# Unit 12: Introduction to Factoring. Learning Objectives 12.2

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1 Unit 1 Table of Contents Unit 1: Introduction to Factoring Learning Objectives 1. Instructor Notes The Mathematics of Factoring Teaching Tips: Challenges and Approaches Additional Resources Instructor Overview Tutor Simulation: Playing the Elimination Game Instructor Overview Puzzle: Match Factors Instructor Overview Project: Making Connections Common Core Standards 1.43 Some rights reserved. See our complete Terms of Use. Monterey Institute for Technology and Education (MITE) 01 To see these and all other available Instructor Resources, visit the NROC Network. 1.1

2 Unit 1 Learning Objectives Unit 1: Factoring Lesson 1: Introduction to Factoring Topic 1: Greatest Common Factor Learning Objectives Find the greatest common factor (GCF) of monomials. Factor polynomials by factoring out the greatest common factor (GCF). Factor expressions with four terms by grouping. Lesson : Factoring Polynomials Topic 1: Factoring Trinomials Learning Objectives Factor trinomials with a leading coefficient of 1. Factor trinomials with a common factor. Factor trinomials with a leading coefficient other than 1. Topic : Factoring: Special Cases Learning Objectives Factor trinomials that are perfect squares. Factor binomials in the form of the difference of squares. Topic 3: Special Cases: Cubes Learning Objectives Factor the sum of cubes. Factor the difference of cubes. Lesson 3: Solving Quadratic Equations Topic 1: Solve Quadratic Equations by Factoring Learning Objectives Solve equations in factored form by using the Principle of Zero Products. Solve quadratic equations by factoring and then using the Principle of Zero Products. Solve application problems involving quadratic equations. 1.

3 Unit 1 Instructor Notes Unit 1: Factoring The Mathematics of Factoring Instructor Notes This unit builds upon students knowledge of polynomials learned in the previous unit. They will learn how to use the distributive property and greatest common factors to find the factored form of binomials and how to factor trinomials by grouping. Students will also learn how to recognize and quickly factor special products (perfect square trinomials, difference of squares, and the sum and difference of two squares). Finally, they ll get experience combining these techniques and using them to solve quadratic equations. Teaching Tips: Challenges and Approaches This unit on factoring is probably one of the most difficult students will spend a lot of time carrying out multi-step, complex procedures for what will often seem to be obscure purposes. At this stage in algebra, factoring polynomials may feel like busy work rather than a means to a useful end. It doesn t help that students may remember having trouble with factoring from when they studied algebra in high school. Encourage students to think of factoring as the reverse of multiplying polynomials that was learned previously. Then, a problem multiplying polynomials was given and students were asked to calculate the answer. In this unit, the answer is given and the students need to come up with the question. Sound familiar? In a way, factoring is like playing the popular game show Jeopardy. Greatest Common Factor Finding the greatest common factor of whole numbers should be reviewed before finding the GCF of polynomials. Then it is a logical step to demonstrate how to factor expressions by using the distributive property in reverse to pull out the greatest common monomial from each term in a polynomial: 1.3

4 [From Lesson 1, Topic 1, Topic Text] Remind your students to pay particular attention to signs as it is easy to make a mistake with them, and also to check their final answers by multiplying. Grouping After your students are comfortable pulling the GCF out of a polynomial, it is time to teach them a new method of factoring factoring by grouping. Begin by introducing the technique on 4-term polynomials. It's fairly easy for students to understand how to break these polynomials into groups of two and then factor each pair. Trinomials are trickier. Indeed, many textbooks do not use grouping for factoring trinomials, and instead use essentially a guess and check method. While factoring by grouping may initially be a more complex procedure, it has many significant advantages in the long term and is used in this course. The hardest part is figuring out how to rewrite the middle term of a trinomial as an equivalent binomial. Students will need to see this demonstrated repeatedly, as well as get a lot of practice working on their own. Even after they grasp the basic idea, they'll often have trouble deciding which signs to use. It will be helpful to supply them with a set of tips like the one below: 1.4

