# Some steps forward, some steps back

ES1 – a thinking mathematically targeted teaching opportunity focused on testing ideas and looking for patterns.

Adapted from Number Visuals on YouCubed.org

## 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.

## Outcomes

- MAO-WM-01
- MAE-RWN-01
- MAE-RWN-02
- MAE-CSQ-01
- MAE-CSQ-02

## Collect resources

You will need:

- something to write on
- something to write with
- a counter or toy.

## Some steps forward, some steps back – part 1

Watch the video to see what to do (2:58).

(Duration: 2 minutes and 58 seconds)

[Text over a blue background: Some steps forward, some steps back. In the lower right-hand corner of the screen is the waratah of the NSW Government logo. Small font text in the lower left-hand corner of the screen reads: NSW Mathematics Strategy Professional Learning team (NSWMS PL team).]

### Speaker

Some steps forward, some steps back.

[Black title on a white background: You will need…

Bullet points below read:

· a paper and something to write with

· a counter of a toy.]

### Speaker

For this task, you will need... paper and something to write with, and a counter. You can also use a toy.

[White text on a blue background: Let’s investigate!]

### Speaker

Let's investigate.

[On white paper, the numbers 1 to 12 are written in a row inside red boxes. The number 9 has a small red underline. Below the numbers sits a plastic triceratops figurine.]

### Speaker

Hello mathematicians. I've got a challenge for you today. I've got the numbers here marked from 1 to 12. They've each got a space, a box around them.

[The speaker picks up the triceratops and places it underneath the number 5. They gesture to the row of numbers, then point to the underlined number 9.]

### Speaker

Now, the triceratops here is going to start on the number five. It's then gonna take some steps forward and some steps back. And then will repeat those moves. The goal is to land on the number nine.

[The speaker points from the triceratops to the number 9, and then to the number 8.]

### Speaker

OK, so let's think. I can't go straight to the nine because I have to then take some steps back. That wouldn't work.

[The speaker picks up the triceratops and walks it four steps from the 5 to the 9. They hover the figurine back over the number 8, then place it down underneath the number 9.]

### Speaker

So if I went, one, two, three, four to the nine, I'd have to then move away from the nine again. I can still try it with some steps back. Let's try that.

[With a pink marker, the speaker writes the number 4 and draws an arrow pointing to the right at the top of the paper.]

### Speaker

Let's go... four steps forward. And instead of writing my sentence, I'm writing how many steps and the direction that I'm moving in.

[The speaker writes the number 3 and draws an arrow pointing to the left. They place the triceratops on the number 5 square. They move it four steps to the right, landing on the number 9, then three steps to the left, landing on the number 6.]

### Speaker

And let's try with three steps back. Let's see if that works. One, two, three, four. One, two, three back. OK?

[The speaker moves the triceratops four steps to the right, landing on the number 10, and then three steps to the left, landing on the number 7.]

### Speaker

One, two, three, four, one, two, three back.

[The speaker draws an X next to their writing at the top of the paper.]

### Speaker

No, I ended up in seven, so that does not work. I think maybe if I take less steps back? Let's try something like this.

[The speaker writes the number 4 and draws an arrow pointing right, then the number 2 and an arrow pointing left. They move the triceratops back to the number 5 square.]

### Speaker

Four steps forward, and two steps back.

[The speaker moves the triceratops four steps to the right, landing on the number 9, then two steps to the left, landing on the number 7.]

### Speaker

One, two, three, four, one, two back.

[The speaker moves the triceratops to the right, landing on the number 11, then to the left, landing on the number 9.]

### Speaker

One, two, three, four to 11, one, two back.

[The speaker places a check mark next to their second row of writing. They move the triceratops back to the number 5 square.]

### Speaker

OK, I've done it, I've landed on nine. So that works. OK. What if I take less steps forward?

[The speaker writes the number 3 and draws an arrow pointing right, then the number 2 and arrow pointing left.]

