Last week I was reminded why I love being a teacher.

I’m currently throwing down a PL gauntlet in my district for all our elementary and middle school teachers. For each grade level I’ve focused on 1 or 2 of the big ideas outlined within CCSS.

In 3^{rd} grade, I really tried to stress the importance of a *Unit Fraction. *Lots of students leave elementary school not understanding the concept of a unit fraction as 1/a. Just as numbers are composed of ones, fractions are composed of unit fractions and it’s imperative that students conceptually get this understanding.

A teacher was about to begin her fraction unit and asked me to model a lesson we did during our PL…I happily obliged. I’m not a big fan of fraction bars, I love Cuisneaire Rods but there’s something about Tangrams that tickle my fancy.

Day 1…First day of fractions…tell me what you notice…

Students replied with:

- I see shapes.
- I see a triangle.
- I see a square.
- I see 5 triangles.
- The are 2 triangles that are the same size as each other.
- There’s another 2 triangles that the same size as each other.
- I see a diamond kinda.
- You mean a rhombus.
- It’s actually a parallelogram.

It was a beautiful conversation and all I did was record students’ *noticings*.

I asked them to cut up their shapes. *What do you notice?*

**Student:** Two small triangles make up the square.

**Me:** How much of the small square does 1 little triangle make? *They quickly responded with a half*. Anyone now how to write that as a fraction?

1/2

**Me: **Great!** **What do the numbers mean?

**Student: **2 means the square is cut into two pieces that are the same size as the triangle. The 1 means that the triangle can cover only one of the 2 pieces.

At this point, I’m not correcting anything. I’m just listening and recording. We want to know what they know.

**Student: **Two small triangles make up this triangle (pointing to the medium one).

**Student: **Two small triangles make up the parrell thing.

Students explored the pieces for a couple more minutes and then I dropped it…

*“Can you find out how many of each shape it would take to make the original square? Like if the square was made from big triangles, how many would it take? If the big square was made from medium sized triangles, how many would it take? Etc…”*

First things first, they forgot the size of the original square so that in itself was a beast. I ended up sharing a piece of paper that matched the original square for groups that struggled.

The end objective was to label each piece of the Tangram as a unit fraction (1/a).

Procedurally, this could have been achieved in one day but that’s not how we roll!

It might seem counter-productive to begin fractions with such a difficult task but students arrive at various levels of understanding. I have lots of take-aways from the beautiful mess of learning that took place in 1 hour.

- We swung for the fences and out came the misconceptions.
- This was a beautiful way to decide our next moves because now
*we*know what*they*know. - Starting with a difficult task and allowing misconceptions to come to the forefront leveled the playing field.
- Flexibly grouping students after this task was easy because the students grouped themselves. We just took notes.

The best part (for me) was an email I received from the teacher after her follow-up lesson…

Great stuff, amigo!

I recently did a task with tangrams and unit fractions with my 7th graders. These same misconceptions were evident. Having the students determine the total number of equal parts was powerful. Some went from thinking about fractions as simply as part of a whole to equal parts of a whole.