# We Got A Problem #6: Useless Rulers

By the way, there’s an interesting problem kicking around Dan Meyer’s blog.  But now for something completely different…

Take a ruler or any stick.  Mark it off into three equal parts.  Then, perhaps in a different color, mark it off again into five equal parts.  This divides the ruler into how many total parts?

(The mathematician’s version: Consider a number line from 0 to 1.  Mark all multiples of 1/3 and 1/5.  How many intervals are produced?)

Start over.  On another ruler, mark it off into four equal parts, then again into ten equal parts.  How many total parts this time?

On another ruler, mark it off into twelfths and thirtieths.  (This is close to what happens with inches and centimeters, but not quite.)  How many parts now?

If you mark off the ruler into M equal parts, then again into N equal parts, how many parts are there in total?

Can you think of ways to extend either this problem or the rectangle-diagonal problem?

We’d love for readers to be able to explore these problems, so resist the urge to provide answers in the comments. Instead, we’d love helpful suggestions and ideas about different ways to think about them, successful or not. If you’d like to provide a full solution, do so with a pingback to your own blog!

Bowen is a mathematics curriculum writer. He is a lead author of CME Project, a high school curriculum focused on mathematical habits of mind, and part of the author team of the Illustrative Mathematics curriculum series. Bowen leads professional development nationally, primarily on how math content can be taught with a focus on higher-level goals. Bowen is also a champion pinball player and once won \$1,000 for knowing the number of degrees in a right angle.

### 2 Responses to We Got A Problem #6: Useless Rulers

1. Matt E says:

“Completely different”… Ha!

2. Ipsquiggle says:

I almost immediately had an insight that, rather than looking at it as dividing the ruler into 5 and 3 //parts//, you are instead adding 4 and 2 //divisions//. This made it trivial to imagine further answers in my head, and come upon a (somewhat messy, but easily simplifiable) generalization.