# 2 truths 1 lie – reasoning in Stage 2

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)

One Two Infinity

## 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
• MA2-RN-01

## Collect resources

You will need:

• something to write on

• something to write with.

## Watch

Watch 2 truths 1 lie Stage 2 video (8:35).

Prove which statements are true using mathematical reasoning.

### Transcript of 2 truths 1 lie Stage 2 video

(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

Two truths. One lie.

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

· eye balls and brains

· something to write on and write with.]

### Speaker

For this task you will need your 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 large white sheet has been laid over a table.]

### Speaker

Hello there mathematicians. I've got a great challenge for you today. Two truths and one lie.

[The speaker lays down a blue paper on the left side of the sheet with a written heading: Two truths. One lie.

· 32 can be represented with 5 MABs.

· 68 can be represented with 41 MABs.

· 45 can be represented with 28 MABs.]

### Speaker

I've got three statements. Two are true, one is a lie. The first statement is, 32 can be represented with five MABs.

Now, you know MABs from school.

[The speaker places 3 10s MABs in the top centre of the table. They are long green sticks that looks like 10 cubes stuck together. She takes them away quickly. She then puts down two 1s MABs, which are small yellow cubes.]

### Speaker

So what I'm gonna do first, is I'm gonna use my mathematical imagination. I learned this technique from (UNKNOWN). And I'm going to think about how I could make 32. So I'm picturing, let's see if you can do it with me. Three 10s and then two ones. Did you picture something like this?

[The speaker places 3 10s in the top centre of the table. Next to them, she puts down two 1s.]

### Speaker

Now, if you look at this, we've got three 10s, we've got two more.

[She points to each MAB.]

### Speaker

But I've got one, two, three, four, five MABs.

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

· 32 can be represented with 5 MABs.

· 68 can be represented with 41 MABs.

· 45 can be represented with 28 MABs.]

### Speaker

So I'm just showing that the first statement is true. Now let's think about the statement, 68 can be represented with 41 MABs…

[The second bullet point is outlined with a pink box.

### Speaker

..so I've decided to draw a table to help me to organise my thinking.

[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

I'm going to start by thinking about 68 in standard partitioning.

[She puts down 6 10s and 8 1s.]

### Speaker

That is six 10s and eight ones.

Now what I want to do here rather than count them out, I'm just going to use direct comparison. I know that eight is two less than 10, so if I line them up-

[She aligns the 1s next to the 10s.]

### Speaker

I can see that that must be eight, so six 10s 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 going to need to change something. So I'm going to think about changing the number of 10s, having less 10 and increasing my ones. So what I'm going to do here is

[She moves the 10s to the left, slightly away from the ones. She takes the 10 closest to the ones away.]

### Speaker

I'm going to take one of these 10s. So I'll only have five 10s.

[Under the Tens column, she writes 5.]

### Speaker

But I'll be swapping this 10, for 10 ones.

[She puts a 10 down on the sheet. She adds 2 more 1s to the column of 1s.]

### Speaker

I might use two more here to complete that 10 here. But now I need to make my ones…

[On the right-hand side of the 10 1s, she lays down 5 1s across 2 columns.]

### Speaker

..and I'm just going to use a double or dice pattern. OK.

[Using her marker as a pointer, she circles the 10s, the 10 1s and the 2 columns of 4 1s.]

### Speaker

So I've got five 10s and 10 and eight, 18 ones.

[Under the Ones column, she writes 18.]

### Speaker

Which gives me a total of 23 MABs, I did 18 and two more is 20 and then another three is 23. I partitioned my five into two and three.

[Under the MABs column, she writes 23.]

### Speaker

OK. That's still not as many MABs as I need for this question. So I'm going to do that process again.

[She takes the 10 closest to the 1s away.]

### Speaker

This time, I know I'm gonna have four 10s…

[Under the Tens column, she writes 4.]

### Speaker

..and I'll do another 10 ones.

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

### Speaker

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

[Using her marker as a pointer, she circles the 2 columns of 4 1s, then the first 10 1s, then the next.

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 10s…

[She points to the numbers down the Tens column.]

### Speaker

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

[She points to the numbers down the One column.]

### Speaker

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

[A 3 appears under the Tens column.]

