Materials:
Algebra Tiles and TI-83 Plus Calculator. AMSCO Math A Chapter 18 Factoring. All
mathematics material and homework as signments are from that textbook .
Description:
This 5-day unit is about factoring. We start out with a review of the
multiplication of
algebraic expressions, and then we move into division of algebraic ex pression as
a
transition to factoring. Algebra Tiles is heavily used and toward the end of the
lesson,
TI-83 plus calculator will also be used. In addition to technology and
manipulatives
being used in this unit, there are several writing prompts for students to
reflect what they
learn and expand their thinking and development of the concepts.
Day 1: Multiplication of algebraic expressions
Day 2: Division of algebraic expressions (Part I)
Day 3: Division of algebraic expressions (Part II)
Day 4: Factoring polynomials (Part I)
Day 5: Factoring polynomials (Part II)
Day Objectives:
1) Student will be able to competently multiply algebraic
expressions.
2) Student will be able to use algebraic tiles to divide algebraic expressions.
Zero
pairs are used is Day 3.
3) Student will be able to factor polynomials, and compare how multiplication
and
factoring are related.
NCTM Standards (9 – 12): Represent and analyze mathematical situations and
structures
using algebraic symbols.
1) understand the meaning of equivalent forms of expressions
2) use symbolic algebra to represent and explain mathematical relationships
3) build new mathematical knowledge through problem solving
4) apply and adapt a variety of appropriate strategies to solve problems
5) monitor and reflect on the process of mathematical problem solving
NYS Math A Core Curriculum: Key Idea 3 Operation
1) Use multiplication and division with real numbers and algebraic expressions.
2) Multiplication of polynomials: powers , products of monomials and binomials
3) Division of polynomials by monomials
4) Factoring: common monomials , binomial factors of trinomials
LESSON 1: PRODUCT OF TWO BINOMIALS
This lesson is designed for students to review the product of two binomials.
Specifically,
students in my school building are term s.html">likely to have been taught to find the
product using
FOIL instead of using algebra tiles. Therefore, this lesson gives the students
an
opportunity to develop ability using algebra tiles. Zero pairs are emphasized is
this
lesson. This lesson is the first of the series, preparing students for the
upcoming lessons
on factoring.
Anticipatory Activity:
This lesson begins with a question/challenge. Consider how to find the area of
the
following extendable square bracket . Before extension, it is a square, and for
the
moment, let’s define the measurement of the side of this square shape bracket x
cm. The
sides of the square bracket can be extended and after extension, the bracket is
rectangular
with its length being maximally extended by 3 cm and its width by 2 cm.
Problem 1: What is, in an algebraic expression, the area of the original
square bracket?
Problem 2: What is the new area of the maximally extended bracket?
How would one represent this area in an algebraic
expression?
Hint: Consider the area of the extended rectangle to be
the total of four smaller areas:
When students are struggling, with the area of the
extended rectangle, the hint is useful to
remind the students. Also, emphasize that rectangles have two pairs of congruent
sides.
After the students work on the two problems, the discussion of the solution is
used to
introduce to them on the product of binomials.
Developmental Activity:
This activity requires students to develop the skills to find the product of two
binomials
using algebra tiles. Teacher will illustrate the following examples to the
students:
Example 1: Express
 |
as a trinomial. |
 |
This example is directly from the problem of the
day in
the anticipatory activity. This example is useful because
students could relate from the previous activity to this
lesson. |
| Example 2: Express
|
as a trinomial. |
 |
This example shows how algebra tiles could
represent
positive and negative numbers. Zero pairs emphasized
here. |
Example 3: Express
 |
as a trinomial. |
 |
This is another example showing how algebra tiles
could represent positive and negative numbers and
how zero pairs play a role here. The commutative
property of multiplication is emphasized in this
example. |
Example 4: Express
 |
as a trinomial. |
 |
This example serves two purposes: 1) is to
acquaint student with the product of two
negative numbers, and 2) is to show how
algebraic tile represents 
term. |
Example 5: Express
 |
as a binomial. |
 |
This example shows that not all product of
binomial
results in a trinomial. |
After the examples are shown to students, students are to
try out, using algebra tiles, the
following problems. Students are given time to work on those problems. Teacher
is to
go around to help out struggling students. The concept of zero pair will be
emphasized.
Selected students are to draw their solutions on the board.
Sample problems for try out: Express the following five
expressions as polynomials:
Closing Activity:
This activity is to mirror the anticipatory activity with different extensions,
with a built-in
writing component for student to reflect on what they have developed in this
lesson.
Consider how to find the area of the following extendable square bracket. Before
extension, it is a square, and for the moment, let’s define the measurement of
the side of
this square shape bracket x cm. The sides of the square bracket can be extended
and after
extension, the bracket is rectangular with its length being maximally extended
by 3 cm
and its width by 2 cm.
