Tag Archives: Standards for Mathematical Practice

MP8: The Medians of a Triangle

How do you give students the opportunity to practice “I can look for and express regularity in repeated reasoning”?

SMP8 #LL2LU Gough-Wilson

When we have a new type of problem to think about, I am learning to have students estimate the answer first.

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I asked for their estimate in two slightly different problems because I wanted them to pay attention to what was given and what was asked for.

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Students then interacted with dynamic geometry software.


What changes? What stays the same?

Do you see a pattern?
What conjecture can you make about the relationship between a median of a triangle and its segments partitioned by the centroid?

As students moved the vertices of the triangle, the automatic data capture feature of TI-Nspire collected the measurements in a spreadsheet.


I sent another poll.

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And then we confirmed student conjectures on the spreadsheet.

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

1 Comment

Posted by on August 15, 2016 in Angles & Triangles, Geometry


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Visual: SMP-1 Make sense of problems and persevere #LL2LU

What if we display learning progressions in our learning space to show a pathway for learners? After Jill Gough (Experiments in Learning by Doing) and I published SMP-1: Make sense of problems and persevere #LL2LU, Jill wondered how we might display this learning progression in classrooms. Dabbling with doodling, she drafted this poster for classroom use. Many thanks to Sam Gough for immediate feedback and encouragement during the doodling process.

Screen Shot 2014-08-16 at 1.21.17 PMI wonder how each of my teammates will use this with student-learners. I am curious to know student-learner reaction, feedback, and comments. If you have feedback, I would appreciate having it too.

What if we are deliberate in our coaching to encourage learners to self-assess, question, and stretch?

[Cross-posted on Experiments in Learning by Doing]


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Unit 11 – Modeling with Geometry

Student Reflections

I can statements:

Level 1: I can create a visual representation of a design problem. 100% Strongly Agree or Agree

Level 2: I can decompose geometric shapes into manageable parts. 94% Strongly Agree or Agree

Level 3: I can estimate and calculate measures as needed to solve problems. 100% Strongly Agree or Agree

Level 4: I can use geometry to solve a design problem and make valid conclusions. 100% Strongly Agree or Agree



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).


Which Standard of Mathematical Practice did you use most often in this unit? In which other Standard of Mathematical Practice did you engage often during the unit?

Students answered that they used make sense of problems and persevere in solving them the most and model with mathematics next.


Think back through the lessons. Did you feel that any were repeats of material that you already knew? If so, which parts?

11A Volumes of Compound Objects

11B 2D Representations of 3D Objects

11C A Tank of Water

11D Popcorn Picker

11E Hot Coffee

11F Two Wheels and a Belt

  • I didn’t think any of these were repeated. We really looked at geometry and used it to solve hard real world problems, and I’ve never had to do that before.
  • I do not feel like this was a repeat. Many of us knew about basic areas and volumes of course, but in this unit we went into depth with them and learned how to get the dimensions of more complex shapes. I think we covered new material, because we have mainly been going over coordinates, trig, and triangles previously and this seemed like a new and fun topic to cover for us.
  • I already knew how to find the volumes of compound objects from unit 10.


Think back through the lessons. Was there a lesson or activity that was particularly helpful for you to meet the learning targets for this unit?

  • 11F Popcorn Picker was very helpful in helping me learn the targets for this unit. It helped me realize that although two objects may have the same surface area, their volumes may not be the same.
  • Popcorn Picker definitely helped me understand how exactly different dimensions affect the volume of a cylinder even though the dimensions are nearly the same. Using the piece of paper to compose a cylinder using 8.5 and 11 as two different circumferences as well as the height helped me see that the volumes will be different.
  • The coffee one helped me because it made me talk with others at my table and look for ways to solve the problem.
  • Although I enjoyed the popcorn activity, I feel like the most helpful activity was the one where we cut out the cards and thought through the gradual shapes of the water as it emptied the top figure and filled the bottom figure.  It allowed me to work with a team to reason out our arguments and working with my peers in a collective effort enlightened me with thoughts and ideas that I had not previously thought of or would have otherwise ventured to discuss.
  • I really liked the Hot Coffee unit. I understood it well, and it was a good problem to work and figure out. It was also really good for me to make sure to use the right units and convert correctly, which I don’t do sometimes.


What have you learned during this unit?

