# Duck's away follow up (combinations of 5)

ES1 – A thinking mathematically targeted teaching opportunity focussed on developing number sense and early additive strategies

## 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

## Before we begin

If possible, watch the reading of ‘Ducks Away’ by Mem Fox available on ABC iview or read the book if you have it.

## Watch

Watch the video to explore how smaller numbers are nesting inside larger numbers (7:20).

Explore combinations of 5 using the story 'Ducks Away'

### Speaker

[Screen reads: Ducks away – follow up. First, watch Emma Watkins read Ducks Away by Mem Fox. You can view this on ABC iView as part of their education program.]

Hello mathematicians! This lesson today follows up from the reading of 'Ducks Away' by Emma Watkins on ABC iView.

It's OK if you haven't seen it. You can still join us, but if you'd like to go watch that first, press pause now and go do that and then come back and join us. See you soon!

Welcome back young mathematicians. I hope you really enjoyed watching the story of 'Ducks Away'.

It's OK if you haven't seen it yet, we can still play around with some really interesting ideas in the story.

[Screens shows wooden toy bridge. The presenter traces along the bridge with her finger.]

What happens is a mother duck goes for a walk with her young ducks and her ducklings and they walk across a bridge and look, here's my representation of the mother duck, “Quack, quack, quack, quack, quack, quack.” And she starts walking along the bridge and along come her little baby ducks, her ducklings.

“Quack, quack, quack…” one baby duckling. “Quack, quack, quack, quack…” a second baby duckling. “Quack, quack, quack, quack…” Her third baby duck, the duckling. “Quack, quack, quack, quack…” a fourth duckling, and… “quack, quack, quack, quack, quack…”, a fifth duckling.

[Screen shows a wooden toy bridge with six yellow counters on it. The first counter, closest to the right of the bridge is larger than the rest and has an eye on it. This represents the mother duck. Following the mother duck, there are 5 identical counters of the same size with eyes on them as well. These represent the 5 baby ducklings.]

And then what happens is these poor ducklings, one of them gets caught up by the wind and splashes into the water, down below, and then all of the other ducklings start to get really curious.

And they look over the edge of the bridge and splash into the water and then the next duck looks over the bridge and you guessed it, splash! Into the water and it happens with the next duck splashes in and finally, the last little duckling left on the bridge loses its footing and tumbles into the water down below.

[Screen shows the presenter picking up the last small counter, twirling it before placing it underneath the bridge. As she tells the story, she does the same with the other 4 counters, showing how they fall into the water, like the ducklings did in the story. The presenter then removes the largest counter from the screen.]

Let's count them to check 1, 2, 3, 4, 5. So she has 5 ducklings, but sometimes she has 5 of them on the bridge and none down below. Sometimes she has 4 of them on the bridge and one down below.

[Screen shows the presenter with the 5 yellow counters on the bridge. She then moves each counter, one by one under the bridge, noting how many counters are above and how many are below.]

Sometimes she has 3 of them on the bridge and 2 down below. Sometimes she has 2 of them on the bridge and 3 down below.

Sometimes she has one of them on the bridge and 4 down below and by the end of the story, all of her 5 ducks, her ducklings, are in the water down below.

[Screen shows presenter moving the final counter below the bridge to join the other 4 counters.]

And what this made me think about is it's a bit like when we count, that even once we know that we have 5, it didn't matter if some of them were on top of the bridge or below in the water, we still know and can trust that there's 5.

[Screen shows the presenter moving 2 counters to the top of the bridge. She then moves the remaining 3 counters to the top of the bridge.]

So, let's have a look at what this tells us about the number 5, 'cause I can now start to think like a mathematician and record what's happening.

So, at the moment there's 5 ducks at the top, and there's none down the bottom, so we know that 5 and zero is equivalent to 5, so I can say 5 is 5 and zero, and then one tips over the edge.

[Screen shows the presenter writing under the bottom right of the bridge. The presenter writes the heading ‘5 is’ and creates a bullet point underneath that says 5 and 0. She then moves one counter underneath the bridge.]

And now I still know that there's 5 ducks altogether, but this time I can see my 5 as 4 and one more, so 5 is 4 and one.

And then another one tumbles over the edge and now I still know that there's 5 'cause it doesn't matter that they have moved, but this time I can see 5 as... yes, 3 and 2. And then what happens next in the story?

Yes! Another duck falls over into the water and how many ducklings are there altogether?

[Presenter creates another bullet point underneath the first one. The new bullet point says 4 and 1. She then moves another counter under the bridge so that there are 3 counters on top and 2 underneath. She creates a new bullet point and writes 3 and 2. The presenter then moves another counter under the bridge so that there are 2 counters on top and 3 underneath. She creates a new bullet point and writes 2 and 3.]

Yeah, 5 because it doesn't matter that they have moved, the quantity of the total number of ducklings is still 5. And there's how many on top of the bridge? 2 and how many down below? 3.

So, 5 is 2 and 3, and then another one falls off and what happens now? How can we see the 5 ducklings?

That's right, they're partitioned, aren't they? There's one on top of the bridge, one, and there's 4 down below.

[Presenter moves another counter underneath the bridge. There is now one on top of the bridge, and 4 underneath. She creates a new bullet point and writes 1 and 4.]

And then the very last duckling tumbles into the water. And what does that tell us now about 5?

[Presenter moves the final counter underneath the bridge. There are now no counters on top of the bridge and 5 below. The presenter creates a new bullet point and writes 0 and 5.]

That's right, there's no ducks on top and then 5 ducks down below. And so, what we could see in our story 'Ducks Away', is it told us some things about 5 that 5 is 5 and zero, here's 5.

[Screen shows the presenter moving the bridge off screen. She moves a pink piece of paper onto the left side of the screen and places the 5 counters onto it. The presenter also places an empty paper plate onto the table.]

There are none. But 5 is also... 4 and one more, but I still have 5 ducks altogether. That 5 is... 3 and 2.

[Presenter moves one counter to the plate. She signals between the single counter, and the group of four counters on the pink sheet of table. She then moves another counter onto the plate so that there are 2 on the plate, and 3 on the pink piece of paper. She circles both groups.]

But I still have 5 altogether, that 5 is 2 and 3. Yes, and I still have 5 altogether. That I can also move one more across. And now I have one and 4.

[Presenter now moves another counter onto the plate, so that there are 3 counters on the plate and 2 on the pink sheet. The presenter points to the bullet point which reads 2 and 3. She then moves another counter onto the plate so that there are 4 counters on the plate and one on the paper. The presenter points to the bullet point which reads one and 4.]

And if I move the last one across. That's right, I could say that 5 is zero and 5.

[Presenter moves the last counter onto the plate and points to the bullet point which reads zero and 5.]

This book really helps us see the small numbers that nest inside the bigger ones. What a great mathematical thing to discover!

So, as we like to ask, what was the maths?

So, the story of ducks away helped us uncover some really important mathematical ideas. It showed us that 5 can be made up or composed in lots of different ways.

For example, you can have 5 ducklings, when one is on the bridge and there's 4 ducklings in the water below, so one and 4 still makes 5 ducklings. We also started to realise that smaller numbers like 4, 3, 2, and one can be found inside of 5.

This is a really tricky idea, and it's something that we will come back and explore as time goes on. Looking forward to the next time we get together mathematicians!

[End of transcript]