# Rekenreks 1 (part-part-whole)

A thinking mathematically targeted teaching opportunity focussed on relationships for adding and subtracting with benchmarks and part-part-whole concepts

## 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
• MAE-RWN-01
• MAE-RWN-02
• MAE-CSQ-01
• MAE-CSQ-02

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

You will need:

## Watch

Watch Rekenreks 1 video (11:48).

Investigate representing numbers on a rekenrek.

### Transcript of Rekenreks 1 video

(Duration: 11 minutes and 48 seconds)

[Title over a navy-blue background: Rekenreks 1. In the lower left-hand corner of the screen is the waratah 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

### Speaker

Hello, mathematicians. To get started today, you'll need a pencil, your workbook and some paper, your imagination, and also your rekenrek. If you don't have those things yet, click pause now and go get them and then come back and get started. OK, here we go. ,

[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 you made. It might look a little bit like this or it might look different…

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

### Speaker

..and here's one I made on the computer, and we're gonna use the one I drew on the computer to help us today. OK.

[Text over a blue background: Last time we noticed…]

### Speaker

So, last time when we were looking at rekenreks, we noticed some really cool things.

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

* Bigger numbers are made up of smaller numbers.

4 is…

A table below shows different combinations of smaller numbers which together make 4. Pictures illustrate how this can be shown using a rekenrek. Beside the number combination 4 and 0, a rekenrek has 4 red beads on the top line pushed to the left; no red beads on the bottom line are on the left. Beside the number combination 3 and 1, 3 red beads on the top of the rekenrek and one red bead on the bottom of the rekenrek are pushed to the left. Beside the number combination 2 and 2, 2 red beads on the top and the bottom of the rekenrek are pushed to the left. Beside the number combination 1 and 3, one red bead on the top of the rekenrek and 3 red beads on the bottom of the rekenrek are pushed to the left. Beside the number combination 4 and 0, no red beads on the top of the rekenrek are pushed to the left; 4 red beads on the bottom of the rekenrek are pushed to the left.]

### Speaker

They helped us see that bigger numbers are made up of smaller numbers. Like we could see four being made up of three and one or two and two or one and three.

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

* When each chunk of colour represents 5…

An image of a rekenrek below shows the 5 red beads on the top row pushed to the left side.

Text: … we can make 4, by thinking about ‘1 left behind’.

An image of a rekenrek below shows 4 red beads pushed to the left, with one red bead remaining on the right.]

### Speaker

And we also noticed that each chunk of colour represents five on the rekenrek and that means that we could think about four, for example, by leaving one bead behind. The one left behind strategy. OK. We're gonna use this knowledge today to help us with today's challenge.

[Text on a blue background: Today’s challenge.

On a white slide, a numeral 7 appears in a blue circle. Below is an image of a rekenrek. All of the beads on both rows are on the right side. Text appears on the slide: 1 or 2 slides only.]

### Speaker

OK. Here is our target number, seven, and we're gonna try to make it in one or two slides only.

So, if you have your rekenrek there, you can have a go at this with me too because here's what I was thinking.

[On the top row of the rekenrek, one red bead moves to the left, followed by another red bead, then another red bead, then another red bead. Text at the bottom of the slide reads: 1 and 1 and 1 and 1.]

### Speaker

I could move across one bead and then another one. So, now I have one and one and one, and oh, yes, nice spotting, mathematicians. I can't do that because that's been four slides and we need to try to do it in one or two slides only. OK, let's push the beads back.

### Speaker

OK, let's have a think together. I know, what about if we slide across one chunk of five? You do it with me. Ready?

[The five red beads on the top row of the rekenrek slide to the left. Text at the bottom of the screen reads: 5. 2 yellow beads on the top row slide to the left. The text at the bottom of the screen now reads: 5 and 2. Additional text on the slide reads: 7 is…]

### Speaker

One chunk of five, and then we would need... Yes, two more. OK. That's right, seven is five and two. Yes. And you might have had five and two on the top row or five on the top row and two down the bottom. Still seven, that's right, because seven is five and two.

### Speaker

Alright, what's another way? Can you think of one? Oh, is this what you were thinking? Let's see, what about double three?

[3 red beads on the top row of the rekenrek, and 3 red beads on the bottom all slide to the left. Text at the bottom of the slide reads: Double 3. The action is repeated.]

### Speaker

Can you make that? Double three in one slide? Do you want to see it again? Ready? Double three and…

[One red bead on the bottom row moves to the left. The text at the bottom of the screen reads: Double 3 and 1 more. Additional text on screen reads: 7 is…]

### Speaker

..yes, one more. Double three and one more, that's seven.

Alright, I wonder if there's another way.

[The 3 beads on the top row slide back to the right. The speaker repeats the action.]

