Extending an Area Task

I just love when students are so excited to extend an activity! During the Notice and Wonder portion of this lesson, a lot of students wondered why those four letters were the ones given. Was it because they are at the beginning of the alphabet? Is it because they have the same area? What would happen with other letters?

Today, Mrs. Sharp gave them the chance to play around with others letters. She asked them to design their own letter and find its area. It is so interesting to see their choice of letters, the way they chose to decompose the shape and their math work all around it. Many of them made multiple shapes because they just wanted to keep going…that is always so AMAZING to hear!

I also see all of this being SO helpful when they find volume of figures composed of two non-overlapping rectangular prisms in 5th grade!

Here are just a few of the creations they came up with in class today:

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Area Task in 3rd Grade

Since the 3rd graders just wrapped up their unit on area, I thought it was the perfect time to do a task that hit on some really important ideas about area, while also encouraging them to move beyond counting squares. I wanted to see how (or if) students broke apart shapes to find the area, how they used addition and/or multiplication to more efficiently count squares, and if they used any other strategies such as subtracting out blank spaces or decomposing and rearranging pieces to find the area.

I chose this task from Illustrative Mathematics.

We started with a notice/wonder activity:

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They had so many great wonders that inspired me to think about a follow up activity about other letters, but I will chat about that later.

Since they wondered if the letters all had the same square units and if they were all the same, I used that as the lead into the activity. Even as I was giving directions, however, I saw a majority of them start to count squares by 1. I paused them, told them I was so excited to see they knew counting the squares would get them the area and knew they all could find the correct area that way so we were going to try something different. I asked them to find the area without counting all of the square by 1.

In looking at the work, I saw them as a bit of a progression of thinking. I put them in order here of how I see students moving through these ideas about area.

As I expected, some counted each row and added them. It was great they know area is additive, but I would love to ask this student if there is a way they could have used multiplication to  make it a bit easier.

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Some added in chunks, to which I would love to ask the same question. I was excited to see them cutting the shapes up into rectangles in places that made sense.

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Some students made some larger cuts and I would love to have them meet with the student above and discuss how they decided on their cuts.

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Some used some of the strategies above but also relied a bit on symmetry.

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Finally, I saw 2 students moving squares from the “bumps” to the empty spaces.

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It was always interesting to me that in 5th grade I would still see students find the area of shape on a grid by counting the individual squares even though I know they had better strategies. I think it is the fact that students jump into doing things without thinking about the things first. This is why I think journal writing is so important. It allows students to be more reflective about their decisions.

I asked them to write about which shape was the easiest to find the area and which was the most difficult. It was interesting to see some focus on the size of the number they were working with while others focused on the shape and how it could be partitioned.

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As a follow-up activity, I am going to ask them to choose a letter where they would use the same strategy they used with C and a letter where they would use the same strategy they used with B.

 

Fraction Talking Points: 3rd Grade

The 3rd grade is starting fractions this week and I could not be more excited. Fraction work 3-5 is some of my favorite stuff. Last year we tried launching with an Always, Sometimes, Never activity and quickly learned, as we listened to the students, it was not such a great idea. We did not give enough thought about what students were building on from K-2 which resulted in the majority of the cards landing in the “Sometimes” pile without much conversation. And now after hearing Kate Nowak talk about why All, Some, None makes more sense in that activity, it is definitely not something we wanted to relive this year!

We thought starting with a set of Talking Points would open the conversation up a bit more than the A/S/N, so we reworked last year’s statements. I would love any feedback on them as we try to anticipate what we will learn about students’ thinking and the ideas we can revisit as we progress through the unit. I thought it may be interesting to revisit these points after specific lessons that address these ideas.

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We were thinking each statement would elicit conversation around each of the following CCSS: 

Talking Point 1CCSS.MATH.CONTENT.3.NF.A.3.C
Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

Talking Point 2CCSS.MATH.CONTENT.3.NF.A.2
Understand a fraction as a number on the number line; represent fractions on a number line diagram.

Talking Point 3CCSS.MATH.CONTENT.3.NF.A.2.B
Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

Talking Point 4: CCSS.MATH.CONTENT.3.NF.A.1
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

Talking Point 5CCSS.MATH.CONTENT.3.NF.A.3.D
Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Talking Point 6CCSS.MATH.CONTENT.3.NF.A.3.C Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

After the activity, we have a couple of ideas for the journal prompt:

  • Which talking point did your whole group agree with and why?
  • Which talking point did your whole group disagree with and why?
  • Which talking point were you most unsure about and why?
  • Which talking point do you know you are right about and why?
  • Could any of the talking points be true and false?

Would love your feedback! Wording was really hard and I am really still struggling with #4.

If you want to read more about Talking Points for different areas, you can check out these posts:

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.

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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).

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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!:)

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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..

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

 

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

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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!

