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Rachel Kluber – 3 Day Unit
CAPSTONE PROJECT
Curriculum and Instruction 403
The following project is a 3 day unit-plan that I have put together involving the
Quadratic Formula and using that formula to find the real or imaginary x-intercepts of a
parabola. This unit would come after students have been introduced to quadratic
functions and addresses a few mathematical standards that you will see throughout the
lessons. I chose to use these lessons because I did actually teach the last two to an
Algebra II class at Centennial High School in Champaign. These formal lesson plans
represent how I would have changed the lessons I taught to better anticipate student
responses and how I could better explain the Quadratic Formula after teaching it once. I
use a few different instructional strategies including filling in structured notes sheets
alongside the students and walking around the room while students are doing group work.
The first strategy of providing a notes sheet and filling it out with the students sets the
standard of how students should be taking notes in my class. This also helps them to stay
organized and on task. Second, allowing students to work in groups as I work with them
more personally and individually allows me to assess my students’ progress.
My unit plan is structured around the fact that I will be providing my students
blank note sheets that I will project on the smart board and fill in along with them. The
warm-up problems will be completed on an overhead so that it can be projected
throughout the entire lesson alongside the smart board. An ELMO will also be used to
project things on a graphing calculator. My lessons will denote the times that I use any of
these pieces of technology. These strategies will benefit the students for the same reasons
that I chose to structure my lessons in this fashion- it will keep the students organized as I
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Rachel Kluber – 3 Day Unit
project and complete the notes on the smart board. Leaving the warm-up problem on the
projector allows us to refer back to it easily throughout the lesson. Using the ELMO too
to project things on the graphing calculator will allow students to continue follow along
with the lesson.
I want my students to not only gain more experience being an organized student
in my class, but also start to develop a sense of how to derive formulas. I feel that
deriving the Quadratic Formula requires a little bit of a higher-level way of thinking
about algebra and mathematically, Algebra II should start to point students in this
direction. My personal goals as an educator to high school students include teaching
students how to think about math and approach solving problems. Students are given the
tool of the Quadratic Formula and asked to find x-intercepts of a parabola. At this point,
they have seen this done in a few different ways (factoring, graphing, completing the
square), and it is up to them after this unit to analyze a problem and decide which method
would work best as they get more into modeling real-life situations with quadratic
functions.
The point of having the notes sheet also extends past just keeping students on task
and organized- since there are a few ELL students and a couple students with IEPs- the
notes sheet allows for students to represent information in a way that they will understand
and remember. Since the notes sheets are blank and not very wordy, the ELL students
will be allowed to fill in the notes sheet in a way that they find most helpful, as well as
the IEP students. They will always be allowed to use a calculator during these three
lessons and at different points in the lesson; I show students how to interpret information
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Rachel Kluber – 3 Day Unit
on the graphing calculator. I will be expecting the Spanish aid to be helping my ELL
students fill out notes during these lessons as well.
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Rachel Kluber – 3 Day Unit
Class: Algebra II
Grade: 11th
Number of Students: 26 (4 ELL, 2 IEPs)
Time Allotted: 50 minutes per class period.
LESSON 1: Introduction to the Quadratic Formula
1) GOAL OF THE LESSON: Derive the quadratic formula, and then use it to find the
x-intercepts of a given quadratic equation.
2) OBJECTIVES:
 Students will review the completing the square method- to change quadratic
equations from standard form to vertex form.
 Students will learn the definition of x-intercepts of a quadratic function
 Students will use the completing the square method on the standard form of a
quadratic equation to derive the quadratic formula.
 Students will learn how to use the Quadratic Formula to find the x-intercepts of a
quadratic equation and apply this to practice problems.
3) MATERIALS:
a) Quadratic Formula Notes Sheet (Attached)
b) Writing utensil
c) Algebra II Textbook (for reference)
4) MOTIVATION (3 minutes): The last few lessons, we have been introduced to and
have worked with quadratic equations. We have seen the general equation in the
form of y  ax 2  bx  c . Using this, we have learned how to graph the quadratic
function, find the vertex, the axis of symmetry, the y-intercept, and the symmetric
point. They then learned completing the square to put the quadratic function in
2
vertex form or y  a(x  h)  k . Now we must find the x-intercepts of the parabola
using the quadratic formula, which many of you might have seen before. Let’s first
explore exactly where the quadratic formula comes from and how we use it to find
the x-intercepts
of a parabola.

