WEBVTT

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[Music] Welcome

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to Teaching Quadratic Functions.

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My name is Wendy Loeb,

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and I teach eighth grade math,

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which is Algebra II, Algebra I,

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and Pre-Algebra,

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at Twin Groves Middle School

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in Buffalo Grove, Illinois.

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I begin introducing functions

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in Pre-Algebra; however,

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these are very basic linear

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input/output problems.

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in Algebra I, we move

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onto linear functions

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and introduce absolute value functions

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and quadratic functions.

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in Algebra II, I review linear

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and absolute value functions.

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Then, I introduce piecewise functions

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and go into much more depth

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on quadratic functions,

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and then I introduce polynomial,

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radical, exponential, logarithmic,

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and rational functions.

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in Algebra II,

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I try to show students connections

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between both linear

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and nonlinear functions

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and how they apply

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to various situations.

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The main concepts involved

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through our functions are graphing,

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solving systems of equations,

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operations with polynomials,

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and making real-life connections.

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Since quadratic functions have various

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forms, we practice factoring techniques,

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completing the square,

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and the quadratic formula.

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When solving polynomial functions,

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I also teach dividing polynomials,

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solving polynomial equations

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by both factoring and graphing,

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and applications

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such as using Pascal's triangle

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when expanding a polynomial.

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So, I try and make a lot of connections

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between functions and other math skills

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and topics of Algebra.

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My goals for this lesson were

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for the students

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to graph quadratic functions

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in standard form,

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by hand and with a calculator,

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and to find the maximum

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and minimum values

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of quadratic functions.

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First, the students were expected

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to find the vertex

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of a parabola-of a quadratic function

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that's written in standard form.

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And what we do is we use the opposite

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of b/2a to find the x coordinate,

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and they substitute that in

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to find the y coordinate,

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and then they are going to make a table

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of values where they are going

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to find an ordered pair on either side

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of the vertex so that they could plot

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the graph.

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And we do this by hand,

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and then we do this

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on the graphing calculator.

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And we also like to show how

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to find the vertex

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on the graphing calculator,

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so that in case a number is a very,

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very complicated fraction,

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they can see how the calculator can

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check their answer for them.

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And after we find the vertex

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and we graph the parabola,

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we try and talk about characteristics

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of the parabola,

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what each part of the equation means.

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What does the a

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in the quadratic function represent?

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How can you tell whether it's going

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to open up or open down?

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How can you tell if it's going

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to be a wider parabola

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or a narrower parabola?

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We talk about when do we use the

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parabola, and what does the vertex mean

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in a real-life situation?

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The first thing

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that I do is introduce the topic

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of what, well, first of all,

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the parts of a standard form

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of a quadratic function

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so that the students knew what the

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different parts of the equation meant.

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The standard form would be

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f(x)=ax squared+bx+c, what the purpose

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of the a is, what the purpose

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of the b is, so that they can understand

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how to find the vertex

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by doing the opposite of b/2a.

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Then, what I do is I give them an

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example of a function in standard form,

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and we take a look at what the graph

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of that would look like.

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So, I would say, "Okay,

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here is the equation y=x squared-4x+3,

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let's find the vertex of this equation."

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And we will find the vertex

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by hand first.

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After we find the vertex by hand,

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we will start to make a table,

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and we'll find two other ordered pairs

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on either side of the vertex

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so that we can make a nice symmetrical

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graph of this parabola.

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And we will plot those points

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on graph paper and graph it.

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We do the problem together.

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Then, I say, "Okay,

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now here is an equation for you to try

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on your own."

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I show them how to do this

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on a graphing calculator.

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We put in the same equation,

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and I show them

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where to find the features

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on the calculator

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so that they could calculate the vertex.

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And then we go to the table

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on the calculators and say, "look,

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this is the same table we created

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by hand on our paper,"

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so they can see the connection

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between the two.

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After that, I am going to apply, "Well,

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why do we need to even graph a parabola?

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Why do we need to find a vertex?

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What's the application of this?"

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And this is when I try

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to give them a real-world connection.

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So, the one that I used

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in class was the one

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where a company was trying

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to sell some unicycles,

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and I gave an expression

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that models the number

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of unicycles it sells per month,

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and I said that p stands for the price

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and the r stands for the revenue,

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and I gave a range of prices

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that it could sell for.

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We come up with an equation

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so that we can figure out the vertex

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of this to come

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up with what the price would be

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to maximize our revenue

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and the maximum revenue.

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And then, we say, "Well,

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in order to find the revenue,

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we need to multiply the price per item

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by the number of items."

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And so, we discuss well,

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what's the price per item?

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We don't know;

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that's our p. What's the number

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of items?

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That's our expression that we have.

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And we multiply those together

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so that we can get a quadratic function.

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We find our vertex,

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and then I ask the class, "Well, okay,

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what does this number

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in the vertex represent?

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That's the price."

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So, the first number we get represents

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the price of one unicycle.

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How can we find

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out what the revenue would be?

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Well, we have to substitute that back

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into the equation.

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So, the students substitute that back

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and they find the revenue.

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So, what does this represent?

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The maximum revenue

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when you have a particular price.

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Then, I will give them a different

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problem and say, "Okay,

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now here is your new problem.

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See if you can find what price will

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maximize the revenue

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in a different situation."

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My students have the most trouble

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with abstract concepts.

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For example, even though they understand

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what an imaginary number is,

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they have a hard time understanding

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when they are going to need to use it.

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Piecewise and logarithmic functions are

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probably the most difficult functions

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for my students to grasp.

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Setting up various word problems are

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usually difficult

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for many students in general.

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One of the ways I help students

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with this is by modeling problems

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for them in class that are similar

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to the ones in their homework.

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I also try to give them multiple-step

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problems that break

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down the procedure step-by-step

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and ties together many

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of the function concepts they

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have learned.

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To learn more about teaching quadratic

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functions, please explore the additional

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resources on the Doing What

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Works website.

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[Music]