Title of Lesson: ____________________Quadratics (Day 1 and 2 combined)_______________________
Taking a function-based approach to introducing quadratics.
Big idea(s) or other content goals (What you plan for students to take away from this lesson)
We hope students understand from this lesson that a graph of a quadratic function is a parabola, and
this is obtained by multiplying two LINES together. Depending on what kind of lines we multiply, our
parabolas will have different orientations, etc. Where the parabolas cross the x-axis are the “roots” of
the parabolas. Connections between these roots and factoring will also be discussed.
TEKS correlations (In words, not just numbers)
Grade Level:
TEKS:
(A.9) Quadratic and other nonlinear functions. The student understands that the graphs of
quadratic functions are affected by the parameters of the function and can interpret and
describe the effects of changes in the parameters of quadratic functions.
The student is expected to:
(A) determine the domain and range for quadratic functions in given situations;
(B) investigate, describe, and predict the effects of changes in a on the graph of y = ax 2 + c;
(C) investigate, describe, and predict the effects of changes in c on the graph of y = ax 2 + c;
and
(D) analyze graphs of quadratic functions and draw conclusions.
(A.10) Quadratic and other nonlinear functions. The student understands there is more than
one way to solve a quadratic equation and solves them using appropriate methods.
The student is expected to:
(A) solve quadratic equations using concrete models, tables, graphs, and algebraic methods;
and
(B) make connections among the solutions (roots) of quadratic equations, the zeros of their
related functions, and the horizontal intercepts (x-intercepts) of the graph of the function.
(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, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology
(including, but not limited to, calculators with graphing capabilities, data collection devices,
and computers) to model mathematical situations to solve meaningful problems.
(2A.6) Quadratic and square root functions. The student understands that quadratic
functions can be represented in different ways and translates among their various
representations.
The student is expected to:
(B) relate representations of quadratic functions, such as algebraic, tabular, graphical, and
verbal descriptions; and
(C) determine a quadratic function from its roots or a graph.
(2A.7) Quadratic and square root functions. The student interprets and describes the
effects of changes in the parameters of quadratic functions in applied and mathematical
situations.
The student is expected to:
(A) use characteristics of the quadratic parent function to sketch the related graphs and
connect between the y = ax2 + bx + c and the y = a(x - h)2 + k symbolic representations
of quadratic functions; and
(B) use the parent function to investigate, describe, and predict the effects of changes in a, h,
and k on the graphs of y = a(x - h)2 + k form of a function in applied and purely mathematical
situations.
(2A.8) Quadratic and square root functions. The student formulates equations and
inequalities based on quadratic functions, uses a variety of methods to solve them, and
analyzes the solutions in terms of the situation.
The student is expected to:
(A) analyze situations involving quadratic functions and formulate quadratic equations or
inequalities to solve problems;
(B) analyze and interpret the solutions of quadratic equations using discriminants and solve
quadratic equations using the quadratic formula;
(C) compare and translate between algebraic and graphical solutions of quadratic equations;
and
(D) solve quadratic equations and inequalities using graphs, tables, and algebraic methods.
(2A.9) Quadratic and square root functions. The student formulates equations and
inequalities based on square root functions, uses a variety of methods to solve them, and
analyzes the solutions in terms of the situation.
The student is expected to:
(A) use the parent function to investigate, describe, and predict the effects of parameter
changes on the graphs of square root functions and describe limitations on the domains and
ranges;
(B) relate representations of square root functions, such as algebraic, tabular, graphical, and
verbal descriptions;
(C) determine the reasonable domain and range values of square root functions, as well as
interpret and determine the reasonableness of solutions to square root equations and
inequalities;
(D) determine solutions of square root equations using graphs, tables, and algebraic methods
Materials (Include everything you and students need. List safety precautions, if applicable.)
TI- Navigator System
Calculators
Projector
Safety precautions: Nothing seems inherently
dangerous… Be mindful not to trip over the various
cords that will be plugged in everywhere.
Handouts, overheads, Powerpoints, web pages (Attach or provide links)
Pre-test
Day 1 points and lines worksheet
Day 1 linear products/quadratics worksheet
Notecards with calculator login/password info
Day 2 worksheet
Day 2 homework
Several overhead transparencies
Day 2 “parabola graphs” transparency
“Parabola graphs” handout
LESSON (In segments. No set number. Aim for 2 pages.)
Time
Segment
title
Lotto
Discussion
5-10
minutes
What teacher is doing (Include specific questions)
We asked the students last time about the probability of
winning the Texas Lottery. We will go over this and answer any
followup questions.
