Category Archives: Perimeter and Area

3rd Grade Perimeter Part II

Last week, I posted about a 3rd grade lesson I planned and taught with Hope and Lori. We did not get to everything we planned so I love that they filled me in on what happened the next day when they continued the work! And when the continuation involves looking at student work, I love it even more! That said, this will be a bit of a student work-heavy post with things I noticed and wondered in steps moving forward from here with the students…

After measuring a piece of paper in the previous lesson, we wanted to ask students how they would find the distance around any-sized piece of paper. In giving them the journal writing, we wanted to have them reflect on the measuring and calculating they did in a more general sense and see how they put the process into words. Most student papers resembled the explanation in piece of work below:

It was really interesting to find most students drew a picture to illustrate their explanation even when not asked to do so. To me, this is a nice mix of show your thinking and show your work. In reading this example below, this student is thinking a lot about conversions and I think, moving forward, the class needs to have a discussion about combining different measurement units.

The mix of units shows up again below. I can see they probably chose centimeters because they didn’t have a smaller unit than the inch and didn’t know how to name the measurement in inches. I love the “not really the size” but I wonder about the border look of the perimeter. Is this student seeing the 6 inches and 3 cm ending where the line is and counting boxes instead of the distance around?

This one is an amazing look at how the formula we all know, and probably had to memorize, arises in third grade. The calculation on the back was equally as nice. This is an example of something during a class share that I would show last in a progression to compare with the previous strategies as it does a nice job of showing the process of finding perimeter in two ways.

This one was so interesting because it involved a square and a circle. The measurements on the back were most intriguing and I have so many questions for this student. Like, how do you know that is a square? (because the sides are not the same length) Where is the 1/2 coming from in your answer? (because I cannot tell where he is stopping and starting his measuring) and Why did you want to cut a circle and a square?

Then, Hope asked them to draw their own ant path and some really interesting things came out that will have to be a blog post in and of itself! There are things we didn’t think about in our question and some things we really need to think about moving forward. Like…

Could this student start thinking about area? Why did the choose to draw a non-rectangular path?

Where are the measurements for each side? Why did you label them where you did? Why did you choose to use inches and centimeters?

When did you choose to use inches and when did you choose centimeters? Could you have measured it all in inches? all in centimeters?

First, the fact the student sent the ant to Walmart is too funny:) I would love to ask this student how he or she added all of those side lengths? and Why was it longer to get home from Walmart than it was to get there? Could the ant have walked the same distance there and back? How?

On this one, we did not anticipate students’ ants taking the same path back that he did out. So this is important to think about distance and versus distance around something.

Oh, an isosceles right triangle, how fun! I would love to ask this student about this perimeter in centimeters because of the diagonal cuts in the boxes. A lot of students counted the diagonals as 1 unit like they did for the sides of the boxes, so would that work out if you measured it with a ruler in centimeters? Why?

My question is where to go with a student who is here? All teachers face this, right? There are some students who conceptually and computationally have a grasp on an idea. This student can obviously find perimeter and is very comfortable with the computation piece of it, so what do you ask him from here? Do you give him things to measure that closer to a quarter and see how he works with the fractions? Do you ask him if his strategy will work for every shape? (I lean toward this one) Do you ask him about non-rectangular polygon areas? Do you do anything with area at this point? So much to think about!

~Kristin

Developing Claims – Rectangles With Equal Perimeters

2 Replies

Yesterday’s math lesson launched based on this student activity book page in Investigations…

Students read the introduction and I first asked them, “If we were to build these in Minecraft, was feet an appropriate unit of measure?” Some thought that feet seemed too small for a garden and instead wanted to use yards, that was, until one student schooled us all on Minecraft. Come to find out, Minecraft uses the metric system with each block representing one cubic meter. We then changed the unit to meters and were on our way.

I gave them 10 minutes, either with a partner or individually, to build as many different rectangular garden designs as they could with 30 meters of fencing. As I walked around, it was interesting to see how students were designing their gardens. Some started building random dimensions and adding/subtracting fence links to eventually hit 30 while others had thought of one rectangle to start and adjusted from there by subtracting/adding from sides. It was fun to see the ones that discovered a pattern in their building and sped through the rest of the rectangles. Some beautiful patterns emerged….

