“Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3).  Explain why the fractions are equivalent, e.g., by using a visual fraction model.” CCSSM 3.NF.3b

Let’s work backward on this one. We know that one way to find an equivalent fraction is to multiply both the numerator and denominator by the same number. In other words, multiply the fraction by a version of one. For example, if we multiply 1/2 by 2/2 (a version of 1), we get 2/4. Multiply it by 3/3 and we get 3/6, etc. The same holds true for division. Divide 4/8 by 2/2 and you get 2/4. The problem is that we often teach students the process without helping them to understand why it works. The process is totally abstract, that is, it is completely symbolic. Let’s take a look at how we can incorporate both concrete (hands-on) and representational (pictorial) activities to build an understanding of the abstract process.

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You can certainly use fraction tiles for this activity, but I want to show it with fraction strips, which you can download for free below.

Provide each student with a halves strip.

Have kiddos lightly shade 1/2 and darken the line between the halves. Engage the students in math talk throughout this activity. For example, asking why is this 1/2? What does the denominator (2) mean? How about the numerator (1)? You get the idea.

Ask students, Talk to a partner. How might we use this same strip to see fourths? After students have had a chance to talk briefly, ask for ideas. Hopefully, at least one group of students suggests folding the strip. Direct students to fold the 1/2 strip in half and then in half again, creating fourths. Have them darken the fold lines. Ask what fraction they see shaded now (2/4). Write 1/2 and 2/4 next to each other on the board and ask students what they notice about the numerators and denominators. They might not notice anything yet, and that’s okay.

Ask students, Talk to a partner. What do you think would happen if we folded it again? After students chat for a minute or so, ask for responses. Just like with a science experiment, it’s always a good idea to have students predict during math explorations. Next, have the kiddos fold the same strip in half again. Again, have students darken the fold lines. Ask, What do we have now? (eighths) How many are shaded? (4)

Write 1/2 and 4/8 next to each other on the board. Also, write 2/4 and 4/8 next to each other. Have students turn and talk to each other about the relationships they see between the numerators and denominators. Hopefully, you’ll have students who start to see the multiplicative relationships. Then you can introduce the written process:

See if they can explain why this works with the strips (When we folded it, we had twice as many shaded parts and twice as many total parts.). See if they can work together to apply this process to 1/2 and 4/8 and 2/4 and 4/8.

Give them practice with the strips and other combinations. Keep it open-ended! Ask them if they can find, show, and describe other equivalencies with their strips.

As with all mathematical concepts, multiple representations are the key to deep understanding. Let them explore this idea with fraction representations other than strips, as shown below.