### Speaker

Let's try with three steps forward, and two steps back.

[The speaker moves the triceratops three steps to the right and two to the left. They repeat the action, landing on the number 7. They draw an X next to their written instructions. They place the triceratops back underneath the number 5 square.]

### Speaker

One, two, three, one, two back. One, two, three, four, well hello nine, but I've got to take two back, so that doesn't work. So we have found one solution to our problem. Over to you. Are there more? Remember that the triceratops, or your counter, or whatever toy you're using, needs to take some steps forward and some back, and then repeat those moves one more time and land on number nine.

[White text over a blue background: Over to you! Are there other solutions to this problem? Can you see any patterns in the solutions that work?]

### Speaker

Now over to you. Are there other solutions to this problem and can you see any patterns in the solutions that work?

[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

- Explore the number visuals and record the different ways you see each number visual made up of other numbers.
- Record your thinking using the visuals worksheet in your student workbook.

## Some steps forward, some steps back – part 2

Watch the video to see what to do next (1:55).

(Duration: 1 minute and 55 seconds)

[On a white paper background, the numbers 1 to 12 are written in a row. Each number is inside a red square, and the number 9 has a small red underline. A plastic triceratops figurine sits underneath the number 5 square.

Writing in red marker at the top of the page in three rows: the number 4 with an arrow pointing right and the number 3 with an arrow pointing left; the number 4 with an arrow pointing right and the number 2 with an arrow pointing left; the number 3 with an arrow pointing right and the number 2 with an arrow pointing left. Beside the first and third row are X marks, and the second row has a check mark.]

### Speaker

Now, I found a couple of other solutions to this problem. I could go three steps forward and one step back. Let's see.

[At the top of the paper, the speaker writes the number 3 with an arrow pointing right, and the number 1 with an arrow pointing left. They move the triceratops figurine three steps to the right along the row of numbers, landing on the number 8, then one step to the left, landing on the number 7. They repeat the sequence of steps, placing the triceratops on the number 9 square.]

### Speaker

Three steps forward and one step back. One, two, three. One back. One, two, three and one back. That works.

[The speaker places a check mark next to their writing at the top of the paper. They place the triceratops underneath the number 5 square.]

### Speaker

I could also do five steps forward, and three steps back.

[The speaker moves the triceratops five steps to the right, then three steps to the left. They repeat the sequence, placing the triceratops on the number 9.]

### Speaker

One, two, three, four, five, forward. One, two, three, back. One, two, three, four, five steps forward. One, two, three, back. So, that also works.

[The speaker writes the number 5 with an arrow pointing right, and the number 3 with an arrow pointing left. They place the triceratops under the number 5 square.]

### Speaker

Was that one of the ones that you got? There was another one that I got really excited about, but I don't think it's allowed. See what you think.

[The speaker moves the triceratops six steps to the right, and four steps to the left, landing on the number 7. They move the triceratops six steps to the right, off the row of numbers onto a blank space.]

### Speaker

So, I was thinking of six steps forward. One, two, three, four, five, six. And then four back. One, two, three, four. Then six step forward. One, two, three, four, five, six. That actually takes me to 13, which is not part of our game.

[The speaker moves the triceratops four steps to the left, back onto the row of numbers, placing it on the number 9.]

### Speaker

But four steps back, one, two, three, four, does land me on nine. What do you think? I think because it ends on 12, that move is not allowed. Did you get some of my solutions? Maybe you got some different ones.

[White text on a blue background: What’s (some of) the mathematics?]

### Speaker

What's some of the mathematics?

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

Bullet points below read:

- Mathematicians test out ideas.
- Mathematicians look for patterns.
- Mathematicians explain their thinking.

Beside the bullet points is an image of the 1 to 12 number set up with the triceratops figurine and some of the instructions written out at the top.]

### Speaker

Mathematicians test out ideas, they look for patterns, and they also explain their thinking.

[End of transcript]