C HAPTER 4: FACTORING ALGEBRAIC EXPRESSIONS

1Introducing the Lesson(5 to 10 min)Introduce the concept of common factoring with the following context: Ask students to imagine that a customer orders two rugs, one with an areaof 10 m2 and the other with an area of 15 m2. Ask: What are the possibledimensions with only whole numbers? You could have students sketch twodifferent possibilities for each rug on grid paper. Discuss how students’ sketches are examples of factoring because the areais expressed as a product of two factors, the dimensions. Have studentscompare sketches and look for a pair of rugs with a common dimension(for example, 10 m by 1 m and 15 m by 1 m, or 5 m by 2 m and 5 m by3 m). Then ask them to identify the greatest possible common dimension,or greatest common factor (for example, 5).2Teaching and Learning(30 to 40 min)Copyright 2011 by Nelson Education Ltd.Learn About the MathExample 1 explores the answer to the lesson question: the sum of the squaresof two consecutive integers is always divisible by 2. Pose the question, andhave students individually choose two consecutive integers to test theconjecture. Then have students work in groups of three to brainstorm whetherthe conjecture is true and how they might convince someone that their answeris correct. To help students understand Abdul’s solution, demonstrate how thetwo groups are formed using concrete materials. The two groups areidentical both include x2, x, and 1 therefore the expression is divisibleby 2.Have students discuss the Reflecting questions in their groups of three. Thendiscuss the solutions with the whole class.Answers to ReflectingA. Lisa used n and Abdul used x because the value of the first integer isunknown. This allows the rule to apply to any integer that is substitutedfor the variable.B. Lisa needed to show that 2 is a common factor for the terms in theexpression in order to show that the expression is even, or divisible by 2.C. Answers may vary, e.g., I preferred Abdul’s strategy because he used algebra tiles, which madethe solution easy to see. I preferred Lisa’s strategy because I prefer strategies that don’t requireme to use materials.4.1: Common Factors in Polynomials 133

Consolidation3(15 to 20 min)Apply the MathUsing the Solved ExamplesIn Example 2, the GCF is represented using two different strategies:concretely (with algebra tiles) and algebraically. Have the students work inpairs, with one student using algebra tiles and the other student workingthrough the solution algebraically. Partners should then explain to each otherwhat they did. Discuss both strategies with the class, having differentstudents explain their work. Invite students to talk about advantages of beingshown different strategies. Ensure that students understand what is meant byfactoring fully, as described in the second Communication Tip.For Examples 3 and 4, point out that the GCF can have more than one term.In Example 3, where the GCF is a monomial, have students check that thefactoring is correct by expanding the answer. The product should match theoriginal expression. Ask students to explain how the distributive property isused in the solution. In Example 4, the GCF is a binomial. To help studentsunderstand the grouping, suggest that they underline each group of terms witha coloured pencil in their notes. Ask students why the factor (x – 2) in part a)is in the answer once instead of twice, as it is in the original expression. Toshow that the original and factored expressions are equivalent, have half ofthe class expand 5x(x – 2) – 3(x – 2) and the other half expand (x – 2)(5x – 3).Then compare the results.The polynomials in parts a) and d) of question 9 have the same trinomial asone of their factors. Students’ answers may look slightly different if they usea negative sign. For example, in part b), the following answer is also correct:–5ac(2a – 4 c2).9. a) dc2 – 2acd 3a2d d(c2 – 2ac 3a2)b) –10a2c 20ac – 5ac3 5ac(–2a 4 – c2)c) 10ac2 –15a2c 25 5(2ac2 – 3a2c 5)d) 2a2c4 – 4a3c3 6a4c2 2a2c2(c2 – 2ac 3a2)e) 3a5c3 – 2ac2 7ac ac(3a4c2 – 2c 7)f) 10c3d – 8cd2 2cd 2cd(5c2 – 4d 1)ClosingHave students read question 18 and discuss why expanding is theopposite of factoring. Emphasize that expanding involves multiplyingand factoring involves dividing, as shown in the In Summary box,under Key Ideas. Ask students to suggest examples with numbers toshow that multiplying is the opposite of dividing. Then have studentsdemonstrate that expanding is the opposite of factoring by verifyingsome of the Practising questions they have completed.134 Principles of Mathematics 10: Chapter 4: Factoring Algebraic ExpressionsCopyright 2011 by Nelson Education Ltd.Answer to the Key Assessment Question

