Unit Plan: Factoring

Subject:        Algebra 1

Unit:             Using Factoring [Chapter 10]

Lesson:        Factoring Trinomials [Section 10-3]

Book:           Glencoe. Algebra 1. McGraw-Hill: New York. 1998. Chapter 10

Materials:   Paper, Pencil, Chalk Board, Calculator, Index Cards, & Tape

Teacher:      Mr. Josh Biber

Grades:        9

 

Chapter 10: Using Factoring (Goals and Objectives of each section in the chapter)

v     Lesson 10.1

Ø      Finding the prime factorization of integers.

Ø      Finding the greatest common factor (GCF)

v     Lesson 10.2

Ø      Using the GCF and the distributive property to factor polynomials

Ø      Using grouping techniques to factor polynomials with four or more terms

v     Lesson 10.3

Ø      Factoring quadratic trinomials

v     Lesson 10.4

Ø      Identifying and factoring binomials that are the differences of squares

v     Lesson 10.5

Ø      Identifying and factoring perfect square trinomials

v     Lesson 10.6

Ø      Using the zero product property to solve equations

 

Standards of Learning [Chapter 10]

A.11  The student will add, subtract, and multiply polynomials and
      divide polynomials with monomial divisors, using concrete
      objects, pictorial representations, and algebraic
      manipulations.
 
A.12  The student will factor completely first- and second-degree
      binomials and trinomials in one or two variables.  The
      graphing calculator will be used as both a primary tool for
      factoring and for confirming an algebraic factorization.
 
A.14  The student will solve quadratic equations in one variable
      both algebraically and graphically.  Graphing calculators
      will be used both as a primary tool in solving problems and
      to verify algebraic solutions. 
 
A.16  The student will, given a rule, find the values of a function
      for elements in its domain and locate the zeros of the
      function both algebraically and with a graphing calculator. 
      The value of f(x) will be related to the ordinate on the
      graph.

Outlook and Overview of Unit Lesson

 

Mathematical Topics, Summary, & Objectives

            The main focal points of this unit in Chapter 10 will focus around the concept of factoring. Students will begin by reviewing what it means to have common factors, and more so, the greatest common factor. This will hopefully guide the student’s thought process into grouping and being able to pull out common terms and factors. From there the students should be able to recognize certain characteristics in factoring such as trinomial factoring and grouping.

Since the concept of factoring often stumps students due to the ability to almost “luckily” see the combination of numbers at first, I have chosen to do my unit plan on that very important skill. One of the main components to this lesson will test the student’s ability to make educated guesses based on the knowledge they have from factors of various numbers and terms. As always, I would also expect my students to provide a check in terms of “FOIL” and other methods to insure success!

 

Materials & “The Bigger Picture”

            The materials needed for this unit of planning should require no extra materials other than those that are (generally) readily available in any classroom. This would include the following: pencil, paper, chalk/dry-erase board, overhead projector, calculator, index cards [for card activity], and tape.

            Although this lesson in Chapter 10 can vary from class to class, I can see this lesson taking about 2-3 block days or 4-6 regular class periods. I have only made a lesson plan for section 10-3 because I feel it is the most important in the section. The factoring in the previous sections should be a review from Pre-Algebra; however, the factoring of trinomial equations should be brand new to the children. Since sections 10-4 through 10-6 correspond to the principles used in 10-3, I chose only to focus my notes for this unit plan on that. [Plus, that was my understanding of the assignment for the unit plan.]


Supplementary Materials, Activities, & Assessments

            There should be no apparent need for extra materials such as transparencies or handouts. The lesson is one which notes will be taken by the student, but more importantly the students will be practicing factoring alone or in small groups.

            The activity, “21 Card Pick Up” is a blessing for those teachers who need a little bit of piece and quite. At least for a small amount of time! The activity can be given for practice or even as a class quiz grade if needed. I liked to use it as a more interactive way of getting students up and out of their seats doing things.

            The formal, written assessment is designed to test whether or not the students understand the concepts and steps in factoring as well as logical thinking. I have attempted to leave ample room for scratch work if I child desires to use it. This assessment would not be effective in a classroom where the teacher does not use a “Scissors Box” even though it’s basically the same concept of guess and check.