5) PROCEDURE:
a) Warm Up (7 minutes): “Good morning everyone! We are going to start today
with a warm up, as usual. While XXXX passes out these note sheets, I am going
to write the following things that we have learned about quadratic equations up on
the over-head and I want you to find these for the first equation on your notes
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Rachel Kluber – 3 Day Unit
sheet in the Warm-Up. Then, I want you to refresh yourself on completing the
square in the second problem to find the vertex of that quadratic equation.”
Here, we will review the prior knowledge that students should have regarding all of the
bolded words from the motivation:
 Graphing a given quadratic equation
 Finding the vertex of the given parabola
 Naming the line of symmetry of the given parabola
 Finding the y-intercept of the given parabola and the symmetric point
 Method of completing the square to put a quadratic equation in standard form
b) The solved Warm-Up Problem is on the attached Teacher’s Version of the Notes
Sheet
c) The students will complete the warm-up problem while I walk around and answer
questions and look at their work. Then, as a class we will go over the two
problems.
d) Lesson (20-25 minutes):
i) The students will have blank note sheets that they will be filling out as the
lesson progresses. The note sheet will be projected up on the smart board and
I will be filling it out during the lesson. My specific examples and flow of the
lesson will be better understood if this lesson is read alongside the
TEACHERS version of the notes sheet attached to this.
ii) X-INTERCEPTS:
(1) “Now that we have seen all the other parts of a quadratic function, let’s
discuss the parts of the parabola called the x-intercepts”. Ask students
where they have seen intercepts before (Students should mention that yintercepts of a function is where the function crosses the y-axis and x =
0). Flip it, and x-intercepts are where the function crosses the x-axis but
y=0.
(2) What does this look like graphically? Draw the parabola from Warm up
Question 1 on the notes sheet and circle where the parabola crosses the
x-axis.
(3) So if we want to find the values for x, where the parabola crosses the xaxis we need to set y=0, but how are we going to solve for x? Have
students mull this over for a minute and then introduce- “This is where
the Quadratic Formula comes in handy, we use it to solve for x when
y=0”
iii) DERIVING THE QUADRATIC FORMULA:
(1) Where exactly does the Quadratic Formula come from? In the warm-up
and in the past few days we have been working with the method of
“completing the square” to alter the general form of quadratic functions
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Rachel Kluber – 3 Day Unit
and solve for x that way. We can derive the quadratic formula by
completing the square on the equation in the form y  ax 2  bx  c :
SEE TEACHER/STUDENT NOTES SHEET
iv) USING THE QUADRATIC FORMULA:
(1) “Now that we have derived our quadraticformula, we use this formula
to find the x-intercepts by plugging in the a, b, and c values from a given
quadratic equation.” Do this with the warm-up problem:
SEE TEACHER/STUDENT NOTES SHEET.
(2) Do the second example on the notes sheet as a class. Students are able to
use their calculators (and should be) to simplify some of the parts of the
Quadratic Formula.
(3) Point out that if there is a negative number under the radical sign in the
Quadratic Formula, students are to assume “no real x-intercepts” since
they have no method yet to take the square root of a negative number!
v) LET’S PRACTICE:
(1) Students have four more quadratic equations on their notes sheet to
practice using the quadratic formula. At this time, I will be walking
around and checking their work, answering their questions, and
encouraging them to work together with peers sitting around them.
(2) If students are obviously struggling with using the formula, I will direct
the attention back to the smart board and continue to go through the
examples as a class, letting students talk through the process of plugging
in constants for a, b, and c and then simplifying.
(3) Students may need to refresh themselves on how to simplify radical
expressions- they saw this in Geometry and it may have been a while for
some. I anticipate that this may be a rough spot.
6) CLOSURE (5 minutes) “Alright everyone I know that you are practicing this but
can I have everyone’s attention up at the front before you leave class today. This
lesson was jam-packed full of information about the Quadratic Formula but you
should have a few things highlighted on your notes sheet.
1. You cannot use the Quadratic Formula unless y = 0,
2. You must memorize the quadratic formula, and
3. It is simply a matter of plugging in your constants a, b, and c and then simplifying
it down to get your x-intercepts.
I am going to be passing out your homework now and it is to be completed for
tomorrow. These worksheet problems are similar to the ones that we just practiced
with. Have a great rest of your day!”
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Rachel Kluber – 3 Day Unit
7) EXTENSION I really do not anticipate the need for an extension since I know that
this lesson is going to be rather long. However, if there is time at the end of the lesson
after the students have completed the last practice problems, instead of having them
work on their homework- I want them to discuss ways that they can remember the
quadratic formula. What tools or methods can they use to remember the formula?
Here is one method that I have heard:
Once upon, there were two twin boys. We will call them b 2
The two twin boys lived in a mansion,
b2
 no longer felt that the
The twin boys had a babysitter, but they were getting older and
babysitter should live in the mansion with them, so they sent her outside. b  b2

The two twin boys, who were getting older, decided that they wanted to go to a party.
There, they met 4 awesome chicks. The 4 awesome chicks wanted to come hang out at

the mansion, but they brought 2 alligators. The boys made the alligators
stay in the
basement.
b  b 2
2a
After a while, the 4 awesome chicks decided that they did not like the boys and they
were sad, but the mansion was so big that they decided to live there anyway.

b  b 2  4ac
2a
The End.

8) ASSESSMENT Students will be assigned a worksheet titled Intercepts of Quadratics
for homework. This worksheet is to be completed the night that it is assigned. The
worksheet is attached to this lesson, as well as an answer key. This assessment will
really tell if students understand how to use the quadratic formula AND if they are
able to simplify their answers correctly.
9) MATHEMATICAL STANDARDS ATTAINED
 A-SSE.3. Choose and produce an equivalent form of an expression to reveal
and explain properties of the quantity represented by the expression.
Students have to make sure that their quadratic equations equal 0 before
proceeding with the quadratic formula to find the x-intercepts. This is an
equivalent form of the expression. Also, simplifying the expression that the
quadratic formula yields for a specific parabola is interpreting the data.
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Rachel Kluber – 3 Day Unit

A-F-IF.8 Write a function defined by an expression in different but
equivalent forms to reveal and explain different properties of the function.
This has to do with the method of completing the square. Students change the
quadratic equation from general form to vertex form using completing the square,
which reveals the maximum/minimum (vertex) of the parabola.

A-REI.4a Use the method of completing the square to transform any
quadratic equation in x into an equation of the form (x – p)2 = q that has the
same solutions. Derive the quadratic formula from this form.
We use the completing the square method in the beginning of the lesson to derive
the quadratic formula. I find this interesting because I did not learn this in high
school, but deriving the quadratic formula is in fact, a high school standard!
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Rachel Kluber – 3 Day Unit
TEACHER’S NOTE SHEET
Quadratic Formula and x-intercepts of Quadratic Equations
Objectives:
 Define x-intercepts of a quadratic equation.
 Derive the Quadratic Formula
 Use the Quadratic Formula to find the x-intercepts of a given quadratic equation.
Warm-up:

1) Find the vertex, axis of symmetry, y-intercept, and symmetric point for the
following quadratic function. Then, graph it.
y  x 2  6x  2
b
Vertex: Students know and should use
to find the x coordinate of the vertex.
2a
(6) 6
  3, Plug 3 in for x to get y = y  32  6(3)  2  7 . Vertex at (3, -7)
2(1)
2
Axis of Symmetry: Vertical line that goes through the vertex, x = 3

y-intercept: When x=0, y  0 2  6(0)  2  2 , y-intercept at

(0, 2)
 Symmetric point: to the y-intercept is (6, 2)





