By William McCallum

“I’m afraid I can’t explain myself, sir.
Because I am not myself, you see?” Alice in Wonderland.

The idea of equivalence in mathematics is tricky for learners, because when we talk about two things being equivalent, for example the fractions $\frac35$ and $\frac6{10}$, we are emphasizing two contradictory things: Continue reading “Fractions: Units and Equivalence”

By William McCallum

In everyday language, $\frac{a}{b}$, $a\div b$, and $a : b$ are all different manifestations of a single fused notion. Here, for example are the mathematical definitions of fraction, quotient, and ratio from Merriam-Webster online: Continue reading “Untangling fractions, ratios, and quotients”

By Melissa Greenwald

You know it is time for a change when half of the students in class are lost by the third lesson of a new unit.

I teach third grade in a charter school in Philadelphia. We use Go Math! and each year I have followed Chapter 8: Understand Fractions, exactly as written. In the first lesson, students name equal parts in pictures such as halves, thirds, and fourths, and then move into finding equal shares. By the third day, when we discuss unit fractions, I feel like I have already lost about half of my students. Despite this, I usually trudge along and move into the fourth lesson where students are asked to identify the shaded fraction of different shapes. By the end of the lesson, my students typically have learned the rote skill of counting the number of shaded and total pieces in order to write the fraction. This becomes incredibly evident when we move to putting fractions on a number line and problematic when problem solving with fractions. Continue reading “Adapting Curriculum For Students to Know, Use and Enjoy Fractions”

By Jared Gilman

As I sat down at my local coffee shop to plan my upcoming 5th grade unit on fractions, a wave of dread spread across my body. I started having flashbacks to last winter, when my students’ frustrations with fractions led to daily meltdowns. Looking back at my lesson plans, I noticed how many reteaching lessons I was forced to add into the middle of my unit. I recalled the painstaking hours of scouring YouTube for videos on the “easiest tricks” and “fastest shortcuts” for adding and subtracting fractions. “My students just didn’t get it,” I thought at the time. This year would be different, I told myself as I gulped down my large iced coffee.

By Kate Nowak
(co-authored with Kristin Gray)

“Ultimately, the goal of this unit is to prepare students to make sense of situations involving equivalent ratios and solve problems flexibly and strategically, rather than to rely on a procedure (such as “set up a proportion and cross multiply”) without an understanding of the underlying mathematics.”
Illustrative Mathematics 6–8 Math, grade 6, unit 2, lesson 12

By Kristin Gray

Recently, our 3rd, 4th, and 5th grade teachers had the opportunity to chat math for 2 hours during a Learning Lab held on a professional development day. It was the first time we had done a vertical lab and it felt like perfect timing as 3rd and 4th grade would soon be starting their fraction unit and 5th would be entering their decimal unit. Prior to the meeting, we read the NCTM article, “Identify Fractions and Decimals on a Number Line” by Meghan Shaughnessy, so we started the meeting discussing ideas in the article. We then jumped into playing around with clothesline number lines and double number lines, discussing what they could look like at each grade level based on where students are in the fractional thinking. Continue reading “Fraction & Decimal Number Lines”