  • This unit helped me to realize how much I’ve learned this year in geometry and how to do many things like finding volumes and areas of different shapes.
  • How to use the least amount of information to find the need item.
  • I’ve ;earned how to divide a complex geometric objects into parts and calculate It’s volume. I can find out the necessary information needed to solve this kind of problem and how to use them to solve the problem. I can apply math to every day life and model with mathematics. I can also make visual representation of a design problem.
  • I have learned to attend to precision. Throughout not only the homework lessons but also in class, I learned to be careful and slow down. I often hurried over the question and did not take into account the measurements. For example, I hardly ever noticed when the question gave dimensions in feet but asked for the answer in inches. It’s not a hard concept, but it requires patience and effort that I was trying to shortcut on.
  • During this unit, i learned a lot about using what i already knew and combining that to solve difficult problems.
  • I’ve learned how to break down 3D shapes into simple 3D shapes so I can get their volume, and how to do actual geometry problems with all of the things I’ve learned this year. I’ve learned what the different cross sections of objects can look like, and I learned that the world’s largest coffee cup help 2015 gallons of coffee.
  • I learned that I need to model with mathematics more often.


And so the journey continues … figuring out how to provide more opportunities throughout the course for my students to model with mathematics.


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Unit 10 – Geometric Measure and Dimension

Student Reflections

I can statements:

Level 1: I can use formulas to calculate area and volume. 100% Strongly Agree or Agree

Level 2: I can identify cross-sections of 3-D objects, and I can identify the 3-D objects formed by rotating 2-D objects. 90% Strongly Agree or Agree

Level 3: I can explain the formulas for area and volume. 100% Strongly Agree or Agree

Level 4: I can calculate the area and volume of geometric objects to solve problems. 100% Strongly Agree or Agree



Geometric Measurement 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.

3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Visualize relationships between two-dimensional and three-dimensional objects

4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.


Which Standard of Mathematical Practice did you use most often in this unit? In which other Standard of Mathematical Practice did you engage often during the unit?

Students answered that they used look for and make use of structure the most. They used make sense of problems and persevere in solving them and reason abstractly and quantitatively next.


Think back through the lessons. Did you feel that any were repeats of material that you already knew? If so, which parts?

10A Length Area

10B Surface Area

10C Volume

10D Enlargements

10E Cross Sections

10F The Best Box

10G Performance Assessment

10H Mastering Arc Length and Sectors

  • I already knew length and area. I had been taught the formulas for surface area, but I didn’t actually understand why those formulas worked until this unit. Cross sections were completely new to me.
  • I felt like I already knew some of the information for calculating volume and area of 3D forms, but in this unit I learned why and how the formulas work.
  • The length and area was a bit of a repeat of what we’ve learned in the past. It was nice to review and be able to bring to memory what we’d already learned as a basis for the new things we were going to learn this unit.
  • This Volume lesson was a tiny bit of something I already knew, but I never got as much into it as we did. Before, I just learned the formula and never understood. I also only ever learned the volume of a rectangular prism.
  • The very first lesson in which we used the basic formulas was a bit of a repeat from past years. However, in the past we had to memorize the formulas, whereas this year we were given the opportunity to understand the why the formula is what it is. I find that understanding it helps more than simply memorizing it.


Think back through the lessons. Was there a lesson or activity that was particularly helpful for you to meet the learning targets for this unit?

  • Yes, Length and Area helped me with Surface Area and Volume by making sense of structure within the problem.
  • When you got the orange peels and separated them into four circles to show how the surface area of a sphere worked, it really helped me understand how surface area works. After that I started thinking about how the formulas worked instead of just memorizing them.
  • 10G Performance Assessment helped me out the most. It summed up most of the chapter and helped my apply what I learned to many different problems.
  • I think the lesson of volume helped the most in understanding the formula. When we talked about the different figures fitting into one another it helped me understand why this certain formula was there. It helped show me that all formulas are taken from a common one.


What have you learned during this unit?

  • In this unit I have learned how to use surface area, lateral area, volume, and cross sections. I now have a better understanding from where the formulas come from by looking at the figures and in a way putting together the puzzle pieces to come up with the final formulas.
  • I learned exactly how area, surface area, and volume are all different.
  • I have learned how to calculate lateral area, surface area, and volume of forms and I also learned why the formulas work and not just to plug in numbers to those formulas.
  • I have learned how to identify cross sections and explain the formulas for surface area and volume.
  • I’ve learned how to use formulas to calculate area and volume and why we use those formulas. I’ve learned how to make sure to convert problems to the same dimensions and same ratios, and I’ve learned what cross sections are and the different ones in different 3D figures.
  • I have learned to understand the formulas and not just memorize, but really understand them. I can use these formulas to solve word problems and figure out tough questions. I’ve also learned about cross-sections and how they differ for each 3D shape. I have also understood the surface areas of 3D shapes and how they work with all shapes.
  • During this unit, I have come to understand the geometric formulas, and I believe that understanding will help me later on in tests like the ACT if I were to forget the formula that I memorized in 7th grade, I would remember the concepts and understandings of why the formula is what it is.


