Tag Archives: geometry

The Surface Area of a Sphere

How do you help your students make sense of the formula for the surface area of a sphere?

We started with an orange that was spherical. We talked about a great circle, and we used string to draw some great circles for the sphere on a piece of paper.

If we peel the orange, and fill the circles with peeling, then how many circles will we fill? I sent a Quick Poll to get their estimates. 20% were correct, which reminds me that I must find a way to work in more estimation. (Or I’ll admit I actually wondered whether someone who taught below me could work in more estimation. I don’t know what to replace!)

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This year we didn’t actually peel the orange. I had been at #NCSM14 on the day we were going to do this in class, so after our conversation, we looked at a picture from another geometry class that their teacher had emailed me while I was out of town.


Four of the great circles were filled for this spherical grapefruit. At this point, I almost said the formula for the surface area of a sphere. My students would have nodded, and we would have moved on. But I stopped myself. Instead, I asked my students to write down on their paper what the formula was. I walked around, monitoring their work. I made myself wait on them. I thought the formula would be obvious, but it wasn’t obvious to many of my students. We talked more about the great circle, the dimensions of the great circle, and what those dimensions have to do with the sphere. A few more wrote down that the area of the surface area of the sphere was πr2*4. One student wrote down that the surface area of the sphere was πr3. I asked him to share what he had written with the rest of the class so that we could learn from this misconception. What would the units be if we calculated πr3? Cubic. What do cubic units represent? Volume. We can often use units to help us make sense of a formula.

Finally, we settled on what we would see in a textbook or on a formula chart if we looked up the surface area of a sphere: 4πr2. But it was hard won. Will my students remember it? The journey continues … with hopes that the image of the orange and 4 circles filled with peeling is imprinted in their memory.


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Circles – Student Reflections

Unit 8 Student Reflections

I can statements:

  1. I can use relationships between angles and arcs in circles to solve for missing measures. (100% strongly agree or agree)
  2. I can use relationships between secants, chords, and tangents in circles to solve for missing measures. (100% strongly agree or agree)
  3. I can use similarity to calculate arc length and area of a sector. (92% strongly agree or agree)
  4. I can prove relationships between secants, chords, and tangent in circles. (96% strongly agree or agree)


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.

4. (+) Construct a tangent line from a point outside a given circle to the circle.

Find arc lengths and areas of sectors of circles

5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.


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 look for and make use of structure second most.


Think back through the lessons. Did you feel that any were repeats of material that you already knew? If so, which parts?
8A Angles and Arcs
8B Tangents and Chords
8C Angle Measures
8D Segment Lengths
8E Pi
8F Arc Length and Sectors
8G Performance Assessment
8H Mastering

  • The only part that was a little familiar was 8E pi.
  • I’ve never gone in depth with circles, so no, none of the material was repeats, except maybe a few spots when we learned what pi was.
  • 8E was a similar of a repeat of what we learned in Algebra because we had to find pi in multiple situations. It made it much easier for me to understand. Before then, I did have a small misunderstanding of what I needed, but after remembering what I had learned in eighth grade and then I was able to understand more. This was a lesson that showed me that algebra and geometry were very identical.
  • The Pi lesson also had some repeated information, but it was nice to learn about the Pi ratio in detail.
  • I have had little experience up until this year with circles, other than memorizing the formulas.


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? If so, how?
8A Angles and Arcs
8B Tangents and Chords
8C Angle Measures
8D Segment Lengths
8E Pi
8F Arc Length and Sectors
8G Performance Assessment
8H Mastering

  • It helped me understand circles more when we used real life situations.
  • Pi was an integral part of the unit and it helped me kind of piece things together. Finding degree measurements in a circle was like a puzzle, when you got one piece you could move on to the next.
  • I believe the day where we thought through ways to find the center of the circle gave me insight into the relationship between the circle and the accuracy necessary to correctly talk about it. It also gave me a chance to apply what I had learned previously to find an accurate solution.
  • All of the lessons were very helpful for the understanding of the chapter, especially 8A, the beginning lesson, which became the “backbone” for the later lessons.
  • I think lesson 8B helped a lot because it introduced tangents and chords, which was a totally new concept for me.