5 [From Lesson, Topic 1, Topic Text] Factoring by grouping has the great advantage of working for all trinomials. It also provides a method to determine when a polynomial cannot be factored. (This is not obvious when students are using the guess and check method.) Sometimes students don t remember to look for the greatest common factor of all the terms of a polynomial before trying to factor by grouping. This isn t wrong, but the larger numbers can make the work more difficult. Plus the student has to remember to look for a greatest common factor at the end anyway. In order to illustrate this, have students factor 9x + 15x 36 without pulling out the greatest common factor of 3 -- they will notice that the numbers are cumbersome. After this, have them try again, this time factoring out the 3 as the first step. They will see the benefits. Once the grouping method is mastered, let your students use it to factor perfect square trinomials. Hopefully they'll soon see a pattern, though you will probably have to nudge them along. Eventually, they should learn to recognize if a trinomial is a perfect square, and be able to factor it without grouping. After the rule for factoring a perfect square trinomial has been developed, set them to finding one for factoring the difference of two squares. This rule is usually very easy for students to figure out. Then have them try to factor the sum of two squares, such as x + 4. Make sure they understand that this cannot be done. 1.5

6 Intermediate algebra students will also need to know how to factor the sum and difference of two cubes. They are sure to have trouble remembering the formulas. Try pointing out that the formulas are really the same except for signs: A binomial in the form a 3 + b 3 can be factored as (a + b)(a ab + b ) A binomial in the form a 3 b 3 can be factored as (a b)(a + ab + b ) The sign in between the two cubes is the same sign as in the first factor in the formulas. The next sign is the opposite of the first sign and the last sign is always positive. Now all they have to remember are the variable parts of the formulas. Easy! Factoring Quadratic Equations The last topic in this unit is solving quadratic equations by factoring and applying the zero products rule. Begin by solving an example where the polynomial is already factored and set equal to zero, such as the following: [From Lesson 3, Topic 1, Worked Example 1] Now give your students a problem like "Solve x + x 1 = 0 for x." Ask them how they would attempt to solve for x. Someone will suggest factoring the left hand side by grouping and they will be on their way. Then pose the problem x + x 1 = 18. Make sure your students know that in order for the principle of zero products to work, the trinomial must be set equal to 0. Sometimes students are 1.6

7 so focused on new techniques, they forget basic principles for rewriting an equation and they may need to be prodded to add (or subtract) something to (or from) both sides so that one side equals zero. Be careful -- once students get into the hang of applying the zero products rule to solve equations, they may start trying it on expressions as well. For instance, if a problem says to factor x + x 1, some will do so and then go ahead and calculate that x = -4 or 3. Remind your students to only do what a problem asks factor when it says to factor and solve when it says to solve. The Sense Test Application problems have an extra requirement that solving given equations do not -- answers have to make sense based on their context. Consider the following scenario: [From Lesson 3, Topic 1, Topic Text] 1.7

9 Summary After completing this unit, students will be more comfortable with factoring any polynomial that is given to them. They'll be able to pull out the GCF and factor by grouping, and recognize special cases such as perfect square trinomials, difference of two squares, and the sum and difference of two cubes. They'll have had experience combining these techniques to solve quadratic equations, and will have gained an appreciation that factoring can be used to solve real-life problems. 1.9

10 Unit 1: Factoring Purpose Instructor Overview Tutor Simulation: Playing the Elimination Game Unit 1 Tutor Simulation This simulation allows students to demonstrate their ability to factor polynomials. Students will be asked to apply what they have learned to solve a problem involving: Factoring out the greatest common factor Factoring by grouping Factoring the sum or difference of perfect squares Factoring the sum or difference of cubes Problem Students are presented with the following problem: Many mathematicians have tricks they use to analyze an expression in order to determine if they can factor it quickly. In this simulation, we are going to focus on techniques for quickly factoring polynomials with two terms (binomials) or with four terms. Recommendations Tutor simulations are designed to give students a chance to assess their understanding of unit material in a personal, risk-free situation. Before directing students to the simulation, Make sure they have completed all other unit material. Explain the mechanics of tutor simulations. o Students will be given a problem and then guided through its solution by a video tutor; o After each answer is chosen, students should wait for tutor feedback before continuing; o After the simulation is completed, students will be given an assessment of their efforts. If areas of concern are found, the students should review unit materials or seek help from their instructor. Emphasize that this is an exploration, not an exam. 1.10