And so the journey continues … still struggling to find a balance between teaching conceptual understanding of mathematics and using mathematics to solve problems … although I do think the students’ language when reflecting on their understanding of geometric measure and dimension is a good sign.


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Unit 9 – Coordinate Geometry

Now that school is out, I am going back through our last few student surveys to make notes for next year.

I can statements:

Level 1: I can represent and use vertices of a geometric figure in the coordinate plane. 100% Strongly Agree or Agree

Level 2: I can use the equation of a circle in the coordinate plane to solve problems. 96% Strongly Agree or Agree

Level 3: I can use slope, distance, and midpoint along with properties of geometric objects to verify claims about the objects. 96% Strongly Agree or Agree

Level 4: I can partition a segment in a given ratio. 81% Strongly Agree or Agree



Expressing Geometric Properties with Equations G-GPE

Translate between the geometric description and the equation for a conic section

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Use coordinates to prove simple geometric theorems algebraically

4. 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).

5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.


Which Standard of Mathematical Practice did you use most often in this unit? In which other Standard of Mathematical Practice did you engage often during the unit?

Students answered that they used look for and make use of structure the most. They used make sense of problems and persevere in solving them and model with mathematics next.


Think back through the lessons. Did you feel that any were repeats of material that you already knew? If so, which parts?

9A Lines

9B Quadrilaterals

9C Triangles

9D Area & Perimeter

9E Circles

9F Translating Circles

9G Polygons

9H Loci

9I Coordinate Geometry Performance Assessment

9J Mastering Coordinate Geometry – Partitioning a Segment

  • Yes and No, for me most of the material was new and different. I learned how to partition, and use loci. I already knew the equations but in the course of Unit 9 I learned how to use equations like the Pythagorean theorem to find the side area and perimeters of a triangle.
  • I already knew how to find the area and perimeter of simple geometric shapes but not all the shapes.
  • Pretty much none of these were repeats. I’ve heard of bits of these lessons, but I learned so much more during all of the units that I felt like those bits don’t count.
  • I already knew how to find the slope and do other things in 9A. I learned it in 8th grade Algebra and it was a nice break from just learning new things everyday and just having to review that day.
  • The lesson about the equations of lines and the lesson about area and perimeter were things that I had been taught before, but I was taught them in different ways that weren’t as in depth as what we went to in class.
  • Yes, I did feel that content from numerous lessons (namely 9A, 9B, and 9C) was derived from lessons that I have previously covered either in this class or a previous math-related class. For instance, dealing with parallel lines and perpendicular lines, as well as distinguishing between the two by looking at the x-intercept of an equation’s standard form, was a very prominent concept in the algebra I course I took last year. Also, as were the concepts of calculating distance, midpoint, and slope, all of which were key concepts last year. Finally, I felt that observing shapes on a coordinate plane, as was practiced in lessons 9B and 9C with quadrilaterals and triangles, respectively, resembled a lesson that we covered in November/December as part of units 4 and 5, in which we specifically observed each shape and its properties.


Think back through the lessons. Was there a lesson or activity that was particularly helpful for you to meet the learning targets for this unit?

  • There was no particular activity that really sticks out in my mind, but simply watching how others worked the problems on the board really helped. Many people brought up ways to do things that I never thought of, and a majority of the time, their way was more effective than mine.
  • Yes, I think in unit 9E when we had to write a definition for circle in quick paw helped me to get a better understanding of what a circle is and to understand why how I defined it was wrong.
  • 9H was one of the hardest ones for me to do but I think it helped me a lot because it made me think harder than the others.
  • Just like in most all the units, the mastering lesson always helps me. This just lets me go back through everything and make sure I understand it.

What have you learned during this unit?

  • I learned how to gather measurements for shapes from their coordinates. Also, though, I learned to really persevere in solving problems. This unit was one of the harder ones for me, and a lot of times I wanted to give up. I kept trying, though, and when I would eventually understand a problem, it was worth it.
  • I have learned how the coordinate plane and geometry are 2 sides of the same coin and that you can use graphs to calculate geometric properties.
  • I learned that you can find the area and perimeter of all shapes not just the simple geometric shapes.
  • I learned a lot in this unit. I learned about all the shapes and how to use equations to solve problems involving them. I learned what loci was which was new and kind of confusing. I also learned how to partition a ratio for a given segment which was cool because it took a lot of different ways to look at the segment that i wouldn’t have thought of before learning this. This unit contained a lot of review from past years, but also a lot of new material that i had never seen before.
  • If this unit has taught me anything, it is that geometry and algebra can be collaborated so that multiple relationships between and details of certain points, lines, and shapes can be proven, all made possible by borrowing the coordinate plane, distance formula, midpoint formula, and slope formula, all in which we became fluent through two years of algebra.