Some students feel like the practice assignments really helped them make sense of the unit as a whole:

  • All of the homework activities helped me understand the unit.
  • I find it wasn’t the lesson itself, but the homework. The homework challenged me to really think and on a few of them I had to search how to do them. It really helped me make sense of problems on the test and on the lessons after some homework.
  • The 8H homework was actually probably the most helpful of any of the things I did. I thought I knew this unit pretty well, until I went and made a 5/10 on my first try. It really helped me go back, review, and relearn the things we’d been doing because I had some very skewed ideas about circles before that lesson.

Some students feel like the Performance Assessment tasks really helped them make sense of the unit as a whole:

  • The performance assessment was very helpful to me because in a sense it was a combination of all the previous lessons. This also was a way to make myself quicker at solving the problems and figuring out short-cuts. This was a great review to prepare for my test, which I believe that really was an asset for me. The test was much better with the performance assessment.
  • The Performance Assessment was very helpful because it let me see which parts of the unit I didn’t understand as well as the others and showed me what I needed to work on before the test.
  • Every lesson this unit was helpful, but the Performance Assessment and the Mastering lessons helped the most. In the lessons before these, we learned a couple things, but in these lessons, we learned how to combine everything that we learned to find the correct answers to challenging problems.

What have you learned during this unit?

  • I learned how to calculate the arc length of a circle using the relationship of it to the whole circle.
  • In this unit I have deepened my knowledge of Pi, which is one of my favorite numbers and how to recognize tangents. I have also learned how to find arc measures and how to find an angle measure in a circle.
  • I have learned the relationships between radii, chords, tangents, and secants in circles. I’ve also learned so many different ways to solve for missing links within circles. I can also find missing lengths outside of the circle. I find that this unit was the toughest to this point.
  • … I’ve also learned that’s circles are more complicated than they seem at first.
  • During this unit, I learned about the relationships between arc lengths and the angles that intercept them. I also learned how tangents and chords relate to circles, and how to figure out their lengths. Another thing I learned was what Pi was and where it comes from. A fourth thing I learned was how to determine the arc length and area of sectors in circles.


And so the journey continues … trying to determine what is best for students. Do I take out the pi lesson next year, since students had some knowledge of pi coming in to high school geometry? Or do I leave it, since several students noted that they enjoyed learning about pi in detail? Eventually, I’ll have students who should have an informal derivation of the relationship between the circumference and area of a circle in grade 7 (7.G.B.4), but until then, I’ll probably keep going over at least part of the lesson.

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Posted by on March 14, 2014 in Circles, Geometry, Student Reflection


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Making Time for Tasks

I co-presented a power session at the T3 International Conference Sunday morning, and I’ve posted the stories that I shared of Running around a Track I and Running around a Track II. I have also posted Running around a Track III during which I processed the feedback we received during and after the sessions.

Several participants asked about the time that rich tasks take, and so I thought it might be helpful to think about making time for tasks in this separate post. I enjoy teaching on a block schedule. I know I don’t feel as rushed as I would otherwise. But a friend asked me how often we do these types of tasks in our classes.

How often do you do this type of rich task?

There are some tasks through which the math content can be initially and naturally be learned. I think of Placing a Fire Hydrant (and last year’s notes here) and Locating a Warehouse (with last year’s notes here). There are some that are more culminating tasks for a unit, especially the modeling tasks. A typical unit for us lasts about 8 days. (See some of the Unit Student Reflections posts to know more about how we organize content: Special Right Triangles, Dilations.) We do the culminating performance type tasks 1-2 of those days towards the end of the unit. On the other days, we are learning the content using practices such as look for and make use of structure and look for and express regularity in repeated reasoning, but not always through tasks.

For example, when our learning goal is

G-C.A.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.

I find that exploring the relationships using technology first, verifying conjectures using formative assessment questions (more skill-based practice), and then moving towards why works best for my students. If you know of a good task for learning G-C.A.2, I would love to know. But for now, at least, we end this unit with a few culminating tasks instead of beginning the unit with them. The culminating tasks are Circles in TrianglesInscribing and Circumscribing Right TrianglesTemple Geometry. I will note that while we didn’t actually do the tasks until the end of the unit, I did show students one of the diagrams from the beginning so that they could keep in mind throughout the unit where we were heading.

