Category Archives: Geometric Measure & Dimension

MP5: The Traveling Point

How do you give students the opportunity to practice “I can use appropriate tools strategically”?

When we have a new type of problem to think about, I am learning to have students give their best guess of the solution first. I’ve written about The Traveling Point before.

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Students sketched the path of point A. How far does A travel?

Students used paper and polydrons, their hands and string.

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I sent a poll to find out what they were thinking about the distance traveled.

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Students then interacted with dynamic geometry software. Does seeing the figure dynamically move help you better see the path?

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Does seeing the path help you calculate how far A travels?

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And so the journey to make the Math Practices our habitual practice in learning mathematics continues …

And the journey for my own learning continues. Thanks to Howard for correcting me. The second two moves do not travel a distance of 6, but the length of the circumference of the quarter circle.

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One student figured that out by the time the bell rang.

I look forward to redeeming this lesson this year, as the journey continues …


Posted by on August 23, 2016 in Geometric Measure & Dimension, Geometry


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Making a Better Question Worse

I recently read a post on betterQs from @srcav with an area question from Brilliant that I added to today’s opener.


I knew something was up when I saw my students’ results.

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My mistyping was a good reminder of the importance of nouns.

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Without showing any results, I sent the corrected question as a Quick Poll.

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(I wonder whether the way the first question was asked prompted the misconception in the wrong answer, but I won’t find out until I have another group of students.)

We are learning to look for and make use of structure.

We are learning to contemplate, then calculate.

And we are learning how to ask better questions, as the journey continues …


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The Surface Area of a Sphere

The Surface Area of a Sphere

I’ve written before about making sense of the surface area of a sphere. The lesson this year unfolded (unpeeled?) a bit differently.

I’m not sure how students might guess that the surface area of a sphere has something to do with the area of a great circle of the sphere. We talked about what a great circle must be, we used fishing wire to measure the circumference of a great circle of the sphere (orange), and I asked them to estimate how many great circles would cover the orange. You can see the huge variety of responses.

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We cut the orange in half. I showed them the surface of the great circle and the act of “stamping” it onto the orange peeling. Do you want to keep your estimate? Or change it?


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We went from 20 responses below π, 9 infinites, and 3 correct to 13 responses below π, 5 infinites and 7 correct.

I hesitated about what to ask next. We were ready to peel the orange to see how many great circles we could cover and figure out what the surface area formula would be, but I was curious about whether students could make sense of the formula before we did that if I told them what the formula was. So we waited on peeling and I sent another poll.

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68% understood that the sphere’s surface area formula, 4πr^2, meant that 4 great circles could cover the sphere. We peeled it to be sure.


And so the journey to figure out what questions to ask when continues …


The Circumference of a Cylinder

We talked about pi earlier this week in geometry, and we used Andrew Stadel’s water bottle question to start.

I’m not one to pull of the wager that Andrew used (unfortunately, my students will agree that I am a bit too serious for that), but we still had an interesting conversation.

Compare the circumference and height of the water bottle.

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Here’s what they estimated by themselves.

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Then they faced left if they thought height > circumference, straight if =, and right if height < circumference. (I saw Andrew lead this at CMC-South year before last … I certainly didn’t think of it myself.) They found someone who agreed with their answer, and practiced I can construct a viable argument and critique the reasoning of others.

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Next they found a second person who agreed, and practiced I can construct a viable argument and critique the reasoning of others again. (By this time, we decided it was easier to raise 1, 2, or 3 fingers based on answer choice rather than turn a certain direction as it was a challenge for some to see someone turned the same direction.) Finally, they found someone who disagreed, and practiced I can construct a viable argument and critique the reasoning of others.

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I sent the poll again.

It didn’t change much.

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So without discussion, I sent a poll with a bit more context … a cylindrical can holding 3 tennis balls. Would the can of tennis balls help them reason abstractly and quantitatively?

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Apparently not.

Here’s what they thought by themselves.

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And here’s what they thought after talking with someone else.

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The clock was ticking. I still wanted us to talk about pi. I asked someone who correctly answered to share her thinking with the rest of the class to convince them. And we used string to show that the water bottle circumference was, in fact, longer than its height.

