## A Fraction Number Line Lesson

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:

1. 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.)
2. Create a number line that shows thirds from 0 to 1.
3. 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.
4. Draw arrows down to the second number line where fractions line up.
5. Solve 1/3 + 1/6 using one of the number lines.
6. Repeat the above steps with a number line showing fourths and then eighths. (See top photo.)
7. Add a third parallel number line showing sixths. (See bottom photo.)
8. 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.

## Teaching Evaluation Without the Teacher

Last year, it was my turn to be observed. After teaching for 27 years, I was ready to take a risk, try something new. So I went to my principal and asked, “Do I have to be in my classroom when you come do my observation?” As you can imagine, she didn’t exactly know how to respond. But since my boss was a good sport, and trusted me, she said, “Well, what do you have in mind?” After I explained, she decided to play along.

When the time came, my principal walked into my room and I walked out. It was a very good year, probably the best group of kids I’d ever had, a class of very mature fifth graders. One girl ran the class. She managed 32 students reviewing a language arts assignment. The students worked in groups, pairs, and held a whole class discussion. According to my principal, my observation went well. I had to take her word for it, since I wasn’t there.

So why would I do such a crazy thing?

About 12 years ago, I began my journey to make my students the most independent they possibly could be. I had read an archived newsletter at learningcentered.org that told the story of a sub who didn’t show up for school. The principal was walking the hallways, noticed the kids in one class working away diligently, but didn’t see any teacher. Upon inquiry of the students, the principal found out that the teacher hadn’t shown up, and neither had the substitute. So the kids just went about their business of learning. This struck me. Could my students do this? Would they?

So to me, the real assessment of my teaching is how my students work without me. And if that’s my goal, then why should I be in the room for my teacher observation?

Am I crazy? Did I get you thinking? Talk to me.

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