How many bales?
A thinking mathematically targeted teaching resource focussed on exploring multiplicative relationships through an investigation based on a classic nursery rhyme
Adapted from a task by Dianne Siemon
Siemon, D. (2013). Launching mathematical futures: the key role of multiplicative thinking. Mathematics: Launching Futures, 34-42.
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, 2021
You will need:
something to write on
something to write with
some counters (you could use pasta or brick building pieces).
Watch How many bales? Part 1 video (1:49).
[Text over a navy background: How many bales? From Professor Dianne Siemon. The NSW Government logo is in the lower right corner of the screen. Small font text at the bottom of the screen read: NSW Mathematics Strategy Professional Learning team (NSWMS PL team).]
Hi there, mathematicians. Welcome back. We have a new problem for you today that comes from Professor Dianne Siemon. And Professor Di is a brilliant mathematician, and we love it when she shares her problems with us.
[A title on a write background reads: You will need…
Bullet points below read:
- something to write on
- something to write with
- some counters (you could use pasta or LEGO pieces)
Small font text on top of the screen reads – NSW Department of Education
An image shows some small red plastic tiles on a table. Next to the tiles on the right-hand side are colour textas.]
Today you will need something to write on, something to write with, and some counters. Now, you could use pasta or LEGO pieces as well.
[Text on a blue background reads: Let’s investigate!]
[A title on a white background reads: How many bales of wool? From Professor Dianne Siemon.
Below the title is an illustration of a black sheep next to three bales of wool. Text on the image reads – Baa Baa Black Sheep.
Under the illustration is text that reads – If there were 5 sheep, how many bales of wool?]
How many bales of wool? So, you may or may not have heard of the nursery rhyme 'Baa Baa Black Sheep'. And it goes like this. Baa, baa, black sheep, have you any wool? Yes, sir, yes, sir, three bags full. One for the master and one for the dame. And one for the little girl who lives down the lane.
But what you may not know is, there's actually some really interesting mathematics in the 'Baa Baa Black Sheep' nursery rhyme, because as we know, maths is everywhere. The nursery rhyme starts by asking the sheep if he has any bags of wool. To which the sheep replies,' Well, yes, sir. Yes, sir. I have three bags full'. And that's where our problem comes from. If there were five sheep, how many bales of wool would there be in total? Over to you, mathematicians. Can you solve Professor Di's problem?
[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]
Watch How many bales? Part 2 video (12:47).
[On a black desktop, a white sheet of A4 paper has a blue header that reads ‘If there were 5 sheep, how many bales of wool?’ On the right of the paper there are four coloured markers, red, light blue, blue and pink.]
Welcome back, mathematicians. How did you go? Did you find out that we would have 15 bales of wool in total if we had five sheep? I hope there was lots of sweaty brains. Professor Di always leaves me with a sweaty brain. So, let's dig into some thinking.
[The speaker picks up the red marker to draw on the sheet of paper.]
And to get started, let's think about what we already know from the problem. And our problem was if there were 5 sheep, how many bales of wool? Yeah, that's right. Good thinking. We know that we have 5 sheep. And we can start by drawing our 5 sheep on the page. And I could spend some time to draw some really fancy sheep, but these sheep don't need to be fancy. So, what I might do is I might use a cloud shape to represent each of our sheep. So, we have 1, 2, 3, 4 and 5.
So, we have our 5 cloud sheep now. And from the nursery rhyme and the picture Professor Di shared with us, we also know that each of our sheep needs to have 3 bales of wool. How might we represent our bales of wool? Hmm. Yeah, we could draw three rectangles under each sheep so that now this sheep has 3 bales of wool. And if we continue to draw these, we can actually represent our bales of wool for each of our sheep.
Now that each of our sheep has its 3 bales of wool, we can start to think about how might we be able to find how many bales of wool there are altogether. Now, I know that there are too many bales to see just by looking and thinking. But what I'm thinking is we could actually use counting or skip counting to help us solve this problem. And because we have 3 bales of wool for each sheep, we can use the counting sequence of 3’s to help us determine how many.
So, we have 3 bales of wool for this sheep. And I know the next number is 6 so now we have 6 bales of wool. And I know the next number in the sequence is 9. So, we now have 9 bales of wool, but I'm actually not too confident in knowing what the next number in the sequence is. So, what I'm going to do is use a counting sequence I'm comfortable and confident with and continue to count by ones. So, I know we have 9.
And then we'll have one more is 10 and another one is 11. And the next number in the sequence is 12. We'll have 13 and then 14, and our last one will make 15. So, let's have another look. At the start, we were pretty confident in counting by 3’s. And we had 3, 6, 9. And then we continued to count on by ones because we were more confident. We had 10, 11, 12, 13, 14, and 15. So, we've solved the problem. If there were 5 sheep and each sheep had 3 bales of wool, how many bales of wool we have? Well, we know that we have 15 bales of wool in total.
[A second sheet of white A4 paper has a red header that reads ‘If there were 5 sheep, how many bales of wool?’]
Another strategy that we could use to find out how many bales of wool in total is to arrange our bales of wool into equal groups.
[The speaker picks up the light blue marker pen to draw on the sheet of paper.]