I intended to follow up with this Quick Poll. But I was in a hurry and forgot. Maybe next year.

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You can find more number sense ideas from Andrew here.

I’ll look forward to hearing about how they play out in your classroom, as the journey continues …


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The Area of a Trapezoid: Differentiating Success Criteria … Not Learning Intentions

I am enjoying our slow book chat on Dylan Wiliam’s Embedding Formative Assessment. (You can download the first chapter here, if you are interested.)

Chapter 3 is called Strategy 1: Clarifying Sharing, and Understanding Learning Intentions

How do we support students who need scaffolding while at the same time pushing students who need a bigger challenge?

I struggle with differentiation. But as we focus more on mathematical flexibility, I am learning to understand what Wiliam means by differentiating success criteria instead of learning intentions.

Consider this learning progression on mathematical flexibility from Jill Gough.

Flexibility #LL2LU GoughWhat if we pair that with a content learning progression on the area of trapezoids?

4: I can prove the formula for the area of a trapezoid more than one way.

3: I can prove the formula for the area of a trapezoid.

2: I can calculate the area of a trapezoid by composing it into a rectangle and/or decomposing it into triangles and other figures.

1: I can calculate the area of a trapezoid using the formula.

Our practice standard for this lesson is “I can look for and make use of structure”.

SMP7 #LL2LU Gough-Wilson

Wiliam says that there are 13 conceptually different ways to find the area of a trapezoid. Some of them are more challenging algebraically than others. Some of them are more challenging geometrically than others.

How many ways can you prove the formula for the area of a trapezoid?

How might we use this exercise to differentiate success criteria for our learners?

I got to try this with 6th-12th grade teachers in a recent Mississippi Department of Education geometry institute. In our Geometric Measure and Dimension session we moved from areas of special quadrilaterals in the coordinate plane to proving the area formulas for a kite and a rhombus. Then we proved the area formula for a trapezoid. We had some teachers for whom it was a challenge to generalize the height of the trapezoid as h and the bases as b1 and b2 instead of using numbers to represent the lengths.


The first instinct for many teachers was to either compose the trapezoid into a rectangle with dimensions b2 × h and subtract the areas of the two extra right triangles

Or to decompose the trapezoid into a rectangle with dimensions b1 × h and add the areas of the two right triangles.

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The algebra can be challenging, especially when deciding how to represent the lengths of the bases of the triangles. Will you call one of them x and the other b2x – b1? Or will you recognize that together, the bases have a sum of b2 – b1?


One of the least instinctive methods in the 200+ teachers in my sessions was to decompose the trapezoid into two triangles using a diagonal. It is also one of the most accessible methods algebraically. A few times I asked a teacher who was stuck what would happen if you drew one diagonal. Then I walked away. I almost always came back later to a successful proof.

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How might we use this exercise to differentiate success criteria for our learners?


Once they were successful with decomposing into two triangles, they were ready to consider decomposing into three triangles. A few teachers breezed through the algebra and were ready for another challenge. (We noted the freedom to connect the endpoints of b1 to a point on b2 that partitions b2 into any ratio, 1:1 or 1:2 or 1:x.)

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Some decomposed the trapezoid into a parallelogram and a triangle.

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Some used rigid motions to make sense of the area of the trapezoid, rotating the trapezoid 180˚ about the midpoint of one of its legs, creating a parallelogram with base b1 + b2 and height h. For others, rigid motions was a challenge. They asked for scissors so that they could cut out trapezoids and physically translate and rotate them.

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Others decomposed the trapezoid into two trapezoids using the median, and then rearranging the top trapezoid into pieces to form a parallelogram with base b1 + b2 and height ½h.

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Or a rectangle with the same dimensions.

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A few used the median to create the “average rectangle” with area equal to the trapezoid.

Or the “average parallelogram” with area equal to the trapezoid.

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One decomposed the trapezoid by constructing a segment from one endpoint of b1 to the midpoint of the other leg, and then rearranging the triangle formed to make the trapezoid into a triangle with base b1 + b2 and height h.

Another did the same from one endpoint of b2.

