Category Archives: Geometry

A Coordinate System

This standard in 5th grade always seemed like so much of a “telling lesson” for me.

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I never thought it was really addressed in the spirit of this standard in our curriculum, so I was typically like, “Here is what we call a coordinate grid. These are axes, x and y. We name the points like this…” and so on. It is not my usual approach so it always felt blah for me, for lack of a better word. I told them, they practiced plotting some points, and we played a little bit of Battleship (which was really fun).

Last week, I was planning with Leigh, a 5th grade teacher, and we spent a lot of time just talking about what we appreciate about the grid and how we can develop a sense of need in the students for it. Since they are in the middle of their 2D Geometry unit, we thought this could be the perfect place to plot points that connect to form polygons and look at patterns in the ordered pairs.

The questions we wanted to students to reason about through our intro lesson was:

  • Why a coordinate grid?
  • Why name a point with an ordered pair? 
  • Why is this helpful?
  • What structure do we see?

So, we created this Desmos activity. This was our thinking on the slides and the pausing points we have planned for discussion:

Slide 1: It is really hard to describe a location without guides or landmarks.

Slide 2: Note how difficult it is. Pause and show class the results.

Slide 3: It gets easier. Still need some measurement tool. Notice the intersection of axes.

Slide 4: Note it is a bit easier this time. Pause and show class results.

Slide 5: Much easier because of the grid. Still need a starting point. See it is the distance from axes.

Slide 6: Now it is much easier. Pause show class results. Would love to show all three choices side-by-side (don’t know if this is possible in Desmos).

~Pause~ Ask, “What names of things on the grid would make it easier to talk about the point’s location?” Give students vocabulary and ask them to revisit Slide 6 to describe the location to a partner.

Slide 7: Practice writing some ordered pairs.

Slide 8: Practice writing some ordered pairs.

Slide 9: Start to see some structure in the four ordered pairs of a rectangle.

We are ending with this exit ticket (with grid paper if they choose to use it):

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While we are not sure this is the best way to intro the grid, we thought it would generate some interesting conversation. Since we are teaching it tomorrow, there isn’t much time for feedback for change, but we would love your thoughts.


Rhombus? Diamond? Square? Rectangle?

It happens every year, in what seems like every grade level…students continually call a rhombus a diamond. Last year, when we heard 3rd graders saying just this, Christopher helped the 3rd grade teachers and me put the students’ thinking to the test with a Which One Doesn’t Belong he created.



This year, at the beginning of the geometry unit, we heard the diamond-naming again along with some conversation about a rectangle having to have 2 long sides and 2 short sides. What better way to draw out these ideas for students to talk more about them than another Which One Doesn’t Belong? We changed the kite to a rectangle this time, hoping we could hear how they talked about it’s properties a bit more.

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Overwhelmingly, the class agreed D did not belong because it had “5 sides and 5 corners” and eventually got around to calling it a diamond, which in their words was “not a real shape.”

While we knew a lot of things could arise, our purpose was diamond versus rhombus conversation, so of course the students had other plans and went straight to the square versus rhombus.We wouldn’t expect anything different!:) For every statement someone had about why the square or rhombus did not belong, there was a counter-statement (hence the question marks in the thought bubbles).


Jenn, the teacher, and I were really surprised at how much orientation of A and B mattered to the name they gave the square and rhombus but did not matter for the rectangle. That was just a rectangle, although one student did wonder if a square was also a rectangle (he heard that from his older sister). The students had so many interesting thoughts that we actually had to start a page with things they were wondering to revisit later! That distributive property one blew me away a bit!:)


We then sent them back to journal because we wanted to hear how they were categorizing a square and rhombus. It ended up being really interesting just seeing them try to explain why they were different and change their mind because they just started turning their journals around!

Some stuck with them being different..



Some thought they were different, but one could become the other…



Some were wavering but the square was obviously the “right way.”


Some argued they were the same…

So much great stuff for them to talk about from here! I left wondering where to go from here? In thinking about the math, is it an orientation of shapes conversation? or Is it a properties conversation? In thinking about the activity structure, would you pair them up and have them continue the conversation? Would you throw the rectangle into this conversation? Would you have some playing with some pattern blocks to manipulate? Would you pull out the geoboards? I am still thinking on this and cannot wait to meet and plan with the 3rd grade team!