We have struggled to know whether it is best to have a separate coordinate geometry unit or whether we should be using coordinate geometry all year. To me, it is kind of like wondering in a calculus class whether you do early or late transcendentals. I like late transcendentals, because students go back through differentiation and integration towards the end of the course, and they really have to think about which one they are using and why. It turns out to be a nice culminating review that wouldn’t happen if we had been differentiating exponential and logarithmic functions along the way.


And so the journey continues … always trying to figure out what works best for student learning.


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Great Tasks: Infinity Pizza

Connie Schrock and I recently presented a webinar on Great Tasks:

What is a “great task”? How is a great task used effectively in the classroom? Published by the National Council of Supervisors of Mathematics (NCSM) with support from Texas Instruments, NCSM Great Tasks for Mathematics 6-12 provides opportunities for students to explore meaningful mathematics by challenging their thinking about essential concepts and ideas. Explore tasks that are aligned to the Common Core content standards, with special attention given to the Common Core Mathematical Practices. Learn how tasks can be enhanced with the TI-84 Plus family and TI-Nspire™ technology to promote mathematical reasoning and active engagement for students of the 21st century.

I shared a story about using one of the Great Tasks with my students, which I have included in this post.

My geometry students are in a unit on Geometric Measure and Dimension. Last week, I gave them the first part of the Infinity Pizza task. Students are asked to create a fair method for cutting any triangular pizza into 3 equal-sized pieces of pizzas. I asked students to work alone for a few minutes before they started sharing what they were doing with others on their team. I walked around and watched. Many students started on paper.

I like seeing the progression of ideas from this student.


Of course at least one student had to decorate her pizza with pepperoni.


My students and I use TI-Nspire Navigator to help manage our classroom. I took a Class Capture to see what students were doing on their handhelds. Last year I read Smith & Stein’s Five Practices for Orchestrating Productive Mathematical Discussions.

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The practices for teachers are to anticipate how students will respond to a task, monitor while students are working on a task to select which student work should come out in a whole class discussion and sequence that student work so that students will ultimately be able to connect the mathematics in the different solutions.

Let’s look at the second class capture I took – I can update the Class Capture as often as I want manually by pressing the Refresh button, or I can set it to update automatically every 30 seconds, which I often do. Whose work might you choose to discuss first if you were leading the class discussion?

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I chose Alex first. Alex proceeded to tell us that he joined midpoints together and ended up creating a pizza with four equal slices instead of three. Alex answered a different question, but he gave us some good mathematics to remember about the triangle formed by the midsegments.

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Next we moved to Kristen. Kristen was in the process of trying to trisect the angle. She hadn’t made it to whether the triangular pieces were equal in size.

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Then we moved to Reagan. I made Reagan the Live Presenter, which meant that her calculator and interactions with the calculator were showing at the front of the room. Reagan told us that she created the medians of the triangle to cut her pizza. You can see from the measurements that she took that the pieces are equal in size. Reagan had hidden the lines that contained the medians to make her pizza look better, but it is by seeing the six triangles formed by the medians that she was able to explain why her three triangular pieces of pizzas have equal area.

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I called on one other student to share his solution – but I have to show two other pictures that I had taken of students working on paper.

Flynt had thought about using the centroid to divide the triangle into three equal areas, and he had also noted that he could partition the base into three equal parts. One of the CCSS Standards for geometry talks about partitioning a directed line segment into a given ratio. We worked on that standard in our last unit, and it really excites me for my students to attend to precision with the language that they are using during class. I’ve been teaching geometry for 20 years, and my students have not used that language in the past.


Benny also talked about partitioning. He suggested that we needed to partition the base of the triangle into a 1:1:1 ratio.


I asked Benny to share what he had done with the rest of the class. He did, and one of the students asked whether it mattered which side of the triangle he used as the base. Benny said no, but he wanted the image rotated (which is quick and easy thanks to our interactive whiteboard) so that the second side he showed as the base would still be horizontal, and on the bottom. That’s the image you see on the bottom right.

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You can see that Katlyn was using the same method as Benny with technology; however, if we look at her file, we can tell that she had not yet figured out how to trisect the base. We always have more to learn.