2) Find the vertex of the following quadratic equation by completing the square.
y  3x 2  24 x 11
y 11  3x 2  24 x
1. Subtract 11 to make room to complete the square
2
y 11  3(x  8x)
2. Factor out -3 so the x 2 coefficient is 1
y 11 (3)(16)  3(x 2  8x 16)
3. Add 16 on the right to complete the square, and
add (-3)(16) on the left to balance the equation.
2
y  3(x  4)  59
4. Factor andthen simplify into vertex form.
The vertex is at (-4, 59)

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Rachel Kluber – 3 Day Unit
NOTES:
What are x-intercepts?
Graphically:
Anticipation: Students should connect this to yintercepts in the quadratic equation, where they set
x=0.
Definition: The value of x when y is set = 0.
Also known as roots to a quadratic or solutions to
the quadratic when y is made 0.
“So, this is where our parabola crosses the x-axis,
let’s take a look at our graph from our first warm-up
problem and circle where the x-intercepts are
located.”
TRANS: “If we set y = 0 and try to solve for x, this proves very difficult. (Let students
try this with the warm-up problem for a minute or two). This is where the Quadratic
Formula (which you may have heard or seen before) comes in handy. We use the
Quadratic Formula to find the x-intercepts of a parabola, but first let’s take a look at
where the Quadratic Formula comes from.”
Deriving the Quadratic Formula: Where does this formula come from?!
Let’s take the general form of our Quadratic Equation, y  ax 2  bx  c where a  0
We want to solve for x when y = 0 to find our x-intercepts. So use 0  ax 2  bx  c and
complete the square to solve for x. Let’s go slowly, step-by-step!


0  ax 2  bx  c
START

c  ax 2  bx
Subtract c from both sides
c a
b
3.   x 2  x
Divide all terms by a to make the x 2 coefficient 1
a a
a
2
 b 2
c  b 
b
4.      x 2  x   
Divide the middle term by 2 and square it to
2a 
a 2a 
a

complete the square on the right hand side, add it to the left to balance.
2
2
c  b  
b 
5.      x  
Factor the right hand side to be a squared expression
a 2a   2a 
1.
2.








2
c  b 
b
6.       x 
a 2a 
2a
(4a)c b 2
b
 2 x
(4a)a 4a
2a
b 2  4ac 
b
8.  
  x 
2
2a
 4a 
7.  
Take the square root of both sides
Make it so we can add the fractions under the radical
Combine the fraction under the radical
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Rachel Kluber – 3 Day Unit
9. 





b 2  4ac
x
b
2a
Simplify
4a 2
b 2  4ac
b
10. 
Simplify
x
2a
2a
b
b 2  4ac
11.  
Get x alone
x
2a
2a
b  b 2  4ac
12.
QUADRATIC FORMULA!
x
2a
TRANS: “Now that we have derived our quadratic formula, we use this formula to find
the x-intercepts by plugging in the a, b, and c values from a given quadratic equation.
Let’s use our warm-up problem to see how this works”
Using the Quadratic Formula
Example
Let’s try this with our problem from
the warm-up.
For a quadratic equation in the general form
of y  ax 2  bx  c where a, b, and c are
y  x 2  6x  2
constants and a  0, we find the x-intercepts
0  x 2  6x  2
Set y=0,
of the parabola by setting y=0 and using the
Quadratic Formula to solve for x.
a 1
b
 6


2
c
b  b  4ac
2
x
2a

(6)  (6) 2  (4)(1)(2)

x
Ask: Why can a not be equal to 0 both in
2(1)
 the
Quadratic Formula and in the Quadratic

Equation?
6  28
62 7
x
x
2
2

x  3 7
(Relate this back to graph in notes)


LET’S PRACTICE TOGETHER
Using the quadratic formula, find the xUsing the quadratic formula, find the x
intercepts of x 2 10x  21  y
intercepts
of y  3x 2  7x  2
Set y=0,
a=1



x 2 10x  21  0
b = -10
c = 21
Set y=0,
a=3

0  3x 2  7x  2
b = -7
c=2
x
(10)
 (10)2  (4)(1)(21)

2(1)
x
2
(7)
  (7)  (4)(3)(2)
2(3)
x
10  16
2
x
7  25
6

x  5  2 , x=3, 7


=
x
75
6
2
11


=x=
1
,
3
Rachel Kluber – 3 Day Unit
PRACTICE ON YOUR OWN
Using the quadratic formula, find the xUsing the quadratic formula, find the x2
intercepts of y  2x  4 x  3
intercepts of y  4 x 2  4 x 1
Set y=0,
a = -2
0  2x 2  4 x  3
b=4
c = -3

Set y=0,
a = -4
2
(4)
  (4)  (4)(2)(3)
x
2(2)


0  4 x 2  4 x 1
b=4
c = -1

2
(4)
  (4)  (4)(4)(1)
x
2(4)
4  8
(UH OH- there’s a negative
4

sign under the radical, what does this
mean?)
x
x
4  16 16
8
 in
Anticipation: Students will begin work
class on this problem and realize that they
cannot have a negative number under the
radical- I will pause here to talk with my
 will
students about what this means and we
refer to this solution as “No real xintercepts”

x
4  0
8
x
4 1

8 2
There is only one x-intercept at x = 0.5
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Rachel Kluber – 3 Day Unit
STUDENT’S NOTE SHEET
Quadratic Formula and x-intercepts of Quadratic Equations
Objectives:
 Define x-intercepts of a quadratic equation.
 Derive the Quadratic Formula
 Use the Quadratic Formula to find the x-intercepts of a given quadratic equation.
Warm-up:
1) Find the vertex, axis of symmetry, y-intercept, and symmetric point for the
following quadratic function. Then, graph it.
y  x 2  6x  2

2) Find the vertex of the following quadratic equation by completing the square.
y  3x 2  24 x 11

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Rachel Kluber – 3 Day Unit
NOTES:
What are x-intercepts?
Graphically:
Deriving the Quadratic Formula: Where does this formula come from?!
Let’s take the general form of our Quadratic Equation, y  ax 2  bx  c where a  0
We want to solve for x when y = 0 to find our x-intercepts. So use 0  ax 2  bx  c and
complete the square to solve for x. Let’s go slowly, step-by-step!
1.




