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[Text over a blue background: Let’s explore! Small font text in the upper left-hand corner reads: NSW Department of Education. In the lower right-hand corner is the waratah of the NSW Government logo.

Two pieces of paper lay side-by-side on a white surface in front of the speaker. To the right, is a red marker pen.]

Speaker

So mathematicians, how did you go exploring with your 29 balloons? I asked my friends to do this activity too, and I've got some of their thinking to share with you today.

[The speaker brings an image into the frame. It depicts 29 playdough balls. They are arranged into 14 groups of 2, with one ball on its own.]

He is one of my friends, Tom. Now I can see that Tom has tried to put his bunches with two balloons in each bunch. However, I noticed that there's a left one here. Let's see how many bunches of two, Tom had.

[The speaker draws circles around the bunches of 2, each time drawing a larger circle to include more bunches.]

He had one. One bunch of two. Now two twos. Three twos. Four twos, five twos, six twos, seven twos, eight twos, nine twos, ten twos, eleven twos, twelve twos, thirteen twos, fourteen twos, and this one left over. So the bunches of two for Tom didn't work. They couldn't be put into equal groups. Now, mathematicians, we know that we can share IDs. But sometimes our brains, like other structures better than others. It makes sense to us. Looking at Tom's group, I find it a little bit tricky to see this one done here. So I'm going to restructure his thinking into an array because arrays really help my brain to understand a collection. OK, I think I'll start with this two down here.

[On a blank sheet of paper, the speaker draws two orange dots, close together. She crosses of a pair of balls in the photo of Tom’s bunches of balloons.]

I've got one two, add this one.

[She draws two more orange dots, above the first two that has drawn. She crosses off another bunch from Tom’s photo. She continues to add more dots to the array, crossing off the bunches as she goes.]

Two twos.

Three two's... four two's... five two's... six twos... seven twos... eight twos... nine twos... ten twos... eleven twos. twelve twos.

Thirteen twos... fourteen twos.

[The array now features 2 rows of 14 dots. The speaker draws a horizontal line below the array. Beneath the line, she draws another orange dot. Beneath the dot, he writes “14 twos and 1 left”.]

And for this one left over, I'm going to record it under this line here. So I have fourteen twos and one left.

OK, now put someone else is thinking to share with you as well.

[The speaker brings another image into frame. It features 4 bunches of 8 balls, and one bunch of 5 balls. The speaker points to each of the bunches of balls.]

So this is my friend Bob. Now Bob's thinking, I can see that he was using groups of eight. But just like Tom, Bob also had some left over. And he's put them here in a group of five. Bob had one eight, two eights, three eights, but then he had these five left over. Now I really like how I put tonnes into an array. That helped my brain to see things clearly with his leftovers. Let's use the same strategy again using Bob's thinking over here. Let's start with this eight.

[Beside the array representing Tom’s bunches of balloons, the speaker draws a horizontal line of 8 dots. She crosses out one bunch of 8 from the image.]

I've got one eight.

[She draws another line of 8 dots above the first line. She crosses out another bunch of 8 from the image.]

Now, I have two eights here.

[She draws a third line of 8 dots. She crosses out the final bunch of 8 from the image.]

Now I have three eights, but I have these five left over.

[She draws a line beneath the 3 lines of 8 dots. Beneath the line, she draws 5 more dots.]

So I can draw a line like this and I'm going to record my left over ones underneath. They didn't quite make a group of eight. One, two, three, four and five.

[Beneath the array, she writes “3 eights and 5 left”.]

So I had 3 eights and 5 left.

[Text over a blue background: What’s (some of) the mathematics?]

Let's have a look at what some of the mathematics sees in this activity.

[Text: We used trial and error to try and share the 29 balloons equally into bunches.

We noticed that 29 can’t be shared equally into bunches, groups or rows. We tried lots of different ways to organise them equally into groups but we always had leftovers.

