Lesson Plan – 1 – Algebra I
Introduction
 Lesson Topic: Factoring Quadratic Polynomials
 Length of Lesson: 90 Minutes
 VA Standards of Learning: A.2.c – The student will factor completely first- and seconddegree binomials and trinomials in one or two variables.
 Context: This is the fifth class in a series of nine (block) classes on polynomials and
factoring. The students will have completed the first four sections on adding and
subtracting polynomials, simple factoring, multiplying binomials and multiplying
polynomials.
 Global Themes: This lesson is part of the chapter teaching that a single quantity can be
represented by many different equations/expressions and how these
equations/expressions can be broken into piece parts by a process known as factoring.
Content Objectives
1. The student will factor a polynomial in one or two variables.
Assessment Aligned to Objectives
1. The student will factor a polynomial in one or two variables.

Formative: Following the first portion of the lecture, the teacher will first work problems
from a worksheet with the students asking the students to contribute to the solutions.
The teacher will check for comprehension among the students as he or she works
through the problems of this worksheet. After the second portion of the lecture the
students will work on a second worksheet independently with the teacher checking
each student’s progress by walking around. Students who are slower to work the
problems will be given assistance and a supplemental factoring worksheet with numbers
only. Students who are able to complete the worksheet without difficulty will be
assigned iPad app SAT prep questions related to polynomial factoring.
Materials/Technology and Advanced Preparation
 Algebra 1 text (used in preparation and homework assessment)
 Computer (for teacher use)
 PowerPoint presentation
 Projector and screen
 Whiteboard
 Worksheet 1 – Teacher Led
 Worksheet 2 – Student Independent
 Worksheet 3 – Factoring Drill
 iPads with SAT Prep application loaded (for student use)
 Before class, computer and projector must be set up, enough copies of worksheet 1 must
be printed, and slope demonstration must be set up at the front of the room.
Algebra I Lesson Plan – Factoring
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Teaching and Learning Sequence
TIME
TEACHER ACTIONS
STUDENT ACTIONS
Introduction/Anticipatory Set – Outline how the lesson will begin. How will you focus student attention on lesson
content, build on prior knowledge, motivate students to learn, etc.?
Ten
minutes
The teacher shows the YouTube
video using algebraic tiles to help
conceptualize factoring of quadratic
polynomials. After the video has
finished the teacher reiterates the
basic concept of the video which is
that the two factors of a polynomial
can be thought of as the measure of
the length and width of a rectangle.
The student gains a different
perspective and understanding of the
factors of a quadratic polynomial that
is analogous to their existing
understanding of finding the area of a
rectangle using the length and width.
Lesson Development – Outline the sequence to be followed in the development of the lesson. Pay particular attention
to concept development and questioning.
Fifteen
minutes
The teacher gives first portion of
lecture on factoring quadratic
polynomials of the form ax2+bx+c
using the possible factors with sums
table approach.
The student sees the approach to
factoring using factors of “c” term
using factors with sums table
approach.
Twenty
minutes
The teacher works a 6 problem
worksheet with student input.
The students, as a group, raise hands
to contribute to each step of the
process in factoring each of the six
problems on the worksheet.
Fifteen
minutes
The teacher gives second portion of
lecture on factoring, this time with a
second variable showing how it does
not matter in the method used to do
the factoring.
The student sees the approach to
factoring using factors of “c” term
using factors with sums table
approach with two variables in
quadratic polynomial.
Twenty
minutes
The teacher distributes worksheet 2
and monitors student work. Slower
students given number factoring drill
sheet. Students who complete the
worksheet assigned SAT test prep
iPad app work.
The student working independently
works the factoring problems on the
worksheet.
Five
minutes
The teacher assigns the homework.
The student writes down the
homework assignment.
Closure – Outline how the lesson will be concluded. How will you summarize, review, reinforce, enrich, and/or
encourage students to reflect on what they have learned?
Algebra I Lesson Plan – Factoring
Five
minutes
Teacher asks students as a group a
closing set of questions about the
approach to factoring. Teacher
assesses how well students
remembered key points.
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Students respond to questions about
the factoring process to show they
understand.
Homework
The homework assigned will be as follows: Page 504 – 32-40, 44-48; Page 505 – 49-60, 71-73 in
the Algebra 1 text. This will be assigned at the very end of the class. The assignment covers
everything that was discussed in class and will be turned in at the beginning of the next class and
graded for completion. The next class material builds on this material and a review of the
homework will be built in to the introduction of the new material that introduces a non-unity “a”
term in the quadratic expression to be factored.
References
Randall I. Charles, Basia Hall, Dan Kennedy, Allan E. Bellman, Sadie Chavis Bragg, William G.
Handlin, Stuart J. Murphy, Grant Wiggins (2012). Prentice Hall Algebra 1.
YouTube video: Expanded and Factored forms using Algebra Tiles - MrAlgebra9
http://www.youtube.com/watch?v=BXyI1nReQjc
Appended Materials
Attach the following forms and resources to the completed lesson plan.
 Lesson Organizer
 Curriculum Framework Document
 PowerPoint Presentation including
o Worksheet 1
o Worksheet 2
 Supplemental Worksheet – Factoring Drill
 Assigned Homework
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Lesson Organizer
Prior Knowledge and NEW Instructional Content



