Lesson Plan #3 for iPod’s in the classroom
Indicators:
KCK10MICA0220: Factor polynomials using GCF, factor by grouping, factoring a trinomial into the
product of two binomials, and factoring special cases (ACT).
Related Indicators:
KCK10MICA0201: Solve quadratic equations using the methods of factoring, completing the square, and
applying the quadratic formula (ACT).
Focus Indicator/Objective:
The student will factor polynomials using a variety of methods.
Level: Application
Class: Intermediate College Algebra
Prerequisite Skills:
Apply the distributive property
Apply the process of FOIL
Apply the order of operations
Apply long division and synthetic division
Classroom discussion:
Discuss the following questions with the class before beginning the lesson.
1. What does it mean for two people to have something in common?
2. What does it mean for two numbers to have a common factor?
3. Write on the board 𝑥 2 + 𝑏𝑥 + 𝑐 followed by 𝑥 2 + 6𝑥 + 5 and ask the following questions.
a. What is the coefficient of x?
b. What is the value of b?
c. What is the value of c?
Note: It may be a good idea to write a few other examples to help them understand how to identify a,
b, and c. Maybe even mix up the equation by putting the x first, etc.
Vocabulary:
Have students define the following terms. Be sure to give an example of each to help the students
visualize what the defined term actually looks like.
1. Absolute value - The distance of a number from zero on the number line; is never negative
2. Binomial - The sum of two monomials
3. Coefficient - A numerical factor in a term of an algebraic expression
4. Distributive property - A mathematical property that allows you to simplify an equation or
expression with grouping symbols
5. Exponent - The number of times the base is used as a factor 8
6. Factor - An integer that divides evenly into another integer
7. Factoring - To write an expression as a product of factors
8. FOIL - An acronym used to help remember the method for multiplying two binomials
together; First, Outer, Inner, and Last
9. Leading coefficient - The coefficient of the highest power term 4x
10. Monomial - An expression that is a number, a variable, or the product of a number and one
or more variables
11. Perfect square - Numbers whose square roots are integers
12. Polynomial - The sum of two or more monomials 5x
13. Term - An entry or element in a numeric sequence, or a number, variable, product, or
quotient in an expression; terms are separated by addition or subtraction
14. Trinomial - The sum of three monomials
Sum and Product Chart
Use the following to help the students understand some of the basic concepts of factoring.
Directions: Find the two numbers that add together to get the sum and multiply to get the product.
Sum
Product
Pairs
_____________________________________________________________________________________
4
4
_____________________________________________________________________________________
9
18
_____________________________________________________________________________________
20
100
_____________________________________________________________________________________
-5
4
_____________________________________________________________________________________
-2
-15
_____________________________________________________________________________________
0
-9
_____________________________________________________________________________________
16
48th
_____________________________________________________________________________________
1
¼
_____________________________________________________________________________________
Examples:
Factor the following. Work the following factoring problems with the students. Demonstrate how to do
the first problem in each different set and then let the students attempt the next one. Be sure to show
different methods of how to factor, particularly with the problems in the form of 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 such as
the Guess and Check Method and the Box Method.
1.
a.
b.
c.
Factor using the greatest common factor.
