The idea of fractions being equivalent is such a key understanding for helping students develop “fraction sense” that it absolutely must be done right. Students need to see a variety of representations of equivalent fractions to help them see the relationships between the numerators and denominators when fractions are equivalent. Beginning in 3rd grade, students should use concrete materials to experience equivalent fractions. The use of concrete materials helps students to develop images in their minds that they can draw on later when manipulatives or drawings aren’t available. Another great visual representation for equivalent fractions is double number lines.

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A common practice for teaching equivalent fractions is to show students to either multiply both the numerator and denominator by the same number or divide both the numerator and denominator by the same number. Essentially, you are multiplying or dividing the fraction by 1, which doesn’t change the amount of the fraction.

Without a visual representation, however, that is an incredibly abstract concept. But we teach the “trick” because it is quick and easy, even though it does nothing to further our students’ understanding of fractions. Now don’t get me wrong, I’m not knocking the mathematical way of finding equivalent fractions. I just think it should be the end result of LOTS of concrete and pictorial practice.

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.

Students can then transfer that knowledge to a pair of double number lines showing the equivalent fractions. The understanding of the multiplicative relationship actually even helped us construct the number lines as I recently worked with a group of 4th graders to compare numbers with different denominators. Consider the following picture as you read through the conversation we had while drawing the number lines you see pictured. You see my drawing, which was projected on a document camera, and the students were showing the same work in their own math journals.

Me: (writing 7/12 and 2/3 in my math journal) *So, we are comparing 7/12 and 2/3. 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!

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

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.

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.

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.

What do you do when you are comparing two fractions that aren’t easily drawn on a number line? For example 2/3 and 4/7.

This is just one strategy and may not be the best in all cases. With 2/3 and 4/7, students might want to find a common numerator. 🙂

I really enjoyed your post. I have just finished teaching this concept to a group of fourth graders. I think that transfer from concrete (fraction pieces) to the number line is such a critical piece to the learning. Students need time to build number lines (part of grade three) using their fraction pieces to see those connections. Even though many fourth graders get the idea of same whole and same part when comparing, if they haven’t truly developed the idea of partitioning on a number line, they struggle to see the same whole, same part. Using a number line that is scaled to the size of fraction pieces is helpful for students who still in that concrete and visual representation connection.