# 2 truths 1 lie – reasoning in Stage 3

Stage 3 – A thinking mathematically targeted teaching opportunity to explore, reason and use our mathematical imagination to partition quantities in standard and non-standard ways

Adapted from Marian Small *What should K-8 math look like? (2016, October)*

## 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
- MA3-RN-01

## Collect resources

You will need:

something to write on

something to write with.

## Watch

Watch 2 truths 1 lie Stage 3 video (9:26).

(Duration: 8 minutes and 34 seconds)

[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

2 truths, 1 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.]

### Speaker

For this task, you will need the eyeballs and brains and something to write on and write with.

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

### Speaker

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:

· 68 can be represented with 41 MABs.

· 32 can be represented with 15 MABs.

· 136 can be represented with 28 MABs.]

### Speaker

Hello there, mathematicians. I've got a nice, sweaty brain problem here for you today. It's called 2 Truths, 1 Lie, and it's an adaptation of the work of Marian Small. So on my screen here, you'll see three statements. Two of them are true, one is a lie. Our job today is to find out which is the lie and to prove it. So let's look at these statements. The number 68 can be represented with 41 and MABs. The number 32 can be represented with 15 MABs. The number 136 can be represented with 28 MABs.

[The first bullet point is outlined.]

### Speaker

Now, let's think about the statement. 68 can be represented with 41 MABs-

[A large white sheet of paper has been divided into 4 columns. The first column is titled: Representation. The second column: Tens. The third column: Ones. The last column: MABs.]

### Speaker

-and I decided to draw tables to help me to organise my thinking. So I'm gonna start by thinking about 68 instead of partitioning.

[Under the Representation column, the speaker puts down 6 10s MABs side by side. Next to the 10s, she places 8 1s].

### Speaker

That is six tens and eight ones. Now what I wanna do here, rather than count them out. I'm just gonna use direct comparison.

[She lines up the 1s up against the 10s.]

### Speaker

I know that eight is two less than 10 by folllowing them up, I can see that, that must be six tens and eight ones.

[Under the Tens column, she writes 6. Under the Ones column, she writes 8.]

### Speaker

Which gives me a total of 14 MABs.

[Under the MABs column, she writes 14.]

### Speaker

Now I'm looking for 41 MABs. So I'm gonna need to change something. So I'm gonna think about changing the number of tens. Having less tens and increasing my ones. So what I'm gonna do here is I'm going to take one of these tens.

[She removes the 10s closest to the 1s. Under the Tens column, she writes 5.]

### Speaker

So I only have five tens. But I'll be swapping this ten-

[She puts down a 10s]

### Speaker

-for ten ones.

[She brings in some ones. She adds 2 1s to the stack of 1s.]

### Speaker

I'm going to use two more here to complete that to here. But now I need to make my ones and I'm just going to use a double or glass pattern.

[She puts down 4 rows of 2 columns of 1s, with spaces in between.

Using her marker, she circles the 10s, the column of 10 1s and then the 4 rows.]

### Speaker

OK, so I've got five tens and ten and eight, 18 ones-

[Under the Ones column, she writes 18.]

### Speaker

-which gives me a total of 23 MABs, I do 18 and two more is 20 and then another three is 23, I partitioned five into two and three.

[Under the MABs column, she writes 23.]

### Speaker

OK, that's still not how many MABs as I need for this question, so I'm gonna do that process again.

[She removes the 10s closest to the 1s. Under the Tens column, she writes 4.]

### Speaker

This time, I know I'm gonna have four tens and I'll do another ten ones.

[She lines up 1s against the 10s to create a column of 10 1s.]

### Speaker

So I've still got my eight ones, but eight, 18, 28...

[Using her marker, she circles the 4 rows of 1, then circles that with the first column of 10 1s, then with the second column of 10 1s.

Under the Ones column, she writes 28.]

### Speaker

-and that gives me a total of 32 MABs.

[Under the MABs column, she writes 32.]

### Speaker

Now I'm starting to notice something on this table. Let's see if you can notice it too. Have a look and see what's happening to my tens-

[She points to each of the numbers under the Tens column.]

### Speaker

-and then have a look at what's happening to my ones.

[She points to each of the numbers under the Ones column.]

### Speaker

You might also like to have a look at what's happening to the number of MABS.

[3 appears under the Tens column.]

### Speaker

I know I'm gonna have three tens because I'm gonna take one of those and rename it as ten ones. What do you think is gonna happen with my ones? I predict it's gonna be 38 because I can see a pattern emerging and this is why tables can be so beneficial.

[She points to each of the numbers under the Ones column.]

### Speaker

Eight, 18, 28. It looks like my ones are going up by tens. So I think I'm gonna have 38 ones-

[Under the Ones column, she writes 38.]

### Speaker

-this time to check. I'm gonna use my mathematical imagination. I'm gonna try to imagine this ten as ten ones.

[She picks up the 10s closest to the ones.]

### Speaker

Let's see if you can do it too.

[Using her marker, she circles the 4 rows of 1, then circles that with the first column of 10 1s, then with the second column of 10 1s.

### Speaker

Eight, 18, 28.

[She circles the 10s that she’s holding.]

### Speaker

If we imagine this is ten ones that's right, 38, 38, and three is 41.

[Under the MABs column, she writes 41.

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

Bullet points below read:

· 68 can be represented with 41 MABs.

· 32 can be represented with 15 MABs.

· 136 can be represented with 28 MABs.

The first bullet point has been outlined.]

### Speaker

So I have shown that statement to be true.

[Next to the first bullet point, the text TRUE appears.]

### Speaker

Now let's think about the statements. 32 can be represented with 15 MABs.

[The second bullet point has been outlined.]

