Jill Gough and I presented a session at #NCSM14 entitled Developing Conceptual Understanding through the Progressions: Fractions, Ratios, and Proportions.

Thanks to the work of Gail Burrill, Tom Dick, Wade Ellis, and Becky Byer, TI has just released the first lesson module in a series called Building Concepts. The first module is on Fractions, and it is available in its entirety on the TI website.

Proportional Reasoning is coming soon. Their work is based on the work of Wu: Teaching Fractions According to the Common Core Standards.

I have been saving two pictures from my daughter’s end of first grade scrapbook that I finally got to share in a PD session.

I love that fractions made Kate’s top 5 things learned in first grade. Kate also learned how to spell big words in first grade (except fractions).

On another page in her scrapbook, Kate notes that fractions are the best thing about first grade.

Look closely at Kate’s visual representation of the fraction 1/3. Kate’s picture is telling of one of the reasons we need to pay close attention to how we develop conceptual understanding of fractions for our students.

The Essential Learnings for our session:

- I can describe a fraction
**a/b** as **a **copies of **1/b**.

- I can construct questions that push and probe student thinking about fractions.

- I can explain the role that technology plays in deepening student understanding of fractions.

While participants were gathering for the session, I sent a Quick Poll with a question from TIMSS 2011, Grade 8. Item M 02_04.

Participants answered very closely to how US students answered.

(And they made it easy for me to tell that 25% of them had it correct!)

US 29%

International 39%

Florida 29%

Massachusetts 47%

Korea 86%

We moved to the first activity, What is a Fraction?

A tweet from Gayle about the fraction 5/3: 5 copies of 1/3. Love this CCSS language.

- When you increase the value of D, how does the number of equal parts in the interval from 0 to 1 change? What happens to the length of those parts?

- If a fraction is made up of more than five 1/5 unit fraction parts, what can you say about the value of the fraction?

- When comparing two different unit fractions, how can you tell which fraction is greater?

I asked participants to think alone about the last question for a minute before sharing with their group. One participant noted that as the denominator of a unit fraction increases, more of them fit in a one-unit interval. Someone else noted that as the denominator of a unit fraction increases, the size of the fraction decreases. Someone else noted that the fraction farther to the right is greater.

True or False. Explain your reasoning.

- A whole number cannot be a fraction.

- A fraction can have many names.

We used Class Capture (taking a picture of each participants’ handheld) to discuss a few examples and counterexamples. (I wasn’t as coordinated during the PD session to take pictures of the Class Capture for this blog post as I sometimes am during class with my students!) A friend overheard one group saying that they knew whether the statements were true or false, but they were not confident that they could explain them.

How can we increase the confidence of our teachers? How can we use visual representations of a fraction to know whether the statements are true or false? Will our students recognize the connection between what they are seeing on the number line and the truth of the given statements? Can the technology help them make their own conjectures about the validity of the statements instead of relying on their teacher for an explanation?

We moved to the second activity, Equivalent Fractions.

What questions could you ask from looking at the first page?

As we moved to the second page of the TNS document, our focus became on how we know whether one fraction is larger than another fraction.

Which is larger, 4/3 or 5/4? I love this question! I love that I can think through which fraction is greater using the idea that **a/b** is **a **copies of **1/b** and not have to evoke any algorithmic procedure that I know for comparing fractions.

We moved to the 3^{rd} activity, Fractions and Unit Squares.

What is the area of the unit square? When the fraction is 1/5, what is the area of one of the rectangles?

One the next page, we took some time to show that 4/16 of a unit square is equivalent to ¼ of a unit square.

I used Class Capture to monitor their work. Several groups showed that the shading for ¼ was the same amount for 4/16.

I noted that none of them showed that 4/16 was equivalent to ¼ the way that I had. I saw each column as 1/4 of the whole. Each column is also 4/16.

My students surprise me every day with seeing things differently from how I see them. What they see is beautiful. I just have to slow down and pay attention!

Our last exploration was to shade ¼ of a unit square and think about what is ½ of ¼.

Our hour session had gone by very quickly. We revisited our essential learnings for the session.

- I can describe a fraction
**a/b** as **a **copies of **1/b**.

- I can construct questions that push and probe student thinking about fractions.

- I can explain the role that technology plays in deepening student understanding of fractions.

At some point, I shared a picture that my geometry students had recently discussed when partitioning a segment.

How might my high school students benefit from learning that **a/b** is **a** copies of **1/b**? Will it help them when they think about partitioning the segment into equal parts?

How might learning that **a/b** is **a** copies of **1/b **help my students when we are studying the unit circle and they are having to make sense of π/3, π/4, and π/6? Or comparing 2π/3 to 5π/6? Or adding π/6 to π/4?

I know I have so much more to learn about teaching fractions through conceptual understanding. The resources for these activities have been a good place to start.

And in case you are looking, another good place to learn is the Fractions Progression Module at Illustrative Mathematics.

Tonight’s IM Task Talk features two 5^{th} grade tasks about fractions:

5.NF Connor and Makayla Discuss Multiplication

5.NF Fundraising

And so the journey continues, both as the learner and the teacher …