Category Archives: Area

Satellites Spinning Round: Circles Part 3 (Tangents and Incircle)

Inspiration

Have you ever wondered how far a satellite is from the planet?  Sometimes people think about these objects flying through the sky and rotating around a planet.  Such wonders have even inspired greats like Isaac Newton and Kepler.

Some good folk over at illustrative mathematics put this whole satellite into some simpler geometry with circles, triangles, and tangents.

https://www.illustrativemathematics.org/illustrations/416

 

Inspired by these I sought out to simplify the idea of tangents and how they fit around a circle.  We played with this:

Applet

http://tube.geogebra.org/student/b115694#material/115596

Questions

Warm up with some questions about the applet,
    • What shapes do you see?
    • Which measures, do you think are congruent?
go into some more typical math questions,
    • What is the perimeter of the triangle?
get a little geeky
    • What is the total area of the triangle (circle included)?
Then get all geeky advanced.
  • How do you calculate the value for the area of the blue shaded area (circle excluded)?

 Student discourse:

Along the way here were some awesome teachable moments:

T: What kind of segments do you see?
S1: Tangents
T: How do you know?
S1: ’cause it says ‘tangent’.
T: I’m still not convinced. Maybe I mislabeled it. Prove it.
S2: They’re perpendicular. Tangents are perpendicular.
T: Anything else? Any other types of segments or is that it?
[crickets]
T: What about inside? Any segments inside the circle?
S3: Radius! There’s a radius!
T: Where?
S3: Here?
T: Where? Pretend I’m blind.
S3: You already are blind Butler, that’s why you wear glasses.
T: [fake smile]
S3: Here at AJ. Ohh, and also AF. And these ones to but you didn’t label them.
T: Do you notice anything else? What if you drag around that red thing?
S3: They’re all the same?
T: What are the same?
S3: The radiussus. Does that mean they’re equal?
T: What do you think?
S3: Yeah, equal. They have to be, cause of the red thing.

S: Hey Butler, is that a kite?
T: It sure looks like it huh? What do we know about kites?
S: There’s stuff that’s the same. Like congruent.
T: For example…?
S: The pieces that match.
T: So how can you use that?
S: They’re equal. Oh wait, they’re equal. Sooo….
And then the student goes on to apply that understanding to fill in the blanks for the missing segment lengths.

The idea of using a few measures, congruent relationships, and some triangle calculations helped the students apply the geometry and calculations further than just one calculation.  As a follow up assessment they completed a static version of the question.  This was easy to create by just taking a screenshot of the applet at some particular balance.

Some day I want to advance this further with some clock math.

Circles Part 1: Similarity Intuition

All circles are similar, right?

Okay, maybe it’s not given.  In fact, it needs to be proven.  This proof is yet another that is so easily demonstrated with dynamic math tools (like desmos and geogebra

Circle Similarity with Desmos
https://www.desmos.com/calculator/vscq2qtp2w

So long as you can move one center onto the other (translate) and dilate one radius to equal the other, similarity is achieved.  This works for every circle.  The perfect proportional balance achieved with circles lays the foundation for most of not all relationships found in them.

Proportional Measures

Similarity gives us a simple system for comparing measures in multiple figures.

2014-07-07_1512

You can even explore this with repetition of congruent triangles:

Dimensional Analysis with Scale
http://ggbtu.be/m127379

Applications

Below are a couple of applications that push further with exploring the measures of circles, proportionality, and relationships in the measures.

Pizza Pi 

This activity was first inspired by a posting on John Stevens‘ website wyrmath.wordpress.com .  It looks at proportions with a different angle (pun intended) and asks user to compare various portions of different sized (but similarly shaped) pizza.

Rolling Tires

Another great one from Andrew Stadel.  This is a 3 act math lesson looking at relationships with circumference of a tire and cumulative rotations.

Coming up next…

So where do we go from here.  Following the basic intuition of similarity and proportions in a circle, we can build into

  1. More Angle Relationships with circles
    • Central angles and/or non-central angles
    • Circles circumscribed around a triangle start to simplify some of these relationships.
  2. Segment Relationships
    • Tangents and Incircle
    • Dynamic Relationship with Intersection of Chords, Tangent-Secant, and Secant-Secant (hint: they’re all a similar relationship)

See you in Circles Part 2: Angle Relationships.