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Tag Archives: #ShowYourWork

Is This a Rectangle?

Is This a Rectangle?

One of our learning intentions in our Coordinate Geometry unit is for students to be able to say I can use slope, distance, and midpoint along with properties of geometric objects to verify claims about the objects.

G-GPE. Expressing Geometric Properties with Equations

B. Use coordinates to prove simple geometric theorems algebraically

  1. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

We recently used the Illustrative Mathematics Task Is This a Rectangle to provide students the opportunity to practice.

We also used Jill Gough’s and Kato Nims’ visual #ShowYourWork learning progression to frame how to write a solution to the task.

How often do we tell our students Show Your Work only to get papers on which work isn’t shown? How often do we write Show Your Work next to a student answer for which the student thought she had shown her work? How often do our students wonder what we mean when we say Show Your Work?

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The Show Your Work learning progression begins to help students understand what we mean when we say Show Your Work. I have seen it empower students to ask each other for feedback on their work: Can you read this and understand it without asking me any questions? It has been transformative for my AP Calculus students as they write Free Response questions that will be scored by readers who can’t ask them questions and don’t know what math they can do in their heads.

We set the timer for 5 minutes of quiet think time. Most students began by sketching the graph on paper or creating it using their dynamic graphs software. [Some students painfully and slowly drew every tick mark on a grid, making me realize I should have graph paper more readily available for them.]

They began to look for and make use of structure. Some sketched in right triangles to see the slope or length of the sides. Some used slope and distance formulas to calculate the slope or length of the sides.

I saw several who were showing necessary but not sufficient information to verify that the figure is a rectangle. I wondered how I could steer them towards a solution without telling them they weren’t there yet.

I decided to summarize a few of the solutions I was seeing and send them in a Quick Poll, asking students to decide which reasoning was sufficient for verifying that the figure is a rectangle.

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Students discussed and used what they learned to improve their work.

It occurred to me that it might be helpful for them to determine the Show Your Work level for some sample student work. And so I showed a sample and asked the level.

But I didn’t plan ahead for that, and so I hurriedly selected two pieces of student work from last year to display. I was pleased with the response to the first piece of work. Most students recognized that the solution is correct and that the work could be improved so that the reader knows what the student means.

I wish that I hadn’t chosen the second piece of work. Did students say that this work was at level 3 because there are lots of words in the explanation and plenty of numbers on the diagram? Unfortunately, the logic is lacking: adjacent sides perpendicular is not a result of parallel opposite sides. Learning to pay close enough attention to whether an argument is valid is good, hard work.

Tasks like this often take longer than I expect. I’m not sure whether that is because I am now well practiced at easing the hurry syndrome or whether that is because learning to Show Your Work just takes longer than copying the teacher’s work. And so the journey continues …

 
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Posted by on April 11, 2017 in Coordinate Geometry, Geometry, Polygons

 

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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.

(1&2)

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?

(3)

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?

(4)

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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(5)

Some decomposed the trapezoid into a parallelogram and a triangle.

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(6)

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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(7)

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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(8)

Or a rectangle with the same dimensions.

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(9)

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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(10)

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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(11)

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.

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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?

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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.

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(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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Structure, Flexibility, and Planning

I set up a recent lesson by asking students to deliberately practice SMP7, look for and make use of structure.

2 SMP7 Number #LL2LU Gough-Wilson

This practice requires us to make visible what isn’t showing. In geometry, that often means drawing auxiliary lines.

We don’t always see structure in the same way or at the same rate, so once you’ve found one way to solve the problem, I want you to also deliberately work on your mathematical flexibility. Find a second way to work the problem.

1 Flexibility #LL2LU Gough

I had 5 questions prepared, the last of which I learned about in Justin’s and Kate’s posts last year about a Five Triangles task. I’ve been thinking a lot this year about not only planning learning episodes but also planning ahead what instructional adjustments I’ll make based on the feedback I get from my students. In my planning, I struggled with which question to use first. Which question would you use first with your students?

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Last year, I had the following results in the following order.

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After this first Quick Poll, I didn’t display the correct answer, asked students to team with someone else in the room, and sent the Poll again.

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After this Quick Poll, we had a student who answered 53˚ share his reasoning with the rest of the class so that we could figure out where the reasoning went wrong.

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The class went fine. But I wondered what would have happened if I had started with a question that required the use of auxiliary lines (even though students struggled with the question that already had them drawn). So I tried that this year.

I could “hear” thinking and I could “see” productive struggle as students started out working the problem individually. Once they started sharing some of the ways that they made visible what wasn’t pictured, I saw evidence of SMP7. Because I had deliberately asked them to work on their math flexibility, they weren’t satisfied with only one way to solve the problem.

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Many wanted to share their way with the whole class.

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They tried another one, and again, you could “hear” thinking. I didn’t even have to suggest individual think time to the class, as they naturally all wanted to try it by themselves first.

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I posed the folded rectangle problem, but the bell rang before students could really dig in to solving it. Maybe next year I’ll be brave enough to start with it, as the journey continues …

p.s. I’m currently reading Ilana Horn’s Strength in Numbers: Collaborative Learning in Secondary Mathematics, and before I was able to publish this post, I happened to read a section entitled “Turning Some Pet Ideas about Mathematics Teaching on Their Heads: Start with Challenging Stuff, Not Easy Stuff”. Her premise is that starting with easy stuff is inequitable, as students who get the mathematics quickly can take over the problem, and those who don’t miss out on the opportunity work with their team. Starting with challenging stuff levels the playing field for all students to contribute and learn.

 
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Posted by on October 26, 2015 in Angles & Triangles, Geometry

 

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Circles and Squares

We started our performance assessment task day for Geometric Measure & Dimension with Circles and Squares, a Mathematics Assessment Project task.

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Several students wondered about the ratios of the areas of the figures.

Without doing any calculations, what is the ratio of the area of the smaller square to the larger square?

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Now practice look for and make use of structure.

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We did the usual and purposeful individual work then team work before sharing with the whole class.

I loved watching students look for and make use of structure to make sense of the relationships in the diagram.

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And then successfully determining the ratio … even though there is always room for more attention to precision.

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