Factors fun

A thinking mathematically context for practise focussed on developing multiplicative strategies and knowledge of factors

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-MR-01

• MAO-WM-01
• MA3-MR-01

Collect resources

You will need:

• 3 pencils

• a paper clip

• 4-6 pink counters (or another colour) and 4-6 blue counters (or another colour).

Watch

Watch the Factors fun video to learn how to play (8:25).

Use factors to fill a number grid.

Hello Barbara.

Hello Michelle.

How are you?

Barbara

I'm very well, how are you?

Michelle

Very good thank you and we hope our mathematicians out there are also very good.

Um, let's play! You know how we played multiples madness recently?

[Screen shows a 5 by 5 game board with 25 squares. On each square there is a number written. To the right of the game board, there is a spinner that is going up by fives starting from the number 5 to the number 50. On the right side of the game board there are pink counters, and on the left side there are blue counters.

Underneath the game board there is a recording sheet with the words ‘Student 1’ and ‘Student 2’ on there. Underneath each student name there are 3 columns. The first column says ‘spun’, the second column says ‘number sentence’ and the third says ‘covered’.]

Yeah.

Michelle

Well, this is a very similar game. In fact, it's just based on reversing the idea of the game.

Oh, okay.

Michelle

Yeah, so that we can practice thinking about division, while we're also thinking about multiplication.

Barbara

Given what we know about multiplication.

Michelle

Exactly right.

So, we've made our spinner, and this game we're not going to play blank. It's a blank version so you can make it what you like, and we're gonna do it based on fives today.

[Michelle circles the spinner with her finger then above the game board where it reads blank, she crosses blank out and writes fives.]

Okay.

Michelle

And so, what I did was write the product of multiplication using fives, up to 10 times 5.

Barbara

So, you started from 5 and then counted by fives.

Michelle

Like 5 ones, 5 twos, 5 threes, 5 fours and so on.

And we spin the spinner like this, and now what I'm thinking about is 10 divided by 5.

[Michelle demonstrates how you use the spinner by putting the pencil in the centre of the paper clip, then placing it in the middle of the spinner and flicking it with her finger to spin. She then demonstrated how you would place the counters over the numbers on the game board.]

Okay.

Michelle

And in the same way you, like, if that would be 2, and so I have choices of where I put my 2.

Okay.

Michelle

And today we're playing, so you get 4 in a row, but you only get 4 counters.

Barbara

So, you need to move around.

Michelle

So, you need to move them once you run out.

And a square counts.

Okay.

Michelle

Okay? Four in a row, in any direction, or you can form a square.

Barbara

But not like this?

Michelle

No, not a square.

Ok not a square.

Barbara

So, do I write for you, and you write for me?

Yes.

Barbara

Or do we write for ourselves.

Okay we're writing for each other. Oops, okay, okay what do think? Spin again?

You choose.

Barbara

Okay I want to go with 35.

[Barbara spins and lands on 35.]

Okay.

Barbara

So, 35 divided by 5.

Yep Yeah.

Barbara

[Michelle writes Barbara’s name on the top of the right-hand side of the game board. In the ‘spun’ column she writes the number 35. In the ‘number sentence’ column she writes 35 divided by 5 equals 7. In the ‘covered’ column she writes 7. She then places a blue counter over the number 7 in the bottom left-hand square.]

Yes.

Barbara

So, I can choose any 7?

Any 7 you like.

Barbara

I've got 2 choices so, both corners, so I'm turning over that one.

Okay.

Michelle

Oh, I got 15 so 15 shared into 5 groups is equivalent to 3 in each group and I think I might go here.

[Michelle spins and lands on 15. She places a pink counter over 3 in the fourth row down, second square across.]

Okay.

Michelle

So, you need to write for me. So, you spun a 15. And then it was 15 shared into 3 groups, shared into 5!

Barbara

5 groups, because we've always got 5 groups.

[Barbara writes in Michelle’s scoring columns on the left side of the game board. In ‘spun’ she writes 15. In ‘number sentence’ she writes 15 divided by 5 equals 3. In ‘covered’ she writes 3.]

Michelle

Which leaves 3 in each group. Your turn.

Come on spinner!

Barbara

I know. Okay, 40.

[Barbara spins and lands on 40.]