What students are doing
Has anyone figured out the odds of winning the Texas Lottery?
Can you explain how you got that? (If not, we’ve got it, and
will go through it.)
Pre-Test
10
minutes
5-10
minutes
Going over
the pre-test
First, just to see where y’all are with equations of lines, which is
going to be big in the lesson today, we’re going to hand out this
pre-test. Don’t worry, it’s not a stressful graded thing, it’s a
way for us to get a feel for how much of this y’all already know.
We think this will be a breeze for you guys, but if not, we’ll hit
on whatever we need to before we go on.
PASS OUT PRE-TEST
We’ll have the pre-test up on the overhead, and ask students to
help us with the answers.
Completing the pre-test.
Raising hands and helping the teachers fill out the pre-test.
Ask why we got the answers we did.
2
minutes
Background
info
TI-Navigator
Setup
10
minutes
What are we used to working with when we add two things
together, or multiply? What exactly are we adding, multiplying,
dividing, etc? (“numbers” or “variables” is what we’re looking
for.)
Right, so far in math we’re used to performing operations on
numbers and variables. And what happens when we multiply
two positive numbers together. What do we get? (a positive
answer.) How about when we multiply two negative numbers
together? (a positive number.) And when we multiply a
positive number times a negative number? (a negative
number.) Keep that in the backs of your minds. You may
notice something interesting later in the lesson having to do
with this concept.
“Today we’re going to use calculators to explore our lesson.
One of the cool things about this technology is that y’all get to
create your own points, lines, equations, etc. and then we can
see everybody’s work combined.
Logging onto their calculators.
First we need to get signed in to the calculators.
PASS OUT CALCULATORS AND LOGIN INFO, then take everyone
through the following steps. Make sure that “Begin Lesson” is
clicked on the computer, or else they won’t be able to sign in!
Step 1: Turn on calculators.
Step 2: Hit the APPS button
Step 3: Scroll down to NavNet, and hit enter
Step 4: Enter your login name and password (make sure you’re
in alpha lock) These logins are found on the your notecards.
Step 5: follow directions on calculators… (press any button to
continue, wait for teacher, etc.)
10
minutes
Drawing
points on a
graph
5
minutes
Introducing
worksheet
10
minutes
Shooting
lines through
the first
point
10
minutes
15
minutes
Shooting
lines through
the second
point
NavNet lines
Allright! We’re gonna start basic, and build up what we know
on here. First off, we’d like you to place a point or points on
your graphs (with the “MARK” button,) where your y-value is
the same as your x-value”.
HARVEST! (i.e. if there are no negative values, ask students
why, etc…) Discuss for a bit, and then go on to the next one.
“This time we want everyone to go to a spot in which your yvalue is half of your x-value.” HARVEST!
“What do you notice? How did you decide where to go?”
“Finally, let’s go to where your y-value is TWICE your x-value”.
HARVEST, if necessary.
We will put the worksheet up on the overhead, and go through
it, describe function machines, making sure everyone
understands the task.
CLICK ON STOP ACTIVITY, to get them out of the network!!!
“Now we’d like you to come up with equations of different lines
that all go through the point (-2,0). Use your graphing
calculators to check your equations, and make sure they all go
through (-2,0) before you write them down. Be creative!”
Plotting points, commenting on the class graph…
This time we want you to come up with five more lines that go
through the point (4,0). Make sure you test ‘em on your graphs
before writing them down!
Putting lines in their calculators.
“Now pick your two favorite equations for each point. Pick the
fun ones, interesting ones, whatever strikes you. For y1 and
y2, we want equations that go through (-2,0). In y3 and y4,
put in equations that go through (4,0).”
Show the lines up on the network. Discuss holes, trends, why
we got those, and ask students what they notice. If necessary,
Commenting on what they notice with the lines, discuss gaps in
the stars, etc.
Going through the worksheet, asking questions if they have
any.
Students are experimenting with lines on their own computers,
not using the network.
5
minutes
10
minutes
Grabbing
others’ lines
Introduce
graphing
parabolas
2
minutes
Notes on
how to enter
the
equations
20
minutes
Exploring
different
graphs
NavNet with
parabolas
20
minutes
ask students to resubmit one of their lines to fill in the gaps, so
we get a better star at each point. HARVEST!
“now choose three lines that go through each point that you did
NOT come up with, but you think are interesting.”
“Here’s where it gets really cool. Now we are going to try
multiplying pairs of these lines to get graphs like the following.
Before we do it, though, does anyone have any guesses on how
how we’d get, for example, this parabola?” (point to one…)
Discuss. Demonstrate how to multiply lines together on the
calculator, and make sure everyone understands how to do it.
