Category Archives: Geometry

Always, Sometimes, Never….Year 2, Part 2

In my previous post: https://mathmindsblog.wordpress.com/2014/11/26/always-sometimes-never-year-2/ the class organized the quadrilateral cards into A/S/N columns and today we continued that work.

I switched the groups up so I had students entering a new set of cards with differing views. Always fun stuff for some great mathematical arguments. I had them discuss differences they saw from their previous table and decide if they want to move any cards.

Having each group do a written proof for all 18 cards seemed overwhelming, so I gave each table 3 of the cards to focus on proving within their group. I had them write their individual thoughts about the 3 cards in their journals before starting to work together. In class tomorrow they will prove the placement of their 3 cards to the class (aka jury). Here are the beginning workings of their proofs:

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Before we left for the day, I asked them to reflect and write about any changes they made because of their discussions or any cards their group was still thinking about. Here are a couple before and afters…LOVE the argument ones!

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Always, Sometimes, Never….Year 2

Last year, I used this activity with my 5th grade students and blogged about it here. This year, I used the same activity, however tried it a bit earlier in the unit than last. Based on our Talking Point activity before the unit began, I found the students had very good understandings that would emerge naturally in the Always, Sometimes, Never activity. I wanted to see how using the activity before a lot of our classification work would affect the outcome, if it would differ from last year’s.

This year, we had played “Guess My Rule” the day before in which students use attributes to choose two quadrilateral cards that fit their rule and one quadrilateral card that does not, while their partner tries to guess the rule. It is great for thinking about classification by sides or angles and vocabulary building. At the end of class, we did a few rounds together, and chatted about some vocabulary that was helpful and talked about our classification by sides, angles, or both.

Being the day before Thanksgiving, you never know how it was going to go, but they were so engaged in the work. Here is a copy of the cards that I used:

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I gave them time to individually read them and underline any words or wording in which they needed clarification. Although I was surprised that some students asked about words we had discussed in prior class periods, I was so much more happy they felt safe enough to ask for clarification. This is a prime example of not making assumptions in what our students know just because we have talked about it before in class.

They cut them out and went to work. As expected, they would have a brief conversation with their group and place them quickly into the appropriate column on their desk. I let them do that on my first round around the classroom and then as I heard some debates starting, I suggested that maybe using their journals to show their proofs may help their argument.  For those who were quickly done, I said, “If it falls in the Sometimes category, you should be able to show when it does happen and when it doesn’t happen, right?” I also pointed out the reasoning for Always never being able to be disproved and the opposite for Never. This had them really go back and take a deeper look at the cards and got their conversations going.

Of course, we did not get to come back together as a group and come to consensus but here were the table card arrangements as the class ended:

Photo Nov 25, 11 23 46 AM Photo Nov 25, 11 23 57 AM Photo Nov 25, 11 24 37 AM Photo Nov 25, 11 24 49 AM Photo Nov 25, 11 24 56 AM Photo Nov 25, 11 25 19 AM Photo Nov 25, 12 11 19 PMI had them do an individual reflection on which card they are still really struggling with and these responses are going to help in framing how I proceed from here on Monday after vacation:

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IMG_8194_2IMG_8195 IMG_8196_2 IMG_8197_2 IMG_8198 IMG_8199_2 1It is really interesting that for some students the orientation of the shape makes the biggest difference, others strugglw with the vocabulary, and, like last year, that rhombus one is blowing their minds. It is so interesting to me that a student can apply shape attributes to make a conclusion that a rhombus is a rhombus, but then to take that reasoning and apply it to another shape, is extremely difficult. This led to a very interesting conversation between a colleague (who was in observing) and myself about students knowing definitions versus descriptions….still wrapping my own head around that one…will probably be a blog post coming soon:)

With all of this information from the students, on Monday, I plan on putting them in groups based on the related cards they were left grappling with. I think rearranging the groups will lead to interesting conversations and more detailed proofs. Each table will get three cards to create an argument for the placement of that card. They will present their argument to the class and we will try to come to consensus as a class. Last year we did this share as a whole class, and I didn’t feel like it “wrapped up” and things were left hanging out there that needed to be a bit more solidified in future classification work, so hopefully this will be change that.