 

 

Subtraction in 2nd Grade

From my own experience teaching 5th grade and a lot of conversations with teachers in grades 1-4, subtraction always seems to be such an area of concern with students. After the introduction of subtraction as take-away, students tend to live in that land forever. This idea was spinning in my head the entire time I read Zak’s article in NCTM’s Teaching Children Mathematics. The article came at an ideal time as I had just planned and taught a subtraction lesson in two second grade classrooms.

As we were planning, the teachers and I discussed two reasons we think students struggle with relational thinking when dealing with subtraction:

  • The lack of variety in contexts we often give students to think about subtraction in different ways.
  • The lack of explicitness in connecting the ways students solve various subtraction contexts.

Based on the problems they were doing in their Investigations unit, we wanted to take the same numbers and create two different contexts for subtraction…one as a take-away context and one as a distance context and then have the students think about what made them subtract in one and what made them add in another. (After reading Zak’s article I want to try another structure where we keep the same context, but change the numbers to see if that impacts their solution process.)

First we did a notice/wonder on the following scenario:

Mike is saving money to buy a game. He already has some money saved.

After collecting the things they noticed and wondered (in black ink), I asked them what question they think we could ask and solve based on this story. They gave the question in red ink and then I asked them what information they would need to solve it and I put a red star next that information. I told them the game cost $22 and he had $8 saved and sent the on their way.

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Every single student added up, as we anticipated.

We shared some strategies and asked students how they were similar and different. We wanted to hear if students were using the number to solve or represent their thinking. We also wanted those students using the number to solve to see how they could also represent their thinking with equations.

Then we gave the students a second problem to solve: same numbers, different context:

Bobby had 22 toy cars in his book bag. He gave 8 of his toy cars to his friend Becky. How many cars did Bobby have left?  

They all took away the 8 except for one student. The variety of strategies was great and we were actually surprised to see a student solve it the same exact way he had used in the previous context.

The lone addition solution…

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We shared some strategies and asked them to compare the two problems. They noticed they were the same numbers and same answer but one student actually exclaimed, “How can they be the same answer when we added in one and subtracted in the other?!” What a great jumping off point for future lessons!

Time was up for this class period, but I made sure I stopped by to chat with the student who added in both because I was interested in his thinking. He said he knew that he had to subtract in that problem because it was taking away and he knew that you can add up to solve a subtraction problem. So, how does that happen? He has been in our school with all of these other students K-2, so how does he know this? What experience did he have, inside or outside of school, that allowed him to decide this was subtraction, know he can add up, and solve it? and then, Is there ever a context where he WOULD take away or has he learned this as to never have to take anything away?

After this lesson and Zak’s article, I am still reeling with my thoughts on subtraction. There has also been a lot of great conversation on Twitter about sequencing student work and I wonder how a set of student work like this could/would/should play in to students seeing subtraction more relationally? Lots of great stuff to think about!

 

Cuisenaire Rods and Balance

We have been having a lot of fun with cuisenaire rods in Kindergarten! The last time I was with this class, they explored ways to build the orange rod using the other lengths. As the students worked, I heard a lot of “A white and a blue make an orange.” and “Two yellows make an orange.”

After reading the beginning portion of Marilyn Burn’s About Teaching Mathematics, I have equality and use of the equal sign on my mind a lot, especially when reflecting on these particular activities with cuisenaire rods. In the last activity, there was a lot of talk that seemed to be about “making” the orange more so than the two of them being the same as one another in length.

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I wondered how we could shift that perspective a bit, or at least add another type of experience to the different rods being of equal length. I grabbed Balancing Act from my Math Reads kit and went to Kindergarten for another round!

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After reading the book aloud, we did a notice and wonder. I was surprised at the ease of use of the word “balance” and the discussion about which different animals balanced one another out in the story.

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My first direction to the students was to go back to their seats and find rods that would balance with one another. After looking around and seeing some rods on their ends with students creating their own teeter-totter, the teacher and I had to do a quick pause and redirection:)! She quickly grabbed rulers and we asked the students to pretend the ruler was the teeter-totter so that they rods could stay flat on the table, much better!

It was so interesting to see their different approaches! These two boys had different ideas. The one student liked to find rods the same length before putting them on the ruler, while the other put the longer rod on and kept adding smaller ones to the other side. He had to keep putting the longer one over the smaller ones each time, but his truly looked like balancing.

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This was so interesting to see multiple rods balancing each side and I especially liked the red rod going vertical on the one ruler.

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The student below was SO interesting because she was the only one balancing different sizes each time but the opposite sides were still staying balanced at the same time!

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This table worked together so nicely, but started to have a great disagreement when I asked if all of the sides were balanced because the one side is shorter than the rest!

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This group made the equal length rods but they was using the ruler to test that all of the sides were the same length:

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I started thinking it may be a neat activity to ask students to take turns balancing out each other’s cuisenaire rods. For example Player 1 could put a rod or combination of rods on their side of the ruler and then Player 2 must balance out what they did, except using different rods. They could take turns being the first one to place the rod each time. If you wanted to be tricky, they could also remove rods on their turn instead of adding them.

I am wondering, however, how to incorporate notation into this? Would you ask the students to write equations for these rods during this activity?