8𝑥 3 + 6𝑥
10𝑦 3 + 20𝑦 2 − 5𝑦
8𝑥 4 + 4𝑥 3 − 2𝑥 2
2. Factoring 𝑥 2 + 𝑏𝑥 + 𝑐 when c is positive.
a. 𝑥 2 + 9𝑥 + 18
b. 𝑥 2 + 10𝑥 + 24
3. Factoring 𝑥 2 + 𝑏𝑥 + 𝑐 when c is negative.
a. 𝑥 2 + 7𝑥 − 18
b. 𝑥 2 − 5𝑥 − 24
4. Factoring 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 when c is positive.
a. 6𝑥 2 + 19𝑥 + 10
b. 4𝑥 2 + 16𝑥 + 15
5. Factoring 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 when c is negative.
a. 3𝑥 2 − 4𝑥 − 15
b. 2𝑥 2 − 7𝑥 − 15
6. Factoring perfect square trinomials.
a. 16𝑥 2 + 40𝑥 + 25
b. 𝑥 2 + 12𝑥 + 36
7. Factoring difference of two squares.
a. 81𝑥 2 − 49
b. 9𝑝4 − 16𝑞 2
8. Factor by grouping.
a. 4𝑥 2 + 6𝑥𝑦 + 6𝑥 + 9𝑦
b. 4𝑥 2 + 12𝑥 + 5𝑥𝑦 + 15𝑦
9. Factor polynomials with a degree higher than 2.
a. 𝑥 3 + 8𝑥 2 + 21𝑥 + 18
b. 𝑥 3 − 𝑥 2 − 10𝑥 − 8
iPod Time:
Homework/Class work. Factor each polynomial completely.
1. 2𝑑3 + 4𝑑
2. 12𝑥 4 − 4𝑥 3 − 8𝑥 2
3. 𝑥 2 − 8𝑥 + 16
4. 𝑚2 + 20𝑚 + 100
5. 𝑛2 + 9𝑛 + 20
6. 𝑦 2 + 7𝑦 − 30
7. 𝑑2 − 6𝑑 − 7
8. 𝑐 2 − 10𝑐 + 24
9. 2𝑥 2 + 11𝑥 + 5
10. 4𝑔2 − 10𝑔 + 6
11. 6𝑝2 − 18𝑝 − 24
12. 12𝑑2 + 7𝑑 − 12
13. 9𝑥 2 + 12𝑥 + 4
14. 100𝑝2 − 25𝑞2
15. 25𝑚3 − 90𝑚2 − 40𝑚
16. 2𝑥 2 − 6𝑥𝑦 − 𝑥 + 3𝑦
17. 3𝑥 3 − 15𝑥 2 + 2𝑥𝑦 − 10𝑦
18. 𝑥 3 + 3𝑥 2 − 𝑥 − 3
19. 𝑥 3 − 3𝑥 2 − 18𝑥 + 40
20. 𝑥 4 + 6𝑥 3 + 𝑥 2 − 24𝑥 + 16
21. 𝑥 4 + 10𝑥 3 + 33𝑥 2 + 44𝑥 + 20
Now it is time for the student to get out their iPod and open up the application EasyFactor Quadratics.
It is an application that will assist them in checking their answers to the above problems. That is why it
is extremely important they work the first set of problems without the use of this application so they will
know how to work the problem on their own. It is really great because it give the students step by step
details of how to find the answer. All you have to do is use the wheel at the bottom of the screen to put
in the numbers/coefficients needed for the problem you are trying to solve. Then you use the far right
side of the wheel to move from step 1, the original problem, to the final step which is some cases could
be step 8. Hopefully, the students will not scroll ahead to final step to find the answer but instead will
attempt to work the problem first or if they get stuck, they can use the steps provided to assist their
efforts. The application also includes some of the terminology used with quadratics such as
explanations of coefficients, trinomials, terms, etc. The only problem with this application I could
identify was that it does not factor by grouping which would be needed to solve #16-21. However, after
showing the students about synthetic division or long division, there may still remain a factorable
expression that could still be used with this application. I think it is also necessary to note that the
customer reviews are not always accurate. I think they may be written by college students who want
the answers immediately without caring about the process of how to actually achieve the answer.
Although, with this application, I found it also did give the answers, except, as mentioned earlier, those
that needed to be factored by grouping or by synthetic division. All in all, a very good and useful
application. Factoring, from my experience in my school district, seems to be one of the more difficult
concepts to master, even if the student has been taught the skill in 2 or 3 previous years. It is also one
of the more important skills for students to master which is why an application like this would be very
good for them have.