However, before I left school today, I went back to the 3rd grade standards to read them more closely:Screen Shot 2017-01-05 at 7.07.05 PM.png

and read the Geometry Learning Progressions, only to find this in 1st grade:

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Would love to hear any thoughts and ideas in the comments!



Rhombus vs Diamond

Every year in 5th grade, when we begin classifying quadrilaterals, students will continually call a rhombus a diamond. It never fails. While doing a Which One Doesn’t Belong in 3rd grade yesterday, the same thing happened, so Christopher’s tweet came at the most perfect time! (On Desmos here:

Of course I had to pop into the same classroom today and try it out! The lower right was so obviously a diamond to me that I was curious to see if students saw the same thing and if it changed their reasoning about the rhombus as a diamond.

Here are pictures of the SMARTboard after our talk:


After great discussions around number of sides, rotations, decomposition and orientation, they finally got to the naming piece. Honestly, I was surprised names didn’t come up as one of the first things. It started with a student saying the square didn’t belong because it is the only one that doesn’t look like a diamond. The next student said the lower left was the only one “that didn’t have a name.” When I asked him to explain further, he named the square, rhombus, and diamond. Because I knew at the end of our talk I wanted to ask about the diamond vs rhombus, I wrote the names on the shapes. Another classmate added on and said the lower left “may not have a name but it is kite-shaped and looks like it got stuck in a tree sideways.” I asked the class what they thought about the names we had on the board and it was a unanimous agreement on all of them. Funny how quickly they abandoned their idea from yesterday, so I reminded them….they were not getting off the hook that easy;)

“Yesterday you were calling this rhombus a diamond, what changed your mind?”

Students explained that the lower right actually looks like a real diamond and the rhombus doesn’t now that they see them together.

“Can we call both of them a diamond?” I asked. I saw a few thinking that may be a great idea. I had them turn and talk to a neighbor while I listened to them.

We came back and they seemed to agree we couldn’t call them both a diamond because of the number of sides. They were really confident in making the rule that the quadrilateral one had to be a rhombus and the pentagon was the diamond. I pointed to the kite and asked about that one, since it has four sides. “Could we call this a rhombus?” They said no because the sides weren’t equal, so not a rhombus. And because it didn’t have five sides, not a diamond either.

Thank you Christopher! All of these years of trying to settle that rhombus vs diamond debate settled right here with great conversation all around!

Next up, this one from Christopher…


Perimeter in 3rd Grade

I am in the unique position over the next few weeks to see perimeter and area work in 3rd, 4th and 5th grade. It is so incredible to see the overlap across all three grade levels and, being a 5th grade teacher for so long, it is great for me to see where this work begins.

After planning with Hope and her student teacher, Lori, last week, we taught the lesson introducing perimeter today. On Friday, the students measured things around the room in different units of measure, having discussions about most appropriate units. For example, when measuring the length of the room would we use the same unit as we would for the width of our pencil eraser? Why?

Since I was not there for the lesson on Friday, I was super curious to just hear what students thought about when they heard the word “measuring.” I wrote the word on the board and away we went. They were very quick with benchmarks, equivalents and different dimensions we can measure. I did a terrible job with my picture, but I got a couple really interesting questions like, “Can we measure anything? Air?” and “Can we measure the corner angles of things like the carpet?” Also, after a student had shared that one yard is the same as your hip to your ankle, students questioned if that was true because of the different heights of people. All of these things are great for students to explore at later times!


Hope introduced the Investigations problem of an ant traveling around the edge of a piece of paper. To be honest, we were not thrilled with the context, but at the time we could not come up with anything snappy or original, so we went with it. We thought it was nice because, in inches, we could see if students measured to the half inch and also how they worked with the half inch when combining to find the perimeter. In hindsight, I am thinking a city map might have worked, however then the scale comes into play, so maybe not?? We let them choose the unit they thought would be appropriate, put them with a partner and they went off to work together. We were surprised to see most students using inches and when asked, thought that it would be “too many centimeters.” They seemed to chose units based on the biggest unit that still fits the object, but not thinking about precision and getting the smallest unit for that.

This is where I am continually amazed by what students know and intuitively do with mathematics.


It was interesting to see some pairs not know how to deal with the half,”not quite 9,” but know they only had to measure one side and then put “11” on the opposite side.


