 sevens equals 0 times 7 equals 0.]

Let's try.... come on 1! Six...and a 9... now I definitely know I can't go here because I've got 1, 2, 3, 4, 5, 6...one row of 6 left that I could use or one row of 2.

[Presenter spins and lands on 6, then spins again and lands on 9. Presenter points to the bottom row on the grid where 6 squares in columns 5 to 10 are available. Presenter then points to the only other squares available on the grid which are 2 squares in row 5, columns 9 and 10.]

So, in this case I have to record 6 nines ...but I couldn't go.

[Presenter writes next to the grid 6 nines, then an arrow, then the words couldn’t go.]

So, they were my 10 goes and I have 8 squares remaining and I covered 92 centimetres squared. How did you go in your game?

[End of transcript]

## Instructions

Players take turns to spin the spinners. If a 3 and 6 are spun, players can enclose either a block out of 3 rows of 6 (3 sixes) or 6 rows of 3 (6 threes).

The game continues with no overlapping areas.

The winner is the player with the largest area blocked out after 10 spins.

Eventually the space on the grid paper gets really small.

Then, you have to think:

What if my 3 sixes won’t fit as 3 sixes or as 6 threes?

Players can partition to help them! So, for example, I can rename 3 sixes as 2 sixes and 1 six (if that helps me fit the block into my game board).

## Watch

Watch the Multiplication toss part 2 video (2:22). This will show you a strategy to help you when your board starts to get full.

### Speaker

Ok. So, I have been playing multiplication toss again and I have found myself in a pickle.

[Screen shows 10 by 10 grid paper, two 0 to 9 spinners made from decagons, and a large paper clip on each spinner.

The grid has areas outlined in colour. There is a purple line around rows 1 to 3 and columns 1 to 6, which is an area of 3 squares down and 6 squares across. The text 3 sixes is written in this area. There is a red line around rows 1 to 2 and columns 7 to 10, which is an area of 2 squares down and 4 squares across. The text 2 fours is written in this area. There is a pink line around rows 4 to 10 and columns 1 to 7, which is an area of 7 squares across and 6 squares down. The text 7 sixes is written in this area. There is a green line around rows 3 and 4 and columns 7 to 9, which is an area of 2 squares down and 3 squares across. The text 2 threes is written in this area. There is an orange line around rows 5 to 7 and column 7, which is an area of 3 squares down and 1 square across. The text 3 ones is written in this area. There is a dark blue line around 1 square in row 10, column 10. The text 1 one is written in this area.

To the right of the grid is a list written in different-coloured markers. The list is:

3 sixes equals 18, in purple marker.

2 fours equals 8, in red marker.

42 equals 7 sixes, in pink marker.

6 equals 2 threes, in green marker.

3 ones equals 3 times 1 equals 3, in orange marker.

1 one equals 1, in dark blue marker.

The two 0 to 9 spinners are below the grid. The paper clip on one spinner has landed on the number 6, and the paper clip on the other spinner has landed on the number 3.]

I have a bit of a mathematical conundrum because this is my game board that I've been playing with and here are the areas I blocked out.

[Presenter points to the game board, then to the written list which shows the areas outlined, or blocked out, on the game board.]

And I've spun on my 2 spinners, this time, a 6 and a 3. I know that I have more than 18 squares over here, but I don't have an array of 6 threes or 3 sixes, exactly, that I could use.

[Presenter points to spinner with the paper clip on the number 6, then the spinner with the paper clip on the number 3. Presenter then points to the area on the grid which has not been blocked out. This area is 20 squares, in 6 rows. In this area, rows 1 to 3 have 3 squares, rows 4 and 5 have 4 squares, and row 6 has 3 squares.]

So, if I had...I almost have it because I have some threes across here, but 1, 2, 3, 4, 5, 6...would mean that this square here is in the way.

[Presenter uses a pencil to count 6 squares down in the area that has not been blocked out. The presenter includes these squares when tracing around an area of 18 squares. However, the square in row 10, column 10 which says 1 one is included in this area.]

And if I did it down here, I have this section up here that's in my way.

[Presenter points to the area that has not been blocked out below the area outlined in orange which is an area of 3 squares down and 1 square across.]

So, what I need to do now is to try to partition or break apart my 6 threes.

So, what I could think about is, I could think about using 3 of my threes here, and the other 3 of my threes down here, and that would fit! So, let's draw that in.

[Presenter outlines 9 squares in pencil. The 9 squares are in rows 5 to 7 and columns 8 to 10, which is an area of 3 squares down and 3 squares across. The presenter then outlines another 9 squares in pencil. The 9 squares are in rows 8 to 10 and columns 7 to 9, which is an area of 3 squares down and 3 squares across. Presenter then outlines the 2 areas of 9 squares in light blue marker.]

And so I have 3 threes. And 3 threes.

[Presenter writes 3 threes in the 2 areas outlined in light blue marker.]

And I know that 6 threes is 18, and I know that actually 'cause I had this turn up here and so 3 sixes is 18, which also means that 6 threes is 18.

[Presenter points to list where 3 sixes equals 18 is written in purple marker and to the area on the grid where 3 squares down and 6 squares across is outlined in purple. Presenter adds to list so it reads 3 sixes equals 18 equals 6 threes.]

