# Investigating playing cards

ES1 – a thinking mathematically targeted teaching opportunity exploring different ways to visualise and represent quantities in a deck of cards

## Syllabus

Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus (2022) © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2024.

## Outcomes

- MAO-WM-01
- MAE-RWN-01
- MAE-RWN-02
- MAE-CSQ-01
- MAE-CSQ-02
- MAE-FG-02

## Collect resources

You will need:

- something to write or draw on
- something to write or draw with.

## Watch

Watch Investigating playing cards video (12:11).

(Duration: 12 minutes and 11 seconds)

[White text on a navy-blue background reads ‘Investigating playing cards’. In the bottom right corner, the NSW Government red ‘waratah’ logo.]

[Black text on a white background read ‘You will need...’ Bullet points below (read by speaker).]

[A pack of playing cards has the ‘King of Hearts’ on the front. A second pack of cards reads ‘Jumbo School Friendly Playing Cards’ in green text.]

### Speaker

Hello there, little mathematicians and welcome back. Today, we thought we'd spend some time exploring the number representations on playing cards. We see these quite a bit in the classroom when we use them, and the structure used to represent each of the numbers is really worth investigating.

[The speaker lifts away the packs of cards and below is a red ‘5 of diamonds’ card.]

### Speaker

So let's start by noticing some things about the representation of 5. What are some things you can tell me about what you see here? Ah, OK. So you notice that 5 here looks like 5 on a dice.

[The speaker briefly shows a small black and white 6-sided dice.]

### Speaker

Great. Now, let's have a think about how the number 5 is made up of smaller numbers. What smaller numbers can you see inside of this representation of 5? Ah, OK, so you saw 3 on a diagonal.

[The speaker circles 3 of the diamonds with a purple marker pen (further steps explained).]

### Speaker

And you may have recognised that because of what 3 on a dice looks like. What else sits inside of 5? Oh, that's right. Some of you said that there was the 3, and then one more, and then one more. Some of you even explained that as 3 and 2. Alright. Let's use another number 5 card to see if we can have a look at how the structure also lets us to see five in a different way.

[The second card is a black ‘5 of clubs’. The speaker uses the purple marker to outline as she talks.]

### Speaker

What else are you noticing? Right! So you saw 4, 2 on the top, 2 on the bottom, and one more in the centre. That's great. That's a really important idea about the number 4 and the number 5 in fact. It helps us notice that 4 and one more is 5. But it also helps us notice that 4 is one less than 5. And we can think about that when we're counting forwards by ones and how each number is one more, and also when we're counting backwards and how each number is one less.

Some other ways that we can talk about the number 5 in terms of its smaller parts might be double 2 and one more or even 5 ones. 1, 2, 3, 4, 5. 5 ones. OK, let's use our flexible thinking to think about the number 8 in terms of its part.

[The speaker places down an ‘8 of clubs’ card.]

### Speaker

Here's a card of the number 8. What clues in the structure here on this card can you see? Oh, great. I can hear lots of great ideas. So let's share. What I'm going to do is use some different colours to share the ideas of 8 that we have across the class.

[The speaker places down a white A4 sheet of paper. The paper has been folded into 4 sections and in each section there are 8 identical circles. The speaker uses a purple marker and a yellow marker to colour in the circles (as explained).]

### Speaker

OK, great, the first one. I noticed that too. The number 8 actually has 5 inside of it. 1, 2, 3, 4, 5. That dice pattern 5. Great, a lot of people noticed that, and then they noticed that they were 3 more. Same, use colour there to show those 2 different parts, the 5 in the purple and the 3 in the yellow. It sort of reminds us of what it looks like in a 10-frame too where 8 is shown as 5 on the top row and then 3 more. OK, let's show those 2 smaller parts in a different way. Oh, good, I can hear some people talking about doubles.

Yeah, you saw double 3. OK, so let's colour that. There's one of the 3s down the side, and there's the other 3. So there's our double 3 and 2 more, fantastic. And you can see that in the purple and yellow represented. Colour does really help us look into numbers really easily and communicate that with other people which is a really important thing in mathematics. Alright, let's keep investigating.

Ah, somebody saw 2s everywhere. Alright, let's use some colours to show those 2s. There's a 2. There's another 2 in yellow. There's another 2 at the bottom. And then 2, down the middle. Wow, alright, now let's count those 2s. 1 x 2, 2 x 2, 3 x 2, 4 x 2. Now, that idea is a little bit different to the other ones we came up with because all of the parts are the same size. So what I might do to communicate my thinking a little bit better is to actually use 4 circles so that you can see that each of the groups has 2 inside of them.

[The speaker uses a blue felt tip pen to circle each of the groups of 2.]

### Speaker

And now we know that we can rename 4 x 2 as 8, and think about 8 as 4 x 2. Great. Now, are there any other ways that this card, 8, helps us to see the smaller parts inside of it? Of course, great noticing.

