# Tag Archives: circles

## Team Sorting – Right Triangles & Circles

We’ve been sorting students into teams on the first day of each unit this year. After the team sort for Dilations, we were motivated to again make the cards challenging and interesting.

We went to the cafeteria so that students would have plenty of room to find their teams. I told them that we have 6 teams and 30 students.

I overhead the following:

• I’ve just got two sides.
• Am I supposed to calculate the side or the angle?
• I’m noticing we both have 60-30 triangles.
• You would have to find the hypotenuse, and you would have to find the leg.
• Do you have 33 degrees?
• We think we have a team: two sides and an x.
• Who has an x, an angle, and a hypotenuse?
• Do you have the included leg or the opposite leg?
• Is everybody here today?

Eventually, all teams sorted themselves.

From the beginning, students began to notice when we can use the Pythagorean Theorem and when we can’t.

Students began to notice the difference between the opposite leg of the acute angle of a right triangle and the included leg.

Students began to notice when we were looking for the angle of a right triangle instead of one of the sides.

Students began to notice special right triangles.

Yesterday, we sorted into teams for our circles unit.

Students began to notice the difference between central angles and inscribed angles (even though they don’t know the names of them yet).

Students began to notice the difference between diagrams with two chords and two secants (even though they don’t know the definition of a chord versus a secant).

Students began to notice the difference between diagrams with tangents only and diagrams with a secant and a tangent.

Just in case you want to use or edit, you can find all of our team sorting cards here.

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Posted by on January 28, 2015 in Circles, Geometry, Right Triangles

## 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)

Standards:

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.