Make sure your pupils understand the algorithm by using this hands-on lesson using modeling clay.
The algorithm for multiplying fractions is easy, especially if you’ve already covered adding and subtracting fractions, which presumably has exposed your kids to lots of practice with reducing and the idea of simplest form. However, the concept of what is actually happening when you multiply fractions is challenging. Sure, they can multiply 1/2 x 3/4, but do they understand what taking 1/2 of 3/4 means? Or 1/3 of 4/5?
Make this concept visual, concrete, and fun by using modeling clay, or Playdough, in an activity that students will not only enjoy, but will help make fractions really clear.
Note: All textbooks will cover this concept with varying degrees of effectiveness. However, my suggestion is to start with this activity, as it is 100% hands-on. Once your kids understand the concept, they can just rely on the standard algorithm. But teaching the algorithm without their understanding the concept behind it, is a big mistake.
- Modeling clay (Plastilina) or Playdough: enough for each person (or pair) to have about a golf ball-sized piece.
- Paper and pencil.
- They should already know the basic algorithm, as well as how to reduce fractions to their simplest form.
Have the kids work in partners (or solo, if you wish), and pass out a golf ball-sized piece of clay for each two kids. My advice is to let them play with it unhindered for two minutes, with the understanding that at the end of that period, they will do exactly as you say and not play with it anymore. This has been a tried-and-true technique for me.
Begin by telling your pupils to form the clay into a ball. Introduce the activity by explaining that the algorithm for multiplying fractions is really easy, but it is necessary to understand the concept of multiplying fractions: what is actually happening.
For example, you could say, "We’re going to start with an example that’s so easy and obvious it’s almost embarrassing, but because it’s so easy, you’ll be able to see exactly what I’m trying to show you."
Explain/review that "of " means multiply, and make sure they understand this concept. Next, write on the board: 1/2 of 1 = _____. Call on someone; they should respond "1/2." Here is where you will explain that one-half of one equals one-half, and that taking a fraction of a fraction will be the same idea. You can say, "You will be taking a fraction OF a fraction. For example, it’s easy to take 1/3 of a whole pizza, but what if you have only half a pizza, and you want to take 3/4 of that 1/2? Or 5/6 of it? What we’re going to do today will help you understand this."
Now you can explain that you will do a few examples together first.
Have your young mathematicians write the first equation, 1/2 x 1 = 1/2, on their papers, and draw a line under it. Explain that for the following problems, until you say so, they are to follow you step-by-step, and not go ahead; no exceptions!
First Example: 1/2 x 1/2
- Instruct them to separate the ball of clay in half and set the other half aside.
- Next, they break the 1/2 piece into 2 halves.
- Now they will also break the other 1/2 piece into 2 halves, but always keep these separate.
- Note: This leaves the original ball separated into four fourths in front of them.
- Now, hold up one of the pieces and say to your class, "Hold up one of those pieces. What fraction of the total clay is that piece?"
- They respond, "one fourth." (Hopefully)
- You say, "Let's write this down on our paper: one-half of one-half, or one-half times one-half equals one-fourth."
- Check to make sure they all write this down accurately.
- Finally, have them put all the dough pieces together into one big ball.
Second Example: 1/3 x 1/2
- Instruct pupils to separate the ball of clay in half and set the other half aside.
- Advise that this time, they will be taking 1/3 of 1/2.
- Explain that when you take 1/3 of something, you will be separating the pieces into 3 pieces, because the denominator always tells you how many pieces, or groups, you’ll have.
- Next, they divide the 1/2 piece of dough into thirds.
- Once they’ve done that, instruct them to also break the other half into thirds (the purpose of this is to be able to compare the individual pieces clearly).
- Now ask, "What fraction of the WHOLE are those three pieces?"
- They should reply, "one sixth" (The six pieces should hopefully make this obvious, but don’t assume; discuss and clarify as necessary).
- Have your class write the equation: 1/3 x 1/2 = 1/6. Walk around and check carefully for understanding!
- Finally, have them put all the pieces together into one big ball.
Third Example: 2/3 x 1/2
- Start same way: break the ball of clay in half.
- Separate both halves into thirds, separately.
- This time, you are going to take 2/3's of 1/2.
- Pick up one piece, and then another, and say "and this gives us..."
- "Two sixths" should be the response you get.
- Have them put the pieces down and write the equation: 2/3 x 1/2 = 2/6.
- Now, you must explain that you always put fractions into simplest form, and you're going to demonstrate with clay so they can visually understand.
- Hold up two of the pieces.
- "Pick up two-sixths, and combine them together into one little ball. Now, put it down, and look at it. Can you see that those 2/6 together equal 1/3? Look at them. Compare them to those other thirds of one-half you made. Do you see they are equal? Therefore, what equation do we need to write to show that 2/3 of 1/2 equals 1/3?"
- Discuss, and then have them write the equation: 2/3 x 1/2 = 1/3.
- Finally, have them put all the pieces together into one big ball.
Final Example(s): Easier - 1/5 x 1/2. Advanced - 4/5 x 1/2.
- This time, ask students to work the problem on their own, as you circulate the room.
- Discuss the results.
- Make the connection with algorithm and stress that from now on, they can use the algorithm. Remind them to keep this lesson in mind whenever they are using the algorithm so they remember what they are doing.
As a treat, and to solidify the concept, I always do this final activity that the kids love. You start with the whole ball of clay. Make sure they have a clean page of paper; they’ll need it. Warn them that it's important to stay with you and not get behind, and that they must write the equation each time.
Here is how the activity should progress:
- Pupils start with their whole ball of clay.
- Then, you tell them to divide it in half, take 1/2 of 1/2, and write the equation. They do so.
- Now, instruct them to take 1/2 of that 1/2 and write the equation (1/4).
- Next, they will take 1/2 of that 1/2 and write the equation (1/8).
Continue in this manner, as far as you can go. They get really excited as the fractions get smaller and smaller. Using clay is the best way to see exactly what fractions look like as they decrease in size. I continue with this at least until 1/512, or at the point where they have not a speck of clay left. But if they have enough clay, you can go a long way, and the kids are usually hard to stop. This year I had a student who kept going all recess, and was down around 1/32,000 or something. I kid you not.
If you feel your kids need it, you can follow this lesson up with another similar one using the same methodology, with a different substance, or maybe just have them cut out paper shapes. You can also do somewhat similar activities with cups of water (the kids each have several empty dixie cups; they start with a full cup, divide into half, take 1/2 of that 1/2, etc.
All in all, this lesson by itself will go a long way towards helping your pupils understand the concept of multiplying fractions, in a really fun way.