Fractions Greater than One, or the Artists Formerly Known as Improper Fractions

The Background

It's confession time: I was a compulsive fraction simplifier. Not in lowest terms? Simplify it. Improper fraction? No question. Change it to a mixed number. No rhyme or reason, except that the simplest form or mixed number was the only correct answer most of the time... and an improper fraction just felt wrong.

As a student I learned the tricks for converting improper fractions to mixed numbers and vice versa as isolated procedures, and as a 5th and 6th teacher B.C.C. (Before Common Core) I taught it that way. There was very little concept or context... as Phil Daro would say, it was answer-getting, not math. Something like this:

If I was lucky, students put all the digits from their division problem into the correct spot in the mixed number. Spoiler alert: I wasn't always that lucky. Why do we convert improper fractions to mixed numbers anyway? (wait time) No, really, why?

Turns out, there's no mathematical reason why we MUST convert an improper fraction to a mixed number, even though the name "improper" certainly implies that we best fix it ASAP! In the CCSS, improper fractions have been rebranded, in a way, as "fractions greater than one". In fact, no where in the standards will you see the term "improper fraction", yet work with fractions greater than one begins in 3rd grade. Students whose concept of a fraction is built from the foundation that all fractions, whether less than or greater than one, are built up of unit fractions (fractions with a 1 as the numerator) can see the two forms as equivalents and flexibly move between improper fractions and mixed numbers. In other words, improper fractions are no big deal and are not even distinguished from "proper fractions" during initial fraction work - turns out, they're all just fractions! The NF Progressions does a nice job of explaining this:

"[Grade 3] students build fractions from unit fractions seeing the numerator 3 of  ³⁄₄ as saying that ³⁄₄ is the quantity you get by putting 3 of the ¹⁄₄'s together (3.NF.1). They read any fraction this way... ⁵⁄₃ is the quantity you get by combining 5 parts together when the whole is divided into 3 equal parts." (p.3)

So,

 

The Lesson

A few weeks ago, I had the chance to work with a 4th grade lesson study team on a fractions lesson. The district is in its second year of implementation with Eureka Math, and we have been working together on getting to know the curriculum and how to navigate it. Lesson studies have given us a chance to really dive into materials while keeping our focus on what it's all about - the math. This 4th grade lesson study team decided they wanted to write a lesson that would build conceptual understanding of conversion between fractions greater than one and mixed numbers in preparation for upcoming work with addition and subtraction of fractions. During our research, we found this gem in the NF Progressions:

"Converting a mixed number to a fraction should not be viewed as a separate technique to be learned by rote, but simply as a case of fraction addition. Similarly, converting an improper fraction to a mixed number is a matter of decomposing the fraction into a sum of a whole number and a number less than 1 (4.NB.3b). Students can draw on their knowledge from Grade 3 of whole numbers as fractions. For example, knowing that 1=³⁄₃, they see  ⁵⁄₃ = ³⁄₃ + ²⁄₃ = 1 + ²⁄₃ = 1 ²⁄₃." (p.8)

We decided to work with a couple lessons from Module 5 in Eureka Math that focused on decomposing and composing fractions greater than 1 to express them in various forms AKA converting between "improper fractions" (I'll put a quarter in the swear jar) and mixed numbers. We began with a fluency exercise called "How Many Ones?" which focused on helping students consider whole numbers in terms of halves and fourths (e.g. "How many wholes are in 4 halves? How many halves are equal to 3?") We then presented students with an application problem:

Shelly read her book for ¹⁄₂ hour each afternoon for 5 days. How many hours did Shelly spend reading in all 5 days? 