Make sure that you’re paying attention to what goes in what
equation in your calculators. For each multiplication that you
do, keep in mind that we want one line in Y1, the second line
you’re using in Y2, the product of those lines in Y3, and if you
feel comfortable simplifying that equation, that in Y4.
Wandering around, checking for understanding.
Highlight different students’ parabolas, and discuss why their
parabola is shaped the way it is. HARVEST!
Writing down other people’s lines that they like.
Multiplying two equations together to get parabolas.
Graphing products of lines, and noting which time they get each
type of parabola. They are writing their findings down on their
worksheets.
DAY 2
5
minutes
Intro.
10
minutes
Algebra Tiles
(Julie)
15
minutes
Going over
F.O.I.L.
(John)
Revisiting
Parabolas on
NavNet (all
of us)
25
minutes
20
minutes
Quadratic
Formula
(Ashley)
Introduce ourselves again, let them know what we’ll be
covering today.
1. How to simplify two expressions multiplied together (FOIL)
2. Back to the parabola investigations on the NavNet
3. How to find the roots of quadratic equations (quadratic
formula)
Going over the algebra tiles in Virtual Manipulatives, which
gives students a visual representation of what is going on when
you multiply together two expressions. Make sure that another
student (or one of us,) is up at the board writing out what the
expressions are that we’re multiplying, and what the resulting
simplifications are.
Explaining what the FOIL method is, how it works, and giving
students practice problems. Once the explanation section is
over, we’ll hand out a worksheet for more practice.
Going back to the revised worksheets, we will ask students to
multiply together two lines at a time and note which kind of
parabola they get. Things that we want the students to take
away from this are:
1. When you multiply two lines that both go through (-2,0), we
get a parabola that only touches the x-axis at that point. Same
for (4,0).
2. When you multiply two lines that go through separate points
on the x-axis, you get a parabola that intersects the x-axis at
both of those points. WHAT DOES THIS MEAN? Harvest…
3. When you multiply two lines that, at that point, are both
positive or both negative, you get a positive value in your
parabola. When you’re multiplying parts of lines that are one
positive, one negative, you get parabola sections below the xaxis.
4. When the two lines you multiply together have the same
sign for their slopes, you get a happy parabola. When you
multiply two lines with opposite slopes, you get a sad parabola.
What does this mean? Harvest!
Go over the four ways to find the “roots” of a parabola, and why
we want the roots. (Factor, Graph, Complete the Square, and
Quadratic Formula.) Go over the general form of a quadratic
equation. y= ax2+bx+c
Teach them the quadratic formula, and how to spot an a, b, or
c. Make sure to cover situations in which there is no b or c.
Teach them the quadratic formula song. Ask what they think it
Listening, asking questions from last time
Taking notes, or just observing, helping us fill out the
equations, etc.
Taking notes, then working on worksheets. Answering our
probing questions, too.
Working on their calculators to find parabolas like each of the 6
kinds on their worksheets, and then discussing what they
notice.
Taking notes
Extra
Completing
the Square
5
minutes
Cleanup and
pass out
homework
means when they don’t get any answers that make sense with
the quadratic formula. (no real roots)
Have them complete the back of the worksheet, using the
quadratic formula to find roots.
If we run out of time, we can go through completing the square,
and put problems to work on on the overhead or board.
Getting together the cords, etc, and passing out the homework.
Things to Think About and Do While You’re Planning
*How will you introduce the lesson and task(s)? Engage students? Find out about prior knowledge?
How will you communicate your expectations?
How will you set up/pose the task(s)?
How will you transition from one lesson segment to the next?
How will you engage all students? For example, consider students who:
-struggle with task
-race through task
-express an idea that is hard to understand
-express an idea that is different from direction of lesson
-are learning English
-are off task
*What questions will you ask to find out how students are thinking about the task(s)? What they are understanding and learning?
What questions will you ask to extend students’ thinking?
What kinds of artifacts will you require? How will those artifacts be used in the lesson?
How will you facilitate a discussion of students’ thinking and the big ideas?
*What are the big ideas and how/when will they be expressed?
*Engage (solve the problem, conduct the experiment, explore, etc.) in the task yourself and predict what students might do. Play
around with it. See if you can find alternative, viable pathways through the task.
*Be sure to include Direct Teach segments for Teach #1.
Yes, there’s most definitely a lot to think about. Focus on what you think is most important. Starred items are the priorities for
now. You will learn to consider all of these things with time and you won’t need to write most of it down.