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Lesson Design
Subject Area: Algebra
Benchmark Period III
Grade Level: 8-12
Duration of Lesson: 5 – 6 days (1 day second degree with
coefficient of 1, 1-2 days second degree with coefficient greater
than 1, 1 day perfect square trinomials and difference of two
squares, and 1 day common factors)
Standard(s): 11.0: Apply basic factoring techniques to second and simple third degree polynomials.
These techniques include finding a common factor for all terms in a polynomial, recognizing the
differences of two squares, and recognizing perfect squares of binomials.
Big Ideas involved in the lesson:
 Using basic factoring technique to factor second degree polynomials with a leading coefficient
of one.
 Use basic factoring techniques to factor second degree polynomials with a leading coefficient
greater than one.
 Recognize the difference of two squares and perfect squares of binomials.
As a result of this lesson students will:
Know:
 Vocabulary: x-intercept, roots, zeros, solutions, like terms, coefficient, degree, constant term,
exponents, variables, monomial, binomial, trinomial, polynomial, factor, common factor,
factorization, quadratic expression and equation, linear expression and equation, leading
coefficient, degree of a polynomial, cubic expression, prime, composite, perfect square, prime
polynomial, area model
 Standard form of a quadratic polynomial: ax2+bx+c, a 0.
 Factored form: a polynomial is in factored form if it is written as the product of two or more
linear factors, such as, for a quadratic, (ax+b)(cx+d) where a 0 and c 0.
 Special products of polynomials
o Difference of two squares pattern a2-b2 = (a+b)(a-b)
o Square of a binomial pattern a2 2ab+b2 = (ab)2.
 Discriminant expression: b2 - 4ac.
Understand:
 Factoring a quadratic trinomial means to express it as the product of two linear expressions.
 Factoring a quadratic expression is accomplished by using the converse of the distributive
property.
 In an area model, when two linear factors are multiplied, the product represents the area of a
rectangle whose length and width are the linear factors.
 All quadratic equations are not factorable over the set of rational numbers.
 The relation between an equation in mathematics and a sentence in English, (symbolically and
verbally), in particular when the situation represents a quadratic function
Be Able To Do:
 Factor quadratic trinomial by using the greatest common factor and the distributive property.
 Factor a quadratic trinomial by grouping.
 Factor a quadratic expression by recognizing special product forms such as the difference of
two squares and the square of a binomial pattern.
 Find the dimensions of a rectangle or a square, given its area as a factorable quadratic
polynomial.
 Translate a word problem that can best be represented by a quadratic model into a
mathematical model, apply the model and determine a solution or solutions to the problem.
1
Lesson Design

Check and verify solutions to quadratic equations, whether within the context of a word
problem or not, for accuracy and reasonableness.
Assessments:
Formative:
CFU Questions:
What will be evidence of student
DWA, classroom assessments,
knowledge, understanding & ability?
ABWA
Embedded in lesson
Summative: CST
Lesson Plan
Anticipatory Set:
Use with lessons for Factoring with a coefficient of 1 and greater than 1
a. T. focuses students
Use area model to multiply two polynomials.
b. T. states objectives
Go to file for Anticipatory Set. Show three different models of multiplication of
c. T. establishes purpose of binomials.
the lesson
CFU
d. T. activates prior
 Is there another way that we could use to solve this problem?
knowledge
 What property did you use to solve this problem?
 Would you get the same answer if you solved it using a different method?
 How do the three different models compare? Which explains the
multiplication best to you? Why?
Use with lessons for factoring perfect square trinomials and difference of two
squares binomials.
 Use PowerPoint 11.0 Factoring slide 1
Use with lesson for factoring with common factors
 Use PowerPoint 11.0 Factoring slide 9
2
Factoring x2 + bx + c; b > 0 and c > 0
LO: Factoring trinomials with a leading coefficient of 1 where b > 0 and c > 0.
Give algebra tiles and x2 + 5x +6 to students. Ask students to use the algebra tiles
to form a rectangle. Have student demonstrate on board their answer.