 

Student’s Expectations

            Throughout the entire lesson (same as for the entire year) the students will be expected to be participating in the lessons and the instruction. I like to leave my lesson plans very vague and open in order to adjust to the flow of the class. I like to allow students the opportunity to modify my lesson if they have a better example; I feel as though this helps the learning process because it gives them a more personalized meaning.

            In terms of formal responses, I feel as though this lesson will have many mistakes from students, and I feel as though that is a good thing. Since students will be experimenting with the different factors of trinomials, I do not anticipate a student getting the correct answer the first time every time. I feel as though checking oneself is crucial in this entire unit (as well as for the year).

An example of this would be having a student factor x2 - x - 12. Upon [most likely] finding an answer of (x-3)(x+4) to be the factors, I would then see their check by “FOILing” back to reveal that the answer they found to not be the correct solution. So I would expect the student to show the checking work and show me that their solution will not FOIL back into the original answer. Then I would like for them to show the correct solution with a FOIL check.


NCTM Standards

  1. Problem Solving – By using “educated guesses” the students will be able to use problem solving skills in order to see the differences in how changing certain numbers and letters will affect the final outcome and solution.
  2. Reasoning & Proof – The students should logically be able to reason through a difficult factoring problem by applying the basic concepts they know about multiplying binomials [i.e. a negative times a negative equals a positive]. By actually multiplying the factors back out, the students will be able to actually prove they are correct.
  3. Communications – Communication becomes crucial when the students will be working in the 21 Card Pick Up activity. Since the students will not be allowed to talk at first, they must rely on effective non-verbal communication skills in order to be successful in communicating exactly what they are looking for.
  4. Connections – The students will need to be able to connect their prior knowledge of composite [having factors] and prime [no factors] numbers in order to have an easier time with the lesson. If a student knows that a certain number only has 2 factors, then that will allow them to solve the problem more quickly.
  5. Representation – The best thing about this lesson is that now students will have another way to represent longer, more drawn out polynomials with their knowledge acquired from this section. This should help the students recognize that although a problems looks long and difficult does not mean that the same problem can be written in a more, easily understood form.

 

 

 

Standards of Learning [SOL]

            The exact SOL’s have been listed at the very beginning of the lesson plan. Please refer to the above sections for more information and detail.


Section 10-3: Factoring Trinomials

v     Review of multiplying like signs gives positive; different signs gives negative.

Ø      Examples

§         (3) (9) = ____

§         (3) (-9) = ____

§         (-3) (9) = ____

§         (-3) (-9) = ____

Ø      (+) (+) = +

Ø      (-) (-) = +

Ø      (+) (-) = -

 

v     Review of FOIL method for multiplying binomials together

Ø      Ex. (x+3) (x-5) = ???

§         First

§         Outer

§         Inner

§         Last

Ø      Ex. (x+3) (x-5) = x2 –2x –15

 

v     Check for any questions or comments.

Ø      Now allow students to FOIL (2x+3)(x-5)

§         Check to examine whether students recognized the different signs made the non-variable term a negative.

§         Make sure students understood 2x * x = 2x2

§         Go over steps with students including 2x2 + 3x – 10x – 15. Then combining like terms to get 2x2 –7x – 15.

 

v      Have a group discussion to discuss WHY the middle term is negative. Ask if a students can change anything about the original binomials in order to make the “7x” term positive.

 

v     Now take the example that the students just completed [2x2 –7x – 15] and break down the factoring process

Ø      Steps to Success in Factoring Trinomials

1. Arrange Trinomial in ORDER from highest degree

2. Draw double curves [D.C.] under the problem. Ex. (     ) (     )

3. Is the last term [NON-VARIABLE TERM] positive or negative?

v     If Negative…

Ø      Put opposite signs in the double curves. Ex. (   +   ) (   -   )

v     If Positive…

Ø      Look at middle [1st DEGREE TERM] for positive or negative

Ø      If that’s positive, then put double +’s in the D.C.’s Ex. (  +  ) (  +  )

Ø      If that’s negative, then put double –‘s in the D.C.’s Ex. (  -  ) ( -  )

 

v     Now check to make sure the students understand. Allow questions and any clarification

 

v     Now we will make a “Scissors Box”

Ø      Here’s an example to start… (x2 + 8x + 15)

Note: Explanation with visual is easier to understand than reading off notes.