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So now that you know that I don’t give a rich task every day, let’s think about what steps we might take to reduce the class time for the tasks. [These questions and comments will make more sense if you have looked at how the lessons for Running around a Track I and Running around a Track II played out with my students.] One of the participants in our Sunday session wondered what would have happened if I had given the students a teaser of what was to come the day before this lesson. What if I had shown them the picture and asked what they wondered then? Or sent a link to a Google form for them to submit their question outside of class?


Or what if I actually showed them the tasks and questions (blacking out information in I that would give away II or vice versa) just so students could begin the process of thinking about the structure of the track before they did any calculations in class? I think these are great ideas – having students spend the “alone” time for processing the questions being asked could definitely make the time spent on the task take less class time.

Another question that tied into the idea of planning lessons with rich tasks was whether we use a textbook. I think of the textbook as a source of information and practice problems for students. We don’t have our units arranged like our textbook, but I do reference page numbers for each lesson on the student syllabus so that they will know where to look for extra help and practice. (Our textbook isn’t CCSS at this point, and I’m not convinced that the new ones have geometry written in the spirit of CCSS. When I’m reviewing a new textbook, I first look at the Table of Contents. When transformations is still the topic of chapter 8, I find myself skeptical.)

Someone else asked about how rich tasks complement some of the skills practice that students need. We don’t get to every practice problem that we include on our student handouts, but I have finally had the time to work through them, and so I post the worked handouts online for students to check problems they do outside of class for additional practice. We also give online practice assignments with two chances to students through Canvas so that they can get immediate feedback on what they know and don’t know. We are trying to teach our students how to use formative assessment.

[I’ve been reading Transformative Assessment and Transformative Assessment in Action, both by James Popham. According to Popham, the first level of formative assessment is when it is used by the teacher to make instructional adjustments as needed to further student learning. The second level of formative assessment is when it is used by the student to make learning adjustments as needed to further learning. The third level of formative assessment is when it is used by the class as a whole to help all students meet the learning goals of the lesson, and the fourth level of formative assessment is a transformation of the school – all teachers are learning about formative assessment in school wide PD and practicing it in their classrooms.]

My students and I have explicitly discussed in class that if you take the online practice assignment and miss every problem, you should make a learning adjustment before trying again.

So if I don’t use a textbook, how do I plan my lessons?

The top three sites that I use are Illustrative Mathematics, the Mathematics Assessment Project, and the Math Nspired lessons at TI’s site. I also follow a lot of bloggers. It helps that I’ve taught geometry for 20 years, and it helps that I’ve been making an effort for students to learn by doing for at least 17 of those years. (I owe a huge apology to all of the students I had the first 3 years.)

But it does take time. I am lucky to work with a great team of geometry teachers who are willing to help and try tasks and use formative assessment with their students. We taught our CCSS Geometry course last year for the first time, and our administrator worked it out so we could have 4 teachers and 30 students in our first block class together. We worked through the lessons together with the students and each other, and then the other 3 teachers had a planning block after that class so that they could correct everything we had done wrong the first time before they taught it in their own classrooms the rest of the day. This year, we have a team of Algebra 2 teachers doing the same thing, and next year, our Algebra 1 teachers will teach one class together. I didn’t know what my administrator would say when I proposed this idea to him, but I’ve learned it doesn’t hurt to ask. It was a total scheduling pain, and some of our other classes were more crowded, but that was worth the sacrifice for what we learned teaching the first class together.

I’ve just told you what works for us in our efforts to include more rich tasks, but it’s only one perspective. What works for you? What additional sites do you use to find rich tasks? Do you start with them more often than end with them? We’d love to learn more from you as the journey continues … easing the hurry syndrome, one task at a time.