Students were given independent think time to work on their whiteboards to solve. We were impressed at the number of mathematical models students used to express their reasoning. We saw number bonds, tape diagrams, equations, and area models - pretty cool. Stuff like this:

Student work recreated from my observation notes

The teacher chose two students to share their thinking, one who had thought in terms of ¹⁄₂ hours to come up with an answer of ⁵⁄₂ hours, and another who had decomposed the ⁵⁄₂ into ²⁄₂ + ²⁄₂ + ¹⁄₂ = 2 ¹⁄₂ hours. The majority of students had come up with ⁵⁄₂ as an answer, so it was purposeful for the teacher to bring both forms of the answer to the forefront for students to consider. They discussed whether ⁵⁄₂ or 2 ¹⁄₂ made more sense in the context of this situation and they agreed that they would probably say that Shelly read for 2 ¹⁄₂ hours, not ⁵⁄₂ hours if they were explaining it to someone.

This problem set the stage for the concept development portion of our lesson. The learning goal was for students to be able to rename fractions greater than one as mixed numbers by showing their equivalence on a number line, a goal that was supported by the models already written into the Eureka Math lesson. We decided on this particular learning goal because it gave us an opportunity to address both standards 4.NF.3.b (Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model) and 4.NF.1 (Explain why a fraction is equivalent to [another] fraction be using visual fraction models).  

"Visual fraction model" is one of my favorite terms in the standards. It connects beautifully to the CRA instructional approach which focuses on building math concepts through a progression of Concrete, Representational and Abstract experiences. Concrete is the hands-on part, and research shows that it's the most "crucial level for developing conceptual understanding of math concepts". The lesson study team recognized that need for a concrete manipulative to accompany the work in this lesson, and chose fraction tiles as they connect nicely to the number line, the highlighted representational model in this lesson. We designed a Build it, Draw it, Write it (BDW) template for students to use to capture their thinking as they converted fractions greater than 1 to mixed numbers. The first fraction was ⁷⁄₃. Students were asked to work in their groups to use fraction tiles to build ⁷⁄₃. This is what they came up with:

This group thought in a linear way and used their 1 whole tiles to show that 3 thirds was equal to 1 whole. We were hoping to see this way of thinking, as it connects nicely to the number line model.

This group organized their tiles into groups of 1 wholes, each made up 3 thirds. Not but we were looking for, but a great way to think about it.  

We asked students to show how they could decompose the fraction greater than 1 into wholes and "some more" by using a number bond and then to capture this decomposition with an equation. Our earlier work in the fluency exercise with identifying whole numbers in terms of fractions proved to be especially helpful for students:

⁷⁄₃ can be decomposed into ³⁄₃ + ³⁄₃ + ¹⁄₃

Next, students were asked to draw a number line to represent their thinking. There was great conversation amongst the lesson study team members between the two teachings of the lesson about whether we should have students draw a number line with endpoints 0 and 3 and partition the number line into thirds, or whether we should have them draw an empty number line, unitize the ¹⁄₃ and iterate thirds until we got to ⁷⁄₃. Both strategies were valid and each had its benefits.

An example of the Build it, Draw it, Write it (BDW) Template in action

By using the number line, students were able to see that ⁷⁄₃ was the same as 2 ¹⁄₃ because they represented the same point on the number line. I heard a student share with her partner, "⁷⁄₃ is equivalent to 2 wholes and 1 third." This is a HUGE idea that is first introduce in 3rd grade with equivalent fractions less than 1 whole. Students went on in this lesson to work with converting ¹³⁄₅, ¹⁰⁄₄ and ¹¹⁄₃, gradually moving away from the need for the fractions tiles. The BDW template (which I will upload here after I stop fighting with Google Drive) played a huge role in moving students meaningfully through the progression of concrete-representational-abstract, and at the conclusion of the lessons, nearly ³⁄₄ of students successfully met the learning objective.

The student debrief was powerful. Students were brought back together as a whole group and asked, "How can using a model like a number bond or number line help you when converting a fraction to a mixed number?" Students said things like, "You can see how many wholes the fraction has," and "They show you the leftovers after you find the wholes." Finally, they were asked "How do you know that a fraction greater than one and a mixed number are equivalent?" They were able to agree that a fraction greater than one and a mixed number are equivalent when they are the same point on the number line, and several students referred to their work on the BDW to justify their thinking.

Did I mention that

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