And this time, we can show our thinking by drawing as well. And what we might do is represent each of our sheep using a circle. So, I'll draw 5 circles and I'm actually going to arrange them in a dice pattern because a dice pattern is a familiar structure for me. And I know that I have 5 groups just by looking and thinking.
Now that we have our 5 circle sheep here, we still need to add 3 bales of wool to each of our circles. And what I'll do is represent our bales of wool like this.
[The speaker draws 3 ‘V’ shapes inside the top left circle and then proceeds to do the same with the other 4 circles.]
And if I arrange them in the circle this way, I know I have 3 bales just by looking because I can see the points of a triangle. And I know that a triangle has 3 points. So, if I continue to put our bales into our circle sheep, in this way, I don't need to count each one to know that I have 3. So now that we have our 5 equal groups of 3, we still need to figure out how many bales of wool there are in total. And looking inside of 5, I know that we have 5 x 3, but 5 x 3 isn't a number fact that I know. But what I do know is that inside 5 x 3, there are 2 x 3 and I can see our 2 3s here.
[The speaker circles the 2 circles on the left with the purple marker.]
And I know that 2 x 3 is 6. And actually, I can see another 2 3s on this side here. And I can use what I know about doubling to double 6. And I know that double 6 is 12. So, so far, we've determined that we have 12 bales of wool, but we still have one more 3, and I can just count on. So, I know we have 12. We have 13, 14 and 15. So, we have 15 bales of wool in total. And we know that 12 and 3 more is 15. And once, again, we found out that if we had 5 sheep, we know that there would be 15 bales of wool.
[A third sheet of white A4 paper has a purple header that reads ‘If there were 5 sheep, how many bales of wool?’ On the left of the paper there is a cluster of small red plastic squares.]
[The speaker briefly retrieves the first sheet of A4 paper from earlier.]
Mathematicians, I was just looking and thinking about our representation and our strategy we use to solve how many bales of wool here. And I remembered that for each sheep, we drew 3 bales of wool. And the way that we drew those 3 bales of wool could actually make a connection. And it made me think of how we could take these 3 bales and form them into an array. So, if we take the 3 bales in that same structure from our first strategy and arrange them like this, we now have one row of 3.
[The speaker arranges three of the red squares into a row.]
And if I can continue doing this, we'll have 2 rows of 3 and 3 rows of 3, 4 rows of 3. And our last one, if we arrange them again like this, we now have 5 rows of 3, and we've been able to reform our bales of wool into an array. And I know by looking at this array that we have 5 rows and 3 in each row. But I don't know what 5 3s is as a number fact. That's not a known fact for me yet.
But I do know that as a mathematician, I can use noticing and wondering to help me solve problems. And when I look at our structure of five threes here, I can see some familiar spatial patterns. And I notice that I can see 3 3s. If I move these up a little bit, we can see our 3 3s here. And I can see 3 3s. But also, when I look at it, I can see that it's 9, like on a domino. So, I know that inside of our 5 3s, we have 3 3s. And I know that is 9. But I also know that we have 2 more 3s, which I know is 6. And now I know that I just need to combine 9 and 6 to be able to find the total of how many. And I know that I can take one from our 6 and add it to our 9 to make one 10. And I can see that there's 5 more. So, I have one 10 and 5 more, which I can rename as 15.
Actually, now that I think about it, mathematicians, I can see another spatial pattern. And maybe you've seen this one too.
[The speaker moves the left column of red squares away to form a gap.]
If I move this column... whoop, doesn't wanna move. If I move this column over, I can see a familiar structure here. And I know that this is a 10 frame. So, I know that I have one 10 here. And actually, when I look at this, I know that I have half a 10 frame, and I know that half of 10 is 5. So again, we have one 10 and 5 more, which is 15. So, if there were 5 sheep, how many bales of wool? Well, we know there's 15 bales of wool.
[White text on a blue background reads ‘What’s (some of) the mathematics?]
What's some of the mathematics?
[A blue text header on a white background reads ‘What’s (some of) the mathematics?’ Further text below reads ‘There are many different ways to solve the same problem’. Below, 3 colour images side by side of the earlier A4 sheets of paper.]
There are many different ways to solve the same problem. And to solve Professor Di's problem today, we used counting sequences. We also created equal groups and we reformed our bales of wool into an array and looked for familiar spatial patterns and structures.
[A blue text header on a white background reads ‘What’s (some of) the mathematics?’ Further text and bullet points below read by speaker.]
We also learnt that we can use what we know to help us solve what we don't know yet. And this happened when we used the counting sequence of 3s. And we're able to count by 3s until we weren't as confident, but we could then use something we were confident in, like counting on by ones, to help us solve the problem.
We also used familiar spatial patterns and structures like when we could see and notice the 10 frame in our array. And we know that one 10 and half a 10 frame or 5 more is 15.
[Blue text on a white background reads ‘Over to you…’ Further text below read by speaker. At the bottom a cartoon image of a sheep with 4 bales of wool above it.]
That’s it from us today, mathematicians. I hope your sweaty brains or crunchy eyebrows are ready for a new problem. Can you use our 3 different strategies to solve this problem? If each of the 5 sheep made 4 bales of wool, how many bales of wool would there be in total? Over to you.
[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 and reflect
- Can you use our three different strategies to solve this new problem?
- If each of the 5 sheep made 4 bales of wool, how many bales would there be in total?