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 Jody Guarino

We know instructional materials play a key role in student learning experiences but how do we ensure our students are learning from coherent high-quality instructional materials that engage them in critical thinking and provide opportunities to “do math?”

Let’s think about this from the lens of a 4th grade standard, 4.OA.A3: Solve multi-step word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing in for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Continue reading “Instructional Materials Matter: Interpreting Remainders in Division”

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 Kristin Gray

As a teacher, curiosity around students’ mathematical thinking was the driving force behind the teaching and learning in my classroom. To better understand what they were thinking, I needed to not only have great, accessible problems but also create opportunities for students to openly share their ideas with others. It only makes sense that when I learned about routines that encouraged students to share the many ways they were thinking about math such as Number Talks, Notice and Wonder, and Which One Doesn’t Belong?, I was quick to go back to the classroom and try them with my students. It didn’t matter which unit we were in or lesson I had planned for that day, I plopped them in whenever and wherever I could because I was so curious to hear what students would say. Continue reading “Warm-up Routines With a Purpose”

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 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”

By William McCallum

Somewhere back in days of Facebook fury about the Common Core there was a post from an outraged parent whose child had been marked wrong for something like this:
$$6 \times 3 = 6 + 6 + 6 = 18.$$
Apparently the child was supposed to do
$$6 \times 3 = 3 + 3 + 3 + 3 + 3 +3 = 18$$
because of this standard: Continue reading “Ways of thinking and ways of doing”

By William McCallum

You may have noticed that I am back to publishing regular blog posts! My goal for now is a blog post every second Wednesday. I am now also trying to answer forum questions promptly. I want to thank the readers who took up the slack for the last year and a half in answering questions in the forums. In particular, I’d like to call out abieniek, Alexei Kassymov, and Lane Walker, whose answers were always spot on. Continue reading “Misconceptions about Multiple Methods”