Problem 1: What is, in an algebraic expression, the area of the original
square bracket?
Problem 2: What is the new area of the maximally extended bracket?
How would one represent this area in an algebraic
expression?
Problem 3: How did you come up with these algebraic expressions?
Write a paragraph explaining your algebra tile model. Does
zero pair apply in this question? Why or why not?
Homework assignments will be given.
Sample Algebra Tiles
LESSON 2: DIVISION OF ALGEBRAIC EXPRESSIONS (PART I)
This lesson is designed to prepare students for factoring through a transition
of division
of algebraic expressions. To develop division, this lesson continues to employ
the
algebra tile model for the reasoning. To not complicate the subject matter, the
division
that will be developed here will not involve any remainder. Zero pairs will not
be
introduced until Lesson 3. By the end of the lesson, student will be able to
divide simple
polynomial, using algebra tiles as a tool.
Anticipatory Activity:
This lesson begins with a review from the previous lesson. This review serves
three
purposes: to reacquaint students with the algebra tile model, to acquaint
students with
three expressions which will be used in the developmental activity, and to
provide an
opportunity for the teacher to informally evaluate students on the previous
lesson’s
material. Consider the following three expressions:
, 
,
and
. How are they presented using algebra tiles?
Developmental Activity:
This activity requires students to develop the division skills using algebra
tiles. Teacher
will illustrate the following examples to the students:
| Example 1: Find the quotient which is divided by
3. |
|
 |
The problem is extremely useful as the first
example because
one can actually divide
into three equal portions. The
discussion of this example is twofold: one is to show the
students that the meaning of division could be extended to
division of algebraic expressions, and two is to reason out that
if 3 represents the width of a rectangle with area of 3x + 6,
then x + 2 must be the length of the rectangle |
Example 2: Simplify
 |
 |
This problem is deliberately worded
differently from the first example so that students
can get used to the variety of instruction. The premise of this problem
is similar to the
first example of this lesson: one is to show the students that the
meaning of division
could be extended to division of algebraic expressions, and two is to
reason out that if 2
represents the width of a rectangle with area of 8x – 6, then 4x – 3
must be the length of
the rectangle.
Example 3: Find the unknown side of a rectangle if one side is x – 3,
and the area of the
rectangle is |
 |
Again, this example uses a different set of
instruction. This
example departs from the original two examples that
division has to do with dividing something into equal
portions, and this example has to do with the area of a
rectangle and given a measurement of one side, the
measurement of the other side can be found through
division. |
Example 4: Divide 
by
How will these tile fit
into the rectangle?
to the question.
Students are to try out the following divisions in class:
1) Find the quotient: 
.
2) Simplify:
3) Find the unknown side of a rectangle if one side is expressed as x + 3, and
the
area of the rectangle is 
4) Divide by
.
The problems for students to try out are similar to the examples given earlier.
The
instructions are varied to acquaint the students that all of the “different”
instructions
shares similar mathematical meanings. Teacher is to help students out at this
portion of
the developmental activities. This is also a time for the teacher to evaluate to
what extent
students have developed the material up to this point.
Closing Activity:
This activity is to bring what is developed in this lesson to a closure, with a
built-in
writing component for student to reflect on what they have developed in this
lesson.
Students are to write in their journal in their own words, answering the
following
questions:
1) What did you learn today?
2) Explain how you come up with the answer in question number 3?
3) Did the algebra tiles help?
4) In what way did you have to organize tiles for question 3 to form a
rectangle?
Homework assignments will be given.
LESSON 3: DIVISION OF ALGEBRAIC EXPRESSIONS (PART II)
This lesson is designed to further prepare students for factoring through
division of
algebraic expressions. In addition to the skills introduced in lesson 2, the
division in this
lesson will include problems dealing with zero pairs. By the end of this lesson,
student
will be able to divide using algebra tiles, including using zero pairs as a
strategy.
Anticipatory Activity:
Student will take a quiz (not included here) as to check the concept development
so far.
Questions are based on division without using zero pairs. Solution will be
discussed to
ensure all students are ready to move on to division with zero pairs.
Developmental Activity:
Example 1: What is the length of the rectangle with an area represented as
and the width is represented by x + 1?
Example 2: Simplify 
Using both examples 1 and 2, a discussion will be held
about
zero pairs. Questions such as the following will be used:
1) In example 1, it seems that there are -4x instead of
-3x. Is the algebraic tile representation correct?
2) Similarly in example 2, it seems that there are -3x instead of -2x. How is
-2x
accounted for?
Students are to try out the following divisions in class:
1) Divide 
by
.