START
2.
Subtract c from both sides
3.
Divide all terms by a to make the x 2 coefficient 1
4.
Divide the middle term by 2 and square it to
complete the square on the
 right hand side, add it
to the left to balance.
5.
Factor the right hand side to be a squared expression
6.
Take the square root of both sides
7.
Make it so we can add the fractions under the radical
8.
Combine the fraction under the radical
9.
Simplify
10.
Simplify
11.
Get x alone
12.

0  ax 2  bx  c
b  b 2  4ac
x
2a
QUADRATIC FORMULA!
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Rachel Kluber – 3 Day Unit
Using the Quadratic Formula
Example
Let’s try this with our problem from
the warm-up.
y  x 2  6x  2

LET’S PRACTICE TOGETHER
Using the quadratic formula, find the xUsing the quadratic formula, find the x2
intercepts of x 10x  21  y
intercepts of y  3x 2  7x  2


PRACTICE ON YOUR OWN
Using the quadratic formula, find the xUsing the quadratic formula, find the x2
intercepts of y  2x  4 x  3
intercepts of y  4 x 2  4 x 1


15
Rachel Kluber – 3 Day Unit
HOMEWORK / ASSESSMENT
Name _______________________________
X-Intercepts of Quadratics
Date ___________ Period ______
Determine the x-intercept(s) of the given quadratic equation. Use the Quadratic
Formula and show work. Calculators permitted!
1) y  2x 2  6x  2
2) y  2x 2 10x 14


3) y  2x 2 14 x 

41
2
4) y  2x 2  6x 

5) y 
1 2
x  3x  7
3

1
7
6) y   x 2  x  2
2
2

7) y  x 2  9x  22

8) y  x 2  5x  3

9) y 

5
2
1 2 5
x  x 1
3
3
1
1
10) y   x 2  x  2
2
2

16
Rachel Kluber – 3 Day Unit
Name ______Answer Key________
X-Intercepts of Quadratics
Date ___________ Period ______
Determine the x-intercept(s) of the given quadratic equation. Use the Quadratic
Formula and show work. Calculators permitted!
1) y  2x 2  6x  2
x


x

No Real X-Int.
1  5

2
2
3) y  2x 2 14 x 

2) y  2x 2 10x 14

41
2
4) y  2x 2  6x 
7
 2
2
x

5) y 
1 2
x  3x  7
3

x

7) y  x 2  9x  22
No Real X-Int.
x

1
5
9) y  x 2  x 1
3
3



5  13
x
2
7  33
2
8) y  x 2  5x  3


5
1
and
2
2
1
7
6) y   x 2  x  2
2
2
No Real X-Int.

5
2
5  13
2
1
1
10) y   x 2  x  2
2
2
x


17
1  17
2
Rachel Kluber – 3 Day Unit
Class: Algebra II
Grade: 11th
Number of Students: 26 (4 ELL, 2 IEPs)
Time Allotted: 50 minutes per class period.
LESSON 2: X-Intercept Types and the Discriminant
1) GOAL OF THE LESSON: Study the different types of x-intercepts graphically, as
well as algebraically using the Discriminant.
2) OBJECTIVES:
 Students will review the quadratic formula during the warm-up
 Students will determine what type of solutions a quadratic equation will have
based on the value of the discriminant.
 Students will determine how many solutions a quadratic equation will have based
on the value of the discriminant.
3) MATERIALS:
a) X-Intercepts & Discriminant Notes Sheet (Attached)
b) Writing utensil
c) Algebra II Textbook (for reference and homework)
4) MOTIVATION (3 minutes): “Hello students! I hope that you have had plenty of
practice finding the x-intercepts of a quadratic equation! Today, we are going to keep
expanding on the idea of x-intercepts. We are going to look more closely at the types
of x-intercepts that a parabola can have, how many x-intercepts it can have, and also
introduce a short-cut to finding out how many x-intercepts (or roots) a given parabola
will have.”
5) PROCEDURE
a) Warm Up: (7 minutes) “Alright so as always, we are going to begin with a
warm-up while we get our notes sheet passed out for the day. The Quadratic
Formula is really important to memorize so for the next 3 minutes, I want you to
all discuss at your tables if you came up with any good ways to memorize the
quadratic formula while you were doing your homework last night. Then, as a
group, work on the warm-up problem on your notes sheet.”
b) I will be walking around the room at this point listening to student conversation.
If I hear of any really useful tips, I will ask that student to share their method of
memorizing the formula with the rest of the class.
c) The solved warm-up problem is on the Teacher’s Notes Sheet at the end of this
listen, however I will ask a volunteer to come up to the Smart Board and complete
the warm-up problem. At this point, I will ask another student to rephrase the
18
Rachel Kluber – 3 Day Unit
solution on the board in their own words. I will also reiterate here that y must be
set to 0 before the Quadratic Formula can be used.
d) LESSON (20-25 Minutes)
i) The first part of our lesson will be a class conversation before we start the
notes sheet. I will graph the warm-up equation on a graphing calculator and
project in on the ELMO. I will ask students to then connect the values they
found for the x-intercepts in the warm-up to the graph. They should pick up
that the x values they found using the Quadratic Formula is the x-coordinates
in the graph where the graph crosses the x-axis. This should be just review
from the last lesson.
ii) TYPES OF SOLUTIONS (GRAPHICALLY):
(1) I will then draw the graph on the notes sheet and point out the values
where the parabola crosses the x-axis: SEE ATTACHED NOTES
SHEET
(2) In our example we will have 2 real x-intercepts. The other two
possibilities are one x-intercept or no real x-intercepts, which will be
demonstrated on the notes.
iii) TYPES OF SOLUTIONS (ALGEBRAICALLY):
(1) After this is demonstrated graphically, we will talk about what it means
algebraically and what kind of answers we could expect when we use
the Quadratic Formula
iv) DISCRIMINANT:
(1) Introduce definition of Discriminant – part of quadratic formula
under the radical, determines what type of roots we will have. Also
talk about it’s purpose, the relationship that it has to the quadratic
formula, and how you would use it to find the types of roots.
v) EXAMPLES:
(1) We will do the two examples as a class and then students will be
working in their groups on the 3rd and 4th example. I will be walking
around helping and listening during this time to check for student
understanding. Before the end of the lesson, we will do these two
examples as a class to make sure that students know how to use the
discriminant and do not confuse this with finding the x-intercepts.
6) CLOSURE (5 minutes) “Alright everyone can I get your attention back on the board
before we close out today? So yesterday we solved the x-intercepts of parabolas by
using the Quadratic Formula and today we learned the different types of x-intercepts
we could have. We also learned how to use a part of the Quadratic Formula to figure
out what type/how many x-intercepts a parabola will have. We are going to continue
to practice this in our homework tonight.”
7) EXTENSION If there is extra time at the end of the lesson, give the students the
following problem:
Use the Discriminant to determine what type of x-intercepts the following parabola
will have: 2x 2  2x  2  x  x 2

