Age / Grade Level:
10th Grade
Subject:
Algebra II
Number of Students:
25
Number of Students with IEP/504 Plans: 3
Major Content Area:
Simplifying Algebraic Expressions
Title (if any):
Multiplying Binomials with the FOIL Method
The students will learn and/or review two procedures for multiplying binomial
expressions—one using real number properties directly and the other using the
so-called FOIL method. In this lesson, the students will be
able to:
Ø Apply
the commutative, associate, and distributive properties to solve simple
binomial expressions.
Ø Evaluate
how the commutative, associative, and distributive properties can be used to
solve binomial expressions that include variables.
Ø Construct
a formula based upon this understanding.
Ø Utilize
said formula to solve binomial expressions.
The objectives listed above are directly observable through a series of
problems sets that asks the students to solve binomial expressions utilizing
real number properties and then to evaluate what has occurred in each of those
problems. As a class, the instructor leads a discussion that builds the
formula, which the students will use in a subsequent (the third) problem set.
These problem sets are also an assessment tool, as is the chapter test that
will follow the series of lessons in this chapter.
Standards
In this lesson, the students are asked to apply the commutative, associate, and
distributive properties to create their own formula for solving binomial
expressions with and without variables. These real number properties meet
MA-H-1.3.2, because the students will use these commutative, associative, and
distributive properties to solve simple binomial expressions without
variables. Through this process, the students will use these mathematical
ideas to solve their own problems—i.e., solving binomial expressions without
variables—by creating their own formula, which pushes them to understand
mathematical structure concepts at a much higher level. This directly
addresses KY KERA 1.5-1.9 and 2.12. In the end, their creation of this
formula will allow them to solve one-variable equations using a specific
procedure known as the FOIL method, thus reaching M-H-A-1.
This lesson is an integral part of linking mathematics concepts without
variables and mathematics concepts with a single variable. While the
warm-up activities will appear elementary to the students—since the
commutative, associate, and distributive properties will have been reinforced
over the several weeks and months prior to this lesson—students nonetheless
struggle with binomial expressions that include variables. As such, this
lesson illustrates how the foundation created by real number properties makes
single variable binomial expressions solvable. This is important for the
students, as solving quadratic equations—the next major topic in this school
year—is essential for topics such as bridge building, object-propulsion, and
the like.
Ø Overhead
projector
Ø Whiteboard
Ø Word
processor document to illustrate the problems as a handout or on the overhead
(See Appendix)
Ø Word-processed
document to illustrate the FOIL method procedure
II.
(15 minutes) After ten minutes, have the
students volunteer to show on the board how they solved each problem.
a. Some
students will have solved the addition in the parenthesis first and then
multiplied
b. Other
students will have distributed the multiplication over the addition expression.
c. If
one of these two methods were not utilized, question the students whether they
can think of any alternative methods. This will lead them to solving the
problems on their own.
III.
(15 minutes) Once the students see the two options
available to them (from a and b above), ask the class if they notice anything
about problems five and six as compared to one through four.
a. Lead
them to the observation that problem five [(2+3)(3+9)=60] is the sum of
problems two [3(2+3)=3(2)+3(3)=15] and three [9(2+3)=9(2)+9(3)=45]. In
other words, 15+45=60.
b. And
that problem six [(3+7)(2+7)=90] is the sum of problems one
[2(3+7)=2(3)+2(7)=20] and four [7(3+7)=7(3)+7(7)=70]. In other words,
20+70=90.
c. In
both situations, write out the equalities above and show how one expression is
being distributed over the other, much like the distributive property may have
been used in the first four problems.
IV.
(5 Minutes) Review problems seven
[(4+8)(3+6)]=4(3+6)+8(3+6)=4(3)+4(6)+8(3)+8(6)=108 and eight
[(5+6)(8+3)]=5(8+3)+6(8+3)=5(8)+5(3)+6(8)+6(3)=121 and illustrate how the
distributive property could have been used to solve them if they students have
not already done so in their words.
V.
(10 minutes) Ask the students how they might
use the associative, distributive, and commutative properties to solve an
expression like (x+7)(x +3).
a. Note
to the students that they do not have the option here to perform the addition
within the parenthesis before multiplying, so they have to use another
method.
b. This
leads to (x+7)(x+3)=x(x+3)+7(x+3)= x2 + 3x + 7x + 21
c. Before
combining like terms, show them how these four terms relate to the FOIL
acronym—F=x2; O=3x; I=7x; L=21
d. Use
the diagram below to help the students understand the relationships.