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There is a part 2 for the Infinity Pizza task that I haven’t had time to try with my students yet. Every year, though, they come knowing more mathematics than the year before. Maybe next year we will get to Part 2!

My goal for my students is not just to provide opportunities for them to learn the content standards and use the math practices while they are working on a task – but for them to begin to recognize when they are using the practices. I gave my students a list of the practices along with a short explanation of each practice at the beginning of the year. I have that same list posted in the front of my room. This language has become part of how we talk to each other about what we are doing in class – and what our goals are for learning in each unit. Which math practice would you expect to see your students demonstrate with the Infinity Pizza task?

I asked my students which practice they think they used the most during this task. They completed their responses on a Google Form.

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AK: “In this challenge, we used many of the Math Practices, but the one that sticks out to me is construct viable arguments and critique the reasoning of others. I think this was most prominent because of the way each student was given the opportunity to share their solution and the class in turn was given the opportunity to compare and support that student’s method or argue for a different approach. Either way it provoked in-depth thoughts to look beyond the simple “find the solution and move on to the next problem”.”

Katie noted, “I chose [reason abstractly and quantitatively] because when you have a problem like this it’s easier to look at numbers and general terms. I first had to think about the question. … It was hard trying to think of where to start, so I thought about cutting the pizza with the lines connecting the midpoints together. This would have worked had the quadrilateral these lines made been a square, but it wasn’t because I was using a right triangle. So I went to look at the lines of medians. Well, the area for each triangle then was not equal. This was trying my patience, but I kept going. I then looked at the circumcenter and the lines connecting that, but it ended up just like the first one, I had three triangles and a quadrilateral. Which was four pieces that definitely did not have the same area. Then I just thought for a few minutes. It then dawned on me that we were dealing with area and the formula for the area of a triangle is 1/2bh(where b=base and h=height). So, I thought how can I get three triangles from one where the base and height never change. I then used 21 for my base of the triangle because 3 will go into 21, 7 times. I had three triangles with a base of 7 and the height couldn’t possibly different because they all came from the same triangle.”

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Chandler: “I felt that I mostly modeled with mathematics throughout this activity because the entire goal of Infinity Pizza was to utilize a certain math practice (or more than one) in order to determine how to evenly split a pizza into three perfectly even pieces. … Since I knew that the centroid would give me the point of central mass, I searched for the centroid by drawing a line from the perfect center of each angle to the exact median on the opposite side of the triangle. … Note that this was conducted on an equilateral triangle. Next, to elaborate, I tried the same procedure on an isosceles triangle and a scalene triangle. The results were conclusive — my method was feasible for any triangle despite any side length differences, even if the pieces’ sides were of different lengths. …”

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Kelsey: In this activity you had to construct viable arguments and critique the reasoning of others the most. The reason I say this is because you had to come up with a way to split the pizza into 3 equal slices and you had to persuade the class that your reason worked everytime. Also you had to listen to other peoples reasons why and critique if their answer was wrong or write.

Roselynn chose make sense of problems and persevere in solving them: “In order for us to solve this problem, we had to have perseverance. This wasn’t a problem where the answer took a minute or two to solve. It took us a good portion of the block to construct our argument. It also took many approaches. We didn’t get a good answer the first try. It took a while to make sense of the problem, and we had to persevere in solving it.”

And so the journey continues, trying to provide students great tasks that help them make sense of mathematics …


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Running around a Track I

I have been at the T3 International Conference in Las Vegas this week. This morning, I had the opportunity to present a Power Session on CCSS: Bringing content and practice standards together. My next few posts will be a few of the stories from this session.

I gave a class of students Running around a Track I recently.

I started by showing them a picture of the start of the 2012 Olympic Women’s 400 M race. What’s your question? (I took a picture while watching the video.)


The students had good questions to get us started in our exploration.

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who won        1

What is the circumfrence of each lane?       1

how far is th last runnr to the circle     1

Why are the not in a line       1

how far away is the first runner from the last runner?          1

which track is longest 1

what are we trying to find      1

what is the ratio from lane 1 to lane 9          1

What’s the circumference of the track?        1

I know everything?   1

why are they not in a line?    1

who gonna win?        1

how far is the outer lane from the inner lane            1

does contestant number 9 have an advantage over contestant number 1        1

What is the ratio of the inner-most lane to the outer-most lane?  1

how much shorter distance does the inner lane runner have to run than the outer runner?         1

how long is the circumference of each circle            1

What are we looking for?     1

Are they all going the same distance.         1

Who won?     1

What is the measure of the intercepted arc?          1

is this the winter olympics     1

Next, I sent out a Quick Poll of a question from an ACT practice test that students should be able answer as a result of the lesson.

