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Speaker

Okay mathematicians, welcome back!

It is time to get your eyeballs ready to do some deep investigation into 2 collections 6 versus 6.

So, you will need some paper and a pencil.

[Screen reads: You will need… a pencil and some paper.]

Speaker

So first of all, what do you notice about these two collections of 6 that's different?

[Screen reads: What’s different? Screen shows 2 arrays of 6 green dots separated by a blue vertical line. The first array of 6 dots is shown in a shrinking pattern with 1 dot in the top row, 2 dots in the middle row and 3 dots in the bottom row. The second array of 6 green dots on the right are set out in a dice pattern.]

Speaker

Can you see something that's different? Great, can you write it down? Take a note of it or draw a picture of what you notice is different between the 2 collections.

Yeah, and if you thought of one thing that's different, can you now think of another thing that's different between these two collections of 6?

And you can write that one down too or draw a picture.

Yeah, you're right. ‘cause mathematicians represent ideas in lots of different ways. Okay, I now have a different question for you.

What's something that you notice that's the same?

Yeah. Okay, so if you've got one thing that's the same, can you think of a second thing that's the same?

Yeah, and write that down too or draw it.

Okay shall we talk together, so we spoke to some young mathematicians, too and these are some ideas that they came up with.

[Screen reads: What’s different? What’s the same?

Screen shows an array of dots set out in a dice pattern with one column being 3 blue dots and the other column being 3 green dots. The text above the image reads ‘1 shows 6 is a doubles fact (2 threes)’.

Underneath is another array of dots shown in a shrinking pattern, with the first column of 3 dots being blue, the second column of 2 dots being green and the third column of 1 dot being red. The text above the image reads ‘and the other shows a shrinking pattern, 3 plus 2 plus one.]

Speaker

So, one of the things they noticed that's different is that one of the collections shows that 6 is a doubles fact. You know 2 threes.

But the other collection shows that 6 in that representation shows a shrinking pattern where 6 is 3 and 2 and 1 more.

Yeah, like the backwards count 3, 2, 1. Uh-hum.

And actually, what we probably should do mathematicians before we go any further is, we could maybe use a table to organize ideas so that it doesn't end up quite messy.

So, let's draw a table. And yeah, you're right mathematicians often use tables to help them organize information and ideas.

So, it's easier to read and also share with other people, Okay!

[Screen now shows a table being drawn over the content with the content being in the first column and the second column is left blank.]

Speaker

What about some things that are the same? Here are some ideas we had. One, both collections had 6 dots and the other is that both of them showed that there's one column of 3 inside of 6.

[Screen now shows in the second column of the table, 2 arrays of 6 dots. The array on the left is the shrinking pattern of 3 blue dots, 2 green and then one green dot and the array on the right side is in a dice pattern with the first column of 3 dots being blue and the second column of 3 dots being green. The text above the arrays reads ‘Both collections have 6 dots. Both show 6 has 1 column of 3’.]

Speaker

Can you see that?

Yeah, the blue column on the, like staircase, shrinking pattern or shrinking representation of 6 and then a blue column and also a green column on the other representation of 6 that shows 6 is double 3.

Okay! Yeah, we also notice that too about being the same that both collections show 6 is 2 threes.

[Screen now duplicates the 2 arrays of 6 dots with the left array being the shrinking pattern of 3 blue dots, 2 green dots and then one green dot and the right array being the dice pattern with the first column of 3 dots being blue and the second column of 3 dots being green. The text above the arrays reads ‘Both collections show 6 is 2 threes’.]

Speaker

Yes, and in one the staircase representation we see 3 as one column of 3 in the blue and the triangle. The triangle in the green.

And yes, you could have had the triangle at the top and then instead of a column you could have had a row of 3 down the bottom and in the other one we can see the 2 threes, and it's also easier to explain.

Yes, you're right, but the 2 threes are different. Yeah, so one of the threes, one of the collections, one that looks like 6 on a dice pattern has the 2 threes as 2 columns of 3 and the other one has 3 like a triangle and 3 as a row or a column, depending on how you're looking at it.

[Screen now shows an additional 2 arrays of dots added into the first column of the table. The array on the left is the shrinking pattern with the bottom row of 3 dots being blue, the middle row of 2 dots being pink and the top row of one dot being pink. The array on the right side is in a dice pattern with the first column of 3 dots being dark blue and the second column of 3 dots being green. The text above the arrays reads ‘Both collections have 2 threes … but the threes look different’.]

Speaker

Yeah, yes, and another thing that we thought of is that 6 when you look at this version of 6, can also show that it's 3 twos, 2 at the top, 2 in the middle and 2 down the bottom.

[Screen now shows an additional array of dots added into the first column of the table. The array is set out in a dice pattern with the top 2 dots orange, the middle 2 dots light blue and the bottom 2 dots pink. The text above the arrays reads ‘This 6 shows 6 is 3 twos’.]

Speaker

Okay, I have something else for you to wonder about then.

[Screen reads: Now we wonder…]

Speaker

When we look at this collection, these two collections, which collection best shows at 6 can be made by doubling 3 in your opinion.

[Screen reads: What collection best shows that 6..? Screen shows 2 arrays of 6 green dots separated by a blue vertical line. The first array of 6 dots is shown in a shrinking pattern with 1 dot in the top row, 2 dots in the middle row and 3 dots in the bottom row. The second array of 6 green dots on the right are set out in a dice pattern. Underneath the dot arrays reads ‘…can be made by doubling 3?’ The right 6 dots flash.]

Speaker

Can you point to it?

[The right 6 dots flash.]

Speaker

Yeah, we thought that one too because it's got 2 Um, 2 columns and they have an equal number in each column in 3 rows of 2.

That's it. Yeah, yeah, it's an array where there's equal rows and equal columns.

Okay, which collection best shows that 6 has 3 inside of it? That inside the bigger number of 6 that we can see 3. Can you point to it?

[Screen adds additional text which reads ‘…has 3 inside of it? Then the left 6 dots flash and then the right 6 dots flash.]

Speaker

Oh, there's some debate! We agree.

This one shows a chunk of 3. And so does this one. Okay so in that case they tie. Okay.

Which collection best shows that 6 can be partitioned into 3 smaller parts, or 3 smaller chunks.

[Screen adds additional text which reads ‘…can be partitioned into 3 smaller parts?’]

Speaker

Uh-huh debate again. Some people are saying this one because of the idea of 3, 2, one or one, 2, 3 if you're reading it from the top, down.

[Screen now shows left group of dots flashing then right group of dots flashing.]

Speaker

And other people are saying this one 'cause you can read it from the top down and say Uh- 2,4,6. Yeah.

Okay, um what about which collection best shows that 6 is one less than 7.

[Screen adds additional text which reads ‘…is one less than 7?’]

Speaker

Oh neither, I agree with you in this case.

Neither of them are very good at saying or showing to us that 6 is one less than 7.

So, mathematicians over to you!

[Over to you! Draw a representation of 6 that shows is one less than 7]

Speaker

Can you draw for us a representation of 6 that shows 6 is one less than 7? Okay! We look forward to seeing the results.

[End of transcript]