Quadratic Formulae

Quadratic Amy
courtesy of Samuel Winter via flickr CC-BY-NC-SA License
How would you graph?   QuadraticToGraph

Would you factor?  Would you complete the square?  Quadratic Formula?  Are those my only options?

The idea of multiple representations  of the same thing, or multiple methods toward the same end, has strong correlation for learning (and teaching) mathematics.  Jo Boaler demonstrates this in her number talks.  Dan Meyer and Robert Kaplinksy discuss these through the open middle problem design. One way we see this in polynomials (including quadratics) is…

Factoring

There’s the famous XBox method, often abbreviated to a formulaic method by teachers.  I like this approach, especially because it incorporates area models, grouping, flexibility for multiple values of the lead coefficient “a”.    James Key (aka @iheartgeo), brings in more of the number talk by using powers of 10 in addition to powers of x
 .

Here’s a quick preview of his vine that relates:

XBox Numerical

 

Transitioning to more algebraic/abstract examples seems simpler when relating quadratics to numerical examples like this vine. He looks at the factors and the relationships between the terms.  Students in his class are prescribed a formula, but asked to make sense of the relationships and then transition to methods with more regularity.

Complete the Square

A method often confusing to Algebra 1 students because of its abstract structure is again made simple with looping short videos.

Courtesy of @NatBanting
Courtesy of @NatBanting

Plenty of teachers try to support their process with visualization, all that is intended here is to start with the visual reasoning and pattern making, then move to abstract.

The hybrid

It’s not exactly completing the square.  It’s not exactly factoring and using the roots.  It’s the compromise between the two.

  • plot the points (0,0) and (-b,0)
  • shift vertical c units.

Let’s make that easier.  Here’s a dynamic version:

https://www.desmos.com/calculator/lhppvh0sh0

Open up the Middle

But wait, the two points (x1,0) and (x2,0) that plot on the horizontal axis can be any combination that adds up to your linear coefficient “b”.  The vertical shift just needs to balance those numbers with c.

So for our example: QuadraticToGraph, the 8x needs to be split, and c needs to be balanced with the shift:

x1 x2 c shift?
8 0 20 20
0 8 20 20
4 4 20 4
2 6 20 8
1 7 20 ?
-2 10 20 ?
3 5 20 ?

 

Now with that bit of intuition, we can open up the middle and have the user do some exploration.  Here’s a question that encourages looking at multiple representations and seeing how the values are connected:

OpenMiddleQuadratic

When thinking of factors, roots, and vertical shifts, the relationship becomes more dynamic.  We could revisit the open middle prompt and ask to define / express the relationship between a,b and c.   Struggling students can simplify the algebra using Jame’s Keys approach with substituting a value for x.  Here’s a Geogebra applet to help visualize the multiple representations:

http://www.geogebratube.org/student/m187821

 Play with the applet and notice that as long as the quadratics share the same line of symmetry, only a vertical shift is needed to overlap.  This also reveals the congruence in some quadratics.  Ask yourself what makes the quadratics congruent?  

Here’s a similar interactive graph built with Desmos.

What now?

I would expect that this more flexible visualization allows for more depth in understanding how a quadratic function works, both visually and algebraically.  Students who are asked to match quadratic algebraic expressions may be little less intimidated by starting with some dynamic pictures.  So now let’s return to the original question (that has multiple possibilities):

How would you graph?   QuadraticToGraph

What other questions might you ask?

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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.

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.