# Number Lines in Disguise

## Problem

*Number Lines in Disguise printable sheet - main problemNumber Lines in Disguise printable sheet - blank number lines*

What do you notice about this number line?

What questions could you ask?

Click below to reveal some questions you might like to think about.

What number would be where the dot is?

How do you know?

Where would $0$ be on the line?

Often if a mathematician is stuck they might try an example with smaller numbers. Click below to reveal another example.

Consider the number line below.

Can you figure out what any of the other numbers are on the number line?

Where would $0$ be on this line?

What strategies did you use?

Below is an interactive number line. Can you work out what number is marked by the dot? Once you think you know, reveal the answer to check if you were right.

Explore the different number lines at different levels. Think about the strategies that you might use if you get stuck on the harder levels.

You could use this worksheet to create your own challenges.

What makes a challenge easier or harder?

We would love to hear about the strategies you used to figure out the missing numbers, and how you persevered and showed resilience if you got stuck!

*You may be interested in the other problems in our Resilience and Perseverance Feature.*

## Getting Started

- Can you find the value of
**any**of the unmarked points on the line? - Can you find (approximately) where $0$ would be?
- Do you have a strategy that works for Level 1? If so, could you adapt it for the higher levels?

## Student Solutions

Bill from Marshall Laing in New Zealand, and Neilesh from MPLS, both used a trial and improvement method to find the value of the dot when the number line showed 9 and 29. Bill said:

For the first one, the answer is 13 because I added 4 to 9, and then 4 again, and again, and I got up to 25, and the next one was 29.

Neliesh also found that:

0 would be between the 5th and 6th marks.

Oliver from The Priory Academy LSST in the UK, Hikari (Karin) from Bangkok Patana School in Thailand and Arnav from Franconian International School in Germany submitted a strategy for finding the dots. This is Oliver's work:

Method to find out the missing value:

1) Count how many intervals there are between the two given values.

2) Find the difference between the two values and divide that by the number of intervals.

3) Keep adding the result from the first value, until you reach the required missing value.

Eg:

_ _ _ _ _ 12 _ ? _ 60 _ _ _ _ _

In this example, there are 4 intervals between the 12 and 60. The difference between 12 and 60 is 48, so we divide 48 by 4. This gives us 12.

The missing value is two intervals ahead of 12, so we must add 12 two times. The answer, therefore, would be 36!

Arnav also included a formula to find the distance between each pair of lines, and a diagram:

If left number would $=l$,

and right number $= r$,

and number of lines from the furthest left number, to furthest right number $= x$,

then formula would be: $\dfrac{r-l}x$.

So, with the challenge that was given, the formula would be in that case:$\dfrac{29-9}5$

Solving:

$$\frac{29-9}5\\

=\dfrac{20}5\\

=4$$

This means that each line $=+4$

## Teachers' Resources

### Why do this problem?

Looking at a familiar context in a much more challenging scenario can pique curiosity. Number lines are used a lot when children are developing number sense and then are revisited when drawing axes for graphs. This is an opportunity to explore number lines in a challenging context.

This problem also gives us an opportunity to show some strategies we might encourage when students are stuck. Students can pick the level they wish to work at and we can use this to demonstrate when it is useful to try simpler examples to help us develop an approach to solving a more difficult problem.

### Possible approach

Begin by showing students this image. (PowerPoint Slide)

"Have a look at this number line. What do you notice?"

Give students a short time to look at the image in silence, and then invite them to discuss with their neighbour.

"Can you determine the value of other points on the line? How do you know?"

If computers are available, this would be a good time to introduce the interactivity.

"Work with a partner to explore some other examples, and see if you can find a strategy that will work for any level." Alternatively, you may find this worksheet useful, which contains a few examples from each level of the interactivity.

*(Thank you to Mr Gray who created the sheet and very kindly shared it with us. He reported that he found it particularly useful as students were able to record their thinking on it.)*

Level 1 = whole number steps, small negative numbers possible

Level 2 = fractional steps (halves quarters and eighths only), small negative numbers possible

Level 3 = fractional steps (any), small negative numbers

Level 4 = fractional steps (any), negative numbers are always included in the number line

Level 5 = like Level 4 but with larger numbers

While students are working, circulate and listen for pairs who are developing strategies or who have particularly useful insights. Once most students have successfully solved a number of examples, bring the class together and invite pairs to share their strategies and insights.

### Key questions

Can you find the value of any of the points on the line?

Where is $0$ on the line?

How do you know?

Did you always use the same strategy?

Can you find a strategy that you can use for any example?

What made particular questions easier?

What made particular questions harder?

### Possible support

For students who need support with using number lines you may wish to play Five Steps to 50.

You may wish to try Shifting Times Tables for some similar ideas that stay within the integers.

### Possible extension

Ask students to create their own challenges using this sheet. Ask them to think about what makes a question harder or easier. They could make a sheet of easier questions and a sheet of harder questions.