# Array bingo – partially covered arrays

Stage 2 and 3 – A thinking mathematically context for practise focussed on building multiplicative strategies such as using arrays and the commutative property

## 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

• MAO-WM-01
• MA2-MR-01

• MAO-WM-01
• MA3-MR-01

## Watch

Watch Array bingo partially covered arrays video (11:24).

Bingo variation using partially covered array representations.

### Transcript of Array bingo partially covered arrays

[Text over a navy-blue background: Partially covered arrays bingo. 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 white waratah of the NSW Government logo.]

### Speaker

Partially covered arrays bingo.

[A title on a white background reads: You will need…

· A set of game cards (with pictures of different partially covered arrays and their matching product)

· somebody to play with.

On the right-hand side of the text are two images: on the left side is an image of a pair of hands holding cards with shapes and dots, on the right side is an image of a pair of hands holding with cards with numbers.]

### Speaker

You will need a set of game cards with pictures of different partially covered arrays and their matching product and someone to play with.

[Text over a navy-blue background: Let’s play!]

### Speaker

Let's play.

[A large white sheet of paper on a blue cardboard.]

### Speaker

Hello there mathematicians. Welcome back. I have Sam with me today. Hi, Sam.

Hi.

### Speaker

OK so, Sam, look here-

[The speaker lays down an array card with 3 columns of 5 dots running down the side on the sheet.]

### Speaker

I've got an array, and this array is showing me five threes and if I turn it this way Sam…

[She rotates the card to the left.]

### Speaker

..It's showing me...

Three fives.

### Speaker

Yes. So, you know, you can play around with arrays and you can also play around with them where we can't see everything, we can't see all the dots and we call those partially hidden arrays.

Cool.

### Speaker

Yeah, so here's five threes…

[She rotates the card to the right.]

### Speaker

..Where you can see all of the dots…

[Next to the array card, the speaker lays down a partially hidden array card with 5 dots running down the side and 3 dots running across, and a blue rectangle covering rest of the space.]

### Speaker

..And here's five threes where some of the dots are covered. And look, you can still tell it's five threes, can't you Sam?

Mmm hmm.

### Speaker

Yeah, because look, we can see the top row here…

[The speaker traces the 3 dots across the array with her finger.]

### Speaker

..where we can see all of them…

[She point to each of the 5 dots down the array.]

### Speaker

..and the second row where I can see the first dot and then the third row and then the fourth row and then the fifth row and even though they're covered…

[She points to the rectangle.]

### Speaker

..We know they have the same number of dots as the first row because it's an array.

[The speaker traces the 3 dots across the array with her finger.]

### Speaker

So it has to be a rectangular shape, right? So, Sam, today we're going to play a game where we have to now use our imaginations and use what we know about arrays and partially covered arrays to play a form of bingo.

OK.

### Speaker

Now, have you played bingo before Sam?

Mmm hmm.

### Speaker

You have? OK, so you know the idea of bingo is to try and...

### Sam

Get all the cards on your board.

### Speaker

Yeah, correct so yeah, to try and cross off, or it might be put it on the counter or something like that. So in this version of partially covered arrays bingo,

[The speaker holds up a partially covered arrays card pack and flips through it. She collects the partially covered array card on the sheet and puts it on the pack.]

### Speaker

You'll need your partially covered array cards and we'll put our five threes or our three fives back into the pile…

[She places the pack outside the top left edge of the sheet.]

### Speaker

..And we won't be needing this one…

[She takes away the card with the 3 columns of 5 dots running down.]

### Speaker

..And we also need our product cards…

[She holds up a pack of product cards with numbers and flips through it.]

### Speaker

..And Sam, we're gonna use these product cards to create our own game boards.

[The speaker puts down a 9 product card in the top right hand corner of the sheet. Sam lays down a 10 in the top left corner.]

### Speaker

So let's put this between us and you can make your game board over here, and I'll make mine over here.

[On the right side of the 9, the speaker puts down a 30 card. On the right side of the 10, Sam puts down a 90.]

### Speaker

And you can choose any six numbers.

[Below the first rows of numbers, the speaker and Sam each place 4 more cards, creating 3 rows by 2 columns of cards on each half of the sheet.]

### Speaker

So looks like we're making a three by two array right, aren't we? Three twos, alright that's interesting.

[She points to the two 10 cards on Sam’s set.]

### Speaker

Oh look, ooh, I reckon we're gonna be fighting for one of our partially covered arrays. Unless, I wonder if there's a different way that you could represent 10 as an array?