[The speaker places a ‘7 of clubs’ card down alongside the ‘8 of clubs’.]

### Speaker

You notice that you can see 7 inside of 8 except there's one more. See that one more? So 7 and one more is 8, fantastic. Let's show that with those two colours.

[The speaker returns to the A4 sheet to colour the final group of 8 circles (as explained.]

### Speaker

But 8 is 7 and one more, fantastic. That's similar to the thinking we were doing earlier about the number before being one less and the number after being one more. Fantastic.

Now, these playing cards are really helping us to think flexibly about the ways these numbers are made up of smaller numbers. So, what you might like to do is pause the video now and have a think about some of the other numbers to 10, using playing cards if you have them, and thinking about the structure and how that helps you to see the smaller parts. You might also like to use colour to communicate your thinking with someone else - similar to what we've done here. Have fun.

[Two rows of 5 playing cards, all red diamonds. The top row stars with the ‘Ace’ through to the ‘5 od diamonds’. The bottom row starts at the ‘6 of diamonds’ through to the ’10 of diamonds’. Further steps explained.]

### Speaker

OK, so, here is a picture of playing cards, one, being the ace, to 10, that you can use to think about those numbers flexibly. So if you don't have playing cards at home, here's a picture that you can use. So pause the video and have a think about some of the other numbers that we haven't investigated yet. Remember that you're looking inside the rectangle.

[The speaker traces her finger around a black rectangle in the centre of the cards that frames the number of diamonds within.]

### Speaker

OK, so how did you go? Great to hear that you had so many wonderful ideas about each number. Some of those ideas sounded similar, didn't they? Even though you might have been thinking about different numbers. So, let's summarise some of those important similarities by sorting through our ideas.

Now, for some of the numbers represented on the playing cards, we could see doubles. So let's sort which structures and which numbers allowed us to see that easily. Ah, OK, so you were thinking about 2 as double one and 4 as double 2 and 6 as double 3. Ah, and 10, of course, as double 5. There, you can see the 2 dot dice patterns.

Now, interesting that 8 wasn't included in our thinking because that's another even number as well, and we know that that's double 4. Now I want you to think about and use your mind's eye to create a picture that we could have put on the playing card for 8 that shows the double 4. Ah, so it might have looked like 4 on the top, like the dot pattern, and 4 on the bottom, a little bit like this.

[The speaker shows 2 6-sided dice pressed together between her fingers. Each dice has 4 ‘pips’ on it.]

### Speaker

This is what I was thinking. Is that what your mind's eye did? Great, OK. Some people were thinking, oh, like double 3, but adding one more at the bottom. That could be a picture of 8 as well.

That could help us see double 4, fantastic.

Alright, now, while we're thinking about doubles, which structures helped us to notice which numbers we could see were made of double a smaller number and one more? Yes, 5. Here it is here. So you could see the double 2 and then the one more in the middle. Great, any others? Yes, 7. You can see the double 3 just like the number 6 card and the one more in the middle. Great thinking. Oh, and how could we forget 9? Look at that. Double 4 like we had...the picture we had in our mind's eye that 8 could look like and the one more in the middle. Great, so these are really helping us think about doubles and one more for our numbers to 10.

OK. Now, you guys also shared a few ideas about numbers that had 5 inside of it and how the representations and the structures on the cards really helped us to see that. So, I'm going to move the numbers one to 5 out of the way because they are equal to or smaller than 5. So we're only gonna be thinking about these numbers, 6, 7, 8, 9 and 10.

Which representations here really helped us to see that these numbers were 5 and some more? Ah! OK, so you were thinking about the number 7 where you can see the 5 at the top, like the dice pattern, and 2 more. A really important idea about 7, great. Ah, and you can also see the 5 and 3 more inside of the number 8. 8 is pretty cool like that. You can either see the 5 at the top or 5 at the bottom, but either way, 5 and 3 or 3 and 5 still equals 8. Great. Oh, and 9. Look at that. The 5’s in the middle. Fantastic. So it's 5 and 2 and 2 or you can say 5 and 4 more. Fantastic.

Oh, no, I haven't forgotten, great. 10 is also made of 5 and 5 or double 5. Really interesting. OK, now, 6 wasn't included in that conversation. What could a playing card picture for 6 look like, if we were trying to show that it's 5 and some more? Can you use your mind's eye to create that image? Fantastic. I can hear you talking about some good ideas.

[The speaker shows 2 6-sided dice pressed together between her fingers. The top dice has 5 ‘pips’ on it and the bottom dice has one.]

### Speaker

It might just have looked something like that, where you had 5, like the dice pattern, like all the other numbers, and then the one more at the bottom or even the other way around. Because we know that it doesn't matter, which way these are shown, that 5 and one more is 6 or one and 5 more is 6. Fantastic thinking. Great work today, mathematicians. I can't wait to investigate next time with 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]

## Instructions

Use your paper and pencils to have a go at investigating playing cards.