How often have you taught fraction division to the students in your middle school math classes, only to find them "flipping" the wrong number? You may have taught them to "skip, flip, flip," "invert and multiply," or "multiply by the reciprocal." You may have listed out the steps, or taught them a nifty song, but somehow they still flip the wrong one or they forget to flip at all.
OR they change a mixed number into an improper fraction and seem to subconsciously think that since they did something to that mixed number, the flipping had already occurred...and then they don't flip anything.
Why does this happen? I'm going to say that it happens because they don't see the sense in it - it doesn't mean anything to them.
So, I started using another way to teach fraction division - perhaps you've heard of it, or you use it. I never learned it this way as a student, but I like it and it makes more sense to some students. I learned this method when I had a student teacher a few years back. She was teaching the fraction unit, and when her supervisor came in to observe and discuss, she asked if I had ever taught fraction division using common denominators. Having only learned (and then taught) to multiply by the reciprocal, of course I said no.
The next time she visited, she brought me a page from a textbook that explained dividing fractions using common denominators. These are the steps:
Step 1: Find common denominators, just as when adding and subtracting and then make equivalent fractions (students are already used to doing this - hopefully).
Step 2: Create a new fraction with the numerator of the first fraction over the numerator of the second fraction...this is your answer.
Done (unless you need to reduce)!
I was shocked - it seemed SO simple!
Check out this example - it's a simple one, for starters:
5/6 divided by 2/3.
1) Find the common denominator of 6 and 3, which is 6. This gives you 5/6 divided by 4/6.
2) The first numerator (5) becomes the numerator in the answer. The second numerator (4) becomes the denominator. Then reduce.
Let's look at another one, with mixed numbers:
1 and 4/7 divided by 1 and 3/4.
1) Convert the mixed numbers to improper fractions, which gives you 11/7 divided by 7/4.
2) Find the common denominator of 28 and make equivalent fractions. This gives you 44/28 divided by 49/28.
2) The first numerator (44) becomes the numerator in the answer. The second numerator (49) becomes the denominator. No reducing, in this case.
I've shown both methods to my sixth-graders. Some really like it. Others stick to the flipping method - but I don't know if this is because they like it better or because it was the first way they learned it.....most of them had been taught something about fraction division in 5th grade.
As far as teaching multiplying by the reciprocal - if students are going to use it, I think it's important that they understand WHY it works. It may be tough for them to understand, but if they learn the common denominator method first, the proof may then make more sense to them. I found a great article on the NCTM website that uses the common denominator method to prove why multiplying by the reciprocal works - check it out!
Recently I made two math wheels, to use to teach both methods of dividing fractions -taking notes will be more fun!
What do you think? Do you see any advantages or disadvantages to teaching fraction division using common denominators?
Grab this free fraction operations math wheel!
I love teaching fraction multiplication in middle school math--particularly multiplication of mixed numbers. Why? Because I have fun explaining why multiplying mixed numbers DOESN'T work a certain way.
Inevitably, when we start multiplying mixed numbers, some students want to multiply the fractions by the fractions and then multiply the whole numbers by the whole numbers. And I can see why they might think that's ok - after all, when you add and subtract, you deal with the whole numbers and fractions separately. Sometimes, I think they don't want to be bothered with making improper fractions, because it's "easier" to just do 2 x 3 and then 3/4 x 1/2, haha.
So, every year, we end up having this discussion about why that just doesn't work. I enjoy showing/explaining that multiplying 2 3/4 by 3 1/2 means that ALL parts of 2 3/4 must be multiplied by ALL parts of 3 1/2. On the board, we make a list of the problems that would need to be completed: 2 x 3, 2 x 1/2, 3/4 x 3, and 3/4 x 1/2.
Now that we have all four products, we go through the process of adding them all together (finding common denominators, equivalent fractions,etc) and then reducing.....quite a bit of work to get to the answer:-)
Then we compare that to what we get when we convert the mixed numbers to improper fractions. The detailed example of completing four multiplication problems and adding the products proves not only that converting to improper fractions is necessary, but also that it's a lot faster! So, Tip #1 is to show students WHY what they're doing isn't correct...show what the multiplication really means.This may also mean bringing out the graph paper and showing what 3/4 groups of 1/2 looks like, etc, in addition to doing the computation.
Canceling, or "Crossing Out"
Tip #2 - Teach students to cancel, or "cross out" (or whatever you might call it), and show them why it makes life a little easier.
I love teaching this aspect of fraction multiplication. It's hard for some students to grasp at first, but when they repeatedly see that if they don't cross out, they have to reduce at the end of the problem (with larger numbers, like 168/12), they start getting excited about finding how much they can cross out. In recent years, I've found that students aren't learning this in earlier grades as often as they used to--for many, the discussion we have in my classroom is the first time they've encountered it.
Once I teach them the idea of reducing first, and we explore why it works, there are some that still want to stick with what they learned in earlier grades and reduce only at the end, while others get super-excited about the concept of making the numbers they're working with smaller at the start. I may be wrong on this, but it seems that the students who embrace it first are those who know their multiplication facts better and can more easily find the relationships between the numbers in the problem....a student who knows that 15 and 24 can both be divided by 3, for example, is more likely to go ahead and divide them by 3 than the student who can't see it because they can't remember/don't know what 15 and 24 are divisible by.
Multiplying Fractions and Mixed Numbers Wheel
Tip #3 - Give students a graphic organizer to help them remember the process. Some will need this and some won't, but it's handy to have in their binders to reference throughout the year. I recently created a fun math wheel, which is a great way to have students take notes about the concept, practice it, and then add their own personal, artistic touches.
Do you have any special methods you use to teach the multiplication of fractions and mixed numbers?
Grab this free fraction operations math wheel!