While another group had the 8 and one half written exactly like they said it “8 and 1 (one).5(half) inches.” Although written incorrectly, they dealt with it beautifully in their computation. However, I would want to bring up the equal sign in future share outs so they 8×2=16+1=17 would be written correctly. Does anyone use arrows in the elementary grades for this? 8×2–>16+1–>17? Or is it more appropriate for separate lines at this age?


When I walked up to this group I asked where the ant was walking because of their lines through the middle of the paper. They said around the outside but it is the same no matter where you draw the line. I asked them to show me the 8 inches and I left them to talk about the 1/2 inch.

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Some students did not deal with the 1/2 inch but seeing the ways they found the perimeter and wrote their equations, I was able to see the formula for perimeter coming to life.

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As students got finished with their first unit choice, we had them find the perimeter in another unit. It was nice to see the multiplication from their previous unit showing up a lot. IMG_1526

When I saw this one, I didn’t really know what to do with it. What do you with a 3rd grader using .5 as half? I asked them what .5 meant and they quickly said one half. They said one can be broken into .5 and .5 just like it can be broken into 1/2 and 1/2. That is so interesting to me and I would have loved to explore that conversation more, but with a whole class that is not ready to go there, I wrote it on the board and moved on.

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As always, there is not enough time in a class period for me to talk about math with the kiddos. Tomorrow morning, students will journal about their strategy to find the distance the ant traveled. Since the majority of the class only measured two sides, we want to make explicit, through student sharing, why they didn’t have to measure all four sides in this case.

They next part of the lesson, which Hope and Lori will continue tomorrow, includes the students creating their own ant path on grid paper and finding the perimeter of that path. We are not going to dictate that the path must be a rectangle, but the ant must stay on the grid lines. We are hoping that this generates the conversation of when we can double the two sides and add them and when we can’t, assuming students draw irregular shape paths.

#PiDay2015…Circle Fun

Some of my students this year were excited to “celebrate” pi day and were very disappointed that it fell on a Saturday, so we decided to have some pi fun on Pi Day Eve. I am not one for “gimmicky” holiday lessons and wanted whatever I decided to do, to not just be definitions of circles and their properties or a formula for how to use pi to find measurements, but instead an activity that allowed students to discover all of the cool things about circles and patterns that arise from that work.

After brainstorming with a colleague, she suggested I just have the students try to create a prefect circle. Loved it. I put out tape, scissors, rulers, paper, string and told them if they thought of other tools they wanted to use, they had to pass my approval first (I wanted to keep the protractors and compasses out of the equation for right now). Off they went! It was soooo interesting to see all of the great approaches and all of the cool ideas that emerged from their work.

I found it so interesting that quite a few first drew a square and tried to find the center. They said they knew that the circle could be made inside of it because a circle is 360 degrees and each angle of the crossed lines was 90. The problem became figuring out how to get the “rounded edges to be the same.”

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Quite a few groups had seen a compass before (but didn’t know what it was called) and tried to recreate one with the available tools. Some started from finding a center and going from there, while others created the center by just placing the scissors on the paper and going around from there. After many attempts, they were starting to realize how important keeping that constant distance in the scissor opening really was.

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To solve the constant distance problem, one group used tape to keep it the same while another group used string (and chopsticks she just happened to have in her lunchbox that day:).


This group solved the constant distance with two pencils attached with string. The funniest part of this one were the trials as the string kept wrapping around the center pencil as they went around and never meeting exactly back at the start. They eventually figured it out after blaming the “center holder” numerous time for “moving the pencil.” Another group kept a constant distance by taping their string to the center of their paper and putting a pencil on the other end.


This group created a center from overlapping rulers and attempted to put string around the the ruler corners to make an arc, but couldn’t agree with how to get them all the same. While another group tried to use the ruler ends as the center but ran into the same problem with the rounded edges.


This idea was interesting to watch evolve. She had seen the group on the floor (in the pic above) and said she realized that any rectangle rotated would make a circle. She then grabbed a ruler, taped two cap erasers to each end and thought the caps would leave eraser marks she could go back and trace after rotating the ruler. That didn’t work, no marks. She then cut her pencil to get some lead and taped that to one end.

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The final product…..IMG_9586_2 After sharing their circles and approaches, I had the students jot down some things that were important when constructing their circles.

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From these, I realized (and was surprised) the students have some circle vocabulary in their toolbox. I decided to get that out so we could be sure everyone in the class had exposure to all of this great stuff. I asked them to share their findings and what measurements they used or could find in their circle.