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I asked those who finished quickly what would happen if they extended the legs of the trapezoid to form a triangle. It took a lot of algebra for them to prove the area of a trapezoid using similar triangle relationships but once they started, they wouldn’t stop.

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I think that these would be considered 11 conceptually different methods for proving the area of a trapezoid. I can’t remember that anyone found 2 others, and I’m sure there’s a site out there somewhere that I can find two more ways. But I’m not going to succumb to Google yet. I’m going to continue working on my mathematical flexibility, and I’m going to keep practicing look for and make use of structure, as the journey continues …


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The Area of a Rhombus and a Kite

I recently participated in the Mississippi Department of Education Geometry Institute. In our session on Geometric Measure & Dimension, we moved from areas of special quadrilaterals in the coordinate plane to proving the area formulas for a kite and a rhombus.

We had practiced look for and make use of structure on two kites, c and f. Participants had shared their thinking on kite c.


Could they transfer what they had done in the coordinate plane with known segments lengths to a rhombus and a kite with diagonals d1 and d2?

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Just like our students would, some teachers struggled with the idea of generalizing the formula. Several used rulers to measure the lengths of the diagonals or made up numerical lengths for the diagonals and calculated the area.

Can you tell how these participants generalized their work?

I gave the kite as an assessment item for my students last year, and I asked them to practice “I can look for and make use of structure” along with “I can show my work”.

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Whose work can you understand without asking for clarification?

What opportunities do you give your students to practice “I can look for and make use of structure” along with “I can show my work”?


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Calculating Area – Looking for and Making Use of Structure

Several months ago when I was preparing a session on Geometric Measure & Dimension for the Mississippi Department of Education Geometry Institute, I noticed a blog post by Kate letting us know about some new IM tasks. Areas of Special Quadrilaterals (and one triangle) caught my attention, and so I took a look. I had no idea at the time how perfect this task was for starting the session.

Jill Gough and I often talk about pairing a content standard with a practice standard.

For this activity, the content standard was 6.G.A.1:

Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

And the practice standard was look for and make use of structure.

SMP7 #LL2LU Gough-Wilson

How would you find the area of each figure, composing into rectangles or decomposing into triangles and other shapes?


Many teachers used color to make their thinking visible. I learned quickly, however, not to assume that they had found the area a certain way just because of a certain auxiliary line. I walked around, asking about their thinking, selecting and sequencing for our whole group discussion.

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We started with figure a.

Some decomposed into a rectangle and two triangles.

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Some decomposed into unit squares.

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Some composed into a rectangle to think about the areas of the two small triangles.

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Others decomposed into a rectangle and two triangles but then rearranged the two triangles into a rectangle.

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Others translated one triangle horizontally to rearrange into one rectangle.

One lady had not taught geometry for a very long time. She said she thought she knew a formula that would work for calculating the area, but she wasn’t confident about her work. Her formula? A=½h(b1+b2). Most participants knew that formula as the area of a trapezoid. Does it work for the parallelogram? They tried it. It worked. Maybe that adds to our reasons for considering the inclusive definition of trapezoid?

We looked next at figure d.

Some composed into a rectangle and subtracted the area of the right triangle.

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Some decomposed into a triangle and rectangle.

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One decomposed into a triangle and parallelogram.

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Some decomposed into unit squares.

Some decomposed and rearranged into a rectangle.

Then we looked at figure c.

So many ways!

Composing into a rectangle. From there, some subtracted the area of each right triangle. Some halved the area of the rectangle.

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Decomposing into triangles. Some into 4 triangles with both diagonals. Some into 2 triangles with the vertical diagonal. Some into 2 congruent triangles with the horizontal diagonal.

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Some decomposed and rearranged.

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Everyone practiced look for and make use of structure in ways they hadn’t thought to before. Everyone worked on their mathematical flexibility to find more than one way to determine the areas of the figures. Everyone learned at least one new way to look at the figure from the others in the room.

What opportunities do you provide your learners to look for and make use of structure and then share what they’ve made visible that wasn’t pictured before?