Happy Thanksgiving all!

-Kristin

Talking Points – 2D Geometry

We are about to start out work with Polygons, so I decided to kick it off with Talking Points. If you have never read about them before you can check out my post or Elizabeth’s post to learn more.

Here were the points my students were discussing:

tpgI had gotten these points by looking back at their fourth grade geometry unit work and thinking about what misconceptions or partial understandings students have each year when we start this unit.

This time, I made a few changes from past experience. In each group I had a facilitator to be sure that everyone got a chance to speak without interruption during Round 1, and a recorder to keep the tally for the group. Also, after the first talking point, I had advise from a math coach in the room filming with me to add individual think time after the reading of the point. LOVED IT! During think time, they were jotting in the journal and getting their thoughts together. I got things like this from just the think time:

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It was nice to see them take ownership with their journal without being told to write anything down. They were working on proofs before they started. After the six talking points, I posed three questions on the board for them to reflect upon individually:

1 – What talking point are you sure you were right in your answer? Explain your reasoning.

2 – Which talking point are you unsure about your answer? Why?

3 – Which talking point did your group agree upon easily? Why do you think it was easy for your group to agree on that one?

Here are their reflections:

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Never anticipated so much “left angle” talk in my life 🙂 I learned SO MUCH about their understandings and wish I had time right now to add my comments to each journal, but I will very soon!

So, I have a moment here at lunch to reflect on what I learned from these talking points:

TP1 – As I anticipated, this one is always a source of confusion. Every year it seems as if the students know the sentence goes one way or the other but can’t remember it because there is little understanding of the WHY piece. Later on in the unit after we have done more classifications, I do more of these statements with Always, Sometimes, Never, so this is a something I wanted to see how students were thinking about it. Most tables said something to the effect of “I remember last year we said a square is a rectangle or a rectangle is a square, but I can’t remember which one.” Another conversation I heard was that a rectangle has to have two short sides and two long sides.

TP2: I loved this question and was really pleasantly surprised to see some trying to draw it and ending up with unconnected sides. One thing I was so surprised about was the “left angle.” They were not thinking the degrees changed so much from the left to the right angle, but more the orientation of the angle (left side, right side). Interesting.

TP3: I got a great sense that most students knew what area and perimeter were and the best part was that if they didn’t remember, someone at their table did and gave an example. Regardless if they knew they could be the same, I was excited to see a great understanding from most here.

TP4: This one was great. I saw some students drawing a square on their paper, showing the group, rotating the paper and saying, “See, now it is a rhombus.” They all seemed to be in the mindset that a rhombus is a diamond shape, but really not reasoning about the attributes that make it a rhombus.

TP5: They did a very nice job with this one. A lot drew examples of combining two shapes, while I heard others asking their group if the “inside connected side counted” when they were trying to name it.  Also realized that the term polygon was not familiar to most students. I am wondering what they called them in earlier grades? Pattern Blocks? Shapes?

TP6: Interesting one here and it is where we start our 5th grade work with polygons, classifying triangles. Again, the left angle reappeared:) I did hear a few struggling with the name of the angles, obtuse, acute, right but then I had some that said there are other 3-sided shapes that aren’t called triangles. Hmmm, can’t wait to find out what they are! Of course, you always have your comedians who say agree because it could be Bob or Fred.

Can’t wait to start planning this weekend!

-Kristin

Volume with Fractional Dimensions

Before I began our volume work this year, I blogged about my planning process here: https://mathmindsblog.wordpress.com/2014/10/20/unit-planning/. 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:

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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…

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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!