11 Unit 1: Factoring Instructor Overview Puzzle: Match Factors Unit 1 Puzzle Objectives Match Factors is a puzzle that tests a player's ability to factor by grouping. It reinforces the technique of factoring a trinomial in the form ax + bx + c by finding two integers, r and s, whose sum is b and whose product is ac. Puzzle play, especially when done by eye rather than with pencil and paper, will help students learn to quickly identify the components of factors. Figure 1. Match Factors players choose the factors of a central polynomial from a rotating ring of possibilities. Description Each Match Factors game consists of a sequence of 4 polynomials surrounded by 8 possible factors. As each polynomial is displayed, players are asked to pick the matching pair of factors. If they choose correctly, the next polynomial appears. If not, they must try again before play advances. 1.11

12 There are three levels of play, each containing 10 games. In Level 1, polynomials have the form x + bx + c. Level polynomials have the form ax + bx + c. Players in Level 3 must factor ax + bxy + cy polynomials. Match Factors is suitable for individual or group play. It could also be used in a classroom setting, with the whole group taking turns calling out the two factors of each expression. 1.1

13 Unit 1: Factoring Student Instructions Instructor Overview Project: Making Connections Unit 1 Project Introduction The main business of science is to uncover patterns. Often we represent those patterns as algebraic expressions, graphs, or tables of numbers (data). Being able to make connections among those various representations is one of the most vital skills to possess. Task In this project you attempt to make precise connections among these three ways of representing patterns. Instructions (See the online course materials for full size graphs and charts) Work with at least one other person to complete the following exercises. Solve each problem in order and save your work along the way. You will create a presentation on one of the four parts to be given to your class. First Problem Connecting Algebraic s and Graphs: Factor each of the following expressions completely, and then compare the factored form with the picture of the expression that is shown in the graph on the right. Describe any connections that you see, and then repeat for the next expression. In the end, formulate a conjecture as to how an algebraic expression in factored form is related to its corresponding graph. Keep in mind that we are not assuming that you have any knowledge whatsoever about graphs. That is what makes this detective work so fun! 1.13

14 Algebraic Graph Possible Relationship 3 a Factored Form 3b 1 Factored Form 6x 54 Factored Form 1.14

15 Algebraic Developmental Math An Open Curriculum Graph Possible Relationship x 6x 9 Factored Form y 10y 5 Factored Form c 4c 6 Factored Form 1.15

16 Algebraic Developmental Math An Open Curriculum Graph Possible Relationship 3 3a 4 Factored Form 3 3a 4 Factored Form x 4x 30 Factored Form 1.16

17 Algebraic Developmental Math An Open Curriculum Graph Possible Relationship 5d 15d 4 3 Factored Form x 4x 30x Factored Form 1.17

18 The following problems are a special challenge. You may have to adjust the relationship you expressed above to accommodate these new examples. x 5x 3 Factored Form 6t t Factored Form Second Problem Connecting Algebraic s and Tables: For each of the same algebraic expressions that you examined above, compare the factored form with the table of values associated with the expression. (For example, if the expression is, then the value associated with it when x=1 will be. ) In the end, formulate a conjecture that describes how an algebraic expression in factored form is related to its corresponding data table. x 5x 3 (1) 5(1)