                        1                      ??         =

                        1                      ??         = ___

                                                                +8

 

Ø       Now this of the different factors for the number 15 [1,3,5,15]

1                      +3          = +3

                        1                      +5          = +5

                                                            +8

Ø      Now just follow the coefficients across while placing the variable in.

(1x + 3) (1x + 5) = (x2 + 8x + 15)

§         Remember to FOIL back out. Also, does this fit the “Steps to Success”?

 

 

Ø      Now here’s the same example showing negatives… (x2 - 8x + 15)

1                      -3           = -3

                        1                      -5           = -5

                                                                  -8

§         Thus, we have (1x – 3)(1x – 5) = (x2 - 8x + 15)

 

Ø      More examples…  (x2 - 2x - 15)

1                      +3          = +3

                        1                      -5           = -5

                                                            -2

Ø      More examples…  (x2 + 2x - 15)

1                      -3           = -3

                        1                      +5          = +5

                                                            +2

 

§         Note: For simplicity reasons the first few examples will not have coefficients in from of the second degree term at first, then slowly coefficients will be added, however, this will not affect the process of the “Scissors Box”

 

v     Comments, Questions, Concerns…

 

 

 


Factoring Trinomials Activity: “21 Card Pick Up”

Objective:      

The goal of this SILENT activity is for students to see how different trinomials only have certain solutions for factoring.

Directions:

      - Students will randomly draw a card from the stack of 21 cards face down

      - Their card will either be of a binomial form or of a trinomial form.

      - Using their knowledge of factoring trinomials, the students must then locate the other two members in the class that make their card fit successfully in a factorization.

      - The catch is 3 cards and ONLY 3 cards fit together! So (for example) only Bill’s card can ONLY be factored into Steve’s and Dave’s card. Caroline’s card times Bill’s card will not equal anyone else’s card but Scott’s. Caroline’s card times Steve’s card will not yield a trinomial given the 21 card pick up!

      - By using their knowledge of characteristics of trinomials students with trinomial cards can soon realize if their classmates can possibly even fit their card. That is, Dave (x2 + 4x + 3) should be able to recognize that anyone with a negative sign in his or her binomial card will not help her factor in any way.

      - Accommodations can be made to fit the appropriate class size.

Completion:

      - Upon finding a set of 3 cards that work together, the group must write their cards up on the board to show a “FOILed” form.

      - After the teacher has checked the solution for accuracy, the three members must tape their cards to their bodies and link arms together. From this point on they must remained linked together as a “Factorized Problem” would appear.

      - Fortunately, this group will NOW be allowed to talk in order to help other people find their solutions (of course they must remain linked, and the other students must still remain silent until their solutions have been found).

      - Upon completing every set of trinomials [7 total] the class will then take their seats and as a group review all of the 7 equations found while answering any questions or thoughts.


Assessment/ Quiz: Factoring

Example of possible answers are provided in blue.

 

I. Complete the following:

 

      a). a2 + a – 30 = (a – 5)(a _+_ 6)  

      b). x2 __ 6x + 8 = (a - 4)(a - 2)     

      c). a2 + 7a + 12 = (a + __)(a + __)

      d). 36 – 12x + x2 = (x - _6_)(x _-_ 6)

 

II. Complete the “Scissors Box”

 

      a). Factor: a2 - 14a + 40

                        1                      __          =

                        1                      __          = ___

                                                                 -14

      b). Factor: 2x2 + x - 21

                        1                      _-3_       = -6

                        2                      _7_        = 7

                                                                 +1               (2x+7)(x-3) = 2x2 + 7x – 6x -21

                                                                                                2x2 + x - 21

III. Factor each trinomial

 

      a). x2 - 11x + 24 =                                                 (x-3)(x-8) =  x2 – 3x – 8x + 24

                                                                                                x2 - 11x + 24

      b). a2 + 3a – 180 =

 

      c). 2x2 - 5x – 12 =                                                  (2x+3)(x-4) =2x2 + 3x – 8x - 12

                                                                                                2x2 - 5x – 12

      d). -35 – 4a  + 4a2 =