Assessment and Differentiating InstructionWhat You Will See Students Doing When students understand If students misunderstand Students determine the GCF of the terms in a polynomial.Students cannot determine the GCF in a polynomial, or theycannot distinguish the GCF from a factor that is not the GCF.Students factor polynomials by dividing out the GCF.Students cannot factor polynomials by dividing out the GCFfrom each term. They might not divide each term by the samefactor, or they might not divide correctly.Students identify a GCF that has more than one term andfactor appropriately.Students may not identify both terms in a GCF with more thanone term, or they may not factor appropriately.Key Assessment Question 9Students identify the GCF of all terms in a polynomial.Students are unsure of the GCF of all the terms in apolynomial, or they identify a factor that is not the GCF todivide out.Students factor a polynomial by dividing out the GCF fromeach term.Students may miss dividing out the GCF from a term, or theymight divide incorrectly.Students recognize when factors are identical.Students may not realize that each term in one factor mustmatch a term in another factor for the factors to be identical.Differentiating Instruction How You Can RespondCopyright 2011 by Nelson Education Ltd.EXTRA SUPPORT1. Remind students that they can check their factoring by using the distributive property to multiply the factors.2. To help students understand how to determine the GCF, encourage them to look at the numerical coefficients of the terms inthe polynomial first, and then at the variable(s) to determine what all the terms have in common.3. To help students factor a polynomial, explain that one of the factors is the GCF. To determine the other factor, they candivide each term by the GCF. (When the GCF is a monomial, they will need to remember how to divide monomials.)4. Provide algebra tiles whenever they might be helpful to give students a concrete model of factoring. Ask students to describetheir use of the algebra tiles and to relate the final result with the algebra tiles to the factored expression.EXTRA CHALLENGE1. If students excel at factoring, ask them to factor (4a2b – 1)2 – 8(4a2b – 1).4.1: Common Factors in Polynomials 135

4.2 EXPLORING THE FACTORIZATIONOF TRINOMIALSLesson at a GlancePrerequisite Skills/Concepts Identify the factors in an area model. Represent a polynomial with degree 2 using algebra tiles. Expand an algebraic expression using the distributive property.Specific Expectation Factor polynomial expressions involving [common factors,] trinomials,[and differences of squares] [e.g., [2x2 4x, 2x – 2y ax – ay], x2 – x – 6,2a2 11a 5, [4x2 – 25]], using a variety of tools (e.g., concrete materials,[computer algebra systems], paper and pencil) and strategies (e.g.,patterning).Mathematical Process FocusGOALDiscover the relationship between thecoefficients and constants in atrinomial and the coefficients andconstants in its factors.Student Book Pages 205–206Preparation and PlanningPacing5 min35-45 min10-20 minIntroductionTeaching and LearningConsolidationMaterial algebra tilesRecommended PracticeQuestions 1, 2, 3, 4Nelson Websitehttp://www.nelson.com/mathMATH BACKGROUND LESSON OVERVIEW In this lesson, students discover how to factor a trinomial using an algebra tile model. First, they explore howto arrange the algebra tiles for each trinomial to form a rectangle. Then they identify the factors as thedimensions of the rectangle. (This factoring strategy using algebra tiles is the opposite of the expandingstrategy using algebra tiles that students learned in Chapter 3.). Next, students use a patterning strategy (looking for relationships) to determine how each trinomial is relatedto its binomial factors. This relationship will be covered more formally in Lesson 3. In Reflecting, students encounter trinomials for which they cannot make a rectangle using algebra tiles. Theyshould begin to realize that not all polynomials can be factored. Students should understand that answers can be checked by multiplying the factors to determine whetherthe product is the same as the original trinomial.136 Principles of Mathematics 10: Chapter 4: Factoring Algebraic ExpressionsCopyright 2011 by Nelson Education Ltd. Selecting Tools and Computational Strategies Connecting Representing