2) Simplify:
3) Find the unknown side of a rectangle if one side is expressed as x + 6, and
the
area of the rectangle is
.
When students complete all three problems, they may present their solution on
the board.
Reinforcement of the concept about will take place during student presentation.
Immediately following the presentation will be a closing activity.
Closing Activity:
Students will be posed with two intriguing problems. The closing is used as a
preview
for lesson 4.
Suppose there are two expressions each representing an area of a rectangle. Find
their
dimensions of the rectangles:
1) 
2)
The problems selected here is to intrigue students to think hard about what is
about to
come in lesson 4. In this case, they are both factoring problems, and they look
very
similar. I expect students to be able to work on problem 1, but they may
struggle with
problem 2. Homework will be to continue working on those two problems.
Additional homework will be assigned.
LESSON 4: FACTORING BINOMIALS AND Trinomial
where A = 1
This lesson is to genuinely develop factoring, without the assisted help on
division.
Problems used in this lesson are binomials and trinomials. The goal for this
lesson will
be that student leaving the classroom with factoring skills, and be able to
check their
result. This is the lesson that achieve the unit goal.
Anticipatory Activity:
A discussion about the previous lessons closing will be conducted. Recalling:
Suppose there are two expressions each representing an area of a rectangle. Find
their
dimensions of the rectangles:
1)
2)
Students present their solutions. Are the solutions “very” similar, just as
similar as the
questions? Why are the solutions so different? This activity is designed to lead
students
to think about how to factor without letting the appearances “fooling” them.
Developmental Activities:
The solutions will be worked out for the above two problems: first, using
algebra tiles,
and second, using the table function in the TI-83 Plus calculator:
At the y= menu: Enter 
and
. Table starts at x = 0
with increasing interval of 1. Student can check with the results that y1 equals
y2.
Repeat the same process for problem 2. Enter y1 as the trinomial form and y2 as
the
factored form. Check with the table function.
Students are to work on the several problems below in two different ways: first,
using
algebra tiles and second, one using TI-83 Plus to check the result.
Factor the followings:
1)
2)
3) 
4)
5) 
6)
I selected those six problems because two of them are
binomials; two pairs are similar to
the extents that the only the signs are different. One pair is a perfect square ,
which serves
as a discussion piece in closing activities.
Closing Activities:
Student will present the problems on the board. Three students per problem,
working on
1) the algebraic solutions, 2) the algebra tile solution, and 3) check the
solution using
the TI-83.
This lesson ends with a writing activity. The 2 prompts are as follows:
1) In the above six problems, questions 2 and 3 are considered perfect squares,
and
questions 1, 4, 5, and 6 are not. Examine your algebra tiles solutions, and
explain
in your own words, why some of them are perfect squares and why some of them
are not.
2) Of the three portions the factoring process, working out algebraic solutions,
working out algebra tile solutions, and checking solutions using a table. Which
method you enjoy the most, and which you enjoy the least ? Give some reasons to
support your response.
Homework will be assigned.
LESSON 5: FACTORING BINOMIALS AND Trinomial
This lesson ends the unit by finally develop student to factor general binomials
and
trinomials. It is not intended to be a heavy lesson as some of the students may
need to
catch up with the skills involving algebra tiles and using TI-83 plus. The
format is
similar to the previous lesson. Students are still working in a group of three,
with each
factoring problems. The only difference is that problems here will involve
trinomials in a
form of where A does not necessary equal to 1. By the end of the lesson,
students will be able to factor skillfully.
Anticipatory Activity:
A reflection to the previous lesson’s writing activity. Student will read out
loud their
writings. This is a time to review the previous lesson and to let students to
hear others
comments. Homework questions and answer session will be held at this point.
Developmental Activities:
In the activity, the students working in groups will be facilitated by the
teacher, working
on the following factoring problems:
1) 
2) 
3)
4)
5)
Setting remains the same as the previous lesson: one student works on the
problem, one
student works on algebra tiles, and one student work the table feature on TI-83
Plus.
When the solution is acceptable to all three students, put the solution up on
the board, and
students are responsible for presenting the solution.
The developmental activity is meant to be not as instructionally heavy as the
other lesson
because some students may need to catch up with various skills.
Closing Activity:
Finally, to finish this unit, students would be exposed to some binomials and
trinomials
that are not factorable. The writing prompt will be as follows:
Consider 
or
, they are not factorable using algebra tiles. If you don’t
believe it, try arranging the tiles in several different attempts. Why aren’t
they
factorable? Can you name 5 factorable and 5 unfactorable polynomials? Be prepare
to
bring them in for the next class as we will play factoring marathon.
No Homework will be assigned. Student will continue working on the prompt.