-1 * 6 = -6
-1 +6 = 5
1 * -6 = -6
1 + -6 = -5
-2 + 3 = +1
-2 * 3 = -6
x2
5x
x + 2
Instruction:
a. Provide information
 Explain concepts
 State definitions
 Provide exs.
 Model
b. Check for Understanding
 Pose key questions
 Ask students to
explain concepts,
definitions, attributes in
their own words
 Have students
discriminate between
examples and nonexamples
 Encourage students
generate their own
examples
 Use participation
x
+ 3
6
Lesson Design
CFU:
 What is the dimension of the rectangle? (x + 2) (x + 3)
 Label each rectangle in the term of area.
 What is the total area of this rectangle? How do you arrive this answer?
 Can we place (x + 2) on the horizontal line and (x + 3) on the vertical line? Why
or why not?
 If you switch the length and width of the rectangle, what property are you
applying?
Now students will be given the following problems to form rectangles using algebra
tiles.
x2+6x + 9
x2 + 7x + 6
x2 + 3x +1
With the dimensions of each rectangle create a table.
Length
Width
2
x + 5x + 6
x+2
x+3
x2 + 6x + 9
x+3
x+3
2
x + 7x + 6
x+1
x+6
x2 + 3x + 2
x+2
x+1
CFU: Look at each expression and its dimensions. How are they related?
For the expression x2 + 5x + 6, to prove that (2)(3) = 6 (product) and 2 + 3 = 5
(Sum), using area model to show solution.
Draw an area model, x2 goes to the top left box, the constant +6 goes to the bottom
right box, and the second term in the middle of the box.
x2
5x
6
x
For the factors of x2 are x times x.
For the factors of 6, starting with 1. 1 * 6 = 6, 1 + 6 = 7. These are not correct
factors because we need the sum to equal 5. Find the other factors of six: 2 * 3 = 6
and 2 + 3 = 5. These are correct factors because the product equals 6 and the sum
equals 5.
x
+ 2
product of 6
sum of 5
x2
2x
1*6=6
1 + 6 =7
5x
2*3=6
2+3=5
6
+
3
3x
3
Lesson Design
So the factors for x2 + 5x + 6 are (x+2) and (x+3).
Continue working on completing the rest of expressions in the table.
For x2 + bx+ c and using the area model, the first term goes in the top left box, the
constant term, c, goes in the bottom right box, and the second term, bx, in the
center of the box. The sum of the two factors equals b and the product of these two
factors must also equal c.
CFU

Does the order matter when all the terms are positive? Why?
Factoring x2 + bx + c ; b < 0 and c > 0
LO: Factoring trinomials with a leading coefficient of 1 where b < 0 and c > 0.
Give algebra tiles and x2 - 5x + 6 to students. Ask students to use the algebra tiles
to form a rectangle. Have student demonstrate on board their answer.
x2
x
-5x
-3
+6
x
-2
CFU:






What is the dimension of the rectangle? (x - 2) (x - 3)
Label each rectangle in the term of area.
Why this section is (-3x) instead of (3x)? Why (-2x) instead of (2x)?
Why is the constant tern (6), not (-6)?
What is the total area of this rectangle? How do you arrive this answer?
Can we place (x - 2) on the horizontal line and (x - 3) on the vertical line?
Why or why not?
 If you switch the length and width of the rectangle, what property are you
applying?
Students will be given the following problems to form rectangles using algebra tiles.
x2 – 5x + 4
x2 –4x + 4
x2 – 8x + 7
x2 – 7x +12
4
Lesson Design
With the dimensions of each rectangle create a table.
Length
Width
2
x –5x + 4
x –4
x-1
x2 –4x + 4
x–2
x-2
2
x –8x + 7
x –1
x –7
2
x –7x + 12
x –4
x -3
Look at each expression and its dimensions. How are they related?
For the expression x2 –5x + 6, to prove that (-2)(-3) = 6 and – 2 + -3 = 5, using area
model to show solution.
