Fraction February - Division of Whole Numbers by Fractions w/ & w/o Remainders

We are learning to model division of a whole number by a fraction on a number line.  The first day went much better than I expected!  Woo hoo!!!

While planning, I had to be VERY purposeful about what number sets I used.  Not all division of whole numbers by fractions turn out so evenly on a number line.  I wanted students to be able to grasp the modeling and understanding first before introducing number sets that would not be so "nice" to them.

Therefore, as I planned for number sets, I determined...

1. what denominators I wanted them to work with.  I'm not one to get all crazy and give them thirteenths!  
2. what numerators with that unit size will go into which whole numbers perfectly.  These will be the number sets I use first.  (Basically, if you invert & multiply before dividing and it comes out as a whole # answer, the number set works nicely for modeling.)  I tried not to use such a big whole number.  Here are just a few:
• 2, 4, or 6 divided by 2/5
• 3 or 6 divided by 3/5
• 4 divided by 4/5
• 2, 4, or 6 divided by 2/3
• 3 or 6 divided by 3/4
• any whole # divided by 1/any unit size works

These are notes we took for dividing a whole number by a fraction (when they are "nice").  Since students did a pretty good job of explaining reasoning for each step when multiplying fractions by whole numbers as mentioned in this post, I decided to add reasoning in their notes this time around.  I noticed a mistake... Step #5's reason should be that I count the number of jumps because that represents the number of meals the dog ate.

One misconception I had to emphasize was that when we MULTIPLIED whole numbers by fractions, the answer was right there on our number line label.  However, in this division scenario, the answer is in how many "jumps" we needed to take, not where we "landed." One student thought the answer was 10/5, which is 2 wholes.  I had to clarify that the number line labels represent pounds, not number of meals.

I also reminded them that in multiplication, we had commutative property where it wouldn't matter what we multiplied or wrote first.  However, in division, commutative property does NOT work, so students need to be extra careful how they determine what number is written first when they divide.


Division of Whole Numbers by Fractions with REMAINDERS!

So I haven't taught this yet, but will after they grasp the "nice" division.  I'm teaching it separately because it is more difficult to model and understand.  The process is the same, but when the fractional parts don't fit into the whole number evenly, students are left with a mixed number answer.  The difficult part is getting the correct fraction for the mixed number.

I came up with number sets to use that would result in remainders.  I anticipated this would be a challenge for students because there would be a remainder, BUT the denominator in their answer would not necessarily be the same as the unit size they partitioned their number line into. Crazy!  It already sounds confusing just typing the explanation.

Here is an example for 2 divided by 3/5:

What's the answer???

• I know the number of FULL meals my dog will eat is 3 because I have 3 full jumps.
• Is the remainder 1/3 or 1/5???  Students will gravitate towards 1/5 because that is the unit size of the original problem and they can see it in their number line, BUT that is incorrect.  The 1/5 does have a meaning, but it's not part of the answer.
• 1/5 represents the number of pounds I have left in my bag, NOT the amount of meals my dog eats.
• The 1/3 means that my dog can only eat 1/3 of his meal that day. I needed to jump three 1/5 pieces, but could only jump ONE 1/5 piece out of 3; therefore, the dog is getting only 1/3 of his meal.
• The answer is 3 and 1/3 meals.

Here are some extension questions I plan to ask and discuss:

Sooo...when they are able to see the pattern for the algorithm, I can then explain WHY we "invert & multiply" and the unit size (denominator) changes.

Students are essentially creating a double number line where the bottom represents pounds and the top represents number of meals.  Each partitioned piece is either 1/5 or 1/3 depending on what information you are looking at.  :)