1Introducing the Lesson(5 min)Write the trinomial x2 4x 3 on the board, and ask students why theycannot factor this trinomial by first determining the GCF. Explain that thistrinomial can be factored another way, which they will discover using algebratile models. Distribute algebra tiles to pairs of students, and ask each pair to model thetrinomial. Then challenge students to rearrange their algebra tiles to form arectangle. Have them sketch their rectangle model. Ask students to arrange the tiles in a different way to represent the samefactors. Discuss why they can represent x 3 along the side and x 1across. Some students might place the x2 tile at a different corner. Ensurethat each student sees different ways to represent the trinomial with thesame tiles.2Teaching and Learning(35 to 45 min)Copyright 2011 by Nelson Education Ltd.Explore the MathHave students continue to work in pairs, discussing their observations andsharing their work. It is important for students to realize that a rectangle canbe arranged in different ways and that the order of the factors does not matter.Guide students, as needed, when they start arranging the algebra tiles.Emphasize that when they are able to form a rectangle with the algebra tiles,the values along the sides show the factors. Ensure that they use differentcolours of tiles to represent positive and negative values.As students work, ask them to describe how all the tiles represent theexpression and how the rectangles represent the factors. Discuss the style ofwriting the x term before the constant term in a factor (for example, x 3, not3 x), but make sure that students realize that both represent the same value.Students may notice similarities in the trinomials in each part. In part A, eachunit tile is positive. In part B, more tiles must be added to make the rectangle,and each unit tile is negative. Ensure that students add pairs of negative andpositive tiles so the value of an expression does not change, as explained inthe Communication Tip. Ask a few students to explain the CommunicationTip in their own words. In part C, the coefficient of x2 is greater than 1, butstudents do not need to add pairs of tiles to create the rectangles.Encourage students to look for similarities as they compare trinomials. Askthem to consider how each trinomial is related to its binomial factors. In thisexploration, students may be more comfortable using informal language todescribe what they observe.4.2: Exploring the Factorization of Trinomials 137

A. i) (x 1)(x 1)iii) (x 2)(x 4)v) (x – 3)(x – 1)ii) (x 2)(x 3)iv) (x – 1)(x – 1)vi) (x – 2)(x – 1)B. i) (x – 3)(x 1)iii) (x 2)(x – 4)v) (x 2)(x – 5)ii) (x – 1)(x 4)iv) (x 3)(x – 2)vi) (x – 1)(x 5)C. i) (x 1)(2x 1)iii) (x – 1)(3x – 1)v) (x – 4)(2x 1)ii) (x 2)(2x 1)iv) (x – 2)(2x – 3)vi) (x 2)(3x – 1)138 Principles of Mathematics 10: Chapter 4: Factoring Algebraic ExpressionsCopyright 2011 by Nelson Education Ltd.Answers to Explore the Math

D. Answers may vary, e.g.,i) The coefficient of x2 in each trinomial is equal to the product of thecoefficients of x in the factors.ii) The constant term in each trinomial is equal to the product of theconstant terms in the factors.iii) The coefficient of x in each trinomial is equal to the sum of these twoproducts: the coefficient in the first factor multiplied by the constantterm in the second factor, and the coefficient in the second factormultiplied by the constant term in the first factor.Answers to ReflectingE. Both factors are binomials.F. The coefficients in the two binomial factors have the product a. Theconstants in the binomial factors have the product c (the constant term inthe trinomial), but there may be several possibilities. To get the correctcombination of constants in the binomials, systematically try differentpossibilities to determine which factors work. The product of the “inside”pair of numbers and the product of the “outside” pair of numbers musthave the sum b.G. A rectangular arrangement of algebra tiles cannot be created forx2 3x 1 or 2x2 x 1. This implies that these trinomials cannot befactored.3Consolidation(10 to 20 min)Copyright 2011 by Nelson Education Ltd. Remind students that since expanding is the opposite of factoring,expanding is a useful strategy for checking their answers. Students should answer the questions in Further Your Understandingindependently.4.2: Exploring the Factorization of Trinomials 139

Prerequisite Skills/Concepts Expand an algebraic expression using the distributive property.Represent a polynomial with degree 2 using algebra tiles.Identify the greatest common factor for a polynomial.Sketch the parabola of a quadratic relation using its key properties(x-intercepts, axis of symmetry, vertex).Specific Expectation Factor polynomial expressions involving common factors, trinomials, [anddifferences of squares] [e.g., 2x2 4x, 2x – 2y ax – ay, x2 – x – 6,2a2 11a 5, [4x2 – 25]], using a variety of tools (e.g., concrete materials,computer algebra systems, paper and pencil) and strategies (e.g.,patterning).Mathematical Process Focus Selecting Tools and Computational Strategies Connecting RepresentingGOALFactor quadratic expressions of theform ax2 bx c, where a 1.Student Book Pages 207–213Preparation and PlanningPacing5 min30-40 min15-25 minIntroductionTeaching and LearningConsolidationMaterials algebra tilesRecommended PracticeQuestions 4, 5, 7, 9, 10, 12, 13a, dKey Assessment QuestionQuestion 12Extra PracticeLesson 4.3 Extra PracticeNelson Websitehttp://www.nelson.com/mathMATH BACKGROUND LESSON OVERVIEW This lesson focuses on strategies for factoring trinomials of the form x2 bx c. Students’ previous explorations with algebra tile models are further developed as students look for patternsto determine how each trinomial is related to its binomial factors. It is important for students to understand that their answers can always be checked by multiplying the factorsto see if the product is the same as the original polynomial. Expanding is the opposite of factoring. Students will encounter some trinomials that have values of a that are greater than 1. To factor this typeof trinomial, they should look for a common factor among all the terms using the skills they learned inLesson 4.1. When the GCF is factored out, the remaining trinomial has the form x2 bx c. Studentscan continue factoring using the skills they develop in this lesson. This common factor method alsoworks when the first term in the trinomial is –x2.140 Principles of Mathematics 10: Chapter 4: Factoring Algebraic ExpressionsCopyright 2011 by Nelson Education Ltd.4.3 FACTORING QUADRATICS:x2 bx cLesson at a Glance

