Tag Archives: student learning opportunities

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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Problem Solving Points

A few years ago, I began requiring my students to earn 100 problem solving points (PSPs) each quarter, instead of requiring each student to complete a project of my choosing. PSPs are a way for students to show me what they have learned or solved outside of class. These have gone through several iterations over the past few years, but here is what I have now that seems to work.

How students earn their PSPs is their choice. Sometimes a student will ask a question in class for which a) we don’t have time to get into the answer or b) I don’t actually know the answer. When that happens, I throw the question back to the class as an opportunity to earn PSPs. Do some research to find out the significance of Euler’s line, or the shape of the faces of a dodecahedron (we were meeting in the cafeteria, and I didn’t have a model with me), or who first start using the radical symbol. Students then email me what they find, along with their source.

What do you do with the calendar insert from the Mathematics Teacher? Those problems provide another way for students to earn PSP. Some students enjoy working calendar problems. I post the current calendar in my room – and have previous calendars in a binder for students to “check out” to work.

On our class Canvas site, I post enrichment opportunities for earning PSPs in each unit – many times they are from articles I’ve read in the Mathematics Teacher – or activities that I read about on blogs and tweets. I link to some Problems with a Point. And I link to some TI-Nspire labs that would be beneficial for students but that we aren’t going through explicitly in class.

I also post general enrichment opportunities for earning PSPs. Someone recently shared about a TED Talk on Fibonacci, so I posted a link to that for students with a note “Provide evidence to your instructor that you have watched this short TED Talk for up to 5 PSP”. Students will send me an email with their reflections on watching the video.

For the Five Triangles Blog, I have a note “Send a solution to a problem on this site to your instructor for PSP”.

We have been paying more attention to Mindset this year in class, so I have an opportunity for students to explore GRIT: Angela Duckworth says that the key to success is GRIT. Watch her TED Talk here. Then determine how much GRIT you have here. Then email your instructor a reflection with a response to at least one of the following prompts:

I like …

I wish …

I wonder …

I will …

After reading Fawn’s post on Mathmunch, I posted the following for my students: is a site with plenty of opportunities for PSP!

You can DO:

-work on a puzzle

-solve a problem

-struggle with a problem

Turn in your work on the puzzle/problem.

You can MAKE:

-recreate a piece of math art

-create your own artwork inspired by the original work

Turn in your artwork.

You can WATCH:

-watch a video & leave a comment on the post about what you learned and/or questions you have

Turn in a screenshot of your comment on the post.

You can READ:

-read about a mathematician and compose a few questions you’d like to ask this person.

Turn in your two questions.

You can PLAY:

-play a math video game, then write a critique of it (likes, dislikes, suggestions, etc.)

Turn in your critique of the game.

Idea from Fawn Nguyen.

The possibilities for PSPs are endless. Some students read an article on mathematics or technology and share what they learned. Some students share a website that they have found which is helpful for learning more about mathematics. Some students do constructions using TI-Nspire. Some students write journal reflections on using a Standard for Mathematical Practice. Some students work cryptograms and problems of the week or do math history research from my school website.

In an effort for encouraging students to take the PSAT, I do give PSP for the math section of the PSAT during the second quarter (not a 1:1 ratio of math score:PSP earned). And for seniors, I give PSP one time for the math section on either the ACT or SAT. Another teacher had this idea, and I have continued it. All students get some choice about what they find interesting enough to explore for PSPs.

Student can earn some of their points in class by answering bellringer questions and Quick Polls correctly. At the end of the quarter, I total up all of their points and multiply by some scale factor that gives each student around 25-50 PSPs, depending on which quarter it is. I usually start out the year with letting them earn up to 50 from class – and decrease that number as the year goes on. I like that they can earn some of these points through class questions, because that gives them some incentive to not only be in class, but also be active participants in class. Using these points for PSPs instead of entering each assignment in the gradebook separately takes off the pressure of having to get every question correct. We are in the process of learning, and we don’t already know it all – there is plenty of leeway for students to earn PSPs and make mistakes.

I’ve wondered from time to time whether I should stop requiring PSPs, but each time I ask students to reflect on their experience with PSP, they insist that I should keep them going. And so I do, as the journey continues ….


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