Reflection: Vertical Alignment Loci and Analtyic Geometry - Section 3: Modeling


In keeping with the theme of this unit as a rites of passage from Geometry to Intermediate Algebra, this lesson is a good opportunity to introduce students to the importance of forms, which play a key role in higher algebra. 

I want my students to understand that forms are not arbitrary. They are chosen because they have utility. Students are used to using the phrase "simplest form" which is often misleading because it isn't always about which form is simplest but rather which form best suits the purpose we have in mind. For example having the equation of a parabola in vertex form would be best if we want to know the vertex, but possibly not if we wanted to know the x-intercepts. 

I also want them to see structural similarities (and differences) between various forms. For example, the vertex form of a parabola is similar to the standard form of a circle in the sense that they both have the (x-h)^2 component.

At the most basic level, though, I want students to memorize the names and structures of the forms and which forms go with which figures. I want them to remember that we will write equations of circles in standard form, that we will write the equation of the perpendicular bisector (since it is a line) in slope-intercept form, and that we will write the equations of parabolas in vertex form. This knowledge will guide students as they progress from the distance formula-based starting points to the final forms of the equations.

For example, when writing the equation of a circle in standard form, we don't square the binomials because the standard form calls for the factored forms (x-h)^2 and (y-k)^2.

By contrast, when we are writing the equation of the perpendicular bisector, we know that we have to get to y = mx + b , which has no quadratic terms. This lets us know that we must square the binomials in order for the quadratic terms to "go away". 

And then there's the case of the parabola which is really a hybrid of the circles and perpendicular bisectors. We know that the form is y = (x-h)^2 + k which lets us know that we don't need to square the binomial that has x in it (similar to circles) but we must square the binomial that has y in it so that the quadratic term will drop out (similar to perpendicular bisectors).

Developing this type of strategic decision-making and awareness of structure (MP7) is much more important to me, as a teacher of mathematics, than having students memorize how to write the equations of these curves. And since the things I think are important are the things I will emphasize, I have to remind myself what is most important.

  Establishing the idea of form
  Vertical Alignment: Establishing the idea of form
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Loci and Analtyic Geometry

Unit 9: Analytic Geometry
Lesson 3 of 7

Objective: SWBAT derive equations of circles, perpendicular bisectors, and parabolas given their locus definitions

Big Idea: Which came first, the algebra or the geometry? After this lesson, students will interplay between these two branches of mathematics.

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