1Introducing the Lesson(5 min)Begin by briefly reviewing multiplication of binomials. Tell students that the dimensions of a rectangle are (x 1) and (x – 5).Have each student determine the area of the rectangle by expanding:(x 1)(x – 5) x2 – 4x – 5. Then have students rewrite the area equation in the opposite order:x2 – 4x – 5 (x 1)(x – 5). Ask students to explain how factoring is theopposite of expanding. Have students model this with algebra tiles.2Teaching and Learning(30 to 40 min)Investigate the MathDraw the area diagram, shown at the top of page 207 in the Student Book, onthe board so that students can refer to it as they work through the lesson.Distribute algebra tiles to pairs of students. Have them work together,discussing their observations and sharing their work.Answers to Investigate the MathCopyright 2011 by Nelson Education Ltd.A. (x 3)(x 4)LengthWidth(x 3)(x 5)Area: x2 bx c(x 3)(x 6)(x 4)(x 4)Value of bValue of cx 3x 4x2 7x 12712x 3x 5x2 8x 15815x 3x 6x2 9x 18918x 4x 4x2 8x 16816x 4x 5x2 9x 20920(x 4)(x 5)B. The value of b is the sum of the constant terms in the two factors. Thevalue of c is the product of the constant terms in the two factors.i) (x 6) and (x 2)ii) (x 3) and (x 7)iii) (x 5) and (x 6)iv) (x 9) and (x 2)4.3: Factoring Quadratics: x2 bx c 141

C. x2 x – 12D. (x – 3)(x 5)Lengthx–3x 3(x 3)(x – 6)Widthx 5x–6Area: x2 bx cx2 2x – 15(x – 2)(x – 2)Value of b(x – 1)(x – 5)Value of c2–152–3–182x – 3x – 18x–2x–2x – 4x 4–44x–1x–5x2 – 6x 5–65Answers to ReflectingG. The signs can be used as follows: If c in the area expression is positive, then the signs in the dimensionswill be either both positive or both negative. If b is positive, both signswill be positive. If b is negative, both signs will be negative. If c is negative, then the signs in the dimensions will be different; onewill be positive, and the other will be negative. If b is positive, then thegreater number will be positive. If b is negative, then the greaternumber will be negative.H. No. For example, in a trinomial, b and c can have any integer values.There may not always be factors of c with the sum b, so it may not alwaysbe possible to factor a trinomial as the product of two binomials.142 Principles of Mathematics 10: Chapter 4: Factoring Algebraic ExpressionsCopyright 2011 by Nelson Education Ltd.E. The value of b is the sum of the constant terms in the two factors. Thevalue of c is the product of the constant terms in the two factors.i) (x – 5) and (x 3)ii) (x 6) and (x – 4)iii) (x – 6) and (x 5)iv) (x – 7) and (x – 1)F. For example, write two binomials that have x as the first term ineach pair of parentheses. Write the second term in each binomialby determining two numbers that are factors of c and have thesum b.

3Consolidation(15 to 25 min)Apply the MathUsing the Solved ExamplesFor Example 1, ask each student to factor the trinomial using algebra tiles.Have students check their factoring by expanding.In Examples 2

C HAPTER 4: FACTORING ALGEBRAIC EXPRESSIONS Specific Expectation Addressed in the Chapter Factor polynomial expressions involving common factors, trinomials, and differences of squares [e.g., 2x2 4x, 2x – 2y ax – ay, x2 – x – 6, 2a2 11a 5, 4x2 – 25], using a variety of tools (e.g., concrete materials, computer algebra systems, paper and pencil) and strategies (e.g .