Two images below show different ways to arrange the 29 balloons into groups. In the first, the balloons are arranged into 4 groups of 6 and one group of 5. In the second, the balloons are arranged into 5 groups of 5 and one group of 4.]

We use trial and error to try and share the 29 balloons equally into bunches. We noticed that 29 can't be shared equally into bunches, groups or rows. We tried lots of different ways to organise them equally into groups, but we always had leftovers.

[Text: Arrays helped us to see the different ways we could arrange our bunches of balloons and also see how many we had left over each time.

Two images below show the 2 arrays drawn in this video. The first, beside the image of Tom’s arrangement of the balloons, features two vertical rows of 14 dots, with one dot left over. The second array appears beside the image of Bob’s arrangement of the balloons, which features 3 horizontal lines of 8 dots, with 5 dots left over.]

Arrays helped us to see the different ways we could arrange our bunches of balloons and also how many we had left over, each time.

[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]

[Text over a navy-blue background: bunches of balloons. (Stage 1). From resolve. Small font text in the lower left-hand corner reads: NSW Mathematics Strategy Professional Learning Team (NSWMS PL team). In the lower right-hand corner is the red waratah of the NSW Government logo.]

Speaker

Bunches of balloons from reSolve.

[Text over a white background: You will need…

• pen and paper
• 29 balloons (these could be made out of play dough or you can use rocks, paper clips or leaves).

An image below the text shows 29 small, colourful balls of play dough and two pens.]

For this activity you will need, pen and paper and 29 balloons. These could be made out of play-dough, rocks, paper clips or leaves.

[29 colourful playdough balls are arranged in 3 groups over 2 sheets of paper, that have been taped together. 2 of the groups of balls are arranged into 2 rows of 5. The other group of balls is arranged into one row of 5 and one row of 4.]

Hey there mathematicians, today you're going to be helping me to solve a problem using these balloons. Can you see them down here on my page? Just by looking and noticing, how many balloons do you think we're going to be working with today?

[The speaker points to the 2 groups of 10 balls.]

Mm-hm, I can see one full ten frame, two full ten frames.

[The speaker points to the empty space at the end of the second row of the third bunch of balls.]

But when I get to my third ten frame, I see one empty space, and we know that nine is less than one, which means we have two full ten frames, which is 20 and nine more, which is 29. Today we are working with 29 balloons and what we need to do is to put the balloons into bunches that have the same number of balloons in each bunch. Now, because we know our total number of balloons and that's 29, but we don't know how many bunches we're going to have yet, and we don't know how many balloons are going to go into each bunch. We're actually using a strategy called trial and error. I think to begin with, I might try bunches of five balloons. What do we think? Let's see.

[The speaker arranges 5 of the balloons into a bunch. She arranges them like the dots on the 5-face of a dice. She arranges 4 more bunches of 5 balls. There are 4 balls left over. She arranges them into a bunch and puts them to the side.]

Do you recognise this spatial pattern? Perhaps you've seen it on a dice before. I have one five... two fives... three fives... four fives.

Now, I have five fives. Oh, but mathematicians, do you notice, I don't quite have enough here to make a sixth five? So, I only have five fives, but then I have four left over. That means that these aren't equal groups. I can't place my 29 balloons into equal groups of five, because I need one more. Let's try something else.

[The speaker arranges 6 of the balls into a bunch. She arranges them into 2 vertical rows of 3. She then arranges 2 more bunches of 6. There are 5 balls left over. She arranges them into a bunch.]

I have one six and I think you would have seen that one on a dice as well.

Now, I have two sixes.

I have... Oops! Now, I have three sixes.

Now, I have four sixes. Oh, but mathematicians what do you notice? I've got one missing, this particular group, it can't be a group of six, it's only a group of five. So, I have four sixes and one group of five. And that means that we can't put our 29 balloons into equal bunches of six. Well, it's over to you mathematicians, now it's your turn to see what numbers you can use to try and make equal bunches using your 29 balloons.

[Text over a blue background: Over to you…

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]