Factoring Quadratic Trinomials TARSIA P310 2

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Factoring Quadratic Trinomials
Student Probe
Factor x 2  3x  10 .
Answer:  x  5 x  2 

Lesson Description
This lesson uses the area model of multiplication to
factor quadratic trinomials. Part 1 of the lesson
consists of “circle puzzles” as a preparation for the
actual factoring lesson in Part 2. While the goal of
the lesson is for students to be able to factor
quadratic trinomials without devices such as circle
puzzles or rectangles, it may take some time before
they develop fluency when factoring. Part 3 of the
lesson uses the graphing calculator to help students
relate the factors of a quadratic trinomial to its x‐
intercepts.

Rationale
Factoring is a valuable tool which helps students
become aware of the connections between the
roots of a quadratic function and its intercepts.
While many, in fact most, polynomial functions are
prime over the rational numbers, students can gain
valuable insight into function behavior by analyzing
functions in this manner.
When students are required to solve quadratic
equations, factoring will be one of the methods
used.

At a Glance
What: Factor quadratic trinomials
Common Core State Standard: CC.9‐
12.A.SSE.3a Factor a quadratic expression to
reveal the zeros of the function it defines.
Matched Arkansas Standard. CC.9‐12.A.SSE.3a.
Choose and produce an equivalent form of an
expression to reveal and explain properties of
the quantity represented by the expression.
(a) Factor a quadratic expression to reveal the
zeros of the function it defines.
Mathematical Practices:
Look for and make use of structure.
Look for and express regularity in repeated
reasoning.
Who: Students who cannot factor quadratic
trinomials.
Grade Level: Algebra 1
Prerequisite Vocabulary: factor, quadratic,
trinomial
Prerequisite Skills: polynomial arithmetic,
factoring whole numbers, distributive property
Delivery Format: individual, small group, whole
group
Lesson Length: 30 minutes per day, over
several days
Materials, Resources, Technology: graphing
calculator
Student Worksheets: none

Preparation
For Part 1 of the lesson, prepare several circle puzzles for students to solve.
For Part 2, prepare several rectangles for students to use to write the areas as products.
Examples of each of these appear at the end of the lesson.
For Part 3, provide a graphing calculator for each student.

Lesson
The teacher says or does…

Expect students to say or do…

Part 1
1. We are going to solve a
“circle puzzle”. Circle
puzzles have this format:

If students do not, then the
teacher says or does…
What does product mean?
What does sum mean?

ab
a

b
a+b

Notice that the product of
two numbers, a and b, is in
the top section of the
puzzle and the sum of the
two numbers is in the
lower section. If any two
sections of the puzzle are
filled in, it is possible to
determine the remaining
two sections.
2. Complete this circle puzzle. a = 2 and b = 1
or
a = 1 and b= 2

2
b

a
3

3. Repeat circle puzzles with
a variety of number
combinations.
(See Circle Puzzles at the
end of the lesson.)

What two numbers have a
product of 2?
Which of those number pairs
have a sum of 3?

The teacher says or does…

Expect students to say or do…

If students do not, then the
teacher says or does…

Part 2
4. Write the area of the
rectangle as a sum.

x2  2x  x  2

What is the area of each part
of the rectangle?

x 2  3x  2

Refer to Addition and
Subtraction of Polynomials.

 x  1

Refer to Multiplying
Polynomial Expressions.
How do we find the area of a
rectangle?
Refer to Multiplying
Polynomial Expressions.

x2

2x

x

2

5. Can we combine any like
terms?
What do we get?
6. What are the dimensions
of the large rectangle?
7. What is its area as a
product?
8. Since the area of the
rectangle is the same
regardless of how we write
it, we can say
x 2  3x  2   x  1 x  2  .
Multiply  x  1 x  2  to
verify our results.
9. Writing an expression as a
product is called factoring.
10. Repeat Steps 4‐8 with a
variety of rectangles.
(See Rectangles at the end
of the lesson.)

and  x  2 

 x  1 x  2 
 x  1 x  2   x 2  x  2x  2
 x 2  3x  2

The teacher says or does…
11. Factor this polynomial by
writing the sum as a product:
x 2  4 x  2x  8
You may use the rectangle to
help you.

Expect students to say or
do…

x2

4x

2x

8

If students do not, then the
teacher says or does…
Model for students.

 x  4  x  2
12. Since 4x and 2x are like terms,
this polynomial can be written
as x 2  6 x  8 .
How do we know?
13. When we did the Circle
Puzzles, we looked for
numbers with a certain
product and a certain sum.
What numbers have a product
of 8 and a sum of 6?
14. So we can say
x 2  6 x  8   x  4  x  2  .
15. Repeat Steps 11‐14 with a
variety of quadratic
trinomials, moving from
writing the trinomial with 4
terms to writing it with 3
terms.
Part 3
16. Now that we know how to
factor a quadratic trinomial,
let’s see what the graph can
tell us.

4 x  2x  6 x

Refer to Addition and
Subtraction of Polynomials.

What numbers have a
product of 8? (1 and 8 or 2
and 4)
Which of these have a sum
of 6?
4 and 2

The teacher says or does…
17. Using your graphing
calculator, graph
y  x 2  6 x  8   x  4  x  2 

Expect students to say or
do…
Answers will vary, but listen
for “the x‐intercepts are ‐4
and ‐2 and the y‐intercept is
8”.

If students do not, then the
teacher says or does…
Model for students. It may
take several examples
before students see the
relationship.

There are not any factors.

Not all trinomials have real
factors.

.
What do you notice about the
graph?
(See Teacher Notes.)
18. Repeat Step 17 with additional
trinomials until students make
the connection between the
factors and the x‐intercepts.
19. Can someone summarize what The x‐intercepts are the
we have discovered?
opposites of the numbers in
the factors.
The y‐intercept is the
constant term in the
trinomial.
20. Use your graphing calculator
Answers may vary, but listen Where does the graph cross
2
for, “there are no x‐
the x‐axis?
to graph y  x  2x  2 .
intercepts”.
What do you notice?
21. What do you think this means
about its factors?

Teacher Notes:
1. It is suggested that this lesson be taught over a number of days. Students seldom develop
fluency in factoring quickly.
2. It is recommended that teachers use the “circle puzzles” in part one as warm‐up activities
for several days prior to the actual factoring lesson. These puzzles will help students think
about numbers in ways that will help them in the actual factoring lesson.
3. Use a variety of circle puzzles, including positive and negative numbers.
4. Help students remember the format of circle puzzles by placing the diagram in Step 1 where
students can refer to it.
5. When solving circle puzzles, it may be helpful for students to list all of the factor pairs for a
number.
6. Once students can solve circle puzzles, move to writing areas as sums and as products in
Part 2.
7. Remind students that the word factoring means to write an expression as a product.
8. It may take several examples of graphing quadratic trinomials before students see the
relationship between the factors and the x‐intercepts. Be patient and let the students
“discover” this relationship.

Variations
Algebra tiles may be used in place of, or in addition to, the rectangles.

Formative Assessment
Factor x 2  5 x  36 .
Answer:  x  9  x  4 

References
Russell Gersten, P. (n.d.). RTI and Mathematics IES Practice Guide ‐ Response to Intervention in
Mathematics. Retrieved 2 25, 2011, from rti4sucess.

Circle Puzzles

5

3

‐6

6

‐4

1

‐8

‐8

9

‐2

2

‐6

Rectangles

x2

5x

x2

‐3x

x2

4x

x

5

‐x

3

‐3x

‐12

x2

‐5x

x2

‐4x

x2

3x

2x

‐10

5x

‐20

3x

9



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