*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?

##### William McCallum

Bill McCallum, founder of Illustrative Mathematics, is a University Distinguished Professor of Mathematics at the University of Arizona. He has worked in both mathematics research, in the area of number theory and arithmetical algebraic geometry, and mathematics education, writing textbooks and advising researchers and policy makers. He is a founding member of the Harvard Calculus Consortium and lead author of its college algebra and multivariable calculus texts. In 2009–2010 he was one of the lead writers for the Common Core State Standards in Mathematics. He holds a Ph. D. in Mathematics from Harvard University and a B.Sc. from the University of New South Wales.

This is such a great article. Putting so many important, related ideas together in such an easy to understand, open manner. Definitely scratched a few itches I had with parts of the mentioned progression. Thank you.

Eleven, twelve, thirteen, fifteen 🙂

Comments about the progressions are welcome here http://mathematicalmusings.org/forums/forum/public/.

The English translation of a common Spanish speaker’s words naming a fraction, (maybe only improper fractions?) is to say “between” as in “among.”

So 15/4 is read “fifteen between four.” This is a notion of sharing 15 things among 4 “people.”

I wonder how this usage influences students’ understanding of fractions.

I often teach beginning algebra to adult students. One of the big ideas is the concept of like terms. We are able to add, subtract, or compare like terms. We can’t two terms if they’re not like terms.

I use a string of examples like the following whenever I am introducing this idea, when seeing it in a new context, or when a student is unsure.

5+3

$5+$3

5 books + 3 books

5 apples + 3 apples

5/17 + 3/17

0.05 + 0.03

(-5) + (-3)

5x + 3x

5y + 3y

5x^2 + 3x^2

5squareroot(2) + 3squareroot(2)

Also, I use language like this whenever I can.

5x+3x

“five ‘exes’ plus three ‘exes’”

(-5)+(-3)

“five negative ones plus three negative ones”

5squareroot(2) + 3squareroot(2)

“five square roots of two plus 3 square roots of two”

5x^2+3x^2+9x+2x+8+6

= 8x^2 + 11x + 14

“Five second powers of x plus three second powers of x is 8 second powers of x. Nine exes + two exes is eleven exes. Eight ones plus 6 ones is fourteen ones.”

I think it’s important in this context to name/say a constant term like 8 as “eight ones.” So often in this context folks say only “eight plus six is fourteen.” Explicitly naming the 8 as “eight ones” reinforces the idea of what we mean when we write 8, and it reinforces the idea that we are adding like terms.

Beth