### Speaker

I know I'm going to have three 10s because I'm going to take one of those 10s and rename it as 10 ones.

What do you think is gonna happen with my ones? I predict it's going to be 38 because I can see a pattern emerging and this is why tables can be so beneficial. Eight, 18, 28. It looks like my ones are going up by 10s. So I think I'm going to have 38 ones.

[Under the Ones column, she writes 38.]

### Speaker

This time to check. I'm gonna use my mathematical imagination. I'm going to try to imagine this 10 as 10 ones.

[She picks up the 10 closest to the 1s.]

### Speaker

Let's see if you can do it too.

[Using her marker as a pointer, she circles the 2 columns of 4 1s, then the first 10 1s, then the next.]

### Speaker

Eight, 18, 28…

[She circles the 10s in her hand.]

### Speaker

..and if we imagine this is 10 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…

· 32 can be represented with 5 MABs.

· 68 can be represented with 41 MABs.

· 45 can be represented with 28 MABs.]

### Speaker

So I have shown that statement to be true.

Now I'm going to think of the last statement. So I've already proven the first two statements, that 32 can be represented with five MABs, and that 68 can be represented with 41 MABs. And because the challenge is called two truths and one lie, I can assume that that statement is a lie. But as a mathematician, I want to prove this.

So let's look at that statement. 45 can be represented with 28 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

So I'm gonna start by using a table. I've already drawn one up. Standard partitioning would be four 10s…

[Under the Tens column, she writes 4.]

### Speaker

..and five ones…

[Under the Ones column, she writes 5.]

### Speaker

..which gives me a total of nine MABs.

[Under the MABs column, she writes 9.]

### Speaker

Now I've started to see the pattern. I'm going to think about reducing my number of 10 and increasing the amount of ones. So let's look at three 10s.

[Under the Tens column, she writes 3. She places 3 10 MABs under the Representation column.]

### Speaker

So now I've got three 10s…

[Next to the 10s she places 10 1s in a column. Next to the 10 1s, she puts down 5 1s.]

### Speaker

..my fourth 10 I'm renaming as 10 ones. And I still have the five one spot from before. So I've got three 10s and 15 ones. I'm still gonna need another 10 ones here now I want to try to just imagine them rather than make it again. Let's see if you can imagine another 10 ones here.

[She points to the area between the 10s and 10 1s.]

### Speaker

OK. So I've got two 10s.

[Under the Tens column, she writes 2.]

### Speaker

I've got my five ones, 15 ones.

[Using her marker as a pointer, she circles the 5 1s, then the 10 1s.]

### Speaker

I'm going to add the other 10 ones that I'm imagining.

[Using her marker as a pointer, she circles the 5 1s, then the 10 1s, then the space in-between the 1s and 10s.]

### Speaker

So now I have five, 15, 25 ones…

[Under the Ones column, she writes 25.]

### Speaker

..and again, I can see the pattern that I noticed earlier. This gives me a total of 27 MABs…

[Under the MABs column, she writes 27.]

### Speaker

..not 28 MABs.

If I reduce my number of 10s, I'm going to increase by way too much…

[On the sheet, 1 has been added under the Tens column, 35 has been added under the Ones column.]

### Speaker

..one 10 and 35 ones, which gives me a total of 36 MABs.

[Under the MABs column, she writes 36.]

### Speaker

So now I've proven that third statement, 45 can be represented with 28 MABs, is indeed false. It's a lie.

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

· 56 can be represented with 29 MABs.

· 45 can be represented with 8 MABs.

· 81 can be represented with 18 MABs.]

### Speaker

Over to you. Two truths. One lie. Here are our statements. The number 56 can be represented with 29 MABs, the number 45 can be represented with 8 MABs, the number 81 can be represented with 18 MABs. Which two statements are true? Which one is the lie and how can you prove it?

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

### Speaker

What's some of the mathematics?

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

· 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 10s and eight ones. It is also four 10s 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?

· 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?

· As mathematicians, we can use tables to help us to organise our thinking and to identify patterns.

Below the point is an image of the table that breaks down the number of Tens and Ones, and the total number of MABs used.]

### Speaker

As mathematicians, we can 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 8 MABs

• 81 can be represented with 18 MABs.

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