# 2 truths 1 lie (whole numbers)

A thinking mathematically targeted teaching opportunity focused on reasoning, using mathematical imagination and place value (renaming numbers).

## 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.

Adapted from Marian Small '*What should K-8 math look like?' *(2016, October) One Two Infinity.

## Outcomes

- MAO-WM-01
- MAE-GM-02

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

## Collect resources

You will need:

- paper
- pen or pencil
- multi-attribute blocks or MABs (if you have them or you can make or draw your own).

## 2 truths 1 lie (whole numbers)

Watch '2 truths 1 lie (whole numbers)' video (4:10).

[Text over a navy-blue background: 2 truths. 1 lie. 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

Two truths, one lie.

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

Bullet points below read:

- Eye balls and brains
- something to write on and write with
- MABs if you have them or you can make or draw your own

On the lower right-hand corner is an image of MABs.]

For this task, you will need your eyeballs and brains, something to write on and write with and MABs if you have them. Or you can make and draw your own.

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

Let's explore.

[A title on a navy-blue background reads: 2 truths. 1 lie. A text below the title reads: The number…

Bullet points below read:

- 13 can be represented with 4 MABs.
- 32 can be represented with 14 MABs.
- 25 can be represented with 10 MABs.]

Hello there mathematicians. I've got a nice, sweaty brain problem here for you. It's called two truths and one lie, and it's based on the work of Marian Small.

So the idea with this task is that there are three statements. Two of them are true and one is a lie. So we need to try and work out which one's which. So let's have a look at our statements. The number 13 can be represented with 4 MABs. The number 32 can be represented with 14 MABs. The number 25 can be represented with 10 MABs.

[A large white tabletop.]

You'd know MABs from school, they look a bit like this.

[The speaker places a 10s MABs in the top centre of the table. It is a long green stick that looks like 10 cubes stuck together.]

These are 10s…

[She places a 1s MABs next to the 10s. It is a small yellow cube.]

..and these are ones. At school, they're probably made out of wood though, but it's the same idea.

[She takes the MABs away.]

So if I'm thinking about the number 13, I can definitely think about 13 in terms of 13 ones.

[She lays out 5 rows of 2 1s, one below each.]

So two, four, six, eight, 10,

[Next to the first column on the right-hand side, she lays out a rows of 2 1s, and 1 1s below.]

12 and 13.

But I can use what I know about 10 in place value, and I could also think of 13 as one 10…

[On the right-hand side of the 1s, she lays down 1 10s.]

..and three ones.

[On the right-hand side of the 10s, she lays down 3 1s.]

So in the first example, I've got 13 MABs, 13 ones.

[She circles the 1s set with her finger.]

In my second example, I've got four MABs.

One is a 10 and three are ones, but I've got four MABs.

[She circles the 10s and 1s set with her finger.]

So we've just proven that the first statement is true.

[A title on a navy-blue background reads: 2 truths. 1 lie. A text below the title reads: The number…

Bullet points below read:

- 13 can be represented with 4 MABs.
- 32 can be represented with 14 MABs.
- 25 can be represented with 10 MABs.]

Let's take a look at that second statement. 32 can be represented with 14 MABs.

[The table has been cleared.]

So to start off, I think I'm just going to place...

[She puts down 3 10s.]

I'm gonna create 32 using standard partitioning, three 10s and two ones.

[On the right-hand side of the 10s, she puts down 2 1s.]

Now here that's only five MABs as you can see. So I'm looking for 14 MABs. So I'm going to have to rethink and rename this number. So I'll still keep two 10s…

[She puts down 2 10s.]

..and I'll still keep these two ones here.

[On the right-hand side of the 10s, she puts down 2 1s.]

But what I'm gonna do is I'm gonna rename this 10 as 10 ones.

[On the right-hand side of the 10s, she puts down 10 1s in a column.]

Now for this, I could count 10 out, or I can use direct comparison which is what I'm doing now. I'm lining them up.

Ok, so now, I have got two 10s. And I've got 10, 12 ones. Now how many MABs is that? Two and 10 is 12, 13, 14. 14 MABs.

[A title on a navy-blue background reads: 2 truths. 1 lie. A text below the title reads: The number…

Bullet points below read:

- 13 can be represented with 4 MABs.
- 32 can be represented with 14 MABs.
- 25 can be represented with 10 MABs.]

So we've just proven that the second statement is also true.

Now, because the challenge is called two truths and one lie, you could assume that that last statement is false.

[Text over a navy-blue background: Over to you!

Under the text are two questions: Can you prove that 25 cannot be represented with 10 MABs? Can you prove that the third statement is indeed false?]

And now over to you. Can you prove that 25 cannot be represented with 10 MABs? That is, can you prove that the third statement is indeed false?

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

So what's some of the mathematics?

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

A bullet point below reads:

- Numbers can be partitioned into standard and non-standard ways.

Below the point is a text that reads: 32 is…

Below the text are 2 sets of diagrams. On the left side is a diagram of 3 blue 10s and 2 green 1s, with text below that reads: 3 10s and 2 ones. On the right side is a diagram of 2 blue 10s and 12 green 1s, with text below that reads: 2 10s and 2 ones.]

Numbers can be partitioned into standard and non-standard ways. 32 is three 10s and two ones. It's also two 10s and 12 ones.

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

A bullet point below reads:

- You can use your mathematical imagination to help you to solve problems.

Below the point is a red thought bubble. Inside the thought is text that reads: .I’m going to imagine that ten as 10 ones. Below the text is an image of 10 yellow 1s]

You can use your mathematical imagination to help you to solve problems. In this problem, I imagined a 10 as 10 ones.

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

A bullet point below reads:

- You can also use direct comparison as a way to determine the size of collections

Below the point is an image of 2 green 10s on their side on top of 10 yellow 1s put together.]

You can also use direct comparison as a way to determine the size of collections. You don't always have to count.

[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]

### Discuss and reflect

Can you prove that 25 cannot be represented with 10 MABs?

Create your own '2 truths 1 lie' problem and challenge a friend, family member or classmate to solve it!