19
Rachel Kluber – 3 Day Unit
Students will need to recognize that one of the sides of the equation needs to equal 0
before they can use the discriminant (since it is part of the Quadratic Formula). This
would be a little more challenging in that it has that extra step of getting the equation to
equal 0.
8) ASSESSMENT: The warm-up problem will really assess if the students are getting
the Quadratic Formula. Especially since a student will be coming to the board to do it
and then another student will be revoicing that work. During the lesson, I will be
anticipating different things (SEE ATTACHED NOTES SHEET), which can be
addressed as they come up through formative assessment. I will be checking for
understanding through questions and walking around the classroom a few different
times in the lesson. Students will be assigned book-work for homework to be
completed the night it is assigned.
9) MATHEMATICAL STANDARDS ATTAINED
 A.SSE.1 Interpret expressions that represent a quantity in terms of its
context.
The Discriminant is an expression that represents a quantity that determines the
type and number of x-intercepts of a quadratic equation. Therefore, this standard
is met when students make the connection between the discriminant and the
quadratic formula.
20
Rachel Kluber – 3 Day Unit
TEACHER’S NOTE SHEET
X-intercept Types and the Discriminant
Objectives:
 Review the Quadratic Formula
 Look at 3 different types of x-intercepts of a parabola (graphically and
algebraically)
 Use the discriminant to determine what type of x-intercepts a parabola will have
Warm-up:
Find the x-intercepts of the following.
1)
a=2 b=3
2) 5  3x 2  2x  5
c = -5

3  (3)2  (4)(2)(5)
x
2(2)

x
3  49
=
4
x
3  7
4
x = -5/2 and 1


NOTES:
Types of X-Intercepts: Review: find the value of x
when y is set = 0. Known as roots or x-intercepts



Graphically:
(From Warm-Up #1)
2 real solutions: Look at first warm-up problem
graph on graphing calculator and let the students
discuss that there are two real roots.
1 real solution: the parabola only crosses the xaxis at one point (vertex)
No real solutions: The parabola never crosses
the x-axis.
x = -5/2 and 1, 2 real solutions
21
Rachel Kluber – 3 Day Unit
Types of solutions (and the number of that type)
Graphically:
Graphically:
Two real solutions
One real solution
(multiplicity 2)
Types of solutions (and the number of that type)
Algebraically:
Algebraically:
Two real solutions
One real solution
-When solving, provides (multiplicity 2)
two different real
-When solving, provides
number answers. Could
one real number answer (or
be rational or irrational
two of the same value).
roots.
Answer is always rational.
Discriminant:
Graphically:
No real solutions (rather two
complex solutions)
Algebraically:
No real solution (rather two
complex solutions)
-When solving, gets imaginary
numbers (square rooting a
negative number)
**Only used on a quadratic set = to zero!
Purpose: Helps to determine the type of solutions even more detailed than graphically to
a quadratic equation without actually doing the solving.
Relationship to the quadratic equation…. It is the piece in the square root.
Anticipation: When students first start using the discriminant, they may include the
square root since it is the “part of the quadratic formula under the square root”, clarify
that the radical is not included!
If the discriminant is zero….
If the discriminant is
positive…
Then quadratic has two
real solutons.
Moreover… if the
discriminant is a perfect
square, then the solutions
are rational. Otherwise the

If the discriminant is
negative…
Then the quadratic has one real
solution.
Then the quadratic has no
real solutions (rather
Note: Vertex because if the
complex solutions that we
discriminant is 0, we are left
will talk about tomorrow)
b
with x 
in the Q.F.
2a
22
Rachel Kluber – 3 Day Unit
solutions are irrational.
Examples: Without solving determine the type (and number) of solutions to each
quadratic equation. Justify your answer
1)
2)
3)
Discriminant is
positive and not a perfect
square,
Therefore, there are two
real irrational solutions
to the quadratic equation
Discriminant is negative,
Therefore, there are no real
solutions (rather two
complex solutions) to the
quadratic equation
Discriminant is positive and
a perfect square,
Therefore, there are two real
rational solutions to the
quadratic equation.
Example: How do you find the x-intercept(s) of the following?
4)
Make y be 0. Then Solve either by quadratic formula or completing the square.
23
Rachel Kluber – 3 Day Unit
STUDENT NOTE SHEET
X-intercept Types and the Discriminant
Objectives:
 Review the Quadratic Formula
 Look at 3 different types of x-intercepts of a parabola (graphically and
algebraically)
 Use the discriminant to determine what type of x-intercepts a parabola will have
Warm-up:
Find the x-intercepts of the following.
1)
2) 5  3x 2  2x  5

