# Rekenreks 2 (doubles)

Rekenreks 2 is a thinking mathematically targeted teaching opportunity focussed on exploring doubles and near doubles using a rekenrek.

## 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

## Outcomes

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

- MAO-WM-01
- MA1-RWN-01
- MA1-RWN-02
- MA1-CSQ-01

## Watch

Watch Rekenreks 2 video (10:40). You will explore doubles and near doubles using your rekenrek.

[Title over a navy-blue background: Rekenreks 1. In the lower left-hand corner of the screen is the waratah and text of the NSW Government logo. Small font text in the upper left-hand corner reads: NSW Department of Education.

Text on a white background: You will need…

- a pencil
- your workbook or some paper
- your imagination
- your rekenrek
- 10-frame cards

On the right of the slide are images of 10-frame cards. There are 6 cards in total. Each card features 4 even rows of 5 squares. Each of the squares have either a black dot, a white dot, or no dot. The first card has 20 dots. The number of dots in each subsequent card lowers progressively by one.]

### Speaker

Hello there, mathematicians. I hope you're having a delightful day today. You will need a pencil, your workbook or some paper, your imagination, your rekenrek, and also these ten-frame cards. If you don't have these ones exactly, that's OK. You can use other cards like it, as long as you have the numbers represented on a ten-frame from zero through to 20. If you don't have them yet, it's OK. We can start together and then later, when we come to the game, you can go get those things, OK.

[On a white background, is an image showing a rekenrek. It is an apparatus with 20 coloured beads threaded along 2 skinny stick. Each of the sticks are held in place, at either end, by 2 pegs. The pairs of pegs at each end are attached to a paddle pop stick. Each row of the rekenrek has 5 red beads and 5 blue beads.]

### Speaker

So, you might have a rekenrek that you made at home or at school. It might look like this. It might look like something different.

[An image of a computer-generated rekenrek appears below. It features 2 blue rungs, each with 5 red beads and 5 yellow beads.]

### Speaker

Today, to help us, we're going to use a rekenrek that's similar to that one but we made it on the computer, OK.

[Text over a blue background: What do you notice? The computer-generated rekenrek appears on a white background.]

### Speaker

So, mathematicians, over to you to start thinking about, what do you notice when we start playing with our rekenreks today? Here we go.

[One red bead on each row slides to the right.]

### Speaker

How many beads do I have?

[Text in a blue star reads: 2 is…]

### Speaker

Two. And how could I describe them?

[Text at the bottom of the slide: 1 and 1.]

### Speaker

Yes. One way I could describe them is one and one. Is there another way I could describe two?

[Text at the bottom of the slide: Double 1.]

### Speaker

As double one. Two is double one. OK.

[The beads return to their original position on the right. 2 red beads on each row slide to the left.]

### Speaker

What about now? How many can you see?

[Text in a blue star: 4 is…]

### Speaker

Four. And how could I describe four? Four is... You say it. Four is... Oh, yeah.

[Text at the bottom of the slide: 2 and 2.]

### Speaker

Some of you said two and two. Is there another way that I could describe four? Four is...?

[Text: double 2.]

### Speaker

Double two. OK. What about now?

[Another red bead on each row slides to the left. Text in a blue star: 6 is… Text at the bottom of the slide: 3 and 3.]

### Speaker

Yes. There are six, and six is three and three. But I can also describe six as...?

[Text at the bottom of the slide: double 3.]

### Speaker

Yes. Double three. So, I think we might have a working definition of the word double.

[Additional text on screen: A ‘double’ describes two collections that have the same quantity.]

### Speaker

A double describes two collections that have the same quantity because there's three at the top, that's one collection, and three at the bottom, that's my second collection. Let's see if that definition stays.

[Another red bead on each row slides to the left.]

### Speaker

How many beads are there now altogether? There are eight…

[Text in a blue star: 8 is… additional text below reads: 4 and 4.]

### Speaker

..and I can see eight as four and four. And that means there's four at the top and four at the bottom, which means eight is…

[Text: double 4.]

### Speaker

..a double. Eight is, in fact, double four. Yeah, OK.

[An additional red bead on each row slides to the left. Text in a blue star: 10 is…]

### Speaker

What about now? Yes, you can see the pattern, ten.

[Text: 5 and 5.]

### Speaker

And ten is five and five, there's five at the top and five at the bottom.

They are my two collections and they both have the same quantity. So, I can describe five and five as…

[Text: double 5.]

### Speaker

..double five, which is ten. OK. What about now?

[One red bead on the bottom row slides back to the right side. Text in a blue star: 9 is… Text at the bottom of the slide: 5 and 4.]

### Speaker

I have nine, and nine is five and four.

[One red bead on the top row moves slightly t the right, away from the others. Text at the bottom of the slide: double 4 and 1 more.]