Dylan Wiliam talks in his book Embedded Formative Assessment about teachers needing time for collaboration. Our administrators take this seriously, and so our geometry team has the same planning block. One of the other geometry teachers had the idea to include the ACT question in with this lesson.

Who already knows how to solve this? Who knows how to solve it quickly, since the ACT is timed? I removed the choices, just to see how students answered without them. We didn’t talk about the results. I told them we would revisit the question at the end of the lesson. Just over half of the students had it correct initially: 14/26.

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We talked about a few of the student questions from the picture of the Olympic race. And then I passed out the student handout for Running around a Track I. I let the students work in groups to answer the questions, and I sent Quick Polls every once in a while to be sure that they were working productively. I sent a QP for question a) and noticed that 20/26 students had it correct. I was able to use the Navigator to determine who had missed it & provide help for them at their desk while the rest of the class worked ahead.

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I didn’t send a poll for question b). Instead I provided students the correct answer so that they could determine on their own whether they needed assistance or were ready to move on to part c).

Most students got a bit stuck on part c. I knew this because I sent a poll to find out student responses. No one had it correct.

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We used this as an opportunity to figure out incorrect thinking. One student came up to share his work, and the others critiqued his argument, figuring out where their own thinking had gone wrong, and correcting their work.

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To close the lesson, I sent the Quick Poll from the start of class:


75% of the students answered it correctly with no choices (up from 50% at the start of the lesson), and 96% answered it correctly with choices.


I think it is interesting to ask students which practices they used when working on a task.


And which was the most used practice:


At this point in the session, we asked the participants to process this part of the story using the protocol “I like …” “I wish …” “I wonder …” for the discussion. What worked? What would you have done differently had these been your students? What would you do differently if you were going to use this task with a group of students in the future? What if you were going to give students Running around a Track II? How would you intend for the lesson to play out?


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Segments in Circles

Circles: G-C

Understand and apply theorems about circles

2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

We used part of the Math Nspired activity Intersecting Lines and Segment Measures for this lesson.

What is the relationship among the four segments created when two chords intersect?

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I sent a Quick Poll asking my students to estimate the value of x before we began our exploration. Just over one-third got it correct.

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We moved to the technology to see if we could look for regularity in repeated reasoning. The relationship between the segments formed by the chords was not evident to students, even using the technology.

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My students are having to learn how to take risks. They enter my classroom afraid of answering incorrectly. When I ask “what do you notice”, they are hesitant to answer because what they notice might not be what they were supposed to notice. We are having to change the mindset of students. It is okay to be wrong; we learn from our mistakes.

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One student finally took a risk: I noticed that the segments were proportional.

Okay. Could you give us a proportion? PA:PD=PB:PC.

Let’s look at that for a moment. Look at your screen. Is it true, with the measurements that you have, that PA:PD=PB:PC?

How would we get a proportion like that?

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Another student took a risk: We’ve talked about proportions using similar triangles.

Do you see triangles in this picture?

I see ∆PBC and ∆PAD.

(I saw ∆PBA and ∆PCD. It happens so often that my students see differently than I, which always makes me happy that I asked and didn’t draw in my own triangles.)

Could we show that the triangles are similar? How do we show triangles are similar?

Another student took a risk. I can’t remember now what method she named, but it wasn’t a method for proving triangles similar – it was a method that worked to prove triangles congruent. But it is okay that she was wrong. We learned something from that mistake.

We made a list of methods we could use to prove the triangles similar: a sequence of transformations that included a dilation, or SAS~, AA~, SSS~.

What do we know in this diagram? We know PA:PD=PB:PC.

Well, actually, that is what we are trying to prove. So we can’t use that in our proof of why the triangles must be similar. Instead, that will be a result of similarity.

Some students recognized that ∆PDA~∆PCB because of a rotation and dilation. Someone else asked me to number the angles. I see that ∠2 ≅∠5 because vertical angles are congruent.

So are any other angle pairs congruent?

I waited.

And waited longer.

∠3 is inscribed in arc AB.

And so is ∠6.

So they are congruent.

The green marks on the diagram came after the noticing.

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So the triangles are also similar by AA~, which makes the corresponding side lengths proportional.

If you look in a textbook, you won’t see PA:PD=PB:PC. Instead, you’ll see PA∙PC=PD∙PB.

A Quick Poll to assess student understanding showed that we were at just over 80% correct.

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What is the relationship among the segments formed when two secant segments intersect outside the circle?

We didn’t take the time for a first guess. We went straight to the exploration.

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After watching our dynamic geometry software move, I sent a poll.