Previously students have studied how to add like terms in polynomials, multiply
binomials using “FOIL” (first, outside, inside, last) technique, and do simple factoring
of a single variable and/or constant within a quadratic expression.
Opening Activity
o YouTube Video – Expanded and Factored Forms Using Algebraic Tiles
o Ask questions after video:
 So if a rectangle has sides measuring (x+1) and (x+2) what is the area of
the rectangle?
 Use whiteboard to show a rectangle with sides x+3 and x+4. Ask: If I
show a rectangle with these sides what is its area?
Lecture (part 1): Approach for Factoring Quadratic Polynomials (using PowerPoint
slides 1 – 5)
o Slide 1
 To factor quadratic polynomials you must be able to factor the constant
or “c” term in the expression ax2+bx+c form.
 Let’s go over a couple examples and look at the sums and differences for
each just to get our math juices flowing
 If we factor 24 in all the ways possible we see that there are 4 factor pairs
and 8 total factors. 24&1, 12&2, 8&3 and 6&4.
 The sums and differences of each of these factor pairs are shown after the
factors: 24&1 give a sum of 25 and a difference of 23; 12&2 sum to 14
and have a difference of 10; 8&3 sum to 11 and have a difference of 5
and finally 6&4 add to 10 and have a difference of 2.
 When we factor a polynomial the “b” term will come from these sums or
differences
 Let’s factor 36 now
 There are 5 factor pairs for 36: 36&1, 18&2, 12&3, 9&4 and finally the
perfect square factor 6.
 Sums for these 5 pairs: 37, 20, 15, 13, 12. Differences 35, 16, 9, 5, 0
 Why do we go over this?
o Slide 2
 Because this skill is critical in factoring polynomials!!
 A quick review of FOIL, remember “FOIL” – first/first, outside/outside,
inside/inside, last/last?
 One quick example: (x+3)(x+4), just like we just did on the board to
show the rectangle made up of length and width with these measures,
gives – first/first: x*x=x2, outside/outside: x*4=4x, inside/inside:
3*x=3x, last/last: 3*4=12. Adding like terms 4x & 3x gives 7x and you
can write the quadratic expression x2+7x+12
 Factoring is the reverse of FOILing!
 So if we find the length and width of a rectangle with area x2+7x+12 we
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get that its sides are (x+3) and (x+4) long.
o Slide 3
 Let’s do some examples
 Our first problem is to factor x2+8x+15
 What are the factor pairs for 15? 15&1 and 5&3
 All positive signs in the quadratic expression so we focus on sums only
here. 5&3 give a sum of 8 so they are our constant terms in the two
binomials we get when we factor.
 Answer (x+5) and (x+3)
 Any questions?
o Slide 4
 Our next example also has 15 for the “c” term in our quadratic
polynomial. However in this case our “c” term has a negative sign so
now we are going to be looking at differences in the factors
 Since one sign must be negative and the other positive let’s look at all the
factor pairs that will give us negative 15 and see what there sums are
(differences really but let’s think of them as adding a positive and a
negative so that we can always write sum above the column)
 Positive 15 and negative 1 sum to positive 14, negative 15 and positive 1
add to negative 14, positive 5 and negative 3 add to 2 (there’s our
solution), and for completion sake, negative 5 and positive 3 add to
negative 2.
 Our factors? (x+5) and (x-3)
 Any questions?
o Slide 5
 One more example and then let’s work some together
 Let’s factor x2-11x+24.
 Here we have a positive “c” term and a negative “b” term. How do we
get factors to make this work?
 Remembering that a negative plus a negative gives a negative and a
negative times a negative gives a positive we realize that both factors
must be negative
 So to get a positive 24 for the “c” or constant term and a negative for the
“b” term in front of x we will be factoring 24 in pairs with both numbers
negative.
 Negative 24 and negative 1 multiply to 24 and their sum is 25, negative
12 and negative 2 multiply to 24 and their sum is negative 14, negative 8
and negative 3 multiply to 24 and their sum is negative 11.
 There is our answer! But to complete the factoring table…negative 6 and
negative 4 multiply to 24 and sum to negative 10.
 Our factors: (x-8) & (x-3).