And so here now I have 9 and 9 and when you join 9 and 9 together, that still makes 18.

[Presenter points to 2 areas outlined in light blue marker which are marked 3 threes. Each area is 9 squares.]

So, the area is equivalent in value. I've just partitioned it slightly, so I'm going to record it by saying something like 6 threes is equivalent to 3 threes combined with 3 threes.

And we could also write that as 3 times 3 plus 3 times 3, which is equivalent in value to 18. I just partitioned it, so it looks a little bit different.

[Presenter adds to the end of the list in light blue marker:

6 threes equals 3 threes plus 3 threes

equals 3 times 3 plus 3 times 3

equals 18.

Presenter then points again to the 2 areas of 3 threes on the grid.]

I wonder if you could use this strategy to help you out with some of your games.

[End of transcript]

## Watch

Feeling skeptical about Michelle's thinking? Watch the Multiplication toss part 3 video (1:51) to see how she proves 6 threes = 3 threes + 3 threes = 18. It's a strategy you can then use to prove your thinking too!

### Speaker

So, we were thinking further about this idea down here that you could partition an array into a different array, and still be able to cover the same area.

[Screen shows 10 by 10 grid paper, two 0 to 9 spinners made from decagons, and a large paper clip on each spinner.

The grid has areas outlined in colour. There is a purple line around rows 1 to 3 and columns 1 to 6, which is an area of 3 squares down and 6 squares across. The text 3 sixes is written in this area. There is a red line around rows 1 to 2 and columns 7 to 10, which is an area of 2 squares down and 4 squares across. The text 2 fours is written in this area. There is a pink line around rows 4 to 10 and columns 1 to 7, which is an area of 7 squares across and 6 squares down. The text 7 sixes is written in this area. There is a green line around rows 3 and 4 and columns 7 to 9, which is an area of 2 squares down and 3 squares across. The text 2 threes is written in this area. There is an orange line around rows 5 to 7 and column 7, which is an area of 3 squares down and 1 square across. The text 3 ones is written in this area. There is a light blue line around rows 5 to 7 and columns 8 to 10, which is an area of 3 squares down and 3 squares across. The text 3 threes is written in this area. There is also a light blue line around rows 8 to 10 and columns 7 to 9, which is an area of 3 squares down and 3 squares across. The text 3 threes is written in this area. There is a dark blue line around 1 square in row 10, column 10. The text 1 one is written in this area.

To the right of the grid is a list written in different-coloured markers. The list is:

3 sixes equals 18 equals 6 threes, in purple and light blue marker.

2 fours equals 8, in red marker.

42 equals 7 sixes, in pink marker.

6 equals 2 threes, in green marker.

3 ones equals 3 times 1 equals 3, in orange marker.

1 one equals 1, in dark blue marker.

6 threes equals 3 threes plus 3 threes equals 3 times 3 plus 3 times 3 equals 18, in light blue marker.

Presenter points to 2 areas outlined in light blue marker which are marked 3 threes, to indicate each area of 9 squares.]

So, we thought we'd use some evidence to show you how this works.

So, I just made a copy of my game board.

[Presenter holds a second game board. This game board is on blue paper and has the same areas of squares outlined as on the first game board. The areas of squares on the blue game board have the same labels and are outlined in the same colours as on the first game board.]

You can see that they're exactly the same, except that we now have run out of white paper, so we're using blue, and so if I cut out this area, which I'm saying is the same... that 3 sixes is the same as 6 threes, which is the same as 3 threes combined with 3 threes more.

It makes sense why you might go: “Oh my gosh! What are you talking about?”

So, there's my 3 sixes.

[Presenter then cuts out the area of squares labelled 3 sixes on the blue game board. This is an area of 18 squares. Presenter places this next to the first game board.]

And then here is 1 lot of 3 threes.

[Presenter cuts out an area of squares labelled 3 threes from the blue game board. This is an area of 9 squares. The presenter then cuts out the second area of squares labelled 3 threes from the blue game board.]

And so, here's my 3 sixes from my game board and here's one of my 3 threes. And here's the other 3 threes. And we can see that they match my game board.

[Presenter puts the cut out of 3 sixes on top of the area of 3 sixes on the first game board. Presenter then puts each cut out of 3 threes on top of each area of 3 threes on the first game board.]

And now if I take them and lay them over the top of each other like this....

[Presenter picks up the cut out of 3 sixes and the 2 cut outs of 3 threes. Presenter puts the 2 cut outs of 3 threes, which amounts to an area of 18 squares, on top of the cut out of 3 sixes, which is an area of 18 squares.]

I can also see that they have the exact same area and so whilst we're naming it differently and it looks a bit different when it's cut up, this is how I can see that 3 sixes is equivalent in value in area of 3 threes and 3 threes.

[Presenter lays the cut out of 3 sixes on the table, then puts the 2 cut outs of 3 threes beneath the cut out of 3 sixes.]

And in fact, what it's making me think about too is how many other ways could I partition 3 sixes and name them so that I still have an area of 18 squares, but I can start to think about all the different ways that that area could be composed.

[Presenter picks up the cut out of 3 sixes to show the area of 18 squares.]

Over to you, mathematicians!

[End of transcript]

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