9.5.1 Another Number Trick

Activity Symbol Activity: Another Number Trick

Try the following trick several times. Think of a number between 1 and 10. This will work with numbers greater than 10, but the restriction is to keep the arithmetic manageable. [ Try making up your own number tricks. What makes a good trick? ]

Multiply by 4. Add 6. Divide by 2. Subtract 3. Divide by 2, and your answer is?

Write down the number you first thought of and your answer. What do you notice? Can you explain why this happens, either by using a diagram or by writing down the expressions for the answer at each stage?

Answer Symbol

Answer

You should find that this time the answer is always the number you chose at the start. The expressions for the answer at each stage are shown below.

InstructionExpression
Think of a number between 1 and 10:
n
Multiply by 4:
four postfix multiplication n
Add 6:
four postfix multiplication n prefix plus of six
Divide by 2:
equation left hand side open four postfix multiplication n prefix division of two close plus open six division two close equals right hand side two postfix multiplication n prefix plus of three
Subtract 3:
two postfix multiplication n equation left hand side prefix plus of three minus three equals right hand side two postfix multiplication n
Divide by 2:
two postfix multiplication n equation left hand side prefix division of two equals right hand side n

Now try the following.

Think of a number. Add 4. If my answer is 11, can you work out what number I was thinking of?

You might have said “What number do I have to add on to 4 to get 11?” or perhaps “If I take away 4 from 11, what number do I get?” In both cases, you should have arrived at the answer 7.

In the second method, “subtracting 4” undoes the “adding 4” in the original instructions.

This process can be illustrated by a “doing-undoing diagram.”

In the doing part of the diagram, start with the number and write down the operations applied in turn until you get the answer. Here there is just one operation, “add 4.”

For the undoing part of the diagram, start on the right with the answer, in this case 11. Then work back toward the left, undoing each operation in turn until you find the starting number. In this case, “subtract 4” undoes “add 4” and 11 minus four equals seven. So, the number first thought of was 7. Notice how the arrows indicate the direction to read the diagram.

9.5 Your Own Formulas

9.5.2 Doing and Undoing