19 Algebraic Table Possible Relationship 3 a Factored Form 3 a a b 1 b 3b

20 Algebraic Table Factored Form Possible Relationship x 6x 54 6x

21 Algebraic Table Factored Form Possible Relationship x x 6x 9 x 6x

22 Algebraic Table Factored Form Possible Relationship y y 10y 5 y 10y

23 Algebraic Table Factored Form Possible Relationship c c 4c 6 c 4c

24 Algebraic Table Factored Form Possible Relationship a 3 3a 4 3 3a

25 Algebraic Table Factored Form Possible Relationship a 3 3a 4 3 3a

26 Algebraic Table Factored Form Possible Relationship x x 4x 30 x 4x

27 Algebraic Table Factored Form Possible Relationship d 5d 15d 4 3 5d 15d

28 Algebraic Table Factored Form Possible Relationship x 4x 30x x x 4x 30x

29 Algebraic Table Factored Form Possible Relationship x x 5x 3 x 5x

30 Algebraic Table Factored Form Possible Relationship t 6t t 6t t

31 Algebraic Table Factored Form Possible Relationship Third Problem Applying Your Findings: For each expression, factor it completely and write the factored form beneath the expression. Then match it to its corresponding table or graph by writing the letter corresponding to the expression on its matching table or graph. 1.31

32 a) 3c b) a 9a 5 3 c) 3d 1d 3 d) y 7 3 e) z 16 f) x +5x-1 7 g) 3a h) t 6t 7 i) 9y 6y 1 1.3

33 Variable

34 Variable Variable

35 Variable Fourth Problem Predicting the Unknown: One of the primary reasons to make connections is to be able to explain or predict previously unobserved behavior. Below we provide you with some tables and some graphs. Based on these alone, determine whether the expression associated with them can be factored. Explain the reasoning behind your decision. [Hint: You should make use of your observations from the problems above to determine what it means for an expression to not be factorable.] 1.35

36 Variable

37 Variable Conclusions With those from another group, compare your answers and your way of talking about the connections between the factored form of the expressions and the graphs and tables. Work to make sure that your explanation is clear and concise. Prepare a presentation which: 1. Explains the connection between the factored expression and the corresponding graphs and tables. 1.37

38 . Describes briefly how you determined this connection (you may want to discuss some of your original ideas and how you needed to refine them as you looked at more examples). 3. Gives a test for determining whether a given expression can be factored if you are given a graph or table associated with the expression. Finally, present your solution to your instructor. Instructor Notes We would stress that nothing in this project assumes that students have any familiarity whatsoever with graphing. In fact, it is precisely because they do not have this familiarity that we can explore this topic. The project is more about developing students abilities to notice connections between the various functional representations before they even understand how these representations work. So, this project can serve both as a culminating project for Unit 1 on factoring as well as very initial preparation for a unit on graphing. Assignment Procedures Problem 1 The relationship that they should be identifying is that the values of the variable that make each factor zero will correspond to the places where the graph crosses the horizontal axis. For the first eleven graphs, the student may very well identify the the negative of the number in the factor as the place where the graph crosses the axis. For example, for c 4c 6 = (c+3) (c-1) encounter the last two, for example at x=-3, but it crosses a second time at, the graph crosses at c=-3 and c=1. However, when they x 5x 3 = (x-1) (x+3), the graph does indeed cross 1 x= and not at x=1, even though x=1 is the negative of the number that appears in the factor. It may be a challenge for them to determine the true connection, although the fact that they have had experience solving quadratic equations by factoring should facilitate the process. Problem The connection is that the value of the variable which makes each factor equal to zero (and therefore the entire expression equal to zero as well) is the one which corresponds to a zero value for the expression in the table. Notice that, since the values of the variable rise in increments of one, the exact values of the variable that correspond to a zero value for the expression do not actually appear in the last two tables. In these examples, the students will have to notice that the value of the expression changes sign and, therefore, must have been zero somewhere in between. Problem 3 The following table shows the correct matching. 1.38

