# Your number is...

## Problem

*Your Number Is... printable worksheet*

**Press the button in the interactivity below, and see what happens:**

Now try again starting with a different number.

Try again.

Try starting with a fraction... or decimal... or negative number...

**Why are your answers ALWAYS the same?**

Can you design instructions for similar "Think of a number" machines of your own?

Test them on a partner to check that they lead to your anticipated solutions.

## Getting Started

Look at the representation on the right hand side of the interactivity.

Can you explain what it means?

## Student Solutions

Leo, Mikki, Jodi, Amy, Cienna, Jedd, Charlie, Freya, Orla, Freya, Merry, Spike and Mo from Reepham Primary School worked on this task. Here is one of their submissions:

This problem was an algorithm. This works because when doing the calculation you are working out the opposite and the middle part in the calculation always equals five. Also we worked out that doing the calculations backwards and using the opposite symbol also gives you five. This proves that you can do it in any order.

For example:

10 x 2 = 20

20 - 4 = 16

16 $\div$ 2 = 8

8 - 3 = 5

Shao from The Australian International School in Malaysia”¨ wrote:

You think of a starting number. For example, 45. When you add 3, you have your original number plus 3. In this case, 48.

When you double it, you have twice your original number plus 6. In this case, 96.

Then you add 4, and now you have twice the original number plus 10. In this case, 100.

Then you halve your number, and you will have your original number plus 5. In this case, 50.

Finally, when you subtract the original number, you will have:

Your original number - Your original number + 5 (which is obviously, 5!)

This is always true whatever number you chose at the beginning because at the end, you're subtracting your original number from your original number. And since whenever you subtract a number from itself, you always get 0.

For example, if we use 99 as the original number, we will still end up with:

Your original number - Your original number + 5 = 0 + 5 = 5.

Fergus from Golspie High School in Scotland explained this idea very simply:

At the end you subtract the original number so if you ignore the original it makes it :

3, $\times$ 2 = 6, plus 4 = 10 then halved is 5

Aiman added the folowing:-

When you enter a number and follow all the instructions, you always get 5 at the end.

The idea is that, if (this is an example) 3 was added at the start to another random number, the number and 3 seem to be added together, but by subtracting the other number at the end, the answer will always be 3.

If you look at the blue line and orange dots that are next to the instructions that the interactivity gives, the blue line (a random number) and the orange dots (the units that the interactive tells a person to add) are separate, even though when the instructions are used, it is one number that a person sees.

While the instructions are performed, the answer from the second last step is (random number + 5). You can see that even though the number goes through a lot of equations, you still end up with (random number + 5).

The next step subtracts the random number, leaving you with 5. Whatever number is chosen, the answer will always be 5.

Anh Minh from British Vietnamese International School Hanoi, Steven from Reading School in the UK and Qianwei from Humanitree in Mexico showed how this works using algebra. This is Anh Minh's work:

Start with $x$

Add $3:$ $x+3$

Double: $2(x+3) = 2x+6$

Add $4:$ $2x +6+4= 2x + 10$

Half: $\frac12(2x+10) = x +5$

subtract first number: $x+5-x= x-x+5= 5$

Answer always be $5$

Elifsu from International School Laren in Holland made some 'think of a number' machines in a fully animated powerpoint presentation. Click here to download the presentation.

Shreya, Sirat, Lelia, Divya, Kirsten, Reetinderit, Ryan, Harman, Shazana and Agam from ”‹Glendenning Public School in Australia also worked on this task. Here are two of their solutions:

For this challenge task, I figured out that the rule given worked for any number that I tried. It also worked for the numbers 0 and 1. The result was always the number 5. I figured out that this might be because if I removed the first and last instruction of the given rule, the answer was always 5.

I also figured out two more patterns that had a similar rule. The first pattern that I worked out required me to change the numbers of the rule slightly:

Instead of 'adding 3 and 4' to your chosen number, I decided to change it and make it so that you would have to 'add 2 and 2' instead. My pattern looked something like this:

Think of a number: 10

Add 2: 12

Double: 24

Add 2: 26

Halve: 13

Take away the number you started with: 3

I continued this pattern with a variety of numbers, I tried nearly all of the numbers from 0 to 10. The result for all of the numbers I tried with was 3.

AND

Instead of always getting 5 in our method you could always get the number 9 once you finish.

Here is the method we discovered...

Think of a number

Add 10

Subtract 5

Add 4

And subtract the first number.

We tried this method with a majority of 1, 2 and 3 digit numbers including... 4

Here is what we got...

4 + 10 = 14

14 - 5 = 9

9 + 4 = 13

13 - 4 = 9

For 23 we got...

23 + 10 = 33

33 - 5 = 28

28 + 4 = 32

32 - 23 = 9

For 101 we got...

101 + 10 = 111

111 - 5 = 106

106 + 4 = 110

110 - 101 = 9

We then tried many other one-, two- and three-digit numbers, and then came to a conclusion that when you finish the method it will always equal 9.

Thank you all for these submissions.

## Teachers' Resources

### Why do this problem?

This problem provides a great way of introducing a very useful way of representing operations on unknown numbers visually and symbolically (algebraically). The interactivity provides a hook to engage students' curiosity, and the visual representation that appears on the right hand side of the interactivity offers a way of explaining what is going on in order to satisfy that curiosity.

### Possible approach

Ask learners to each think of a number and read out the following instructions:

Add 3

Double

Add 4

Halve

Take away the number you started with

What did you end up with?

Look surprised, when all of them reveal they have ended up with the same number.

"Did you all start with the same number then?"

"Perhaps none of you started with a fraction, or decimal, or a negative number - shall we try it again with one of those?"

More surprise that they still end up with the same number.

Now show the interactivity, or alternatively draw the representation that appears on the right hand side in the interactivity, and ask learners to think about how the visual representation helps us to understand what is happening. Give them a couple of minutes to discuss in pairs, and then draw the class together to share their insights.

Next, suggest that the visual representation might be too time-consuming, particularly if we'd added and multiplied by larger numbers. Introduce a parallel representation using algebra:

$$x$$ $$x+3$$ $$2(x+3) \text{ or }2x+6$$ $$2x+10$$ $$x+5$$ $$5$$

Now invite three members of the class up to the board to each use a different representation simultaneously for some other "think of a number" sets of instructions.

For example:

Think of a number

Multiply by 4

Add 2

Halve it

Add 7

Divide it by 2

Take away the number you first thought of.

This is what the board might look like when the three learners have finished:

Finally, challenge the class to come up with their own examples of "Think of a number" instructions, which will lead to an anticipated solution. Encourage them to record their working using both the visual and the symbolic representation.

At the end of the lesson, a selection of students can read out their instructions for the class to try, and then reveal the anticipated solution (written in advance) with a flourish.

### Key questions

How can I use the visual or symbolic representation to explain the "Think of a number" puzzle?

### Possible support

Spend time working with the visual representations before introducing symbolic notation.

### Possible extension