-Kristin

Volume Student Work…

So, I couldn’t resist the urge to blog about my students’ first day of work with volume since I had posted the other day about my planning,even though I have tons of other things that need to get done. We are sharing their work tomorrow to elicit strategies and make connections. With the 5 Practices in mind, I am heading off to sequence the shared work to make those connections.Tons of things to work with here!

As always, I love your thoughts on the order and items you would share!

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FOLLOW UP – For homework, I had the students try to generalize a strategy for finding the volume of any box and here are a few that I got and plan to build upon…

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Unit Planning

This week, I am starting our next unit in Investigations, Prisms and Pyramids: 3D Measurement and Geometry. Every time I start a new unit, I always like to read the Teacher Notes in the back of the teacher’s book because Investigations has done such a phenomenal job explaining students’ work prior to that unit, the big mathematical ideas, student misconceptions (with dialogue boxes), and offering examples of student work for both the activities and assessments. It is an invaluable resource that often gets overlooked.  After reading the teacher notes, I typically like to look at the Ten Minute Math Activities, build my Number Talks, and then dive into planning out the lesson activities.  I was going to devote this entire post to thinking about this unit on Volume, however I tweeted out suggestions for a blog topic on this flight and Michael gave me this:

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I feel like I just finished up such an exciting (in my mind at least 🙂 blog post about student moments with multiplication of fractions that I wanted to do something a bit different and this tweet gave me an idea. Since my students have done such amazing work with fractions, I started thinking about how I could embed that into our volume unit. So, after reading the Teacher Notes, instead of jumping into the number talks right away, I began reflecting on those great fraction moments and putting them in the context of volume.

Our first activity in the unit is building boxes, estimating how many cubes will fill the box and checking that estimate by physically filling the boxes with cubes. We then share and discuss strategies for finding the number of cubes as a class. I find every year, I have over 1/2 the class who can see the layering and arrive at an efficient strategy including length x width x height fairly early in the lesson, while others cannot visualize how to organize the cubes. They can easily find the number of cubes in a rectangular prism when the box is constructed and filled, however moving to a net or just dimensions is a very tough abstraction.

I think one of the hardest aspects of teaching is meeting every student’s needs on a daily basis. This is one of those lessons in which I feel I struggle a lot. It is beyond the numbers and context and more about the students who have the spatial reasoning to abstract volume vs those who need to always build the box to arrive at an answer. For those who need concrete models, I feel comfortable meeting their needs through using cubes and building on that concrete work to form generalizations for finding volume.  However, what can I do for those who just “see it” and building those boxes is unnecessary and a tedious waste of time? They are done the lesson in like 10 minutes and are waiting to share while others build. Bigger numbers won’t challenge them, they can prove it always works, doubling/tripling volume is a class lesson after that one…..hmmmm….

This is where Michael’s tweet made me think about our fraction work. I have never done my fraction unit before volume unit before. Typically we do volume to start the year and then move into fractions. I think it is so interesting to think about those students who can reason about fraction multiplication and how they would think about a volume problem in which there are fractional dimensions. What does the picture look like? How can you have a 1/2 in x 1/2 in x 1/2 in and end up with 1/8 cubic inch? How do you end up with a smaller fraction than all of your dimensions? That is a tough concept to grasp and I THINK THAT WOULD BE SO COOL for them to think about!

Now the even harder part comes into play, planning to launch that with them. Questions I have to think about (and any/all thoughts are always welcome):

– Do I build on the problem with whole numbers (2 x 3 x 4) and ask, what would happen if that was 4 1/2 tall instead of 4? Would it increase the volume? How do you know?

– Does it make it harder or easier to manipulate a certain dimension, i.e. length, width, height?

– Do mixed numbers feel like a good starting point vs fractions less than 1?

– What will their recording look like? What are good questions to ask them when they have found the answer?

– Is there a better volume model aside from cubes (which you cannot break in fractional parts to prove your point)?

