Fractions: 1/5 = 1/10 = 1/80 = 1?

[Feature photo is a screen shot from the video “the sausages sharing episode,” see below.]

How in the world can ¹/₅ be the same as ¹/₁₀? Or ¹/₈₀ be the same as one whole thing? Such nonsense!

No, not nonsense. This is real-world common sense from a couple of boys faced with a problem just inside the edge of their ability — a problem that stretches them, but that they successfully solve, with a bit of gentle help on vocabulary.

Here’s the problem:

• How can you divide eight sausages evenly among five people?

Think for a moment about how you (or your child) might solve this puzzle, and then watch the video below.

What Do You Notice?

Here’s What I Noticed

I saw five things worth remembering when you talk math with your kids:

• How patiently the man waited, giving the boys time to think. After more than a quarter century of teaching, I still have trouble with that.

• How the boys looked away, moved their fingers, grimaced, and mumbled — all signs of hard thought.

• How the boys alternated between speaking and thinking. While they were speaking, their thought couldn’t advance very far. Then the other boy, the one who had been quiet, would make the next connection — but as he tried to explain his thought, he would get to a point where he was stuck. And then the first boy, who had been thinking, could speak up and take the next step.

• How important the quiet time was for each boy, as they were struggling to understand what the other person had said and to consolidate that with their own ideas.

• How the following quote from W. W. Sawyer could have been written just for this video clip:

Most remarks made by children consist of correct ideas badly expressed. A good teacher will be very wary of saying ‘No, that’s wrong.’ Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching.

— W. W. Sawyer
Vision in Elementary Mathematics

I also noticed that the intuitive method of counting the boys invented at first was the same way Egyptian fractions worked. In Egyptian, the exact, correct answer would have been written as a mixed number:

1 ¹/₂ ¹/₁₀

Cool!

One More Thing

And finally, I noticed how wonderfully many “ones” the boys used to make their fractions: one sausage, one half-sausage, one plate of sausages, and one “bit” of sausage. Each of these was used as the unit at least once during the discussion.

Which brings one more video to mind:

For more about the many meanings of one, see the questions and activities at this TED-Ed video page. For tips on talking math with your kids, please see Christopher Danielson’s blog. And for more on Egyptian fractions, read The Secret of Egyptian Fractions featuring the resourceful Alexandria Jones.

Please Share!

Share your ideas in the comment section below. How would YOU have divided the sausages? Did you ask your kids? What did they say? I wonder how many different ways can we think of to do it….