NOTES:
Types of X-Intercepts:
Graphically:
(From Warm-Up #1)
1.
2.
3.
24
Rachel Kluber – 3 Day Unit
Types of solutions (and the number of that type)
Graphically:
Graphically:
Two real solutions
One real solution
(multiplicity 2)
Graphically:
No real solutions (rather two
complex solutions)
Types of solutions (and the number of that type)
Algebraically:
Algebraically:
Two real solutions
One real solution
(multiplicity 2)
Algebraically:
No real solution (rather two
complex solutions)
Discriminant:
If the discriminant is
positive…
If the discriminant is zero….
25
If the discriminant is
negative…
Rachel Kluber – 3 Day Unit
Examples: Without solving determine the type (and number) of solutions to each
quadratic equation. Justify your answer
1)
2)
3)
Example: How do you find the x-intercept(s) of the following?
4)
26
Rachel Kluber – 3 Day Unit
HOMEWORK / ASSESSMENT
Students are assigned the following work from their textbook*:
Page 187, 21-29 odd, 31-35 all.
Answer Key:

Find the discriminant and name the type
of x-intercepts the function will have:
Find the x-intercept(s) and name the type
of intercepts they are:
21. 3x 2  5x  6  0
35. y  x 2  2x  3
D = -47, imaginary roots
x = 1 and -3, 2 real roots (rational)
23. 2x 2  3x 15  0

D = -111, imaginary roots

25. 10x 2 19x  7  0
x = -1 and 5, 2 real roots (rational)

D = 81, 2 real roots

27. 3x  5x  2  0
2
29. x 2  4 x  4  0

38. y  3x 2  7x  2
x = 2 and 1/3, 2 real roots (rational)

D = 0, 1 real roots

37. y  2x 2  7x  3
x = -3 and -0.5, 2 real roots (rational)
D = 1, 2 real roots

36. y  x 2  4 x  5
39. y  4 x 2  4 x 1
x = ½, 1 real root

40. y  x 2  6x  9
x = -3, 1 real root

* Foerster, Paul A. "Quadratic Functions and Complex Numbers." Classics Education
Algebra and Trigonometry. Upper Saddle River: Pearson Prentice Hall, 2006.
187.
27
Rachel Kluber – 3 Day Unit
Class: Algebra II
Grade: 11th
Number of Students: 26 (4 ELL, 2 IEPs)
Time Allotted: 50 minutes per class period.
LESSON 3: Imaginary and Complex Numbers
1) GOAL OF THE LESSON: Discuss the imaginary number i and use it to write
complex roots of quadratic functions.
2) OBJECTIVES:
 Students will identify the imaginary number i
 Students will simplify and write complex numbers
 Students will define and identify complex conjugates
3) MATERIALS:
a) Imaginary and Complex Numbers Notes Sheet (Attached)
b) Writing utensil
c) Algebra II Textbook (for reference and homework)
4) MOTIVATION (3 minutes): “Good morning everyone. Please put everything away
so we can begin today with our warm-up as usual. While XXXX passes out our notes
template for today, I want to brief you on our goals. Somebody tell me, if we use the
quadratic formula and there’s a negative number under our radical sign, what do we
say about our x-intercepts?” Students should note that we write that there are no real
x-intercepts and then we stop there. “Today we are going to talk about imaginary
numbers and how we can use them to write imaginary roots of quadratic functions!”
5) PROCEDURE
a) Warm Up: (7 minutes) The Warm-Up is on their notes sheet and they will
complete it as a group. The students are asked to find the roots of a quadratic
function but they will find that there are no real roots, and then they will need to
justify their answer. Possible answers: Discriminant is negative, can’t take the
square root of a negative number, doesn’t cross the x-axis when graphed on a
calculator, etc. The warm-up will be completed on the overhead and then left up
during the lesson on the smart board since we will be referring back to the warmup problem frequently.
b) LESSON (20-25 Minutes)
i) IDENTIFYING AND WRITING THE IMAGINARY NUMBER
(1) The blank notes sheet will be projected onto the smart board and I will
be filling it in as students fill it in from their desks.
28
Rachel Kluber – 3 Day Unit
ii)
iii)
iv)

(2) The students will first be introduced to the definition of i but it is
anticipated that most students have seen i or have at least heard of it
before. After the definition is introduced, we will take a look back at
our warm-up problem and see how we would use i to write a negative
square root.
(3) After we do the example, the students will do the next 6 practice
problems at their desks while I walk around and assess their
understanding. I will talk to a few students who may be struggling, but I
definitely anticipate that students have seen and done this before.
SIMPLIFYING AND WRITING COMPLEX NUMBERS
(1) Now that we have practiced simplifying negative square root numbers to
have i, and we have started to simplify our warm-up problems. I will go
back to the overhead and complete the warm-up problem. We will take a
look at how to write complex numbers in the form a+bi by simplifying
the roots. *SEE ATTACHED NOTES SHEET. This will be done on the
overhead so that it can be left up as we fill in our notes sheet and when
students go to practice writing complex x-intercepts.
(2) ANTICIPATION: When the definition of complex numbers is
introduced as a+bi where a is the real part and bi is the imaginary part:
It should be noted that the a and b here are different than the a and b
coefficients used in the quadratic formula!!!! It is important to note that
their textbook will use a and b, but that any two variables could be used
here. If it makes students feel better to use m+ni instead, they could do
that.
(3) When writing the complex x-intercepts on the warm-up- I will write
them separately (so it will be easier to see the conjugate part in the next
definition)
WRITING AND IDENTIFYING COMPLEX CONJUGATES:
(1) Introduce the definition on the notes sheet and then use the warm-up
problem on the overhead to write the complex conjugates for the warmup problem.
(2) Check for understanding: verbally ask the class for the complex
conjugate of 7 16i . Give students a few minutes to respond but this is
fairly easy. Students should also realize that the conjugate only changes
the + or – on the imaginary part. Ask students for the conjugate of
8  2i . They should not change the sign on 8.