Worksheet 1 worked together:
 Problem-by-problem worked out on the SmartBoard calling on individual
students to tell the teacher how to solve the problems.
 Take time here to work through each problem systematically by first
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examining the signs of the “b” and “c” terms, then factoring the “c”
constant term and checking the sums of each factor pair. Ask each time:
“B-term is positive or negative? C-term is positive or negative? How do
we factor the c-term to make this work out? (Answer: for c-term to be
negative it must be a negative times a positive, for it to be positive it can
either be positive times a positive or a negative times a negative; for the
b-term if both factor pairs are positive the b-term will necessarily be
positive, if both factor pairs are negative the b-term will necessarily be
negative. When the c-term is negative then for the b term to be positive
the larger of the c-term factors must be positive and the other smaller
factors must be negative. For the b-term to be negative when the c-term
is negative the larger of the c-term factors must be negative and the other
smaller term must be positive.)

Lecture (part 2): Approach for factoring Quadratic Polynomials (using Power Point
slides 7-10)
o Slide 7
 Now that we have done a few examples together when we are dealing
with just one variable in the quadratic expression let’s add another
variable. Hey, this is really no big deal! Don’t sweat it one bit.
 Let’s factor x2+xy-42y2
 Let’s review our b and c terms first. The c-term is negative 42. The bterm is what? (Answer: positive 1)
 How do you get the final term to have a y-squared? (answer: y*y, a y
ends each of the factored binomials)
 So to get a negative c-term we need one positive and one negative factor.
Negative 42 and positive 1 add to negative 41, positive 42 and negative 1
add to positive 41, negative 21 and positive 2 add to negative 19, positive
21 and negative 2 add to positive 19, negative 7 and positive 6 add to
negative 1, and finally, positive 7 and negative 6 add to positive 1.
There’s our answer!
 So to write the factors for the quadratic expression with two binomials
we have (x+7y) and (x-6y).
 A quick check using FOIL…x*x is certainly x-squared, x*(-6y) is -6xy,
and 7y*x is 7xy, and 7y*(-6y) is -42y-squared. Combining like terms
with xy and we get positive xy. We are good!
o Slide 8
 Okay let’s do one where we use something besides x&y as our variables
 Let’s factor s2-3st-54t2.
 C-term is negative, b-term is negative. What will our factor constants
have for signs? (Answer: One term is positive and the other negative
because that is the only way you can get a negative with multiplication,
because b-term is negative the larger constant will be negative.)
 Factoring with this in mind we really only need to look at the first, third
and fifth factor pairs right? (Look for understanding in the classroom)
 Ask: So what is the form of the factored expression? (Answer: two
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binomials multiplied together with s and t as variables and factor pair that
works as constants, one each, with proper sign.)
o Slide 9
 Okay, one more example and then I will set you free to practice and show
me how well (or not) you understand. For those of you who are having
trouble factoring the constant I have an additional worksheet that simply
focuses on the numerical part of this skill. For those of you who get it
and can complete the worksheet within the 20 minutes I will ask you to
take out your iPads and work on the SAT prep in this area. If you
complete a prep quiz I will ask that you raise your hand and let me know
your score.
 Alright, here we have an expression using p and q as variables. P216pg+48q2.
 Help me with the b- and c-term signs. (should get from students that bterm is negative and c-term is positive so that the factors must both be
negative)
 Factoring -48: -48/-1, -24/-2, -16/-3, -12/-4 (correct answer), and -8/-6
 So we write our factor binomials…(p-12q)*(p-4q)
 Check it. p*p is p-squared, -4pq+(-12pq) is -16pq, and -12q*-4q is 48q2.
All is well!
 Pass out paper worksheet of Slide 10 for students to work independently.

