By William McCallum

In one of our professional development workshops, there is an activity in which the facilitator asks teachers to skip count by $\frac34$. The facilitator records the count, $\frac34$, $\frac64$, $\frac94$, . . . and then asks for patterns they notice in the recording. In a recent workshop, a group of grade 5 teachers noticed that the numerator increased by 3 each time but that the denominator remained unchanged. When the facilitator asked why, they could easily explain the number of pieces were increasing in the numerator, but couldn’t really give an explanation for the denominator other than “it is just always out of 4.” The funny thing is, they weren’t saying “3 out of 4, 6 out of 4, 9 out of 4” when they skip counted. They were saying “3 fourths, 6 fourths, 9 fourths” and writing it in fraction notation. The key to understanding why the denominator stayed the same was hidden in plain sight, in the very language they were using to name the fractions.

To understand this, recall that in the standards fractions are built up out of unit fractions. In grades 1–2 students learn that numbers can be built up out of ones, tens, hundreds, etc. They learn to see 43 as 4 tens and 3 ones. They see ten as a single unit made by bundling together 10 ones, and they learn to unbundle that unit when necessary for addition or subtraction.

In grade 3, students start to see the number one itself as a bundle of smaller units. Their rulers become subdivided into halves or fourths of an inch. On their number line diagrams they see the length from 0 to 1 subdivided into smaller equal lengths, say thirds or fifths. They learn to  see $\frac13$ not as “one out of three” but as a piece of a certain size, three of which make a whole. And then they build other fractions out of pieces; $\frac53$ is literally five thirds, the number you get put putting 5 pieces called $\frac13$ together on the number line.

The work in geometry in grades 1–2 has prepared them for this thinking. To quote from the latest version of the fractions progression:

“. . . first graders might put two congruent isosceles triangles together with the explicit purpose of making a rhombus. In this way, they learn to perceive a composite shape as a unit—a single new shape, e.g., recognizing that two isosceles triangles can be combined to make a rhombus, and simultaneously seeing the rhombus and the two triangles. . . . [they] use fraction language to describe partitions of simple shapes into equal shares—halves, fourths, and quarters in grade 1, extending into thirds in grade 2.”

Where does the “out of” language come from? It is fairly natural language to use when you are describing fractions less than 1. You divide a rectangle into 4 equal parts and shade 3 out of the 4 to represent the fraction $\frac34$. It is also natural when using fractions to describe sets: 3 out of the 4 roses were red, so $\frac34$ of the roses were red. The danger here is that students might not see beyond the formal process: count the red roses, count all the roses, then put the two numbers together. It is not a coincidence that the standards do not use set representations to introduce fractions, but rather use measurement representations (length or area) where the whole is more easily seen as one thing.

So perhaps one way to understand fractions better is to simply listen to ourselves when we name them. When we say “five sixteenths” we are not only naming a particular fraction, we are giving a constructive definition for it: you can figure out how big $\frac{5}{16}$ is by taking a unit called a sixteenth, 16 of which make a whole, and putting 5 of them together. It doesn’t always happen that our names for things describe their mathematical meaning, but this is one case where it does.

Here’s the record of the skip counting from the beginning of this post.

What patterns do you notice? How would you explain why these patterns occur? Do the names for these numbers help reveal the mathematical underpinnings?

### Next Step

Can you think of an example where the words we use for a number get in the way of its mathematical meaning?

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”