LET’S PRACTICE:
(1) The first practice problem on the notes sheet will be completed by me so
students can see the process of finding the complex x-intercepts one
more time (the warm-up problem process will be projected on the
overhead so we will follow the same steps and refer back to that as we
do the first problem as a class)
(2) Students will complete the next three practice problems in their groups
as I walk around and check for student understanding informally by
talking to each group. Students will hopefully have time to complete all
29
Rachel Kluber – 3 Day Unit
three of these as a group before we come back together as a class to
discuss the answers that I saw while walking around.
6) CLOSURE (5 minutes) “Okay class, before I let you go for today I just want to
make a comment. Some of your homework is going to be just more practice finding
the imaginary x-intercepts using the Quadratic Formula and then simplifying the
complex numbers. There are a couple other discovery questions on your homework as
well so just try your best. We will be doing an in-class activity tomorrow so you will
not need your textbook and I will supply the materials. Have a wonderful rest of your
day!”
7) EXTENSION If there is extra time at the end of the lesson, give the students the
following problem:
Today we saw that the definition of i  1 which makes i 2  1 and
i 3  i 2  i  (1)i  i . Use this information to:

a. Evaluate i 4 to i10

b.Describe the pattern you see
c. Show that i and i 0 fit the pattern
d. Quick! What will i100 be
e. What will i 2001 be? i 37 ? i 50 ?

This will show students the difference between raising i to an even power or an odd

power and the pattern associated with i raised to an exponent. This may not come in
 that is important to know for other
handy with quadratics- but thisis something
mathematics. If this problem is too long but there is extra time- at least talk about
i 3  i 2  i  (1)i  i and part a.

8) ASSESSMENT: There are multiple times during this lesson that I walk around and
informally assess students of their understanding as I listen to group conversation,
answer questions as I walk around the room, and take a look at their work on the
practice problems. This is a good way to gauge how students are reacting to the new
content (are they stumped? Is it too easy?) Listening to student conversations would
be a good indicator. Also, students will be given textbook homework again which
will give them more practice on using the quadratic formula for one, and also
simplifying complex numbers.
9) MATHEMATICAL STANDARDS ATTAINED
 A-REI.4b Solve quadratic equations by inspection, taking square roots,
completing the square, the quadratic formula and factoring, as appropriate
to the initial form of the equation. Recognize when the quadratic formula
gives complex solutions and write them as a ± bi for real numbers a and b.
30
Rachel Kluber – 3 Day Unit
Students have been practicing using the quadratic formula for the last 3 days to
find the x-intercepts of a quadratic equation. However, this specific standard has
been completely met after this lesson since students can now incorporate their
new knowledge of complex numbers to completely solve quadratic equations and
identify the complex conjugates.
31
Rachel Kluber – 3 Day Unit
TEACHER’S NOTES SHEET
Imaginary and Complex Numbers
Objectives:
 Identifying and writing the imaginary number i
 Simplifying and writing complex numbers
 Writing and identifying complex conjugates
Warm-up:
2. Justify your answer to Question 1.
There are no real roots because there is a
negative under the radical- students
should note that this indicates no real
solutions, as we are about to learn how
to write imaginary solutions
1. Find the roots for
Set y = 0, 0  x 2 10x  34
Quadratic Formula: a= 1, b= 10, c=
(10)  (10) 2  (4)(1)(34)
34 x 

2(1)
10  36
x
2


Objective 1
What is i?




16

Identifying and writing the imaginary number i
How do we write numbers in terms of i?
Since now, we have been talking
about how a negative under the
radical sign cannot be simplified.
Instead of giving up and saying “no
solutions”, we introduce i
The imaginary number i is a
number whose square is -1.
In other words, i 2  1 and further,

i  1

36  (1)(36)  1 36  i 36
which we know is more simply,
PRACTICE: write in terms of i:
16
50
50


1 16  1 16  4i


Take a look at our warm-up problem:
We got
. We can write this in terms
of i by doing the following:
1 50  1 50  i 25 2  5i 2

49
360

1 49  1 49  7i
1 360  1 360  i 36 10  6i 10




32
Rachel Kluber – 3 Day Unit
77
121
1 77  1 77  i 77
1 121  1 121 11i




Objective 2
What is a complex number?



Simplifying and Writing Complex Numbers
Example:
So now, we see in terms of imaginary
After we go through the example on numbers,
= . In our warm-up
the right, the general form of a
problem, we found that our roots are
complex number can be introduced
, plugging in
gives us
a + bi where a is a real part and bi is
the imaginary part
10  6i 10 6i

  5  3i which contains a
5  3i and 5  3i are both forms of a
2
2 2
complex number with a real and
“real part” (5) and an “imaginary part” (6i)
imaginary part.
 Objective
3

What is a complex conjugate?
Complex numbers in the form a+bi and
a-bi are called complex conjugates of each
other.

Let’s Practice!
Find the imaginary roots of
Identify the complex conjugates



x
(3)  9  20
2(1)
x
3  11
2
x
3  i 11
2

Writing and Identifying Complex Conjugates
Example:
5  3i is the complex conjugate to 5  3i
and vice versa.
Practice: What is complex conjugate to
7 16i ?

