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Special Numbers

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

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Your Number Is...

Think of a number... follow the machine's instructions. I know what your number is! Can you explain how I know?

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First Forward Into Logo 6: Variables and Procedures

Learn to write procedures and build them into Logo programs. Learn to use variables.

Your Number Was...

Stage: 3 Challenge Level: Challenge Level:1

Why do this problem?

This problem provides a great opportunity to introduce the concept of representing operations on unknown numbers algebraically. It leads to work on inverse operations and solving simple linear equations.

Possible approach

You may wish to work on the problem Your Number Is... before trying this.
Either work through a few examples using the interactivity, or, even better, read out the instructions:
Think of a number
Add 4
Subtract 7
Go round the class asking learners what they finished with and then surprise them by revealing their starting number!
As we don't want the class to end up thinking maths is a strange magical mystery, it's time to explain what's going on. One way to begin thinking about this is to collect together some outputs and their corresponding inputs, for example:
Finishing Number Starting Number
11 5
17 8
3 1
23 11
14 6.5
This is not merely an exercise in pattern spotting - once learners have figured out HOW the machine is working out the starting numbers, they need to understand WHY this works.
It would be good to introduce a variety of methods for explaining what's going on:
  • Represent the instructions as a function machine, and explore what happens when the functions are inverted.
  • Represent the chosen number as $x$ and write an expression for the final number $y$ in terms of $x$. Then rearrange to get $x$ in terms of $y$.
  • Discuss simplification to make the functions easier to invert (so representing it as $y=2x+1$ rather than $y=2(x+4)-7$.
Now challenge the class to come up with their own examples of "Think of a number" instructions, which they can invert to deduce someone's starting number from their final number. Encourage them to record their working using function machines and algebra. Set challenges such as having to include particular operations, at least 6 different operations, and a set of instructions that's really quick to invert.

At the end of the lesson, a selection of students can read out their instructions for the class to try, and then work out starting numbers from the final numbers.

Key questions

How can I use function machines and algebra to explain the "Think of a number" puzzle?
How can I use function machines and algebra to create a "Think of a number" puzzle?

Possible extension

Think of Two Numbers is a challenging extension.

Possible support

Start introducing algebraic notation gently through the problem Your Number Is...