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Cavalieri’s Principle

Geometric Measure and Dimension G-GMD

Explain volume formulas and use them to solve problems

1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

How do you provide students an opportunity to make sense of volume formulas? I’ve written before about how we use informal limit arguments to make sense of volume formulas for the cylinder and prism and then Power Solids to make sense of volume formulas for the cone and pyramid.

Using a slinky, we briefly discuss Cavalieri’s principle.

Solids: equal height, cross sections for each plane parallel to and including the bases are have equal area.

What are the implications of Cavalieri’s principle here? (the two solids have the same volume)

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And here? (none, as the conditions aren’t met)

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When we get to the volume of a sphere, I’ve always told my students they’ll have to wait until calculus to make sense of the formula.


(I sneak in this exercise in calculus and wait for someone to notice the result.)

If I ever made sense of the volume of a pyramid or sphere using Cavalieri’s principle while I was in school, I don’t remember. (Surely I’m not the only one.) This year, though, I’m determined to do better. I’ve been saving Pat Mara’s TI-Nspire documents to think this through.

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How can you use these images along with Cavalieri’s principle to make sense of the formula for the volume of a square pyramid compared to the volume of a square prism with base and height equal to the pyramid?

When I got out the play dough to make more sense of the dissection of the cube, my coworker joined me. Our solid isn’t beautiful, but we get why the three square pyramids have the same volume and why one square pyramid will have a volume that is one-third of the square prism with base and height equal to the pyramid.

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What can you say about this square pyramid and cone?

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I like the visual image of seeing cross sections that aren’t congruent but have equal area.

Now for the sphere.

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What do you see?

On the left, a hemisphere.

On the right, a cone cut out of a cylinder.

What’s the same about the solids?

The sphere and the cylinder have equal radii and equal heights. Since the “height” of the sphere is its radius, the cylinder has height equal to radius.

What are the horizontal cross sections?

On the left, a circle. The radius decreases as the cross section slices go from bottom to top.

On the right, a “washer” (or officially, an annulus), where the outer radius is always the radius of the cylinder (constant) and the inner radius is equal to the height of the smaller cone formed with the inner circle of the slice and the center of the base (shown by similar triangles).

Dynamic geometry software shows us that the cross sections have the same area. Convince yourself that they do.

I convinced myself here:

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And then I looked at the next page, which allowed me to move the cross sections and see the similar triangles change.

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So what does this tell us about the volume of a hemisphere?

According to Cavalieri’s Principle, it has the same volume as the solid on the right.

How can you calculate the volume of the solid on the right?

Subtract the volume of the cone from the volume of the cylinder.

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And then what about the height of a sphere for which this hemisphere is half?

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The Illustrative Mathematics task Use Cavalieri’s Principle to Compare Aquarium Volumes could be helpful for exploring Cavalieri’s Principle. I’ve had it tagged for several years now. Maybe this will be the year we take time to try it.

And so the journey as both learner (student) and Learner (teacher) continues, with gratitude for those who share their work and those who are willing to pause their work long enough to learn alongside me …


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Experiencing Dandy Candies as learner and Learner

Back in April, I had the pleasure of attending a CPAM Leadership Seminar with Dan Meyer on mathematical modeling, where he lead us through Dandy Candies. Dan wrote about this 3-Act recently here. I’ve used several 3-Acts with my students, but this was my first time to participate in one from a “lower-case l” learner’s perspective. I’ve read about “purposeful practice” and “patient problem solving” for several years now, and I know that I have some understanding of what they mean, but seeing them in action from the learner’s perspective is powerful.

A few things struck me during the seminar. We don’t do 3-Acts just for fun. (I knew this, but Dan made it very clear that this isn’t just about engaging students in doing something; it’s about engaging students in doing math. I’m not sure I’ve made that as clear to other teachers with whom I’ve discussed 3-Acts.) As Smith & Stein point out in 5 Practices for Orchestrating Productive Mathematics Discussions, there is pre-work for the teacher: identifying the math content learning goal for the lesson and then selecting a task that is going to provide students the opportunity to engage in that math content. Even with a 3-Act, where we let our students’ curiosity develop the question, we do have an underlying question that will engage students in the math content we want them to know. What they ask might not be worded exactly the same, and it might extend the mathematical thinking in which we want students to engage, but the math is there.