Worksheet 2 students work independently
o While students work this worksheet the teacher is walking around looking at the
speed at which his/her students are solving and writing the factor binomials.
o Those students who are moving quickly and getting the answers correct will be
instructed to complete the worksheet, hand it in, and then work on the SAT prep
quizzes on their iPads using the SAT Test Prep app.
o Those students who are not progressing and are having trouble factoring the cterm will be given the supplemental worksheet on factoring and told to work on
it first before returning to Worksheet 2.

Homework assignment
o Teacher gives out the homework assignment: page 504 – 32-40, 44-48; page 505
– 49-60, 71-73.
o Tell students that they have 3 multiple choice that essentially require that they
FOIL to get the right answer, 21 factoring problems all similar to what was
covered in class, 2 reasoning problems that check for understanding of method,
and 3 problems that require simple factoring that will help prepare them for the
next lesson. None of these problems individually should take more than a
minute to do if you are not distracted and focused. I expect this assignment to be
completed within a half-hour. If you are not able to do these factoring problems
this quickly I want to talk with you during our next class.

Closing Activity (PowerPoint slide 11)
o Review of approach to factoring
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o Preview of next lesson when a-term is not 1 and how that one difference requires
just a single additional step.
o Ask if there are any questions about the homework or about anything done in
class today.
o Class dismissed.
Instructional Modifications to
ASSIST Students
Main Events of Instruction
Instructional Modifications to
CHALLENGE Students
Anticipatory Set:
YouTube video
Lecture: Factoring ax2+bx+c
format quadratic polynomials.
Worksheet 1 worked through
with student input
Lecture: Factoring
ax2+bx+cy2 format (two
variable) quadratic
polynomials
Supplemental worksheet of
factoring common multiples
given out.
Worksheet 2 worked by
students independently.
Homework assigned.
Closing activity:
Summarizing Q&A
SAT Test Prep on iPads
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STANDARD A.2.c
ESSENTIAL UNDERSTANDINGS

The laws of exponents can be
investigated using inductive reasoning.

A relationship exists between the laws
of exponents and scientific notation.

Operations with polynomials can be
represented concretely, pictorially, and
symbolically.

Polynomial expressions can be used to
model real-world situations.

The distributive property is the unifying
concept for polynomial operations.

Factoring reverses polynomial
multiplication.

Some polynomials are prime
polynomials and cannot be factored
over the set of real numbers.

Polynomial expressions can be used to
define functions and these functions can
be represented graphically.

There is a relationship between the
factors of any polynomial and the xintercepts of the graph of its related
function.
ESSENTIAL KNOWLEDGE AND
SKILLS
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections, and
representations to

Simplify monomial expressions and
ratios of monomial expressions in
which the exponents are integers, using
the laws of exponents.

Model sums, differences, products, and
quotients of polynomials with concrete
objects and their related pictorial
representations.

Relate concrete and pictorial
manipulations that model polynomial
operations to their corresponding
symbolic representations.

Find sums and differences of
polynomials.

Find products of polynomials. The
factors will have no more than five total
terms (i.e. (4x+2)(3x+5) represents four
terms and (x+1)(2x2 +x+3) represents
five terms).

Find the quotient of polynomials, using
a monomial or binomial divisor, or a
completely factored divisor. Factor
completely first- and second-degree
polynomials with integral coefficients.

Identify prime polynomials.

Use the x-intercepts from the graphical
representation of the polynomial to
determine and confirm its factors.