[Sam picks up an array card from the pile.]

### Speaker

OK, Alright, let's figure it out. Alright Sam. So turn over the first one, let's put it down here so everybody can see.

[Sam lays down a card with 5 dots running down and 5 dots running across below the pile.]

### Speaker

What have you turned over there Sam, how would you describe that array?

Five fives.

### Speaker

Five fives, and do you know what five fives is equivalent to?

25

### Speaker

25. Is that a known fact for you Sam?

Yeah.

### Speaker

Yeah, OK, alright, Oh.

[Sam turns the 25 product card over, facing it down.]

### Speaker

Very nice, OK. Nice, alright, so we'll keep that down there…

[The speaker picks an array card, flips it over and places it over Sam’s.]

### Speaker

..And I'll flip over the next one.

[Her card has 4 dots going down and 5 dots running across.]

### Speaker

OK, so that array is four fives, well because you told me that five fives is 25, then four fives must be 20, oh yes!

[She turns the 20 product card over, facing it down.]

### Speaker

Looky, looky. Nice. OK, Sam.

[Sam picks up an array card from the pile.]

### Sam

Three threes.

[Sam lays down a card with 3 dots running down and 3 dots running across, on the pile.]

### Speaker

Alright, let's put that there so everybody can see, three threes, what's three threes, Sam?

Nine.

### Speaker

Known fact for you?

Yep.

### Speaker

Yeah, OK, nice. When I think about three threes I like to think about two threes, which we know is six and then one more three, which is nine.

[The speaker turns the 9 product card over, facing it down.]

### Speaker

Thank you, Sam. That was a really nice turnover for you, I really appreciate that.

[The speaker picks up an array card from the pile.]

### Speaker

Alright…

[The speaker lays down a card with 3 dots running down and 2 dots running across.]

### Speaker

OK, now, this one is three twos. And when I think about three twos, I don't know about you, Sam, but I like to think about it as two threes, which I know is double three, which is six. OK, any of us got six?

No.

### Speaker

[Sam picks up an array card from the pile. He lays down a card with 10 dots running down and 6 dots running across.]

### Sam

This one's a big one.

### Speaker

That is a big one. And so we could read it like this or we could read it like that as well, so it's really up to you.

[The speaker turns Sam’s card to the left.]

### Sam

I'm just going to go and assume that's 10.

[Sam counts the number of dots running down the card.]

### Sam

Yep. So it's 60 because its six 10s.

### Speaker

OK, six 10s or 10 sixes, yep, yes they're equivalent, aren't they? OK, and at 60, neither of us have got 60. Alright, here we go...

[The speaker picks up an array card from the pile.

Text over a navy-blue background: A little while later…

On the game board: Sam’s set has 10 and 15 facing up – both are in the middle row. The speaker set has 50 and 5 facing up – both are in the bottom row.

The speaker picks up a card and lays down it down. It’s a card with 5 dots running down and 2 dots across.]

### Speaker

This looks promising. Now, that's five twos or...

BOTH:

Two fives.

[Sam turns the 10 product card over, facing it down.]

### Speaker

Yes, this time you do get to turn it over. Alright, now you're talking. OK, what about here?

[The speaker picks up an array card from the pile. She puts down a card with 5 dots running across.]

### Sam

Five. Ooh, you've got that.

[Sam turns the 5 product card over, facing it down.]

### Speaker

Yes, one five. Ooh, this is a close game.

[Sam picks up an array card from the pile.]

### Sam

That's a 10 again. Two fives or five twos.

[Sam lays down a card with 5 dots running down and 2 dots running across.]

### Speaker

[The speaker picks up an array card from the pile.]

### Sam

Oh! three fives is fifteen.

[The speaker lays down a card with 5 dots running down and 3 dots running across.]

### Speaker

Ooh, Sam well done! OK. So, Sam is our winner so over to you Mathematicians to play your own game of partially covered arrays bingo. Have fun.

[Text over a navy-blue background: Here’s another way to play…

The sheet or game board has been cleared.]

### Speaker

Now, Sam, you know, there is another way that we could have played partially covered array bingo. Should we have a go of that version?

OK.

### Speaker

So in this version, instead of making our game board out of the product cards-

[The speaker holds out the product cards, then the array cards.]

### Speaker

We're going to make our game board out of the partially covered arrays. So we'll turn over our product cards here…

[She places the product cards outside the top left edge of the sheet.]

### Speaker

..And let's make our partially covered array bingo board, shall we?

