Fractions can really give students fits! How can we develop conceptual understanding with fractions?

I taught this lesson with a group of struggling 5th grade math learners during an after-school tutoring class. Each student stood at a whiteboard on the wall of my classroom. My goal was to help students conceptually understand why we must find a common denominator when adding fractions. To do this, I asked the students to construct their own number lines. All steps beyond #1 are directions I spoke to students as they wrote on their boards. These were the steps in my lesson:

- I modeled how to make a fraction number line: arrows on both ends showing numbers go infinitely in both directions; small dashes pinpoint exact locations of numbers; begin with zero; space each mark evenly; starting at zero, build your number line and work across the number line rather than starting with a 0 and 1 and subdividing the line into equal size lengths. I wasn’t modeling fraction concepts per se; just creating the number line. (Previous number line work showed students struggling with this.)
- Create a number line that shows thirds from 0 to 1.
- Draw a parallel number line below the thirds number line. Put 0 and 1 directly under the 0 and 1 in the top number line. Now subdivide the new number line into sixths.
- Draw arrows down to the second number line where fractions line up.
- Solve 1/3 + 1/6 using one of the number lines.
- Repeat the above steps with a number line showing fourths and then eighths. (See top photo.)
- Add a third parallel number line showing sixths. (See bottom photo.)
- Use the proper number line to solve 1/4 + 3/8.

Each time I asked the students to create something, I redirected with small hints and asked students to peer tutor until all students had a perfect model. We constantly discussed the models to help students make sense of their work. In step 7, we discussed how very few fractions aligned with the number line above (sixths and eighths.)

Students made excellent progress as shown in their work on the boards.

*Inspiration for this lesson came from my esteemed colleague, Kristian Quiocho. Kristian is one of the most knowledgeable and passionate teachers I know.*

Would it be possible to question the students in the beginning? Maybe – How can you show thirds or sixths (or which ever the desired fraction is) on a number line? instead of modeling it for them?

Do you have cuisenaire rods? As someone who has also struggled with fractions, rods helped me understand fractions a lot more & then connect those to the number line. The number lines are more abstract and the rods provide a concrete model. I think Kristian has some.

I love the use of the parallel number lines. It’s kind of a scaffold for when they move to the problem onto one number line.

Questioning is always where the money is at, and it’s the hardest part. I wonder what questions Ryan and Kristian would ask?

I was thinking about your goal. How do we know whether or not students understand at the end of this why we must find a common denominator? I wonder if you could show them some incorrect answers to these questions? Say Student last year answered these questions this way… Open it up and ask them to prove or disprove why they are correct or incorrect. You may get some good discourse going there. My kids learn a TON from disproving things. Tells them more about what they are doing.

Thanks for sharing! I love learning and thinking about math in the upper grades. I’m itching to get out of first.

Wonderful thoughts, Jamie! Some responses to your questions/comments:

Would it be possible to question the students in the beginning? Maybe – How can you show thirds or sixths (or which ever the desired fraction is) on a number line? instead of modeling it for them?Great point. Actually, I presented this lesson in isolation, but we had already been doing some number line work in previous lessons. I only modeled the proper way to make a number line. I did not discuss how to properly place certain fractions on the number line.

Do you have cuisenaire rods?I have been using concrete models before this lesson on number lines. I have these fraction cube connector things. Not sure of their exact name. I definitely want to get into using Kristian’s cuisenaire rods.

I wonder if you could show them some incorrect answers to these questions?LOVE this! I will be stealing that idea.

Thanks again, Jamie.

I stole that one from Andrew Stadel from NCTM last year.

There’s a lot to think about with fractions on the number line, which is also incredibly an important tool for kids to deeply understand fraction equivalence, and hence fraction operations. Most classrooms avoid it and stick with area models, which has much to do with why middle and high school students continue to be lost when dealing with fraction concepts. We recently did a 4th grade lesson study on the topic of fractions on the number line, researched, planned, taught, observed and debriefed. The write up our shared understandings. In short, kids generally see fractions as 2 numbers (b/c of area models) and need a length model. However, they need to understand that fractions are represented as a LENGTH, not a mark. The name of that place on the line is named that (e.g. 3/4) because it is the length of three units of length 1/4 (called 1/4 because it takes four of those lengths to make 1, which is described by ITS (1) distance or length from zero) placed end to end or concatenated – like measuring. Here’s the summary document…

http://www.lcsc.edu/media/4890900/Cuisenaire-Rods-on-the-Number-Line-shared-understandings.pdf

Thanks so much for sharing, Ryan. I am very excited to read about your lesson study! The “2 numbers” insight is huge. Very helpful.

I actually failed to mention a step that I did at the beginning of the lesson with the kids. I first took the fraction connector manipulatives and laid them sideways. I used them to create a number line to show the kids how they related. Also, I circled the segments on the number line showing them that the fractions were actually those segments rather than the dashes along the number line.