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Factoring Trinomials Part 1
TEACHER NOTES
ALGEBRA NSPIRED: QUADRATICS
Math Objectives

Factor trinomials of the form x2 + bx + c, where b and c are
positive integers

Relate factoring a quadratic trinomial to an area model

Generalize the process for factoring trinomials of the form
x2 + bx + c, where b and c are positive integers
Vocabulary

trinomials
TI-Nspire™ Technology Skills:
 Download TI-Nspire document
About the Lesson

 Open a document
This lesson involves factoring trinomials using a dynamic
 Move between pages
area model. As such, it is implicit that x represents a positive
 Grab and drag tiles for factoring
quantity.

As a result, students will factor trinomials of the form
Tech Tips:
x2 + bx + c, where b and c are positive integers, recognizing
 Make sure the font size on your
that in general the factors of c that sum to b are those needed
TI-Nspire handhelds is set to
to correctly factor the trinomial.
Medium.
 In Graphs and Geometry you can
Lesson Setup

hide the function entry line by
The introduction to this lesson needs to be led by the teacher
pressing /G.
so students will understand how to use these visual tiles. The
rest of the activity can be used with the student worksheet
allowing students to work individually or with a partner or
Lesson Materials:
Student Activity
group.
Factoring Trinomials Part 1.PDF
Related Lessons

Factoring Trinomials Part 1.DOC
After this lesson: Factoring Trinomials Part 2: Overlapping
TI-Nspire document
Areas
Factoring Trinomials Part 1.tns
Visit www.algebranspired.com for
lesson updates and tech tip videos.
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Factoring Trinomials Part 1
TEACHER NOTES
ALGEBRA NSPIRED: QUADRATICS
Discussion Points and Possible Answers:
TI-Nspire Problem/Page 1.2
Tech Tip:
To drag a tile to a new location, simply grab the
point at the top left vertex and move the tile. Notice
how the pieces “snap” into place and how their
orientation changes depending upon where you try
to place them.
If students experience difficulty dragging a tile,
check to make sure that they have moved the arrow
close to the point (vertex). The arrow should become a hand (÷) getting ready to grab
the point. Be sure that the word point appears. Press /
x to grab the tile and the
hand will close. Drag the tile. After the tile has been moved, press d to release the
point.
Teacher Tip:
Discuss page 1.2, and then go to page 1.3. Demonstrate how to factor the trinomial
x2 + 5x + 6 using the tiles. Be sure students see that the pieces on the page are
determined by the given trinomial expression: one square, 5 rectangles, and six unit
tiles. Students might complete a rectangle with pieces left over; for example: (x + 1)(x +
2) makes a rectangle, but the expression on the page is really (x + 1)(x + 2) + 2x +4.
This is a good opportunity to review what it means to "factor"- that is to rewrite the
entire expression as a product rather than a sum. It might be easier to move the x tiles
before the unit tiles are moved, but it is not necessary. Tiles should not be placed on
top of other tiles in this file. It is a good idea for you to explore incorrect arrangements
of tiles so students understand that they must use all pieces and that the pieces must
create a rectangle.
Discuss why the dimensions of the rectangle are the factors of the trinomial. Students
can create two rectangles that look different but are actually congruent because (x +
2)(x + 3) is the same factorization as (x + 3)(x + 2) by the commutative property of
multiplication. This would be a good discussion to have with students.
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Factoring Trinomials Part 1
TEACHER NOTES
ALGEBRA NSPIRED: QUADRATICS
1. a. Go to page 1.2, move the cursor
up to highlight trinomial (1,5,6), and
press · to copy it onto the next
line. Use the backspace . to
erase the 5 and 6. Edit the line to be
trinomial (1,3,2), and press ·.
b. Move to page 1.3, and arrange
The dimensions of the rectangle are x + 1 and x + 2. The two factors
the tiles to form a rectangle with
can be in any order.
2
area x + 3x + 2.
Teacher Tip: Students should be reminded that they are working with an
What are the dimensions of the
area model and that by placing additional x tiles, they are adding area to
rectangle?
the x2 tile. You should circulate throughout the classroom to help
struggling students with the placement of the tiles.
2. a. Multiply the dimensions you
(x + 1)(x + 2) = x2 + x + 2x + 2
= x2 + 3x + 2
found for the rectangle to prove that
x2 + 3x + 2 is the area of the
rectangle.
b. How do the dimensions of the
1 + 2 = 3 and 1 x 2 = 2.
rectangle relate to the numbers
3 and 2?
3. a. Return to page 1.2, move the
cursor up to highlight
trinomial(1,3,2), and press · to
copy it onto the next line. Use the
backspace . to erase the 3 and
2. Edit the line to be trinomial(1,7,6),
and press ·.
b. Move to page 1.3, and arrange all
The dimensions of the rectangle are x + 1 and x + 6.
of the tiles to form a rectangle with
area x2 + 7x + 6.
What are the dimensions of the
rectangle?
c. How do the dimensions of the
1 + 6 = 7 and 1 x 6 = 6.
rectangle relate to the numbers 7
and 6?
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Factoring Trinomials Part 1
TEACHER NOTES
ALGEBRA NSPIRED: QUADRATICS
4. Press /
¡ to return to page 1.2,
and change the command for each
trinomial below. Factor each
trinomial by arranging all of the tiles
on 1.3 to make a rectangle. Verify
that your answer for each one is
correct by finding the product.
a. x2 + 5x + 4 =
(x + 4)(x + 1)
b. x2 + 4x + 4 =
(x + 2)(x + 2)
Teacher tip: You might want to ask students what is special about
the rectangle in b. Could you find other trinomials from the set
possible on page 1.2 whose factors would make a square?
2
5. The trinomial x + 6x + 4 cannot be
factored.
a. How does the .tns document
It is impossible to build a rectangle with the tiles shown.
illustrate that it cannot be factored?
b. How can the constant term be
Answers will vary. Depending on how they started to build the
changed so that the trinomial can be
rectangle, they might arrive at different but correct answers:
factored?
Change the constant to 8
Change the constant to 0
Change the constant to 5
Change the constant to 9
c. State your new trinomial, and
Answers will vary:
show its factors.
x2 + 6x + 8 = (x + 4)(x + 2)
x2 + 6x = x(x + 6)
x2 + 6x + 5 = (x + 1)(x + 5)
x2 + 6x + 9 = (x + 3)(x + 3)
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Factoring Trinomials Part 1
TEACHER NOTES
ALGEBRA NSPIRED: QUADRATICS
6. a. Find the factors of each of the
Trinomial
following trinomials.
2
x + 4x + 3
Factored Form
(x + 3 )(x + 1)
2
(x + 5)(x + 3)
2
(x + 4)(x + 5)
x + 8x + 15
x + 9x + 20
2
x + 12x + 20
(x + 10)(x + 2)
b. Explain how you know you have
Teacher Tip: Other than the first one, the rest of these problems can be
factored each trinomial correctly.
directly modeled on the handheld. You should circulate throughout the
room to help those students who are struggling either with the
mathematics or with the technology. You might also want to assign these
as homework problems. Be sure that students can articulate the fact that
the factors of the constant term must sum to the coefficient of the linear
term.
2
7. Suppose x + bx + c = (x + m)(x + n),
m + n = b and m  n = c
how are m and n related to b? How are
m and n related to c?
Wrap Up:
Upon completion of the discussion, the teacher should ensure that students are able to understand:

How to factor trinomials of the form x2 + bx + c with positive integer coefficients.

The relationship between an area model and factoring trinomials.

The relationship between b and c and the constant terms in the factors of a trinomial of this form.
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