Michelle

Okay, so you spun 40, what are you thinking?

[In Barbara’s scoring columns, Michelle writes 40 in ‘spun’, 40 divided by 5 equals 8 in ‘number sentence’ and 8 in ‘covered’.]

Barbara

Okay so I would, if I had tens, I'd need 4 tens.

Oh yeah.

Barbara

That means I need 8 fives.

Michelle

Yeah, that's a nice strategy, I like that, using tens.

Barbara

To help me with my fives.

Yeh.

Barbara

I just find tens quite easy, so.

[Barbara places a blue counter over 8 in the bottom row, second square across.]

Michelle

Cuz of the place value?

Barbara

Yeah so, I thought I'd use that to help me with the 5 facts.

Michelle

Forty shared into 5 equal groups is 8 in each group.

Okay.

Michelle

Oh, 2 in a row already! Watch out.

I got 35. And so, 35 shared into 5 groups, well I know that 40 is 8.

[Michelle spins and lands on 35.]

Yeah.

Michelle

So, one less 5 is 35, so that must be 7.

Barbara

And you also did it in the first round.

Michelle

Oh, and I did do in the first round. For you, so, there's my only other choice of 7 actually.

[Michelle places a pink counter over 7 in the bottom row, far right square.]

Barbara

So, 35 shared into 5 equal groups means 7 in each group.

[Barbara writes in Michelle’s scoring columns. She writes 35 in ‘spun’, 35 divided by 5 equals 7 in ‘number sentence’ and 7 in ‘covered’.]

Michelle

You can totally see why mathematicians invented that symbol.

Barbara

It's really actually useful!

Michelle

Imagine having to write 35 shared into 5 equal groups is equivalent to 7 in each group.

Barbara

Yeah, I like that mathematicians have developed, um, symbols.

Barbara

Okay, okay thank you, historical mathematicians.

That’s right!

Barbara

Okay, 20.

[Barbara spins and lands on 20.]

Yep.

Barbara

Okay so, this one I know.

Michelle

Yeah?

[In Barbara’s scoring columns, Michelle writes 20 in ‘spun’, 20 divided by 5 equals 4 in ‘number sentence’ and 4 in ‘covered’.]

Barbara

Twenty shared into 5 equal groups means 4 in each group. Now, I can't go this way. [Barbara points along the bottom row.]

So, oh, I don't, you know what, I like the middle.

[Barbara places a blue counter over 4 in the third row down, third square across.]

Michelle

Okay, I don't like that you like the middle of the board, cuz it's gonna be harder for me to win. But it'll also be hard for me to win if I use the bottom of the pencil to try to spin.

I see.

Michelle

Okay, 15. Well I actually know this too. I know 3 fives is 15 so 15 divided by 5 is 3. Um, and I am going to go, oh there's only one other 3 left, so I have to go there.

[Michelle spins and lands on 15.She places a pink counter over number 3 in the second row,the third square across.]

Barbara

Oh, and you got 3 twice.

[Barbara writes 15 in ‘spun’, 15 divided by 5 equals 3 in ‘number sentence’ and 3 in ‘covered’ in Michelle’s scoring column.]

I did. Okay. 50.

Barbara

50. Okay.

[Barbara spins and lands on 50.]

Michelle

That’s a nice one.

Barbara

Yeah, I know that 5 times 10 is 50.

Yeah.

Barbara

So then 50 shared into 5 equal groups is 10.

Michelle

Yeah.

[Michelle writes 50 in ‘spun’, 50 divided by 5 equals 10 in ‘number sentence’ and 10 in ‘covered’ in Barbara’s scoring column.]

Barbara

Umm, now then, I might put it here because things could go my way, this way.

[Barbara places a blue counter over 10 in the first square of the second row.]

Michelle

Oh yeah, it could.

Yeah, more so than, oh.

[Michelle spins and lands on 40. She places a pink counter over 8 in the third row, 5 squares across.]

Barbara

Oh no!!! [Barbara points to another 10 in a better position.] That was silly, wasn’t it?

[Barbara writes in Michelle’s scoring columns. She writes 40 in ‘spun’, 40 divided by 5 equals 8 in ‘number sentence’ and 8 in ‘covered’.

No sound. The video speeds up while they take turns, additional scores are added to Barbara’s and Michelle’s scoring columns.]