### Speaker

Oh, hey, I just noticed something, mathematicians, look. As those beads slide back, yes, it's making me think of another way to make seven. Look, seven is three and four. Can you make that too on your rekenreks?

[3 red beads on the top row of the rekenrek and 4 red beads on the bottom row slide to the left. Text at the bottom of the slide reads: 3 and 4. Additional text reads: 7 is…]

### Speaker

Yes, seven is three and four. OK.

[Text over a blue background: Let’s imagine and think!]

### Speaker

Now we're gonna use our imaginations. So, before we touch the rekenrek, we're gonna imagine it first in our mathematical imaginations, in our mind's eye. So, let's think first.

[On a white slide, is an image of a rekenrek below a numeral 4. Additional text on screen: 1 or 2 slides only.]

### Speaker

Our target number is four, and we wanna try to move it across in just one or two slides.

### Speaker

So, imagine your rekenrek, mathematicians, and imagine you got your big pointy finger up in the air, and imagine moving across four beads in just one or two slides. OK, and show me what that looks like in the air. OK, and I'm going to try to read your mind. Did you make four like this…

[2 red beads on the top row of the rekenrek 2 red beads on the bottom row slide to the left at the same time. Text below reads: Double 2. Additional text: 4 is…]

### Speaker

..as double two in one slide? No? Yes. For some of you yes, but for some of you, I have to keep trying.

[The beads return to their original position. 3 red beads on the top row of the rekenrek slide to the left. Text at the bottom of the screen: 3. One red bead on the bottom row of the rekenrek slides to the left. Additional text at the bottom of the screen reads: and 1.]

### Speaker

OK, did you make four like this, as a slide of three and a slide of one?

Yes, I read some of your minds, but some of you, not quite yet.

[The beads return to their original position. The 3 beads on top slide to the left again, and then back to the right.]

### Speaker

Alright. I know, and it's an actually amazing three. Look, if I slide this back here, three is really quite amazing because it's a quantity I can subitise, which means I can see how many it is without having to count.

[Everything on screen disappears except for a group of 3 red beads.]

### Speaker

Look, here in my screen, yeah, you can see three circles.

[4 more red beads appear on screen.]

### Speaker

But if I put out this many, our brains go, "Oh my gosh, I'd need to count those to work out how many." So, we can subitise three. It's nice for us to be able to use that.

### Speaker

OK, is this how... Yes, I think I've read some of your minds.

[4 red beads on the top row of the rekenrek move to the left. Text below: 1 less than 5.]

### Speaker

Some of you thought about four being one less than five and you used the one left behind strategy.

[The remaining red bead on the top run of the rekenrek slides to the left, then slides back to the right.]

### Speaker

Look, because if I bought all the red ones over, that would be five, but the number before five is four. So, you just left one behind. Four is one less than five.

[The 4 red beads on the top row slide back to the right.]

### Speaker

Oh, did I do a good job at reading your minds, or are there still some I missed? I know, there's lots of different ways. OK, let's try another number. Ready?

[A numeral 11 appears on screen. The rekenrek partially fades away.]

### Speaker

Alright, this time we're going to try to move 11.

So, we're imagining this in our mind's eye. So, imagine the rekenrek in your brain and imagine moving across 11 beads in just one or two slides. Oh, OK. And use your finger, and you might even talk out aloud or describe your thinking to someone who's home with you. And how are you moving the beads? OK, now show me on your rekenrek in one or two slides. OK, let's have a look together. Let's see if I could read your mind. Ready, mathematicians?

[Text appears on screen: 11 is… All of the beads on the top row of the rekenrek slide to the left. Text at the bottom of the screen reads: 1 ten. One red bead on the bottom row slides to the left. Additional text at the bottom of the screen: and 1 more.]

### Speaker

Did you think about 11 as one whole row of ten and one more? Oh, some of you did think about it that way. OK.

[The beads return to their original position and the text at the bottom of the screen disappears.]

### Speaker

Some of you thought about it differently again.

[All of the red beads on both rows slide to the left. Text at the bottom of the screen reads: Double 5. One yellow bead on the top row slides to the left. Additional text at the bottom of the slide reads: and 1 more.]

### Speaker

Some of you thought about 11 as double five and one more.

### Speaker

Oh, yes, I got those ones, but there are still some mystery brains out there to me. OK, let's see if I can get you on the next number.

[A numeral 18 appears on screen. The rekenrek partially disappears.]

### Speaker

So, here comes our next target for show me, 18. OK, mathematicians, imagine your rekenrek now. Close your eyes, imagine it in your mind's eye, and imagine moving 18 beads across. It's a lot of beads. I wonder how you could do that in just one or two slides.

When you think you've got an idea, imagine your finger moving the beads along. And if someone's home with you, talk to them out aloud, tell them what you would do. OK. Now check your strategy on your rekenrek. Does it work? Oh, some of you are revising your thinking. Nice work, mathematicians. One or two slides. OK, I'm gonna see now if I can read some of your mathematical minds.

