Multiplication toss

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 labeling it from 0 to 9.

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

And that will give me the numbers that I'm going to use. And in fact I could start with five, and I now also have a zero which is no good for me because what I know is that 5 times 0 or 0 fives is 0.

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

So fingers crossed my next go is more lucky!

Ahhh 0 and 2, so this time I could say 0 twos is equivalent to 0 times 2, which is also 0. Okay, third time lucky!

Come on, spinner!

Excellent, so this time I got an 8 and ahhh...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.

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. 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. 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.

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.

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 zero, so 3 zeros is equivalent to 3 times 0, which is zero. OK.

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. And I might actually do that. I'm going to use a different color 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.

I realized I didn't actually have to count those 'cause I know my board is 10 by 10. 5 fours I am I going to record that, over here. 5 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. We could just keep going, but we won't.

I could write that 0 sevens or 7 zeros. 0 x 7 which equals 0. Let's try.... come on one! 6...and a 9... now I definitely know I can't go here because I've got 1, 2, 3, 4, 5, 6...1 row of six left that I could use or one row of two. So in this case I have to record 6 nines ...but I couldn't go.

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

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

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

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.

And so I have 3 threes. And 3 threes. 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 is 18. And so here now I have 9 and 9 and when you join 9 and 9 together, that still makes 18. So the the 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 x 3 + 3 x 3, which is equivalent in value to 18. I just partitioned it so it looks a little bit different. I wonder if you could use this strategy to help you out with some of your games.

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. So we thought we'd use some evidence to show you how this works. So, I just made a copy of my 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. And then here is 1 lot of 3 threes. 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. And now if I take them and lay them over the top of each other like this....

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. 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.

Over to you, mathematicians!