# 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

# Area and Perimeter of Squares – Student Noticings

This will be a quick post because I have a student-posed math problem that I need some time to reason through!

Today, students found the area and perimeter of squares that increase in side length by one each time. Students used a variety of models when building their squares from Minecraft carpets, to Geoboards to graph paper. Here is the completed activity sheet from their work: I then gave them a few minutes to talk to their tablemates about things they notice in their work. Here are the answers they shared as a class and I recorded on the board:

“An even dimension by even dimension = an even area”

“An odd dimension by odd dimension = an odd area”

“The perimeter goes up by 4 every time the square gets bigger”

“The areas are square numbers.”

“The areas go up by odd skip counting: +3, +5, +7…”
I was pretty excited because they really pulled out some great noticings and my next step was for them to choose one and find out why that was happening.

AND THEN THIS HAPPENED…

WOW, what a noticing!

Each pair of students chose one noticing from the board and worked on figuring out why that was happening. I had groups share the even dimensions = even area and perimeter going up by four. The tables that chose area going up by “odd skip counting” and the last one, left with no answer but excited to keep trying to “figure it out.”

Now, if you know why this last one works, please let me know that you know, but keep it a secret from me for right now! I want to sit and work through this one but I also need to know who to run to if I don’t get it!

I have found that you have to add the odd dimension area to the even dimension perimeter and if you do it the other way, it does not work. Why in the world does this work every time?

Had to share because it was such great conversation and I left having the hunger to sit and work thru the math….better yet, the students did too.

Enjoy and please let me know if you know why that is working because I may be reaching out!!

-Kristin

**Follow up comment: Thanks to my Twitter buddies, I worked my way to the visual of this problem. It was much easier to make sense of this algebraically, but the “why” took a lot of square drawings and scribbles! It was hard to make the connection between perimeter being the distance around to it being one side or a square tile. Here is part of my working on my Geoboard app…

So the area of a 1×1 + the perimeter of a 2×2 = the area of a 3×3.

# Always, Sometimes, Never – Quadrilaterals

One of my “go to” questions for students when they are working through math in my classroom is, “Will that always be true?” I find it pushes the thinking to another level where students are looking for examples and/or non-examples.

On Twitter one evening I found a blog post by @lisabej_manitou that was the embodiment of my go-to question: http://crazymathteacherlady.wordpress.com/2013/11/20/always-sometimes-never/ . Can you say perfect timing, as my class is in the midst of quadrilateral properties/classifications?

I gave each group of students Lisa’s sheet, clarified any key vocabulary questions and the conversations started rolling!

We have been doing a lot of work with classifications and discussing all of the classifications polygons can have, but this activity took that to a great new level.  The “Sometimes” column has to be my favorite because it requires to think of both cases, true and not true.

One group had a very “heated” debate on the “Square is a Rectangle” card, which if you asked me ahead of time, would not have been the one I expected to hear such debate (at least in the respect that it was). I know that often students come into this unit having formed or memorized some form of the statement that “Squares are rectangles, but rectangles aren’t squares.” Whether it is taught or formed on their own, it is put to test when faced with the always, sometimes, never. Those are the words that are key in the misconceptions built around that statement. That was the conversation I expected to hear when I walked over to the group and looked at the card in question…however it was actually quite different reasoning!

One member of the group was literally “Starting to sweat” (her words) from this conversation. She was trying to explain to her group why a square is SOMETIMES a rectangle. Her reasoning was this: (I had to have her write it down so I could use it later in class and she needed a breathing moment away from her group)

She made an interesting point for students to reason about. If a rhombus can be a square, and rhombuses (or is it rhombi?) are not rectangles, squares can’t always be rectangles.

I pulled the class together to discuss this point because there were others agreeing with her reasoning. They SO wanted me to tell them who was right and who was wrong….um no way! I asked them what would prove or disprove this argument to them. One group said they would need her to show them an example of when a square was not a rectangle because if it is sometimes, it has to be a case of when it is and isn’t.

And, class dismissed. They left wanting to continue: creating arguments, critiquing the reasoning of other, making mathematical models, looking for patterns in their reasoning….I would say it was a great day in math!

I am SO glad they didn’t finish yet bc I am planning on recording some conversations on Monday to post.

Thanks Lisa for the great lesson!

-Kristin