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!