### Speaker

And I do have double four. Can you see that? And one more.

[One red bead on the bottom row slides to the left, pauses, then slides back to the right. Text: double 5 and 1 less.]

### Speaker

I could also say, double five and one less. But it's no longer a double, is it? Because there's not exact same amount in both collections. One has five and one has four, or, yeah, one has five and one had five and one was taken away to leave four.

[The text, “A ‘double’ describes two collections that have the same quantity” disappears from screen. The right side of the rekenrek disappears from screen, leaving only the 9 red beads on the left.]

### Speaker

So, it no longer is a double. Now I can think about this as an almost double.

[Text: A near-double is ALMOST a double. It’s like the double is hiding… there’s just 1 more or 1 less than a double. The right side of the rekenrek returns to screen.]

### Speaker

We call that a near-double because a near-double is almost a double. It's like there's a little double in there and it's hiding. There's just one more or one less than a double. Yeah. OK. 'Cause we can see the double four and one more. And we could also see the double five and one less.

[2 red beads on the top row slide all the way to the right. One red bead on the bottom row slights slightly to the right, away from the others on the left side of the rekenrek.]

### Speaker

Let's see. What about this one? How many beads are there now?

[Text in a blue star: 7 is…]

### Speaker

Yes, there's the seven.

[Text at the bottom of the slide: double 3 and 1 more.]

### Speaker

And I can see the double three hiding in there. Double three and one more.

And I can also imagine...

[A faded red bead slides from the right side of the rekenrek to the left.]

### Speaker

Yes. Now I would have double four and one less. OK. What about this one?

[All of the red beads on the left slide to the right. 5 beads on the top row and 5 beads plus one yellow bead on the bottom row all slide to the left.]

### Speaker

Can you make this one on your rekenrek? Make the same, so it matches. And how many beads are there that we have? 11.

[Text in a blue star: 11 is…]

### Speaker

And how could we describe 11?

[Text at the bottom of the slide: double 5 and 1 more.]

### Speaker

It's double five and one more. What's another way that we could describe it as a double, or in relationship to a double?

[On the top row, a faded yellow bead slides to the left.]

### Speaker

Double six and one less…

[The bead slides back to the right.]

### Speaker

..'cause you can imagine that other dot, if it was there, that bead, and then it going away. OK.

[The beads all slide back to the right of the rekenrek. The text, “A near-double is ALMOST a double. It’s like the double is hiding… there’s just 1 more and 1 less than a double” disappears from the screen.

Text over a blue background: I noticed something…]

### Speaker

You know, when we were doing this, I think I noticed something. Let's see if you noticed it, too.

[Over a white background: You can describe doubles in 1 way…

Below are two rekenreks. Text in a blue star above the rekenrek on the left: 10 is… On the rekenrek, 5 beads on both the top and bottom row are pushed to the left. Text below: double 5. Text in a blue star above the rekenrek on the right: 8 is… On the rekenrek, 4 red beads on both the top and bottom row have been pushed to the left. Text below the rekenrek: double 4.]

### Speaker

That when we describe doubles, we can describe them in one way. Like ten is double five. That's one way. Eight is double four.

[Text: You can describe near-doubles in 2 ways…]

### Speaker

But when we're talking about near-doubles, we can describe them in two ways.

[A rekenrek appears below. Text in a blue star above the rekenrek: 7 is… On the rekenrek, 3 red beads on the top row and 4 red beads on the bottom row have been pushed to the left. Text below: double 3 and 1 more, double 4 and 1 less.]

### Speaker

Look, seven is double three and one more, or it's double four and one less.

[Another rekenrek appears beside the previous one. Text in a blue star above the rekenrek: 9 is… On the rekenrek, 5 red beads on the top row and 4 red beads on the bottom row have been pushed to the left. Text below: double 4 and 1 more, double 5 and 1 less.]

### Speaker

Nine. Yes. Double four and one more. Double five and one less. OK.

[Text over a blue background: Your challenge…]

### Speaker

So, here's our challenge today, mathematicians.

[On a white slide, is an image of a stack of blank 10-frame cards and a rekenrek.]

### Speaker

This is where you need your ten-frame cards.

You need them cut up and in a pile and you need your rekenrek. And we're going to turn over a card on our pile…

[A frame, with 2 rows of squares, appears on the top card. The top row has 5 black dots, and the bottom row has 2 white dots and 3 blank squares.]

### Speaker

..and what quantity does that card represent? Seven. Yes, I can see five at the top and two on the bottom, and seven is five and two. And now you're going to have to think about representing seven on your ten-frame, on your rekenrek, as either a double or a near-double. Can you make it now on your rekenrek? Make a collection of seven as a double or near-double. OK. Let's do it together.

[A numeral 7 appears on a blue star. On the rekenrek, 4 red beads on each row move to the left. One red bead on the bottom row slides back to the right.]