39 Graph: f Graph: b Table: c Graph: d Table: g Graph: h Graph: i Table: e Table: a Problem 4 The table below shows the solutions. Graph: not factorable since the graph does not cross the horizontal axis. Graph: not factorable since the graph does not cross the horizontal axis. Table: factorable because somewhere between x=-1 and x=0 it is equal to zero. Graph: factorable since the graph crosses the horizontal axis somewhere between 4 and 6. Table: not factorable since nowhere does it appear that the expression changes sign or is zero. Note that this is only speculative since the table shows data only for values of x that are integers. However, the students at this stage need not be attentive to this nuance. At this stage it can be helpful to tell the students in each group that for each question, you will randomly choose one person in the group to present the group s answer. This provides motivation for the group as a whole to ensure that each member has a thorough understanding of all of the topics and gives the instructor feedback on how well each individual understands the work that was completed. Recommendations Have students work in teams to encourage brainstorming and cooperative learning. Assign a specific timeline for completion of the project that includes milestone dates. Provide students feedback as they complete each milestone. Ensure that each member of student groups has a specific job. Technology Integration This project provides abundant opportunities for technology integration, and gives students the chance to research and collaborate using online technology. The students instructions list several websites that provide information on numbering systems, game design, and graphics. 1.39

40 The following are other examples of free Internet resources that can be used to support this project: An Open Source Course Management System (CMS), also known as a Learning Management System (LMS) or a Virtual Learning Environment (VLE). Moodle has become very popular among educators around the world as a tool for creating online dynamic websites for their students. or Allows you to create a secure online Wiki workspace in about 60 seconds. Encourage classroom participation with interactive Wiki pages that students can view and edit from any computer. Share class resources and completed student work. Allows students to collaborate in real-time from any computer. Google Docs provides free access and storage for word processing, spreadsheets, presentations, and surveys. This is ideal for group projects. The leading open-source office software suite for word processing, spreadsheets, presentations, graphics, databases and more. It can read and write files from other common office software packages like Microsoft Word or Excel and MacWorks. It can be downloaded and used completely free of charge for any purpose. 1.40

42 The solution addresses none of the mathematical components presented in the task. There were so many errors in mathematical procedures that the problem could not be solved. figures, diagrams, graphs, tables, etc.). There is no use, or mostly inappropriate use, of mathematical terminology and notation. Your project is missing graphics and uses little to no color. 1.4

43 Unit 1 Correlation to Common Core Standards Unit 1: Factoring Learning Objectives Common Core Standards Unit 1, Lesson 1, Topic 1: Greatest Common Factor Grade: 8 - Adopted 010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. Grade: Adopted 010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-SSE. Seeing Structure in s STANDARD EXPECTATION A- SSE.1. GRADE EXPECTATION A- SSE.1(a) Interpret the structure of expressions. Interpret expressions that represent a quantity in terms of its context. Interpret parts of an expression, such as terms, factors, and coefficients. Unit 1, Lesson, Topic 1: Factor Trinomials Grade: 8 - Adopted 010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. Grade: Adopted 010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-SSE. Seeing Structure in s STANDARD EXPECTATION A- SSE.1. GRADE EXPECTATION A- SSE.1(a) Interpret the structure of expressions. Interpret expressions that represent a quantity in terms of its context. Interpret parts of an expression, such as terms, factors, and coefficients. 1.43

45 CATEGORY / CLUSTER A-SSE. Seeing Structure in s STANDARD EXPECTATION A- SSE.3. GRADE EXPECTATION A- SSE.3(a) STRAND / DOMAIN CC.A. Algebra Write expressions in equivalent forms to solve problems. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Factor a quadratic expression to reveal the zeros of the function it defines. CATEGORY / CLUSTER A-REI. Reasoning with Equations and Inequalities STANDARD Solve equations and inequalities in one variable. EXPECTATION A-REI.4. Solve quadratic equations in one variable. GRADE EXPECTATION A- REI.4(b) STRAND / DOMAIN CC.F. Functions CATEGORY / CLUSTER F-IF. Interpreting Functions STANDARD Solve quadratic equations by inspection (e.g., for x^ = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a plus-minus bi for real numbers a and b. Analyze functions using different representations. EXPECTATION F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. GRADE EXPECTATION F- IF.8(a) Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 1.45

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