Factors fun
A thinking mathematically context for practise focused 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
Outcomes
- MAO-WM-01
- MA2-MR-01
- MAO-WM-01
- MA3-MR-01
Collect resources
You will need:
- 3 pencils
your student workbook
a paper clip
4-6 pink counters (or another colour) and 4-6 blue counters (or another colour).
Factors fun
Watch the Factors fun video to learn how to play (8:25).
Michelle
Hello Barbara.
Barbara
Hello Michelle.
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’.]
Barbara
Yeah.
Michelle
Well, this is a very similar game. In fact, it's just based on reversing the idea of the game.
Barbara
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.]
Barbara
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.]
Barbara
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.
Barbara
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.
Barbara
Ok not a square.
Michelle
Okay, your go.
Barbara
So, do I write for you, and you write for me?
Michelle
Yes.
Barbara
Or do we write for ourselves.
Okay we're writing for each other. Oops, okay, okay what do think? Spin again?
Michelle
You choose.
Barbara
Okay I want to go with 35.
[Barbara spins and lands on 35.]
Michelle
Okay.
Barbara
So, 35 divided by 5.
Michelle
Yep Yeah.
Barbara
And the answer is 7.
[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.]
Michelle
Yes.
Barbara
So, I can choose any 7?
Michelle
Any 7 you like.
Barbara
I've got 2 choices so, both corners, so I'm turning over that one.
Michelle
Okay.
Barbara
Ok, your turn.
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.]
Barbara
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.
Michelle
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.
Michelle
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.
Barbara
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.]
Barbara
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.
Michelle
Your go.
Barbara
Okay, okay thank you, historical mathematicians.
Michelle
That’s right!
Barbara
Okay, 20.
[Barbara spins and lands on 20.]
Michelle
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.
Barbara
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.]
Michelle
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.
Michelle
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.
Barbara
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.
Barbara
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.
Okay, your turn.
[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.
Barbara
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.
Barbara
Okay.
Michelle
Can you record my move please?
Barbara
Sure can.
Michelle
Okay, so you really need 30.
Barbara
Yeah, I wish I had...
Michelle
It's your only chance now.
Barbara
Yeh, I think it is.
Michelle
It's also still my only chance.
Barbara
Yeah.
Michelle
Okay.
Barbara
And there's a one in 10 chance of it happening. Okay.
Michelle
Your go.
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.