### Speaker

So, we know there's seven and we might move across four, because double four is eight, and get rid of one to be seven.

[Text at the bottom of the slide: “7 is a near-double. You can double 4 and take 1 away.”]

### Speaker

Seven is a near-double. You can make double four and take one away. Oh, you made it a different way. Yes. You could see the double three hiding in there and the one more to make seven.

[Text: “7 is a near-double. You can double 3 and add 1 more.”]

### Speaker

So, we could also describe it as seven is a near-double and you can double three and add one more.

[The image on the 10-frame card disappears. The red beads slide back into place on the right side of the rekenrek.]

### Speaker

OK, move your beads back across. Let's turn over our next card.

[A frame appears on the top car. It features 4 rows. The top row has 5 black dots. The second row has 5 white dots. The third row has 2 black dots, and 3 blank squares. The fourth row has 5 blank squares.]

### Speaker

Oh, what quantity do we have here?

One whole ten-frame and two more. And one ten and two we rename, yes, as 12.

[A numeral 12 appears in a blue star.]

### Speaker

OK. Can you make 12 on your rekenrek, and represent it as either a double or a near double? Yes. Let's have a look together.

[On each row of the rekenrek, 5 red beads and one yellow bead slide to the left. Text below: “12 is a double. It is double 6”.]

### Speaker

Did you have double six? Yes. 12 is a double. It is double six.

[The image on the 10-frame card disappears. The beads on the rekenrek slide to the right.]

### Speaker

OK, let's try one more together. Ready?

[A frame appears on the card on top of the pile. It has 2 rows. The first row has 5 black dots. The bottom row has 4 white dots and one blank square.]

### Speaker

What quantity does this represent?

[A numeral 9 appears in a blue star.]

### Speaker

It is nine, 'cause there's one less than ten, which is nine. Alright, and can you make nine as a double or near double on your rekenrek? Did you think about it like this?

[On the rekenrek, 4 red beads on each row slide to the left. One red bead on the top row slides to the left.]

### Speaker

Nine is double four and one more.

[Text: “9 is a near-double. You can double 4 and add 1 more.”]

### Speaker

Oh, you thought about it as nine is…

[On the bottom row of the rekenrek, a faded red bead slides to the left, then returns to the right. Text: “9 is a near-double. You can double 5 and take 1 away.”]

### Speaker

..double five and take one away. Yeah. So, again, there were two ways to describe a near-double. Alright, mathematicians, over to you to continue playing this game.

[Text on a blue background: Over to you!

Text: What’s (some of) the mathematics?]

### Speaker

So, what's some of the mathematics here?

[Text: *Doubles’ are when there are 2 collections that have the same quantity. Below, are 2 rekenreks. On the rekenrek on the left, 5 red beads on each row have been pushed to the left. On the rekenrek on the right, 3 red beads on each row have been pushed to the left.]

### Speaker

So, a double is when there are two collections that have the same quantity. So, double five and double three.

[Text: * Near-doubles are ‘almost doubles’. There is a double hiding in the number… you just need 1 more or 1 less to see the double. Below, are two rekenreks. On the rekenrek on top, 3 red beads on the top row and 2 red beads on the bottom row are pushed to the left. On the rekenrek on the bottom, one red bead on the top row has been pushed to the left; no red beads on the bottom row have been moved.]

### Speaker

A near-double is almost a double. There's a double hiding in the number. You just need one more or one less.

[Text: * You can describe near-doubles in two ways…]

### Speaker

And you can describe near-doubles in two ways. That's cool, isn't it?

[A rekenrek appears. Text in a blue star above the rekenrek: 7 is… On the rekenrek, 3 red beads and 4 red beads have been pushed to the left. Text below: double 3 and 1 more, double 4 and 1 less.]

### Speaker

You can describe it like seven as double three and one more. You could also describe it as double four and one less. What are two ways to describe nine as a near-double? Are you imagining? Good strategy. Or you might even be making it on your rekenrek. That's a good strategy, too.

[Another rekenrek appears. Text in a blue star above the rekenrek: 9 is… On the rekenrek, 5 red beads and 4 red beads have been pushed to the left. Text below: double 4 and 1 more, double 5 and 1 less.]

### Speaker

Yes. Double four and one more or double five and one less. OK, mathematicians. Until the next time we gather. Have a great day.

[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

Follow along with the video, watching carefully and using your mathematical imagination to explore doubles and near doubles using a rekenrek.

Create doubles and near doubles:

Choose a card from a pile of shuffled 10-frame cards.

Using your rekenrek, represent the number displayed on the card as a double or near double.

Think about how you can describe the number. For example:

4 could be described as 'double 2'.

5 could be described as 'double 2 and 1 more' or 'double 3 and 1 less'.

Record your thinking.