Alright.

### Speaker

Alright, I'll make mine over here.

[The speaker and Sam each lay down the array cards. They arrange 3 rows by 2 columns of cards.]

### Speaker

Alright Sam shall we play this version?

Sure.

### Speaker

Alright, OK, you do the honours.

[Sam picks up a product card from the pile. He turns it over and puts it below the pile. It’s a 100 card and it’s upside down.]

100

### Speaker

OK, so what could 100 be composed of?

[The speaker turns the 100 card the right-side up.]

### Sam

A 10 by 10 array.

[Sam points to a card on the bottom left-hand corner of his set. It has 10 dots running down and 10 dots running across.]

### Speaker

Yes, it could be a 10 by 10, what else could it be composed of?

### Sam

It could be a two by 50.

### Speaker

Yeah it could be two 50s.

Four 25s.

### Speaker

Or four 25s. Alright, do what either of us said, yeah that one looks, you're looking at that one.

[Sam picks up the card on the bottom right-hand corner of his set. It has 10 dots running down and 10 dots running across.]

### Sam

I think this one.

### Speaker

Yeah, can you see the 10 10s?

[Sam counts the dots across the card, and down.]

### Sam

They are two, four, six, eight, 10.

### Speaker

Yes, and it looks like a very square looking array as well, so nice Sam. OK, so you can turn that over.

[Sam turns the 10 by 10 array card over, facing it down.]

### Speaker

That's right, OK.

[The speaker picks up a product card from the pile. She turns it over and puts it down on Sam’s card.]

### Speaker

25. OK so, I know that 25, an array could be five fives. This is looking promising,

[The speaker points to the array card on the bottom left-hand of her set. It has 5 dots across down and 5 dots running across.]

### Speaker

Yes it is. Awesome, OK.

[The speaker turns the 5 by 5 array card over, facing it down

Sam picks up a product card from the pile.]

### Sam

10

[Sam drops the card.]

### Speaker

OK, so what could that array look like?

### Sam

It could be this one right here.

[Sam points to the card on the left-side of his middle row. It has 5 dots running down and 2 dots running across.

The speaker moves the 10 product card to the pile.]

### Speaker

It could be. It could be five twos.

### Sam

Or it could be that one right there.

[Sam points to the card on the left-side of his top row. It has 2 dots running down and 5 dots running across.]

### Speaker

Or it could be that one, ooh do you get to turn over both?

This one?

### Speaker

No, can only turn over one.

[Sam turns the 5 by 2 array card over, facing it down.

The speaker points to the card on the left-side of her middle row. It has 10 dots running across.]

### Speaker

Ooh look, it could also be one 10. Nice.

[The speaker turns the 10 by 1 array card over, facing it down.]

### Speaker

OK, keep going.

[Sam picks up a product card from the pile.

Text over a navy-blue background: A little while later…

On the game board: Sam’s set has 1 array card facing up in the middle row. It has 10 dots running down and 2 across. The speaker set has 4 cards facing up: both cards in the top row (the one on the left has 3 dots running across, the one on the right has 2 dots running down and 3 across), the right card in the middle row (it has 6 dots running down and 10 running across) and right card in the bottom row (it has 3 dots running down and 2 across.]

### Speaker

OK Sam so, your hope, what, what number are you hoping you're gonna flip over?

20

### Speaker

20, OK.

[The speaker picks up an array card from the pile and turns it over. It’s 10.]

### Speaker

10. Nope, OK.

[Sam picks up an array card from the pile.]

### Sam

That looks promising.

[Sam turns it over. It’s 20.]

### Speaker

Oh, and he's done it again.

[Sam turns the 10 by 2 array card over, facing it down.]

### Speaker

Sam, I quite like that version, it makes you think about differently because you got to, you've got to think about what numbers are composed of, right? And you got to think about the product, so you're really using division, aren't you? Yeah, so over to you mathematicians. Have a go at playing that version of the game.

[Text over a navy-blue background: And another way!

The game board has been cleared.

The sheet now has an array card pack and a product card pack in the centre.]

### Speaker

Now, mathematicians, if you want more of a challenge and you'd like to extend the range of numbers that you're working on, you can use our expansion pack…

[The speaker holds out another pack of array cards and product card. She puts down the product card pack and flips though the array card pack.]

### Speaker

..And put in some extra partially covered arrays and their matching products…

[She puts down the array card pack, picks up the product card pack and flips through it.]