When multiplying two binomials, multiply the (F)irst terms, then the
(O)utside terms, then (I)nside terms, and finally the (L)ast terms.
( x + 3 ) ( x + 2 )
F
First terms
x
( x )
x2
O Outside
terms
x ( 2 )
2x
I
Inside terms
3
( x )
3x
L
Last terms
3 ( 2 )
6
( x + 3 ) ( x+2 ) = x2 + 2x + 3x + 6 = x2 +5x + 6
VI.
(10
Minutes) Present the students with the “Guided Practice Problems” (from the
Appendix) and ask the students to work through them on their own.
VII.
(5 Minutes) Query the students as to how they
answered the questions and go over the answers.
a. If
they did not choose to use FOIL, after they have solved the problem their way,
use FOIL= to solve the same question and compare the answers and amount of work
involved.
VIII.
(5 Minutes) Wrap-up
a. Explain
that FOIL is simply a process to help them remember the different products
necessary when multiplying binomials through the distributive property of
equality.
b. When
the terms within the expression are all numbers, it was easier or less work to
do the addition first and then multiplying the sums, thus leaving less reason
to use the distributive property or FOIL.
c. When
dealing with binomials, though, the addition cannot be done within the
parenthesis, and therefore we must distribute. FOIL helps in that process.
d. Handing
out the “Independent Practice Problems” as homework
The early parts of this lesson, while requiring active individual investigation,
force the students to be actively engaged in the discussion, else it is not
possible to perform the resultant steps in the guided and independent practice
sections. Additionally, the most disruptive students often enjoy
board-work, as they enjoy the attention, which will help keep them actively
involved. Furthermore, this lesson addresses several learning styles,
including the verbal-linguistic, logical-mathematical, and bodily-kinesthetic
intelligences. One could add the inter- and intra-personal intelligences
by doing any of the problem sets as groups. Those who explain their
thought processes, however, are automatically engaging this learning style.
This lesson is unique as throughout there are several ways to solve these
problems, and those at different levels of mathematics mastery are able to
participate freely. However, it allows those who are more advanced, to
help develop a formula that those who are less advanced will likely just
memorize. Nevertheless, the exposure to and ‘building’ of the ideas
behind the FOIL method is clearly a higher-level activity that forces students
to apply what they have already learned in a unique manner.
Throughout this process there are several ways to question the students.
Some include:
Ø What
properties do you already know that can be used to solve these problems
(Knowledge)?
Ø What
alternate methods can be utilized to solve these problems (Synthesis)?
Ø Solve
the following problems using any technique we have discussed in class
(Application)
Ø Examine
the differences and similarities between problems two and three (Analysis).
Ø Create
or design a step-by-step process to solve similar problems (Synthesis).
Finally, it is important to note that LEP, IEP, ESE, ESOL students will be
allowed more time to answer questions during the discussion and will be given
extra time to complete their five individual seatwork and their homework
problems.
As stated in the earlier sections, the students’ progress is evaluated through
several formative assessments. The instructor will be constantly taking
anecdotal notes on the students’ progress as he/she moves around the room
during the student’s guided practice and warm-up activities. As the
discussion begins, the instructor will have additional opportunities to do
so. The guided and independent practice sets are formative
assessments. At the end of this chapter, there will be a summative
assessment including this and related content.
2 ( 3 + 7 )
3 ( 2 + 3
)
9 ( 2 + 3 )
7 ( 3 + 7 )
( 2 + 3 ) ( 3 + 9 )
( 3 + 7 ) ( 2 + 7 )
( 4 + 8 ) ( 3 + 6 )
( 5 + 6 ) ( 8 + 3 )
Guided Practice
Problems: Simplify each of the following binomial expressions.
( 2 + 2 ) ( 3 + 2
)
( 7 + 6 ) ( 8 + 3
)
( 9 – 4 ) ( 10 + 2
)
( 11 - 3 ) ( 7 - 4
)
( x + 4 ) ( x + 3
)
( x + 6 ) ( x - 2
)
( x - 5 ) ( x - 4
)
( x - 4) ( x + 7
)
Independent Practice
(Homework): Simplify each of the following expressions.
( 8 + 6 ) ( 11 – 3
)
( 7 + 2 ) ( 13 + 3
)
( 9 - 2 ) ( 13 - 7
)
( 5 - 7 ) ( 7 + 6
)
( x + 5 ) ( x + 2
)
( x + 5 ) ( x - 7
)
( x - 3 ) ( x + 8
)
( x - 11 ) ( x – 2
)
( 2x - 6 ) ( x + 5
)
( x + 3 ) ( 3x - 5
)