….and there are probably a million more, but the flight is about over….so much planning left to do. Not to mention the actual typing of my lesson plans and 9 million other things I should be getting done while I type this blog! I just love to write about this stuff…cannot help myself! I apologize for any typos, no time to re-read before saving and putting the laptop away!

-Kristin

iPad Garden Problem

The Garden Project:

Over the past couple of weeks, my 3rd graders have been working with our new set of iPads on a Garden Project. Since our school has put in learning gardens this year, I thought it would be an applicable project for them.

The premise of the problem: The school wanted to build a garden with the most space to plant our vegetables. Each group was given 18 feet of fencing (18 toothpicks) to use as their perimeter. They were to design each garden, record the dimensions, and take a picture to save in their photos. After they designed all possible rectangular gardens, they had to create a presentation in Numbers to show me which garden they wanted to build.

The instruction page looked like this (Since this was our first project, I put the app pic next to each direction to help them along the way):

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I am always amazed how fast the students can pick up the technology and run with it! They were collaborating and discussing perimeter and area like pros! We are sharing presentations tomorrow and then we will be extending our thinking by looking at the relationship between area and perimeter!

Hard at work:

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I will post their presentations when I get them uploaded tomorrow! Here is the GardenProject pdf if you want to try it out!

Mathematically Yours,

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: IMG_3140I 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…
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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!

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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)

IMG_3129She 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

Math in a Movie Trailer

Last Wednesday at a PLC meeting, our district instructional technology specialist did a presentation on Blended Learning.  She did a beautiful job of demonstrating apps and web-based activities at various entry levels, so each teacher could participate. One of the fourth grade teachers expressed an interest, and a bit of fear, in trying to use ipads as part of her classroom routines. Since I had been in her room doing some math coaching the previous week, I offered to help her design an activity and give her a hand in the classroom with the ipad piece if she was not comfortable.

We met the next day to start our planning! She was just ending her 3D math unit in which students had been identifying 3D shapes by their silhouettes and attributes and finding volume of a rectangular prism. As a culminating activity, we decided to have the students create a movie trailer in iMovie that “told a story” about the unit. I sent the teacher home with one of the ipads to “play around” with iMovie, since she was not very familiar (or comfortable) with it.  I was so excited to come in the next day to see a trailer she had created at home that night! I LOVE when people jump right in!

This is how our lesson played out over the next two days…

– We created a room in “Todays Meet” on their ipads and had students go in and do a test post.

– We posted the question, “What is the purpose of a movie trailer?” in the TM room and let them type as we showed two movie trailers (Percy Jackson 2 and Despicable Me) on the SMARTBoard. When the trailers were over, we switched back to TodaysMeet on the SMARTBoard to go through their comments and have them expand on them. Here is a clip of the conversation:

TodaysMeet– Next we asked them to continue chatting about things they learned during this math unit. We noticed they were just writing one or two word things so we asked them to expand a bit and use more of their 140 characters. Sample clip:

TodaysMeet2– As a class we scrolled back through and had them stop and ask questions of each other if they didn’t understand what someone had posted. They were so engaged and they all kept asking if they could do this at home?!? YES! Next time I will leave the room open for a longer time frame so students can post as they think of things at home! What a great way to open class the following day!

– We took them through a brief “tour” of iMovie and let them move to a place in the room to look through the themes and storyboards and start brainstorming ideas for their trailer.

– To help them organize their thoughts, I had put a template of the storyboards: http://tinyurl.com/c3g5r2e in the Dropbox that was on each ipad. The students exported the PDF to UPad Lite: Upad

and let them play around with how to write on the document with pen width and different colors.

– The following day, students got in their groups (of 2-3 students) to plan out their storyboard and decide on pictures they need for their trailer.

When we meet on Monday, we are taking them around the school and outside to take pictures they need for their trailer. They are working this week finishing up the project, so this story will have  To Be Continued…

Mathematically Yours,

Kristin