Michelle

And we only get 2 more spins to see if someone's a winner. Otherwise, it's a tie.

Oh really?

Michelle

Yeah.

[Michelle spins and lands on 20.]

Barbara

Okay, so that was a 20, wasn't it? That you spun?

[Barbara writes 20 in ‘spun’, 20 divided by 5 equals 4 in ‘number sentence’ and 4 in ‘covered’ in Michelle’s scoring columns.]

Michelle

So sometimes, if you want, you could play with a fifth counter because that allows you a bit more freedom in moving things around.

Yeh.

Michelle

Because the 4 makes it, does make it harder.

Barbara

It does make it quite hard.

Michelle

Yeah.

So, we could introduce the fifth counter if you want on this spin?

Barbara

Well, if you've only got 2 spins left that's probably a good idea.

Michelle

Yeah, let’s do it. Okay. You'll probably get a 6 anyway!

Oh no, a 3.

Barbara

A 3. Okay, we just know the answer. 15 divided by 5 is 3.

Okay, oh okay but I can introduce a counter. So, I'm just going to introduce a counter.

[Barbara spins and lands on 3. Michelle writes in Barbara scoring columns. She writes 15 in ‘spun’, 15 divided by 5 equals 3 in ‘number sentence’, and 3 in ‘covered’.

Michelle

I would have put it there, cuz then you needed a 9 or 6 to form another square.

[Barbara places a blue counter over 3 in the second column, third square across.]

Barbara

That is totally a much better option!

Michelle

Umm, oh, I think I'm gonna re-spin that. Sorry, cuz I wasn't really paying attention and it...

Uh, 40. Okay, so that's 8, cuz I know umm, 5 eights is 40 so I'm gonna put it here.

Barbara

Okay, so.

[Barbara writes in Michelle’s scoring columns. She writes 40 in ‘spun’, 40 divided by 5 equals 8 in ‘number sentence’, and 8 in ‘covered’.]

Michelle

Oh no, I'm gonna put it there.

Okay.

Michelle

Or there? Or there?

Barbara

Take your time Michelle. Quick game's a good game you know!

Michelle

Umm, well if, I because if I was thinking if I go there, I can't like, oh there's one more spin, right?

[Michelle spins and lands on 40. She places a pink counter over number 8 in the third row,fifth square across. Michelle then moves the counter around and changes it to the bottom row, second square across.]

Barbara

Only one more spin.

Michelle

So, I can't get 4 in a row there. But if I go there I still also can't 4 in a row so I might just go there.

Okay.

Michelle

Can you record my move please?

Sure can.

Michelle

Okay, so you really need 30.

Barbara

Yeh, I think it is.

Michelle

It's also still my only chance.

Yeah.

Okay.

Barbara

And there's a one in 10 chance of it happening. Okay.

Barbara

Come on, lucky last go! Come on 30! Oh, just short! Okay, so that's a 4.

So, 20 divided by 5 is 4.

[Barbara spins and lands on 20.]

Where are the fours? Oh, hold on, I can't go.

Michelle

Oh, so you miss a turn.

[Michelle writes in Barbara’s scoring columns. She writes 20 in ‘spun’, 20 divided by 5 equals 4 in ‘number sentence’ and a cross in ‘covered’.]

Barbara

I was hoping I'd get another spin because I can't go.

Michelle

Okay come on 6! Now what I'm gonna do is start my spinner from a different spot.

[Michelle spins and lands on 20.]

Oh, still no go. And I would get a 4 and I also can't move.

So, in this game we have tied, both losers.

Oh, equal losers!

Michelle

Over to you mathematicians, to have fun with factors!

[End of transcript]

How to play

• Get your game board, spinner, recording sheet, counters, and pencils ready.

• Take it in turn to spin the spinner and divide the number by the chosen divisor (for example, 5).

• Players work out the solution and explain their thinking to their partner.

• The partner records their thinking and if they agree, the player is able to place one of their counters on the number on the game board, claiming that place.

• If the number is taken, students miss a turn.

• If there are no new counters that can be added to the game board, players have to move an existing counter to a new place.

• Players win by getting four counters in a row (in any orientation, including a square).

• If preferred, students can use 5 or 6 counters, looking for 4 in a row.