Category Archives: Geometry

Segment Similarity: Circles Part 4

So this happens in geometry:

Proportional Triangles: http://goo.gl/RDkMQo

It relates back to dilations.

Transformations Applet: http://ggbtu.be/m125888

The big idea: similar triangles have proportional measures.

Scene Change

And then we see this:

Segment Measures in Circles

And formulas like these:

Circle Segment Theorem 1
Circle Segment Theorems 2 and 3

Wait, did you see that?  Down at the bottom with the big red arrow.  It’s fine print but it looks like a proportion of some sort.

Notice and Wonder

Max at MathForum shares the awesome of the notice and wonder practice in a math classroom.  This is what I was experiencing when I saw (a)(b)=(c)(d) and some occasional references to a proportional measure of  .  I was noticing the proportions and wondering about  similar triangles, dilation and those proportional relationships.

So This was my first date with Geogebra.  I was hooked, and I never looked back.  Geogebra allowed me to visualize my intuition and confirm my conjecture.  Eventually I came up with an applet.

Applets

Flipping the Triangle: APPLET

It wasn’t enough to just show how flipping a triangle could reveal the proportional relationship with similar triangles.  We need to also build those proportions using actual measures.  I made it so that the user could drag text boxes that correlated to the needed measures from the circle.

Where’s the similarity?

But wait, are all the relationships the same then?  Not exactly.  I would guess that this is also why most texts/reference guides give 3 separate theorems.  What each instance does have in common is there are two triangles, one similar to the other.  We just need to find the reflection that reveals the proportional relationships.

Lo-Tech

I can imagine others thinking right now, “That’s cool and all, but I don’t have tech for each of my students,” or “I’m not really a techie person.”

So use paper. (Patty paper works great).  Or transparent stuff like sheet protectors.

Paper Practice

Let’s be real.  This is great for conceptual understanding and visualizing the underlying relationships, but we need to practice.  And practice often works hand in hand with paper.  So we work with this concept for a half, maybe 3/4 of a period in the applets and then we practice fluency.  I actually go back to something like the Kutasoftware sheet from the beginning.  The difference this time is that we color code the segments, redraw them as embedded similar triangles, and use proportions.

And not once did we need those similar, yet not so similar theorems for circle segments.

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Clock Tangents: Circles Part 3b

Recently, there was a post looking at how to use the relationships of segments that are tangent to a circle to calculate more than just congruence.  Somewhere, and I’m not sure where, I remember Regan Galvan putting out there a tweet/post/comment about tan-tan angles on a clock.  Upon searching for it I found this:

http://www.visnos.com/demos/clock
http://www.visnos.com/demos/clock

I wanted to know how this thing worked.  So I built my own.

http://goo.gl/Y6lCxN

I purposely left out a lot of the information. I didn’t give any measures for angles or segments. This would be something that is developed in the classroom. The user and/or facilitator should be developing questions like:

  • How do I measure the angle between the clock hands?
  • How far away is the intersection point?
  • Is the intersection point always outside the circle?
  • What time is it when the two hands are perpendicular?
  • What time is it when the two hands are opposite rays?
  • Besides 12:00, what time is it when the two hands overlap?

Getting students to investigate these and asking to show evidence and create an argument to support their claim would be so much more fun than just calculating pre-made samples.  For students wanting more feedback and support, they can use the visnos interactive application.

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 2: Angle Relationships

Once similarity intuition has been built with circles, we can start getting into more specific relationships with angles and segments.  This post will look at using visual information from central angles and inscribed angles.

Pi Charts

http://wp.me/p464yb-2D

Students sometimes lack intuition for the measure of something.  Andrew Stadel has developed this idea into a thorough curriculum on estimation. In my classes we started reasoning through similar exercises.  Once we had a decent understanding of circle parts and whole, we moved on to other types of angles.

Non-Central Angles

At this point most students have the common sense that a circle has 360 degrees, and a triangle is half that at 180 degrees.  Built with this intuition in mind, we look at a triangle created by inscribed angles.

http://tube.geogebra.org/material/simple/id/115590

The next day we get to see the formula that collapses 3 ideas down to 1.

Dynamic Angles in Circles

So many of us have tried to organize all the formulas that go with circles.  It’s nice when textbooks or other curriculum organizers dom something like:

AngleRelationshipsInCircleTable

 

This is nice, but it still implies that all these relationships are discrete.  Instead it would be better to look at them as connected.

http://tube.geogebra.org/material/simple/id/117280

I originally thought students would look at this and see how the measures relate.  It wasn’t obvious, so I interjected with a simple exercise to the students.

Congruent Overlap

I saw two half sheets (8.25×5.5) of paper on my desk.  I held them up and asked the class, “Are these congruent?”  The response was something like, “Uh, Duh Mr. Butler.”  So I challenged them further, “Would you bet $1,000 on it?”

Now the silent sound of a gambler’s contemplation…

A more valuable decision requires some precision and accuracy.  As they were thinking I allowed additional information, “You can’t touch them, but you can ask me to do anything you want.”  They took the bait.  “Just overlap them, Butler.”  Now they’re headed in the right direction.  Overlapping layers can reveal congruence or equal measure.  Anyone could ask a similar question for an image like these puzzle pieces:

Congruent Puzzle Pieces

To support the visual information, students need to look at the measures involved and start building the relationships (SMP7 and SMP8).  After multiple true versions of the math relationships, we can return to something like the original table.

Students start to see that the table is simply 3 snapshots of this dynamic relationship. Then it is easier to organize the information in their relative schemas.  From here we go to paper practice with simple examples and eventually into a more complex question with multiple parts.

Assessment

http://www.geogebratube.org/student/m132524

I took a discrete snapshot from this applet and used it as a paper assessment.  Students were solving for complex relationships with multiple layers, and not all with the same process.  Common Core math like this requires utilizing big picture relationships while at the same time being able to articulate the specifics within those relationships.

Next time:

This series of dynamic relationships within a circle continues next looking at segments, first with only tangents, and then into various combinations with secants and/or tangents.

 

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.

Sum it up, Angle Edition part 1

Teaching Notes

http://mathbutler.wordpress.com/2014/03/14/sum-it-up-angle-edition-part1/

Applet

AnglePairs
http://www.geogebratube.org/student/m92695

Questions

  1. If angle BAE = ______, what are all the other angles.
  2. Which angle sums equal 180o?
  3. Which angle sums equal 90o?
  4. Which angle pairs/groups are equal?
  5. Find which angle pairs are supplementary, complementary and vertical
  6. Does it matter if angle BAE is acute or obtuse?