IMG_9535_2One group had finished their circle early, so I asked them to find some of these measurements. They found the diameter and circumference with the ruler and string they had used in the construction. It was so interesting to see the intuition students have around finding the diameter. They knew it had to go through the center and that no matter where they measured from, it would be the same. It makes me wonder why we, as teachers, sometimes think that we need to give students definitions for things before they get to demonstrate their intuition around these very ideas. I could have told them “diameter is distance across the circle through the center” before the lesson started, but they already knew that, love it.

After testing a few circles, this group started to see pi emerge…

IMG_9563_2 IMG_9564_2 IMG_9562_2For the the last circle in this list, they measured the diameter of their large circle they created and I asked them to estimate the circumference. After seeing that each circumference was “about 3 times as much,” they estimated 46 x 3 to be circumference. They haven’t had a chance to test it on the actual circle yet because we ran out of time, but that will be some fun on Monday!

Happy Pi Day 2015!


Pre/Post Assessment Reflection

We started our 2D Geometry unit with Talking Points:  This was the ultimate pre-assessment in which I could hear what the students were thinking around mathematical concepts while at the same time, they had a chance to also hear the thinking of their peers. After the talking points activity, I had the students reflect on a point they were still unsure in their thinking.

We are now wrapping up our Polygon unit, and I thought it would be interesting for them to reflect back on what they were unsure about in the beginning, and get their thoughts now. I have a class full of amazing writings, but here are just two of the great reflections (the top notebook in each picture is the pre-unit and the bottom is post-unit)….

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Looking at the class as a whole, it was so interesting to see their math language develop and see them laughing at things they had written before. I loved that the student above wrote, “I am smarter!!!” How amazing they can see their own learning!  During their reflection time, it was so fun to also hear students exclaiming, “See, I KNEW I was right!”

This is the first pre/post assessment I have ever done where I think the students enjoyed it as much as I did! They were as proud of themselves as I was of them!


Volume with Fractional Dimensions

Before I began our volume work this year, I blogged about my planning process here: As anticipated, I had many students who quickly developed (or already had) strategies for finding volume and could articulate a conceptual understanding of what was happening in the prism. In my previous post, I was throwing around the idea of giving those students dimensions with fractional length sides, so the other day I thought I would try it out. I did this Illustrative Task as a formative assessment of student understanding. Many students were done in a couple of minutes, with responses for part b that looked like this:

IMG_7972As I walked around the room and saw they were finished quickly, I asked them to revisit part b and think about a tank with fractional dimensions. Because of the great work they had done here I thought they would have some interesting thoughts. These are a few of the responses I got:


So, what did I learn from this work?  I saw they had some great understandings about taking a fraction of one factor to make a number that they knew they needed to multiply by a third whole number factor to get 240.  In the first two pictures, there is a great pattern happening that I want to explore further with the whole class. I also loved seeing that a student took the question “fractional length sides” to include decimals in his work. In my question, however, I had wanted them to consider more than one side in fractional lengths, however not being more explicit, they took it and ran with one side being fractional.  In the next lesson, I thought I would push them a bit with this.

In the following lesson, students were finding the volume of an unmarked prism in cubic centimeters. They had rulers, cm cubes, and cm grid paper available to them, and went to work. Every year this happens, the Investigations grid paper works with the box to be whole number dimensions, however the cm are a bit “off” when using a ruler or cm cubes. I knew this, however, I do love the discussions that evolve from students who used different tools. I also thought this is the perfect opportunity for my students who were beginning to think about fractional sides. What transpired in the whole class lesson is a blog post in and of itself, however this is what came about from the fractional sides work…



Sooo much great stuff here! I had a group who was using the cubes, coming out with halves, but not wanting to round because it was “right in the middle” of the cube. I let them go and came back to see they were multiplying whole numbers, multiplying the fractions, and then adding them together to get their product. I asked them to think about another multiplication strategy to see if they got the same product, then came the array. Another student in the same group solved mentally to get the products. Unfortunately, the class had to leave me to go to their next class, also leaving me with so many things to think about. From here, I want to be sure students start to think about reasonableness of their solutions, compare their fraction multiplication strategy to whole number multiplication strategies, and think about how we multiply three numbers (Associative property). So much to do, I need full day math classes!