Intro to Quadratic Equations
Name: Eric Reyes
Title of
lesson: Intro to Quadratic
Equations, Part 2.
Date of
lesson: Monday Week 3
Length
of lesson: 1 Day
Description
of the class: Mathematics
Name of course: Algebra
Grade level: 9
Honors or regular: Regular
Source of the lesson:
http://www.purplemath.com/modules/grphquad2.htm
(a)
Basic understandings
(4) Relationship between
equations and functions. Equations arise as a way of asking and answering
questions involving functional relationships. Students work in many situations
to set up equations and use a variety of methods to solve these equations.
(5) Tools for algebraic
thinking. Techniques for working with functions and equations are essential in
understanding underlying relationships. Students use a variety of
representations (concrete, numerical, algorithmic, graphical),
tools, and technology, including, but not limited to, powerful and accessible
hand-held calculators and computers with graphing capabilities and model
mathematical situations to solve meaningful problems.
(d) Quadratic and other
nonlinear functions: knowledge and skills and performance descriptions.
3) The student understands
there are situations modeled by functions that are neither linear nor quadratic
and models the situations. Following are performance descriptions.
(A) The student uses
patterns to generate the laws of exponents and applies them in problem-solving
situations.
(B) The student analyzes
data and represents situations involving inverse variation using concrete
models, tables, graphs, or algebraic methods.
Students will be able to:
· Solve quadratic equations.
IV. Supplementary materials, handouts.
· Handouts and questions
Five-E Organization
Teacher Does Probing Questions Student Does/ Response
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Engage: Have students draw different types of parabolas that cross all the axis. |
1. what are the characteristics of these parabolas? 2. what are the points where the graph crosses the x axis called? |
1. They can either cross the x axis: zero times, once, or twice 2. roots |
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Explore: What are methods in which we solve the values where a parabola crosses the x axis? We want to solve: Y=ax^2+bx+c We want to find factors of c that add up to b. Now from multiplying polynomials, quadric is fromed from mulitpyling two factors of the form (x+m)(x+n). The answer is ax^2+bx+c=(x+m)(x+n). Solve x^2+5x+6=0. |
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1. Factoring |
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Explain: Solving by Square-Rooting Let's take another look at that last problem: Solve x2 – 4 = 0. When you have "squared part minus a number", you can put the number over on the other side, like this: x2 – 4 = 0 x2 = 4 Remember that, when solving an equation, you can do whatever you like to it, as long as you do the same thing to both sides. On the left-hand side, I have x2, and I need just x. To turn an x2 into an x, I square-root both sides: x = ± 2 Then the solution is x = ± 2. Solve: x^2+5x+6=0 |
Why the "±" ("plus-or-minus") sign? |
Students read and go over problems on worksheet. Because it might have been a positive 2 or a negative 2 that was squared to get the 4. |
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Extend / Elaborate: Use completing the square to solve x2 – 4x – 8 = 0. As we noted before, this does not factor, so we can't solve the equation that way. And they haven't given us the quadratic in a form that is ready to square-root. But there is a way to manipulate the quadratic to put it in a form that we can square-root. It works like this: First, you put the loose number on the other side: x2 – 4x – 8 = 0 x2 – 4x = 8 Then you look at the coefficient of the x-term; this is –4, in this case. Take half of this (including the sign); this is –2. Then square this value to get +4, and add this to both sides: x2 – 4x + 4 = 8 + 4 x2 – 4x + 4 = 12 This process created a perfect-square quadratic on the left-hand side. That is, if I factor the left-hand side, I get a square: (x – 2)2 = 12 Now I can square-root both sides and solve: (x – 2)2 = 12 |
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Students go over hand out and present their understandings of the material. |
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Evaluate:
-3x2 + 300 = 0 Solve each equation by factoring
Solve each equation by completing the square..
5x2 - 9x + 4 = 0 |
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Students work on the problems and present to the class. |