I never tire of conversations about the 3rd – 5th grade fraction progression because after each one, I leave with the desire to reread the Standards and Progressions with a new lens.

A few weeks ago, a conversation about 3rd grade fractions sent me back to the Standards with a #pairedtexts type of lens. Unlike the hashtag’s typical MO of pairing contrasting texts, I was looking for standards that connected in a meaningful, but maybe unexpected way. By unexpected, I don’t mean unintentional, I mean the two standards are not necessarily near-grade or in the same strand, so the connection (to me) is not as obvious as one standard building directly toward another.

The conversation focused on this standard:

With that standard in mind, imagine a 3rd grade student is asked to locate 3/4 on a number line on which only 0 is marked.

I expect a student would mark off the 1/4’s starting at 0 and write 3/4 above the point after the third 1/4 segment. What exactly is the student doing in that process?

Is the student adding?

Is the student counting?

Is the student doing both?

How does adding and counting look or sound the same in this scenario? different?

This is where I find pairing two standards fun and interesting to think about because it demonstrates how important seemingly unrelated ideas work together to build mathematical understandings. It is also really fun to think about how a standard in Kindergarten is so important for work in grades 3-5 and beyond.

In this scenario, I think we instinctively believe students are adding unit fractions when asked to place 3/4 on the number line because the standard is in the fraction strand and therefore we consider all of the work to be solely about fractions. We also sometimes impose our thinking on what students are actually doing in this task. For example, you could imagine the student marking off the fourths, stopping after the third one, writing 3/4 and say the student was adding 1/4+1/4+1/4 to get to the 3/4 because they moved along the number line. If this is the case, then the standard would pair with this 4th grade standard:

Don’t get me wrong, those standards definitely pair as students move from 3rd to 4th grade, however, since the scenario is about a 3rd grade student, pairing it with a higher grade level standard doesn’t seem to make sense in terms of what students are building on. Right here, it is really interesting to pause and think about how building fractions from unit fractions, locating a fraction on a number line, and adding unit fractions are slightly different things a progression.

When I think about the student locating 3/4 in 3rd grade, I hear counting (with a change in units) and would pair that 3rd grade standard with this Kindergarten counting and cardinality standard:

However, because the 3rd grade work is on a number line and the arrangement and order *does* matter, I would have to add this 2nd grade measurement standard into the mix, but take off the sum and differences part:

So, instead of a #pairedtext, I now think of it more as a #CCSSMashup to create this standard:

With that mashup in mind, I went back to the progressions documents to look for evidence and examples of this.

In the 3rd Grade NF Progression these parts jumped out at me as being representative of this standard mashup:

*The goal is for students to see unit fractions as the basic building blocks of fractions, in the same sense that the number 1 is the basic building block of the whole numbers; just as every whole number is obtained by combining a sufficient number of 1s, every fraction is obtained by combining a sufficient number of unit fractions.*

*The number line reinforces the analogy between fractions and whole numbers. Just as 5 is the point on the number line reached by marking off 5 times the length of the unit interval from 0, so 5/3 is the point obtained in the same way using a different interval as the basic unit of length, namely the interval from 0 to 1 /3 .*

There is also a great “Meaning of Fractions” video on the Illustrative Mathematics site that explains this idea with visuals.

There are so many of these great mashups in the standards, especially in the fraction strand, that I find incredibly helpful in thinking about how students coherently learn mathematics.

I look forward to hearing your favorite #CCSSMashup!

AnnaNo mashups to offer, but I love fractions on the number line! Without the number line representation I often see kids thinking of fractions as a totally separate idea from “regular numbers.” I once tutored a middle-schooler, introduced him to fractions on the number line, and then he started solving problems by saying “I saw the number line in my head!” 😀

I had the privilege of working on the Learning Mathematics through Representations curriculum while in grad school, and I love how it builds that understanding by moving from integers to rational numbers using the number line. Check it out! http://www.culturecognition.com/lmr/mathematically-coherent-curriculum

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