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I’m trying to ease the hurry syndrome, but the seconds are ticking away. We spent a long time making sense of the proof with the chords, and we only have a few minutes left before the bell will ring to end class.

I stopped the poll once half of the class has gotten it correct.

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We formalize our conjecture. We look for and make use of structure. What auxiliary lines do we need to show the similar triangles for this diagram?

And then a visual proof of what we mean when we say that the product of one whole secant segment and its outer secant segment is equal to the product of the other whole secant segment and its outer secant segment.

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The area of a rectangle with one whole secant segment as base and its outer secant segment as height is equal to the area of a rectangle with the other whole secant segment as base and its outer secant segment as height.

And when one secant segment turns into a tangent segment?

The area of a rectangle with the whole secant segment as base and its outer secant segment as height is equal to the area of square with the tangent segment as its side length.

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Students had practice for this last diagram outside of class – and on the bell work at the beginning of the next class.

And so the journey to ease the hurry syndrome continues, sometimes successfully, and sometimes not …


Posted by on February 20, 2014 in Circles, Geometry


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Hopewell Geometry – Misconceptions

Last year I posted student work from the Mathematics Assessment Project task Hopewell Triangles. This year I want to talk about the conversations that we had.

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Students worked on the task for 15-20 minutes by themselves. For those who didn’t need that long, I had two other tasks from Illustrative Mathematics on the back of the handout: Seven Circles I and Shortest Line Segment from a Point P to a Line L.

After students worked by themselves, I collected their responses in the TNS document I had sent. They started talking about their results with their groups. Apparently I didn’t make it clear to the whole class that I would be collecting their responses, because everyone had not keyed in their responses, but I looked at the responses that I did get to monitor student progress while they were talking with their groups. I noticed that one student had answered question 4 incorrectly.

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So I went to his group and listened to them as they constructed a viable argument and critiqued the reasoning of others. Alex explained how he had arrived at his wrong answer. Can anyone figure out Alex’s misconception?

In our whole class discussion, I made sure that Alex shared his thinking. We all have something to learn from other’s misconceptions and we have created a classroom where it is okay to learn from incorrect thinking. It turns out that Alex used inverse tangent to calculate the acute angle measures. He then concluded that the triangle was right because it had a right angle (since the two acute angles were complementary), not realizing his circular logic in assuming that the triangle was right to use right triangle trigonometry for his acute angle calculations.

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Can we show that Alex’s thinking doesn’t work in conjunction with what we know about the other triangles?

Triangle 3 is a 45°-45°-90° triangle, and so its acute angles are 45°. At least half of the students recognized that they had already calculated one of the acute angles for Triangle 1 in problem #2. Then we can show by the Angle Addition Postulate that we have a problem when the shaded triangle is right since 53°, 90°, and 45° don’t sum to 180°.

Most students used the Pythagorean Theorem to show that the shaded triangle was not right, which was our next whole class conversation. One student told us that the hypotenuse of Triangle 3 was 5√2, which he knew because of our work with Special Right Triangles. Another student used the Pythagorean Theorem on Triangle 2 to get its hypotenuse of 7√5. And then another showed us that (5√2)2+(7√5)2≠(15)2. This would have been the end of the conversation in my class when I first started teaching. But because our focus is not only on correct answers but also on logical arguments and understanding, a student stopped us from continuing by asking a question. Why is the hypotenuse of Triangle 2 7√5 instead of 7√3? What a good question. Why is the hypotenuse of Triangle 2 7√5 instead of 7√3?

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How did you get 7√3? The triangle has side lengths 7 and 14, so I assumed the triangle was a 30°-60°-90° triangle. Another student cleared up that misconception. Side lengths of x and 2x are for a 30°-60°-90° triangle only when 2x is the length of the hypotenuse.

The last whole class conversation happened because I overhead a group talking about how they knew Triangle 1 was similar to Triangle A. I had asked them to remember that part of the conversation because I was going to ask them to discuss it with the whole class later. One girl said that she knew the triangles were similar because she saw that 3-4-5 was proportional to 9-12-15. But she was having a hard time visualizing how the triangles were similar. How could we show that the two triangles are similar to each other?

How do we define similarity? Two figures are similar if there is a dilation (and if needed, a sequence of rigid motions) that will map one figure onto the other. Can we show that to be true for Triangle A and Triangle 1?

We need several rigid motions along with the dilation to show that Triangle A and Triangle 1 are similar:

Reflect Triangle A about the side that is 3 units.

Rotate Triangle A’ 270° about the vertex of the angle opposite the side that is 3 units. Let’s call that point X’.