Find the imaginary roots of
Identify the complex conjugates
x
(6)  36  40
2(2)

x
6  4
4

x
3i
2

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Rachel Kluber – 3 Day Unit
Find the imaginary roots of
Find the imaginary roots of
Identify the complex conjugates
Identify the complex conjugates
7x 2  3x 19  0
(4)  16 120
x
2(3)


x
4  104
6
2  i 26
x
3





34
x
3  9  532
2(7)
x
3  523
14
x
3  i 523
14
Rachel Kluber – 3 Day Unit
STUDENTS’ NOTES SHEET
Imaginary and Complex Numbers
Objectives:
 Identifying and writing the imaginary number i
 Simplifying and writing complex numbers
 Writing and identifying complex conjugates
Warm-up:
1. Find the roots for y  x 2 10x  34
2. Justify your answer to Question 1.

Objective 1
What is i?
Identifying and writing the imaginary number i
How do we write numbers in terms of i?
16

50

49

360

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Rachel Kluber – 3 Day Unit
77
121


Objective 2
What is a complex number?
Simplifying and Writing Complex Numbers
Example: y  x 2 10x  34

Objective 3
What is a complex conjugate?
Writing and Identifying Complex Conjugates
Example: y  x 2 10x  34

Let’s Practice!
Find the imaginary roots of x 2  5  3x
Identify the Complex Conjugates

Find the imaginary roots of
2x 2  5  6x Identify the Complex
Conjugates

36
Rachel Kluber – 3 Day Unit
Find the imaginary roots of 3x 2 10  4 x
Identify the Complex Conjugates
Find the imaginary roots of
7x 2  8x  25  5x  6
Identify the Complex Conjugates


Remember: If you are still having trouble with setting up/simplifying the quadratic
formula- please get extra help! This is a very fundamental idea concerning Quadratic
Functions!  
37
Rachel Kluber – 3 Day Unit
HOMEWORK / ASSESSMENT
Students are assigned the following work from their textbook*:
Page 193, 1-11 odd, 21 and 23
Answer Key:
1. x 2 14 x  58  0
21. The following quadratics differ only
in the constant term:
x = 7  3i




3. x 2 10x  26  0
x= 5i
5. 9x 2 12x  68  0
x=

1 8
 i
3 3





x = 2.5, -1
9. x 2  3x  21  x 12
11. 3x(x  5)  2x 2  8x 11
x=

7  i 171
10

23.
x= 0

x = 2  5i

g(x)  x 2  6x  9
h(x)  x 2  6x 13
7. 2x  3x  5  0

x = 5, 1

2

f (x)  x 2  6x  5
x = 3  2i
Find the x-intercept of each
Look at the graph of each on a

calculator or sketch it
What is true about a graph of a
Quadratic Function if the xintercepts are both real numbers?
Both complex numbers? Both equal
to each other?
o Notice that the y-intercept
changes and this changes the
position of the graph- it is
getting shifted up every time
until it no longer touches the
x-axis and the roots become
complex.
a. Write the complex conjugate of 4  7i . 4  7i
b. Write the complex conjugate of 3-8i. 3 + 8i
c. Multiply (7+3i)(7-3i). 
58 Whatdo you notice? It’s a real number
d. Add (11+5i) and (11-5i). 22 What do you notice? It’s a real number
e. Subtract (6+10i) and (6-10i). 20i What do you notice? It’s imaginary
f. Generally, what kind of number will you get when you add complex
conjugates? Multiply? Subtract?
See above!
38
Rachel Kluber – 3 Day Unit
* Foerster, Paul A. "Quadratic Functions and Complex Numbers." Classics Education Algebra and
Trigonometry. Upper Saddle River: Pearson Prentice Hall, 2006. 193
FOURTH DAY ASSESSMENT ACTIVITY
MOTIVATION: Students have learned all of the following thus far, and should now
create a way to organize their information so that they can refer back to it to do problems
quickly since one of the next lessons in quadratic functions is Modeling real-life
problems (story problems). Students will need to be able to identify different parts of a
quadratic function quickly and use them to solve real-life scenarios.
WHAT WE KNOW ABOUT QUADRATIC FUNCTIONS

















General Form of a Quadratic Function
Degree of a Quadratic Function
Vertex (maximum or minimum)
Axis of Symmetry
Graphing a parabola using a table of values
Finding the y-intercept and a symmetric point
Graphing a parabola using the vertex, y-intercept, and symmetric point.
Vertex Form of a Quadratic Function
Completing the Square to find the Vertex Form
Graphing a parabola using the Vertex Form
X-intercepts of a Quadratic Function
The Quadratic Formula and why we use it
3 types of x-intercepts
Discriminant
Imaginary number i
Complex numbers / Conjugates
Simplifying complex x-intercepts of Quadratic Functions
LESSON:
a) Students will have the entire class period to create a booklet containing all the
above information. The booklet will be made from sheets of computer paper
folded in half and stapled. All the materials will be provided to the student.
39
Rachel Kluber – 3 Day Unit
b) The point of this booklet is to allow students a reference to go back to when they
start modeling problems, where they will need all of these pieces of information
readily available and in one place.
c) The students are given free-reign to design the booklet how they want but each
bullet point needs to be
i) TITLED
ii) DEFINED
iii) EXPLAINED in 1 sentence.
iv) EXEMPLIFIED either in a problem or a picture.
d) Students should be using this for their own benefit. It will not be collected and
assessed until the unit is over so the more detailed their booklet is, the better it
will be for them when the modeling problems get tough.
e) This booklet will be a project grade for the unit on Quadratics and will be
assessed alongside their test. This will be a good way for me to look at how each
student chooses to organize information, as well as show me how much each
student knows and understands about each section in our unit.
f) If the students do not finish this assessment item during the class period they
have to work on it, they will have to finish the activity outside of class (but they
have until the day of their unit test to do this). They will be given the option then,
from here on out, to work on the booklet at the end of class instead of homework
if they are ever given work time.
40