### Speaker

..Or we could remove the product cards…

[She takes both packs of product cards away. She rolls 2 dice with 10-sides]

### Speaker

..And roll two 10-sided dice instead.

Over to you, mathematicians.

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

### Speaker

So, what's some of the mathematics?

[A title on a white background reads: What’s (some of) the mathematics?…

· Games provide us with the opportunity to practice our mathematical skills and understanding.

o We used what we knew about the structure of arrays to imagine the dots hiding in the partially covered arrays. For example, we imagined dots hiding underneath the rectangle in this partially covered array to know there were 5 threes.

Below the points are two images: on the left side is an image of a partially covered array card with 5 dots running down and 3 across. On the right side is an image of 3 columns of 5 dots running down, with a pink thought bubble over it.]

### Speaker

Games provide us with the opportunity to practice our mathematical skills and understanding. We used what we knew about the structure of arrays to imagine the dots hiding in the partially covered arrays.

For example, we imagined dots hiding underneath the rectangle in this partially covered array to know there were five threes.

[A title on a white background reads: What’s (some of) the mathematics?…

· We were also practising making connections between different forms of representations.

o In this game, we made connections between diagrams and numbers. For example, we made connections between an array that showed 5 rows with 5 in each row (a diagram) and the product 25 (a number).

Below the points is an image of the game board with the 2 sets of product cards. Each set has 3 rows of 2 cards. The set on the left has a 25 product card in the bottom right-hand corner, which has been highlighted.

On the bottom left side of the game board is an array card with 5 dots running down and 5 across which has been highlighted.]

### Speaker

We were also practising making connections between different forms of representations.

In this game, we made connections between diagrams and numbers. For example, we made connections between an array that showed five rows with five in each row, a diagram and the product 25, a number.

[A title on a white background reads: What’s (some of) the mathematics?…

· We also used our knowledge of spatial patterns to quantify the number of rows and the number in each row. For example, we saw that we had three rows of two or three twos.

On the right-hand side of the points is an image of an array card with 3 dots running down with 2 dots across. The dots running down has a red outline. There is an arrow above the 2 dots across.]

### Speaker

We also used our knowledge of spatial patterns to quantify the number of rows and the number in each row. For example, we saw that we had three rows of two or three twos.

[A title on a white background reads: What’s (some of) the mathematics?…

· We also used what we knew about the commutative property to rotate the array. For example, we took an array that was structured as 5 twos and rotated it, so it became 2 fives. We knew that rotating the array wouldn't change the total, the product was still 10.

Below the points are 2 images. The image on the left is an array card with 5 dots running down and 2 across. The image on the right is an array card with 2 dots running down and 5 across.]

### Speaker

We also used what we knew about the commutative property to rotate the array. For example, we took an array that was structured as five twos and rotated it, so it became two fives. We knew that rotating the array wouldn't change the total, the product was still 10.

[A title on a white background reads: What’s (some of) the mathematics?...

· When we were playing the other way to play, we used what we knew about products having different factors to make a match. For example, turning over a product of 10 meant that the matching array could have been two fives, five twos, one 10 or 10 ones.

Below the point is an image of the game board with the 2 sets of array cards. Each set has 3 rows of 2 cards. On the bottom left side of the game board is 10 product card which has been highlighted.

On the game board, 3 cards have been highlighted: an array card with 2 dots running down and 5 across, another with 5 dots running down and 2 across and a card with 10 dots across.]

### Speaker

When we were playing the other way to play, we used what we knew about products having different factors to make a match.

For example, turning over a product of 10 meant that the matching array could have been two fives, five twos, one 10 or 10 ones.

[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]

## Instructions

1. Each player creates a gameboard using 6 array cards. Set aside the remaining array cards.

2. Place the descriptor cards in a pile, face down.

3. Turn over a descriptor card. If a player has the matching array card on their gameboard, they may turn the array card over.

4. If both players have the matching array card, they can both turn over their matching cards.

5. If neither player has the matching array card, turn over the next product card in the pile.

6. The winner is the first player to turn over all their cards and say ‘bingo!’

## Other ways to play

• Swap how the piles of cards are used in the game.

• Make a gameboard from the descriptor cards and turn over the array cards.

• Extend the number range by adding in the expansion pack (PDF 438.8 KB) and/or replacing the product cards with 2 x 9-sided dice.

## Discuss/reflect

• What strategies did you use to determine how many dots there are in the partially covered arrays?

• Were there any arrays which were known facts for you? Which ones?

• What strategies did you use for the arrays that weren’t known facts for you?