Dividing Fractions

This picture came from one of my favorite days teaching math this year, a couple of weeks before summer vacation:

"Why is the answer smaller?"

Dividing fractions is a completely bizarre procedure for students.  Only the adventurous think to ask why.

The last few weeks at this school tend to devolve into chaos.  Field trips.  Dances.  Award ceremonies.  Teachers tacitly encouraging students to stay home.  For all of the reasons listed above, on this day there were only six students in attendance.  I had just finished showing them that when you divide 12 by 8/9, you come up with 13 and 1/2.

C: If you’re dividing, why is the answer bigger than 12?

Ten years of teaching this skill and nobody had ever thought to ask that question.  I was so excited!

Me: What does it mean if we divide 12 by 4?

J: If divide 12 by 4, we get 3.  3 groups of 4.

Me: Could we subtraction to show how that works?

Class: (blank stares)

Me: Yes, we could!  If I start with 12 and subtract 4 over and over, how many times can I subtract 4?

I do this on the board and the students see that you can subtract 4 three times.

Me: So division is the same thing as repeated subtraction.  What would happen if we subtracted 8/9 from 12 over and over again?  Is it okay if we try this together?  I’m really excited about this!  

Class: (humoring me) Yeah, Mr. Young, let’s do it.

Me: Alright.  What do I have to do if I want to take 8/9 away from 12?

T: Find a common denominator!

Me: What would 12 be rewritten as ninths?

T: 108/9!

Me: Yes!  So 108/9 minus 8/9 would be. . .?

Class: 100/9!

Me: And 100/9 minus 8/9. . .?

Class: 92/9!

We keep subtracting 8/9 over and over again.  My students tell me the difference faster than I can write it.  Each time we subtract, I circle 8/9.

A: Mr. Young, just look at the pattern.  0, 2, 4, 6, 8.  It keeps repeating.

Me: Thank you, A.  That makes it easier for me to subtract.  This is so much fun!

We come to the end.  Only 4/9 left.  We can’t subtract 8/9 anymore.

Me: How many times did we subtract 8/9?

Class: 13!

Me: And how much is left?

Class: 4/9!

Me: What is 4/9 compared to 8/9?

T: 4/9 is half of 8/9.

Me: Yes!  So when we divide 12 by 8/9 and get 13 and 1/2, what does that 13 and 1/2 mean?

C: It means if we divide 12 by 8/9, we get 13 groups of 8/9 plus 4/9, half of 8/9.

Me: So why is the answer greater than the dividend – the 12?

C: Because 8/9 isn’t very big.  It’s less than a whole and you can get more of them out of 12 than just 12.

Me: How close is 8/9 to one whole?

J: Real close.  Only 1/9 away.

Me: If we divided 12 by one, how many groups of one would we have?

Class: 12!

Me: So, C,  does it make sense that if we are dividing 12 up into pieces that are less than one then we would have to end up with more than 12?

C: Okay.  I get it now.

Me: I think that was the most fun I had all year!  I’ve been teaching division of fractions for ten years and nobody has ever asked me a question like that before.  Thank you for letting me go through all of that work with you and for paying attention while we did it.

A: We should have six people in our class every day.

Me: Maybe you’re right.  That would be nice, wouldn’t it?

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