# 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.

## 11 thoughts on “Area and Perimeter of Squares – Student Noticings”

1. jmccarth

Hi Kristen,

This is so cool. I found an algebraic solution but that wasn’t much fun ….. then I went to a visual justification ….. so awesome! What a great pattern to discover!

Janice

Like

1. mathmindsblog Post author

Thanks Janice! I agree, the algebraic felt so much easier for me but definitely not as much fun! The visual model of this took me a while to wrap my head around but I loved it! This is definitely not something I was expecting them to see. I am excited to see if they can work through this in some way on their own!

Like

2. xiousgeonz

Okay, who’s gonna post the visual? Taking that “distance around” and recognizing it as a number, not a distance, and turning it into tiles and wrapping each tile around the smaller square Mathematically it might be important to mention that “adding” area and distance has its issues.

Like