Translate Triangle A’’ by a vector that goes from X’’ to the left vertex of the given rectangle.

Dilate Triangle A’’’ about point X’’’ by a scale factor of 3.

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I was amazed at the mathematics in this lesson and the opportunity for students to make connections. What a great task! If I had written the task, it wouldn’t have occurred to me to include a triangle that students might mistake for a 30°-60°-90° triangle. Instead, the students were given the opportunity to construct a viable argument and critique the reasoning of others. It might not have occurred to me to include a calculation in an early question (#2) that students could use again in the final question (#4) – I would have likely changed the angle measures to have them perform two different calculations. Instead, the students were given the opportunity to look for regularity in repeated reasoning. If I had written the task and included a triangle similar to one of the others, I doubt it would have occurred to me to make one of them have a different orientation than the other. Instead, the students were given the opportunity to look for and make use of structure.

And so the journey continues, with great tasks like this one to make student misconceptions evident and to correct those misconceptions …


Posted by on February 2, 2014 in Geometry, Right Triangles


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Unit 6: Dilations – Student Reflections

CCSS-M Standards:

Similarity: G-SRT

Understand similarity in terms of similarity transformations

G-SRT 1. Verify experimentally the properties of dilations given by a center and a scale factor:

a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

G-SRT 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

G-SRT 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove theorems involving similarity

G-SRT 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

G-SRT 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Circles: G-C

Understand and apply theorems about circles

G-C 1. Prove that all circles are similar.

I can statements:

Level 1: I can identify, define, and perform dilations. G-SRT 1

Level 2: I can determine the similarity of two figures using similarity transformations. G-SRT 2, G-SRT 3

Level 3: I can prove theorems about triangles. G-SRT 4, G-C 1

Level 4: I can solve for and prove relationships in geometric figures using similarity criteria. G-SRT 5


6A Dilations

6B Similarity Theorems

6C-6D The Similarity Ratio

6E Pythagorean Relationships

6F Altitude to the Hypotenuse

6G Dilations Performance Assessment – Dilating a line and a circle

6H Dilations Performance Assessment – Bank Shot

6I Mastering

In which Standard for Mathematical Practice did you engage most often during the unit?

Most students chose MP1, make sense of problems and persevere in solving them, and MP 6, attend to precision.

I asked students whether any of the content seemed like repeats of previously learned material.

  • Aside from problems that were solved by utilizing the Pythagorean Theorem, all of the content from this unit was new to me.
  • I’ve gone over pythagorean theorem multiple times in the past year, however the rest of this is new to me, and frankly quite difficult.
  • I can honestly say I learned everything as new in this unit. This was entirely new for me. I feel I had an easier time learning this material than the other material in other units. I caught on very quickly in this unit compared to others
  • the only thing I already knew was that for a figure to be similar all the side lengths had to be proportional.
  • The only thing i knew about unit 6 before we learned it was that dilations have to do with making an object larger or smaller.
  • There was nothing that was repeated in this lesson. I knew what a dilation was, but I’d never done anything with it, so basically I knew the first 15 minutes of this unit, and the rest was completely new to me.

Which lesson helped you the most in this unit?

  • There is not one lesson or activity in particular that stands out to me, but I do think the extra homework helped, even though the thought of extra homework is not a very appealing thought.
  • 6C-6D was very helpful. If I hadn’t understood similarity ratio, I would have not understood any of the material we learned this unit. I have never gone into so much detail about similarity and ratios. I’m glad I understand them more.
  • The day when we used wax paper to decide if objects were similar or not helped me because it let me visually understand it.
  • The homework was extremely helpful. It provided practice for me, and showed me what I needed to work on. It deffinitely made the lesson more clear.
  • I think that lessons 6A and 6B, dilations and similarity theorems, really prepared me for all the material that was taught after that. Learning these two skills taught me the basics for learning the rest of the unit.

What did you learn during this unit?

  • you have to be very precise and make sure you evaluate your answer
  • I have learned about the similarity ratio and how to dilate a line and a circle.
  • I have learned my favorite transformation: Dilations. I love dilations now and they are my favorite. I figured out how to tell if figures are similar through a rigid motion and dilation or just through a dilation. I learned that the hypotenuse altitude is the geometric mean between the two sides it divides, which helped a ton in figuring out measurements of the sides of a right triangle. I think this was my favorite unit and I learned so many new things.
  • I have learned how to use, apply, and perform dilations to geometric figures. I have also learned that listening to the opinions and ideas of others can lead you to find the answer or the reasoning behind certain answers.


And so the journey continues …

1 Comment

Posted by on January 28, 2014 in Dilations, Geometry


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