After 10 minutes, I asked them fill in the table on their activity page and chat with their table to see if anyone had any dimensions they didn’t have. As a table, they came up with some noticings based on their table, Minecraft builds, and process of building.

The class ended after this group work. Talking with another colleague about the lesson at the end of the day, we chatted about what claims could come out of this lesson. It was so interesting to me to think about not only the geometric claims, in terms of rectangles and dimensions, but also the numeric claims that can evolve from these conversations as well. And to think that these number claims are grounded in a visual connection is pretty awesome to me!

I was so I started today by pairing up every two tables to share with one another. One table read their noticings while the job of the other table was to ask clarifying questions. That gave me time to circulate, listen, and choose the claims for our class share out. After the table shares, we reconvened as a whole class and chatted about a few of the claims as a group.

I asked them to think about similarities and if we could combine some of them to form one claim based on our work.

Class ended with a journal entry in which they worked independently to begin to form a claim or the beginnings of one. Here is where I will start tomorrow…

How fun!

As a follow up on this post, this is the assessment on their work with perimeter and area. During the class period, only one student started working on finding the square that would give them the largest area with a perimeter of 30. After this idea came out in our class discussion, using fractional dimensions was something that others were thinking about during the assessment. Here is a student’s (different than the initial student) work that I thought was pretty fantastic:

-Kristin

Area/Perimeter of Squares…Help.

3 Replies

Through my work each year with area and perimeter, I learn more and more about how I learned (was taught) math. I can work out a “proof” algebraically, however I struggle in connecting it conceptually to the problem. When this happens, I am so hesitant to reach out bc, truthfully, it is a bit embarrassing. I mean, I teach this stuff, right? But I finally hit a point, after I started blogging, when I learned that I will never learn more if I am not open to where I am. Since I encourage my students to write honestly about their understandings, I thought it only fair to throw my thoughts out there into the wonderful #mtbos. So here it is…

Here was last year’s example that I worked my way through: https://mathmindsblog.wordpress.com/2014/01/27/area-and-perimeter-of-squares-student-noticings/

And now here is this year’s: https://mathmindsblog.wordpress.com/2014/12/20/areaperimeter-my-homework-over-vacation/

I am finding the same thing is happening again…algebraically I have it, but struggling with the conceptual connection. I have a bunch of theories scattered on my papers this morning, but nothing that is satisfying me mathematically and would appreciate any thoughts….

So when thinking about area related to perimeter in squares, I know that n^2 x 4/n = 4n, but I am stuck at that 4/n. I marked off what I thought 4/n looked like in my squares, I messes around with ratios, found some patterns, but still not seeing (or putting together) what I want. … so I went to this drawing bc the side was increasing by 2 every time…. Then I went to this…

And while I would love to play with this for a bit longer, I have so many other things to do to get ready for school tomorrow! I feel like I have it somewhere here, I just cannot make a connection that works for me. Would love any pieces to the puzzle put together for me:)

Thanks!!

-Kristin

Area/Perimeter – My homework over vacation

2 Replies

It is always fun when I can look back at my past blog posts, see what I learned from a lesson, and reflect on student reasonings before I teach that same lesson again. This past week we were working on this lesson from last year: https://mathmindsblog.wordpress.com/2014/01/27/area-and-perimeter-of-squares-student-noticings/.

The lesson unfolded in much the same way, with the class patterns looking like this:

I anticipated all of them, however, like last year, there always has to be one that throws me a curve ball and leaves me math homework over Christmas vacation:)

The 5th statement looked like this in his math journal:

With these other noticings below it…

His explanation focused on the pattern of the fraction the perimeter is of the area. The numerator stayed a two and the denominator went up by one on every even dimension. I honestly didn’t know what to ask him because the question “Why is that happening?” seemed way to broad. He played around with building the squares and was not getting anywhere. I am thinking, after chatting with Christopher (@trianglemancsd) on Twitter, that focusing on the meaning of the fraction first may be the way to go??

Basically, I first have to sit down and reason about his on my own…gotta love math homework on vacation!

MathMinds

by Kristin Gray.