[All of the beads on the top row of the rekenrek slide to the left. Text at the bottom of the slide reads: 1 ten.]

### Speaker

So, some of you, I think, thought about 18 as one slide of ten and then leaving two behind.

[8 beads on the bottom row of the rekenrek slide to the left. 2 beads are left behind. Text at the bottom of the screen reads: 1 ten and 2 less than 10.]

### Speaker

So, and then two less than ten to make eight. One ten, and eight.

[The 10 beads on the top row slide back to the right.]

### Speaker

Yes. And look, see, if we brought those...

[Text on the bottom of the screen reads: 2 less than 10. The 2 leftover beads on the bottom row join the others on the left. The 2 beads return to the right side. The other 8 beads also return to the right side of the rekenrek.]

### Speaker

It's two less than ten here because if we brought those beads across, that would be ten, but we left two behind so that makes it eight. OK, some of you thought about it differently, though. Some of you thought about 18…

[9 beads on the top row of the rekenrek slide to the left. Text at the bottom of the screen reads: 1 less than 10.]

### Speaker

..as one left behind on the top row…

[9 beads on the bottom row of the rekenrek slide to the left. Additional text at the bottom of the screen reads: 1 less than 10.]

### Speaker

..and one less than ten on the bottom row, one left behind. And some of you thought about that same movement but called it a double nine.

[The text at the bottom of the screen reads: Double 9.]

### Speaker

And that is 18, yes. And I know that because from 18, I need just two more to get to 20.

Yes, because from eight I need two more to get to ten and 18 is just eight and one ten more.

### Speaker

[8 beads on the top row slide to the left. Text at the bottom of the screen reads: 2 left behind… All 10 beads on the bottom row slide to the left.]

### Speaker

So, two left behind in the top row and then none left behind in the bottom row. That's 18 because it's two less than 20.

### Speaker

Was I able to read your mathematical minds? Sometimes. I like this. Sometimes is a good success rate for me.

[Text over a blue background: Your challenge…]

### Speaker

OK. So, mathematicians, here is your challenge.

[Text on a white background: Your challenge…

* Draw pictures to show how you can make 9, 6, and 13 in just 1 or 2 slides. This of two different ways for each number.]

### Speaker

Can you draw pictures to show how you can make 9, 6, and 13 in just one or two slides and think of two different ways for each number.

[An image appears below. It depicts a hand-drawn rekenrek within a table. The first column is headed ‘Number’. In the cell below is the number ‘9’. The second column is headed ‘first slide’. The cell below features a rekenrek with 5 beads on the top row pushed to the left. Below the rekenrek is the number ‘5’.]

### Speaker

So, for example, here's my first slide for making the number nine, and I moved five across.

[Another column appears in the table. It is headed ‘second slide’. Below is an image of a rekenrek with 5 beads on the top row pushed to the left, and 4 beads on the bottom row pushed to the left. Below the rekenrek is a number 4.]

### Speaker

And in my second slide, I moved across four. And that's one way to make nine.

[Another row appear in the table. In the first cell, is the number ‘9’. In the second cell is a rekenrek, with 9 beads on the top row pushed to the left. Text below reads: 1 less than 10, 9 = 10 – 1. In the third cell is another rekenrek, which has been crossed out.]

### Speaker

A second way that I could make nine is to think about one less than ten. Yeah, and that was only one slide. And if you'd like to add different numbers than 9, 6, or 13, feel free.

[Text over a blue background: Over to you!]

### Speaker

OK, mathematicians, over to you.

[What’s (some of) the mathematics?]

### Speaker

So, what's some of the mathematics here?

[Text on a white background: What’s (some of) the mathematics?

* Thinking of different ways to slide the beads across helps us think about different relationships..

Text in a blue star: 4 is…]

### Speaker

Thinking of different ways to slide the beads across helps us think about different relationships.

[A rekenrek appears below. 2 beads on each row have been pushed to the left. Text below: Double 2.]

### Speaker

So, when we were looking at four, we knew things about four like it's double two.

[Another rekenrek appears below. 4 red beads on its top row have been pushed to the left. Text below: 1 less than 5.]

### Speaker

We also knew that four is one less than five…

[Another rekenrek appears below. 3 red beads on its top row, and one red bead on the bottom have been pushed to the left. Text below: 3 and 1.]

### Speaker

..and we could also talk about four being three and one more. Yeah.

[Text: * We can think about numbers in lots of different ways.]

### Speaker

And so, this really helped us see that we can think about numbers in lots of different ways. OK, mathematicians, until we meet again. 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 imaginations!

• Use a rekrenrek to make the numbers 9, 6 and 13 in just one or two slides.

• Think of two different ways for each number.

• Draw pictures to record the ways to represent each number.