Creating a prompt

posted 2 Sep 2012, 09:10 by Unknown user
Elysia Hole (a teacher in Brighton) recently sent me an arithmetic procedure that she thought had the potential to develop into an inquiry. I think an appropriate prompt from her explanation (below) would be something similar to:
With this prompt students can re-trace the procedure for themselves and proceed to questions about other numerical cases and the algebraic representation.
 
Elysia's explanation is as follows:

Pick any digit and write it down three times. You'll now be looking at something like "333" or "888". Add those three digits together: 3 + 3 + 3 = 9 or 8 + 8 + 8 = 24. Then divide the three-digit number by the sum of its digits... always gives 37 as the answer.

How it works
It works because 37 is a third of 111. When you type the same digit three times, you’re inadvertently multiplying it by 111. So, 888 is really just 111 × 8. Then, when you add the digit together three times, you are effectively multiplying it by three: 8 + 8 + 8 is just 3 × 8 = 24. Now when you divide 888 by 24 – or whatever your numbers were - that is actually (111 × 8) ÷ (8 × 3) and the eights cancel each other out. 111 × {any number} ÷ {the same number} × 3 is the same as 111 ÷ 3 = 37.

 

You can do the same trick with four digits and always get the less pleasant answer 1,111 ÷ 4 = 277.75. The next whole number answer is typing the same digit nine times and then dividing it by the sum of its digits. This has the rather pleasing answer of 111,111,111 ÷ 9 = 12,345,679.


This answer is missing an “8” but that can be fixed by squaring 111,111,111 which will give you the answer 12,345,678,987,654,321. That’s probably a whole new pattern to investigate that I'm not sure yet how it works.