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Unit 3 – Angles

We learned our lesson last year when we tried to cram Angles & Triangles together in a unit that we needed to take a step back & slow down…at least until we have students who have actually experienced the geometry standards in the lower grades as prescribed by CCSS-M. Since this is the first real year of 6-8 implementation by our teachers, we still have a few years before we might be able to make more adjustments. For now, this one unit covers what has traditionally been covered by 3 chapters of our geometry textbook. 3 chapters have been reduced to one CCSS-M standard. The 3 chapters? Points, Lines & Planes; Deductive Reasoning & Logic; Parallel Lines.

Level 1: I can use inductive and deductive reasoning to make conclusions about statements, converses, inverses, and contrapositives.

Level 2: I can use and prove theorems about special pairs of angles. G-CO 9

Level 3: I can solve problems using parallel lines. G-CO 9

Level 4: I can prove theorems about parallel lines. G-CO 9

Congruence G-CO

G-CO 9

Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

I know that I have read statements about de-emphasizing the structure and building of a deductive system in CCSS-M. I wish I could remember in which document I read that. I’ve only been emphasizing the structure and building of a deductive system for about 8 years now – ever since I took a graduate course on geometry and realized its importance. So last year, we tried to jump right in to proving theorems without thinking about logic, conditionals, converses, inverses, etc. It was a disaster. Even though conditional statements aren’t explicitly listed in CCSS-M and hence will not be assessed, I believe they are important. I think it is a big deal that we accept the following as a postulate in our system: If two parallel lines are cut by a transversal, then corresponding angles are congruent. That’s our equivalent to Euclid’s 5th Postulate, and our geometry would look differently if we didn’t accept it without proof. The Triangle Sum Theorem relies on this postulate, as do other important results, and I want my students to know that it is our choice of accepting this postulate that brings about results that wouldn’t be true without it.

So we started with a lesson on logic. I asked students to create a word morph – go from WARM to COLD, changing one letter at a time, creating a real word each time. This short exercise emphasizes that we all start with WARM and we all end with COLD, but we don’t all get there the same way.

I read someone’s tweet about having students create a conditional statement for which the statement, converse, inverse, and contrapositive were always true, which we did. I wish I could remember whose it was. Thanks for the idea.

We also gave student groups cards with a conditional statement, its converse, inverse, and contrapositive. The students were to choose a card as the conditional and then decide which of the other cards was the converse, inverse, and contrapositive. Once they have those statements straight, then I choose one of their other cards (like their inverse) and tell them it is now their conditional statement. If it is now the conditional, then which of the other cards are now the converse, inverse, and contrapositive? This short exercise emphasizes that it doesn’t matter how the conditional is written, from that conditional, you can still create the converse, inverse, and contrapositive.

Then we thought about the game of basketball being a deductive system. What would be the undefined terms? Defined terms? Postulates? Theorems? Monopoly and soccer also work for this.

Then we played The Letter Game as an introduction to proofs. I wish I knew the source of The Letter Game. A quick search on the internet did not produce any results (except my Canvas notes from class), so I’ll include it below.

The “Letter Game”

Undefined terms: Letters M, I, and U

Definition: x means any string of I’s and U’s

Postulates:             1. If a string of letters ends in I, you may add U at the end.

2. If you have Mx, then you may add x to get Mxx.

3. If three I’s occur, that is III, then you may substitute U in their place.

4. If UU occurs, you drop it.

Example. Given: MI

Prove: MIIU

1. Given: MIII

Prove: M


Prove: MIIU

3. Given: MI

Prove: MUI

4. Given: MI

Prove: MIUIU

5. Given: MIIIUII


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I like The Letter Game because it introduces students to the idea that we start with some given information, we have a goal of what to prove, and we use the rules (postulates, theorems, and definitions) in our deductive system to get there. I do not care that students produce a formal two-column proof. I care that students can create a valid, logical argument. I also care that they notice that their valid, logical arguments don’t have to look exactly the same as everyone else’s in the class. There is almost always more than one way to get from the given information to what we are trying to prove. And more than a few students admit they like figuring out a Letter Game proof.

Then we do a few jumbled proofs about segments and angles before we prove our first theorem: Vertical angles are congruent.

And so, right or wrong, the journey continues…

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Posted by on October 20, 2013 in Angles & Triangles, Geometry


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