As students begin using fraction tiles to explore equivalent fractions, they notice that it takes, for example, four twelfths to make a third, because the twelfths are smaller pieces. There is a multiplicative relationship–for every third, four twelfths are required.

Me: *(writing 7/12 and 2/3 in my math journal)* What relationship do you notice with the denominators here?

Student: Times 4.

Me: Explain.

Student: 3 times 4 equals 12.

Me: *(drawing the arch between the denominators and labeling it)* Oh! I saw it as dividing by 4, because 12 divided by 4 equals 3. I guess that’s the same thing, right?

Students: Yes!

Me: *(drawing two number lines and labeling them with 0 and 1) *Okay, let’s draw two number lines and see if that will help us decide on a way to compare these two fractions. Remember, you have to carefully line up your 0 and 1 on the number lines to make sure the wholes are the same size.

Me: *(after both number lines are drawn) *Let’s show 2/3 on the top number line. I picked that one because I think it’s the easiest one to draw, right?

Students: *(laughing) *Yes!

Me: *(dividing the number line into thirds and labeling the parts)* Okay, I know it won’t be perfect, but let’s try to make our three parts as equal as possible. By the way, how many lines will I have to draw to make three parts?

Student: Two!

Me: That’s right. Okay, I think they look pretty good, don’t you? Now, on our bottom number line, we need to show twelfths. That’s a little trickier. Let’s think about that times, or divided by, relationship. Right here at one third *(drawing a line on the bottom number line)*, how many twelfths will I need?

Student: Hmmmm, four?

Me: *(chuckling) *You don’t sound too sure. Why do you think four?

Student: Because it’s times 4. It takes four twelfths to make one third.

Me: Sounds good to me. Let’s label it 4/12. Now how many parts will we need between 0 and 4/12?

Students: Four.

Me: So how many lines will we need to draw?

Students: Three.

Me: *(dividing the space between 0 and 4/12 into four equal parts) *Let’s do it.

Me: Okay. Looks pretty good. *(drawing a line right below 2/3 on the bottom number line) *Hmmm, what fraction would go right here?

Students: 8/12!

Me: Why?

Student: Because it will take another four twelfths.

Me: Makes sense to me. Can you guys draw and label the parts?

Students: Yep!

Me: Okay, can we finish out this number line to show all the twelfths?

Students: Yes.

Me: *(after completing both number lines)* Okay, so now we see the relationship between thirds and twelfths. Let’s get back to our comparison. We were trying to compare 7/12 and 2/3. Can our number lines help with that?

Student: 2/3 is the same as 8/12, so it’s greater than 7/12.

Me: Let’s write that down and be sure it makes sense. *(writing 7/12 under 7/12 and 8/12 under 2/3) *So you’re saying that 2/3 is equivalent to 8/12, and 8/12 is greater than 7/12?

Student: Yes.

Me: Give me a thumbs up if you agree with that.

Students: *(all thumbs up)
*Me: Think we can do another one?

*Students: Yes!*

Okay, I know that was really long, and I thank you for sticking with it. But I just thought it was important for you to “hear” the conversation we had while drawing the number lines.

Remember that finding common denominators is only one of the strategies for comparing fractions. Be sure to check out this series of blog posts for information on all of the other methods. There are several freebies sprinkled throughout the four posts.

I’d love to hear your comments on this lesson!

Kari says

Donna – love your article. I think there is an error in your top photograph. I believe you mean to have 3/5 on one side and 9/15 (not 8/15) on the other in the third pair of equivalent fractions. Thanks for all you share. Kari

Donna Boucher says

Actually, we were comparing 3/5 and 8/15. If you look below those two fractions, you’ll see where we converted 3/5 to 9/15, proving that 3/5 is greater than 8/15.

Michelle says

I agree that there should be plenty of opportunities for young learners to work with the pictorial representations of fractions. You referred to the algorithm for 1/4 x 4/4= 3/12 as a ” trick” which I have disagree. This algorithm follows the Multiplicative Identity Property for fractions. Technically you are multiplying by 1 by using the scale factor 4/4. Also, this skill also vertically aligned to dilation and reduction which is taught in middle school.

Donna Boucher says

I referred to it as “trick” when it is being shown to students without teaching the underlying understanding, and unfortunately that is often what happens.