Draw an area model, x2 goes to the top left box, the constant +6 goes to the bottom
right box, and the second term –5x in the middle of the box.
x2
-5x
+6
With a T-table we need to determine the factors for +6 where the product is +6 and
the sum is negative –5.
Product
of +6
Sum
of -5
x
2*3=6
2+3=5
3
1 + 6 =7
-
1 * 6= 6
x
-2
x2
-2x
-5x
- 3s
6
-2 * -3 =6
-2 + -3 = -5
So the factors for x2 –5x +6 are ( x –2) and (x –3).
Continuing working on completing the rest of the expression in the table.
Factoring x2 + bx + c ; b < 0 and c < 0
LO: Factoring trinomials with a leading coefficient of 1 where b < 0 and c < 0.
x2 –5x -6
x2
5
-5x
-6
Lesson Design
Students will probably place the ones unit with the –x’s. A negative multiplied by a
negative is not negative. Ask student what can they do to place the negative units.
(zero pairs)
CFU:
 What are the differences of this expression (x2 – 5x –6) from the previous
expressions we had been working with (x2 + 5x + 5) and (x2 –5x +6)?
 Can you create a rectangle using only these tiles? Why or why not?
 If you create a rectangle with only these tiles, does your model match our
expression? (x2 – 5x –6)
 Which section does not match? (The section does not match is the
constant, the product of –2 and –3 is 6 not –6 as indicated in our
expression)
 We can’t change the constant value of the expression, so what can we
change in order to form a rectangle?
Teacher’s note: Students might not think about creating zero pairs. Teacher may
remind them the warm-up exercises they did today.
 What would happen if we add just one zero pair? Is that enough to create a
rectangle for x2 –5x –6?
 What is the dimension of the rectangle? (x - 6) (x +1)
 Label each rectangle in the term of area.
 Why this section is (-6x) instead of (6x)? Why (x) instead of (-x)?
 Why is the area of the constant tern (-6), not (6)?
 What is the total area of this rectangle? How do you arrive this answer?
 Can we place (x - 6) on the vertical line and (x +1) on the horizontal line?
Why or why not?
 If you switch the length and width of the rectangle, what property are you
applying?
x
-6
x
+1
x2 – 3x –10
x2 –13x –14
x2 – 6x –7
Length
( x – 5)
( x – 14)
( x – 7)
-6
x2
– 3x –10
–13x –14
x2 – 6x –7 x
x2
x
-6x
x2
-5x
+1
6
+1x
-6
Width
( x + 2)
(x +1)
(x +1)
Lesson Design
Create a T-table, we need to determine the factors for –6 and the sum is -5
Product of -6
Sum of -5
-1 * 6= -6
1 + -6 = -5
1 * -6 = -6
1 + -6 = -5
So the factors for x2 –5x – 6 are ( x +1) ( x – 6)
Factoring x2 + bx + c; b > 0 and c < 0
LO: Factoring trinomials with a leading coefficient of 1 where b > 0 and c < 0.
x2 + x - 6
+x
+3
x2
x
-6
x
-2
x2 + 2x – 8
x2 + 8x – 20
x2 + 3x –10
Length
( x – 2)
( x – 2)
( x – 2)
x2 + 2x – 8
x2 + 8x – 20
x2 + 3x –10
x
-3
x
-
x2
3x
Width
( x +4)
( x + 10)
( x + 5)
+1
Product of -6
-2
-2x
7
-6
Sum of +1
Lesson Design
So the factors for x2 + x – 6 are ( x –2 ) ( x + 3).
CFU:
 Can you make a model of the rectangle using the given quantity (x2 + x –6)
of algebra tiles?
 What do we need to do in order to make a rectangular model?
 How many zero pair do we need in this case?
 How does this change the original expression?
 Why won’t this change the value of the original expression? What property
are we applying? (Identity property of addition)
 What is the dimension of the rectangle? (x - 2) (x +3)
 Label each rectangle in the term of area.
