# Geometry Is Worth The Extra Time…

As I am sure many teachers can attest, there is a constant struggle each year between covering content and the precious amount of time we have to engage the students in learning. Prior to the past two years in the classroom, this guilt always seemed to creep up most during our geometry units. I used to feel that once the students could find area, perimeter, and volume, we would move back into our fraction and decimal work because that always took SO much time to develop a deep, foundational understanding. While geometric representations such as an area model support the fraction and decimal work, it is still not the 2D or 3D unit work.  Right or wrong, I felt I had to prioritize to make use of the little time I had for the best of my students. Over the past two years, however, my geometry units have been taking longer and longer because I have started to see things evolve in my geometry units that has me  wanting to kick myself and go back in time to give my past students a different learning experience. From the connections to number and operations to the development of proofs and generalizations have been eye-opening.

As all of these math connections were going through my head, I see this tweet from Malke (@mathinyourfeet)…

ahhh, it felt like validation in some weird way.

After this Twitter conversation, I started to dig back into my students work to find examples that makes these connections visible.

After doing a dot image as our Number Talk one day, I asked students to see if they saw any connections between the image and our volume work that day. This work shows how students see the commutative property in both, multiplication as groups (like layers in volume) and most importantly puts a visual to how multiplication and its properties “look” in both 2D and 3D.

Then volume led into some great generalizations about how multiplication “works” through looking at patterns, which is extremely important in mathematics in and of itself.  In keeping constant volume (product), students realized they could double one dimension (factor) and half the other. In doubling the volume (product), the students realized they double one dimension (factor) and leave the others the same. T

This volume discoveries later let to this claim on our claim wall:

The students extended this area and volume work to fractions/decimals that showed that fractions/decimals act as numbers in operations as well, supporting the structure of our number system.

While we classified polygons, I saw my students develop proofs for angle measures and our always, sometimes, never experience was invaluable. This work in connecting reasonings through visuals of the polygons explicitly supports the Mathematical Practices of using models, perseverance, and repeated reasoning.

Then our work with perimeter and area solidified the importance of students creating a visual in building number is so important. In a problem with equal perimeter and different area (moving into greatest area), students created a beautiful visual for the commutative property as well as supported students in seeing the closer two numbers (with the same sum), the greater the product.

-Kristin

# Modeling Mathematics – Developing the Need

Today we were talking about things we noticed as we worked with finding a fraction of 1/2. Students are noticing things I expected: that the denominator doubles each time, the numerator is staying the same as the fraction you are dividing the half up into, some are starting to notice that the numerators are multiplying and so are the denominators, and some are just flat out complaining that they have to model it on the fraction the bar.

So the fraction of 1/2 was pulling some great noticings, however I wanted the students to feel the value of being able to model the mathematics, to show what was happening, so I asked them what would it look like if I had 3/4 of a candy bar and wanted to split it with two friends. What fraction of the whole bar would I get?

I was excited that some labeled the 3/4 on a fraction bar with 6/8 and then split that in half and labeled 3/8. They said they “knew half of 6/8 was 3/8” I asked why didn’t they work with 3/4, they explained that splitting the three was not working and 6 was easier because it was even.

Some said they had split the 3/4 in half and it looked about like a 1/3, so it was 1/3. I appreciated the estimations, but looking for them to dig further after they estimated.

I would say that a third of the class had written 3/4 of 1/2= 3/8 with a fraction bar 3/4 shaded and then split in half and labeled 3/8. When asked how the fraction bar modeled their answer, they told me that they didn’t “need” the fraction bar to find the answer, they noticed that you multiply the numerators and denominators. “Can’t we just give you the answer?” “It’s the right answer, right?”

We don’t “need” the fraction bar. Huh.

Then an interesting thought hit me….they see pictures as a tool they don’t need rather than a model of a mathematical situation. It almost seemed as if they viewed the bar as “baby-ish” to use. You know how certain things hit you as WoW?!? That is completely what that comment did. I immediately started to reflect on how I had made the fraction bar sound…did I just make it sound like a way to solve? Did I even use the word model? Am I placing too much emphasis on the modeling piece?

I can see why students view diagrams as a way to solve….when they learn to add, they draw pictures. When they first work with “groups of” they draw pictures. When they first work with arrays, they draw things in row and columns. Once they have learned how to add, the pictures aren’t necessary. When they have learned how to multiply, the arrays and groups of aren’t seen as a necessity.

In that moment, I wanted the students to appreciate how important (and difficult) modeling is in mathematics.  I pushed them to explain how to name that line drawn at half of the 3/4 and we had some great conversations about why this was more difficult than a fraction of 1/2.

In the end, I assured them that I sit with adults all of the time and we struggle (and find MUCH enjoyment) in making models of mathematical situations. They felt ok with knowing it wasn’t just tough for them and I felt ok that they could see a “need” for their fraction bars!

-Kristin