### Speaker

I'm going to imagine-

[A large white sheet of paper.]

### Speaker

-32 as three tens and two ones. I want to draw these.

[She draws 3 long rectangles and 2 squares next to each other.]

### Speaker

That gives me a total of five MABs.

[On the right side of the squares she writes an equals sign and 5 MABSs.

### Speaker

OK, so then if I can imagine that I only have 2 tens-

[She draws 2 long rectangles.]

### Speaker

- yeah, I'll still have my two ones-

[On the right side of the rectangles, she draws 2 squares.]

### Speaker

-but now I'm imagining ten more ones, so that gives me a total of two tens.

[On the right side of the squares she writes an equals sign and 2 tens.]

### Speaker

And then my ten ones that I'm imagining and two more.

[On the right side of the 2 tens, she writes + 12 onesThe first bullet point has been outlined.]

### Speaker

Which gives me a total of 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:

· 68 can be represented with 41 MABs.

· 32 can be represented with 15 MABs.

· 136 can be represented with 28 MABs.

The second bullet point has been outlined.]

### Speaker

So after is proven that that statement is false.

[Next to the second bullet point, the text FALSE appears.]

### Speaker

So I've exposed the lie. So when looking at my third statement, 136 can be represented with 28 MABs.

[The third bullet point has been outlined.]

### Speaker

I can assume that that's true. But as a mathematician, I wanna prove it. I wanna be sure.

[A large white sheet of paper has been divided into 4 columns. The first column is titled: Representation. The second column: Tens. The third column: Ones. The last column: MABs.]

### Speaker

136 standard partitioning would be 100.

[Under the Representation column, she places a 100s MABs – which looks like a blue square, 3 10s and 6 1s].

### Speaker

Three tens and six ones.

[She writes under: Hundreds column – 1, Tens column – 3, Ones column – 6, MABs column – 10.]

### Speaker

And that gives me a total of ten MABs. So that's not the 28 MABs that I'm after. OK so what am I gonna have to do is change my number of hundreds because the only way that really that I can increase my number of tens.

[Under Hundreds, she writes 0.]

### Speaker

So I've zero hundreds. What I'll do instead is...

[Under the other MABs in the Representation column, she puts down 10 10s (measuring it against the 100s) and 3 10s.]

### Speaker

I'll make up my 136 using tens instead. OK, so here's my ten tens, which make up 100, plus and that is three tens.

[She puts down 6 1s.]

### Speaker

And my six ones.

[She touches the 10 10s, then the 3 10s.

Under the Tens column, she writes 13.]

### Speaker

So now I've got 10, 3, 13 tens-

[Under the Ones column, she writes 6.]

### Speaker

-and six ones which gives me a total of 19 MABs.

[Under the MABs column, she writes 19.]

### Speaker

OK. At this time rather than physically swap one of these tens ones for ten ones-

[She pulls the 10s closest to the 1s a little bit down.]

### Speaker

-I'm gonna try to imagine what that would look like.

[Under the Hundreds column, she writes 0.]

### Speaker

This time I've still got zero hundreds. I'm gonna say I've got 12 tens-

[Under the Tens column, she writes 12.

She holds up the 10s she pulled down.]

### Speaker

-but I'm gonna use my imagination to imagine this is ten ones, which gives me 10 and 6, 16 ones.

[Under the Ones column, she writes 16.]

[Under the MABs column, she writes 28.]

### Speaker

And together that's 28 MABs. So I've just proven that the third 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:

· 68 can be represented with 41 MABs.

· 32 can be represented with 15 MABs.

· 136 can be represented with 28 MABs.

The third bullet point has been outlined.]

### Speaker

-136 can be represented with 28 MABs.

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

Bullet points below read:

· 56 can be represented with 29 MABs.

· 45 can be represented with 19 MABs.

· 117 can be represented with 36 MABs.]

### Speaker

Over to you! Two truths, one lie. Here are your three statements. See if you can work out, which is the lie and which two statements are true.

[Text on a navy-blue background reads: So 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?

A bullet point below reads:

· Quantities can be partitioned in standard and non-standard ways. For example, 68 is…

Below the text are 2 sets of diagrams. On the left side is a diagram of 6 blue 10s and 8 green 1s, with text below that reads: 6 10s and 8 ones. On the right side is a diagram of 4 blue 10s, 4 sets of 5 green 1s, and 2 sets of 4 green 1s with text below that reads: 4 10s and 28 ones.]

### Speaker

Quantities can be partitioned in standard and non-standard ways. For example, 68 is six tens and eight ones. It is also four tens and 28 ones. In this example, we've used our knowledge of place value paths when partitioning.

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

A bullet point below reads:

· As mathematicians, we can use our mathematical imagination to help us to solve problems.

Below the text are 2 sets of diagrams. On the left side is a diagram of 3 blue 10s and 2 green 1s. On the right side is a diagram in a thought bubble of 2 blue 10s and 12 green 1s, with text that reads: I can imagine 32 as 2 tens and 12 ones to help me solve this problem.]

### Speaker

As mathematicians, we can use our mathematical imagination to help us to solve problems.

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

A bullet point below reads:

· Mathematicians use tables to help us to organise our thinking and to identify patterns.

Below the point is an image of 4 10s and 11 1s. The 10s are aligned in a row and the 1s are across 5 rows by 2 columns, spaced slightly apart, with an extra 1 in the bottom .]

### Speaker

Mathematicians use tables to help us to organise our thinking and to identify patterns.

[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

Can you figure out which of these statements are true and which one is the lie?

The number...

- 56 can be represented with 29 MABs
- 45 can be represented with 19 MABs
- 117 can be represented with 36 MABs.

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