 Why this section is (3x) instead of (-3x)? Why (-2x) instead of (2x)?
 Why is the constant tern (-6), not (6)?
 What is the total area of this rectangle? How do you arrive this answer?
 Can we place (x - 2) on the horizontal line and (x + 3) on the vertical line?
Why or why not?
 If you switch the length and width of the rectangle, what property are you
applying?
Factoring perfect square trinomials and difference of squares binomials
 Use Riverdeep Course 2, Module: Powers and Polynomials, Unit: Factoring
Polynomials, Tutorial 3: Special Cases Screens 1 and 2 only
 CFU questions attachment 11.0 Special Cases Riverdeep

PowerPoint 11.0 Factoring slides 2-3 (Difference of squares) and 5-6
(Perfect square trinomials)
Factoring using a common factor
 PowerPoint11.0 Factoring slides 10-13 and 15-18 odds (common factor)
 Use Riverdeep Course 2, Module: Powers and Polynomials, Unit: Factoring
Polynomials, Tutorial 3: Special Cases Screens 3 only
 CFU questions attachment 11.0 Special Cases Riverdeep
Guided Practice:
a. Initiate practice
activities under direct
teacher supervision –
T. works problem stepby-step along
w/students at the same
time
b. Elicit overt responses
8
Factor with coefficient of one.
x2 + 4x + 3
x2 + 8x + 7
x2 + 7x +6
Students can work independently or with a partner. Answer can be written in notes,
on whiteboards, and be verbally expressed.
CFU (change numbers for each expression)
 What thing that I want to do to factor this trinomial? (draw area model)
Lesson Design
from students that
demonstrate behavior
in objectives
c. T. slowly releases
student to do more
work on their own
(semi-independent)
d. Check for
understanding that
students were correct
at each step
e. Provide specific
knowledge of results
f. Provide close
monitoring
What opportunities will
students have to read,
write, listen & speak about
mathematics?
Closure:
a. Students prove that
they know how to do
the work
b. T. verifies that students
can describe the what
and why of the work
c. Have each student
perform behavior





Where is x2 located on the area model? 3? 4x?
What are the factors of x2?
What is the next step? (making a T-chart to find the product of 3 and then
sum of 4)
What are the products of 3? If you add them up, what will it equal to?
What are the factors of the trinomial x2 + 4x + 3?
Factoring perfect square trinomials and difference of squares binomials
 PowerPoint 11.0 Factoring slides 4 (Difference of squares) and 7 (Perfect
square trinomials)
Factoring using a common factor
 PowerPoint11.0 Factoring slides 14 and 15-18 even (common factor)
Pair share, whiteboards, writing in notes with summary, draw with algebra tiles and
explain each step
Use with lessons for Factoring with a coefficient of 1 and greater than 1
Draw the algebra diagram and explain in detail how you found the factors
for x2 + 7x + 10.
Draw the algebra diagram and explain in detail how you found the factors for x 2 - 5x
+ 6.
Draw the algebra diagram and explain in detail how you found the factors for x2 5x 6.
Draw the algebra diagram and explain in detail how you found the factors for x 2 + x
- 6.
Use with lessons for perfect squares trinomials and differences of squares
binomials.
 Use PowerPoint 11.0 Factoring slide 8
Independent Practice:
a. Have students
continue to practice on
their own
b. Students do work by
themselves with 80%
accuracy
c. Provide effective,
timely feedback
Resources: materials
needed to complete the
lesson
9
Use with lessons for factoring with common factors
 Use PowerPoint 11.0 Factoring slide 19
Use with lessons for Factoring with a coefficient of 1 and greater than 1
 Independent practice day 1, 2, 3, and 4
 alg 11 ip coeff greater 1
Use with lessons for perfect squares trinomials and differences of squares
binomials, and factoring with common factors
 St11 ind prac perfect diff wksht
Algebra tiles, whiteboard, markers, graphic paper, line paper, rulers, pencils, color
pencils worksheet sheets for independent practice .
Lesson Design
10