I have said before that I use technology to give every student a voice – from the loudest to the quietest, from the fastest to the slowest. When Dan solicited questions we could explore from the group, I was never going to volunteer mine for the list. (I am not criticizing Dan’s move here … just noting that I find it challenging, both as Learner and learner, to establish trust in a short session with participants that I’m likely not going to see again.) I *might* have participated had I been asked to submit my question somewhere anonymously.

And finally, I really like the opportunity that we had before each question to answer before performing any calculations. I’ve been working on providing this opportunity for my students, but it still isn’t automatic. I have to remind myself to ask students to use their intuition first. As I heard from Magdalene Lampert, “Contemplate then Calculate”.

We were working on Modeling with Geometry (G-MG) when I returned to class after the seminar last year, and so I tried Dandy Candies with my students.

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are the heights the same     1

what are the surface areas of the boxes    1

the similarity between how thevolume stays the same and te cross sections change1

Do all the solids have the same volume?    1

are the surface area and volume the same throughout the same changes?     1

do all the boxes have the same volume     1

how many cubes       1

what shapes could be made            1

how does the surface area change 1

same surface area?   1

could the volume make an equal ratio       1

whats the volume of each cube that makes each shape  1

how many different shapes can be made with those boxes        1

Do they all have the shme volume  1

Is the area of any gift formed by the candies the same? 1

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What do you *think*? Which package(s) use the least cardboard?

(No one answered more than one.)

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What do you *think*? Which package(s) use the least ribbon?

(No one answered more than one.)

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What do you *think* are the dimensions for each box?

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I enjoyed watching students use appropriate tools strategically while they were working.

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And then I sent the polls again.

Which package(s) use the least cardboard?

(Two answered B and D.)

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Which package(s) use the least ribbon?

(15 answered B and D; 2 answered B and C.)

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Some mistook “better” for “best”, and others are apparently going to cut the candies in halves.

Modeling with Geometry G-MG

Apply geometric concepts in modeling situations

  1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
  2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
  3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Content-wise, students had the opportunity to learn more about modeling with geometry. And they were able to engage more steps of the modeling cycle than just computation.

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I am not *yet* writing 3-Acts, but as the journey continues, I am grateful for those who do and share …


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Hot Coffee + Show Your Work

I’ve written about this lesson before, but I wanted to write again because of several observations from last spring.

I know that I need to find a way to provide students the opportunity to engage in math modeling more often and earlier in my geometry course. I’m having a hard time finding a way to do that. (Ideas for providing students the opportunity to engage in math modeling while proving theorems about congruence and similarity are welcome!) For now, we focus on modeling during the last unit of the course.

I decided last year to show students the modeling cycle from CCSS at the beginning of each lesson so that students would recognize what I am asking them to do differently and why I’m not giving them all of the information they need up front.

Our learning goals: I can model with mathematics, and I can show my work (leveled learning progression from Jill Gough).

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Once we decided what questions to answer after watching Act 1 of Dan’s World’s Largest Hot Coffee Three-Act, students estimated responses.

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And then teams made a list of the information they needed. I gave them information only as they requested it. Most teams realized later rather than sooner that they would need some type of conversion for cubic feet into gallons.

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When they decided they needed to know how much coffee a regular cup would hold, two of the girls remembered that the teacher with whom I share the classroom always had a cup of tea. They asked to borrow her cup so that they could come up with an agreed upon amount for a regular cup of coffee.

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At one point, a student asked whether getting the right answer mattered. I asked why. She and her teammate didn’t have the exact same calculation.

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It struck me that what we were really working on today was identifying a problem, determining what was essential to know, and creating a model to answer the problem. It’s not that the calculations aren’t important, but for this lesson, the questions were more important. By the time I got back around to that team, they had resolved their computational issue because of a conversion error. Even so, I’m glad I was asked whether it mattered that everyone got the same answer, as it helped shape how I launched our remaining modeling lessons.

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And so the journey to provide students the opportunity to engage in all steps of the Modeling Cycle continues …


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