2.4 Factoring Quadratic Expressions - Mona Shores Blogs

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SummarizeCall on different students to discuss their strategies for factoringexpressions. Refer to the large posters of the student work. Relate thesub areas of the rectangle to the parts of the expanded expression andthen relate the factored form to the dimensions and area of the wholerectangle.In Questions D and E, ask students to explain their method and whyit is correct. For example, x 2 + 7x+12=x 2 + 3x+4x+12=x(x+3)+4(x+3)=(x+3)(x+4).Describe the steps you used to factor the expanded form of aquadratic.• How can you check that your answers are equivalent to the originalexpressions?Materials• Student notebooksACE Assignment Guidefor Problem 2.4Core 29–39Other Connections 56, Extensions 62–64;unassigned choices from previous problemsAdapted For suggestions about adapting ACEexercises, see the CMP Special Needs Handbook.Connecting to Prior Units 56: Moving StraightAheadAnswers to Problem 2.4A. 1. Possible drawing:3x3x3xx 2 9x3Since the lengths of the two segmentsmarked with question marks seem to be thesame length, students may argue that 3 and3 are the only factors of 9 that will work,while others will claim that 1 and 9 arepossibilities as well. Either case isacceptable.2. Possible answers:Factored form: (x+3)(x+3);Expanded: x 2 + 3x+3x + 9 orx 2 + 6x+9;Factored form: (x+1)(x+9);Expanded: x 2 + x+9x+9 orx 2 + 10x+9.1xx x 2 9xxNote: There are infinitely more factor1pairs that will work for 9 such as and 1821or and 36. For this problem we can focus4on the whole-number factors.B. 1. If b=9, then x 2 + 9x+8 in factoredform is (x+1)(x+8).If b=6, then x 2 + 6x+8 in factoredform is (x+2)(x+4).If b=-9, then x 2 - 9x+8 in factoredform is (x-1)(x-8).9972 Frogs, Fleas, and Painted Cubes
If b=-6, then x 2 - 6x+8 in factoredform is (x-2)(x-4).Note: Not every b value that students pickwill result in a quadratic expression that isfactorable at this time by students. Forexample a b value of 10 gives an expressionof x 2 + 10x+8. This expression cannot befactored by students unless they know thequadratic formula which they may notlearn until high school.In this problem, we want students to realizethat they are looking at the factor pairs of 8and finding a pair whose sum is the valuefor b that will make x 2 + bx+8 anexpression that can be factored. Restrictingthe factor pairs of 8 to integers, we have(1 and 8), (-1 and -8), (-2 and -4), and(2 and 4). So the values for b are 9,-9,-6 and 6, respectively, as illustrated above.For the case where b=6, a possible studentexplanation may include the use of an areamodel such as the one shown below:The factor pair consisting of 2 and 4 yieldsa b value of 6 and the factored form of(x+2)(x+4) for x 2 + 6x+8.2. Students will probably find that not all oftheir classmates found the same expressions.However the expressions that students willhave will be either (x+2)(x+4),(x+1)(x+8), (x-1)(x-8) or(x-2)(x-4).C. 1. r=44x4x2. r=3, s=8 (or r=8, s=3)2xx 2 83. r=1 and s=24 or r=24 and s=14. A common strategy is to compare the “c”term of the x 2 + bx+c form to thex2constant terms in the factors; r and s mustbe factors of c. Meanwhile, the sum of r ands must make b.D. Yes; Students can make an area model tocheck that x 2 + 10x+16=(x+2)(x+8).They may also reason that Alyse applied theDistributive Property. In step 1, she separated10x into 2x+8x and in step 2 she used theDistributive Property to rewrite x 2 + 2x asx(x+2) and 8x+16 as 8(x+2). In step 3,she used the Distributive Property to see that(x+2) is multiplied by both x and 8 so shecan factor the (x+2) term out and multiplyit by the remaining (x+8).E. 1. Using the Distributive Property:x 2 + 5x+2x+10=x(x+5)+2(x+5)=(x+5)(x+2).Note: Students may also write this as(x+2)(x+5) which is equivalent becauseof the Commutative Property ofMultiplication. In the answers forQuestion E, parts (1) and (4) the commonfactor has been pulled out in front since itseems the most natural. For part (1), thefactor of x+5 was pulled out in front toobtain (x+5)(x+2).2. Using the Distributive Property:x 2 + 11x+10=x 2 + 10x+x+ 10=x(x+10)+1(x+10)=(x+10)(x+1).3. Using the Distributive Property:x 2 + 3x-10=x 2 + 5x-2x-10=x(x+5)-2(x+5)=(x+5)(x-2).4. Using the Distributive Property:x 2 + 16x+15=x 2 + 15x+1x+15=x(x+15)+1(x+15)=(x+15)(x+1).5. Using the Distributive Property:x 2 - 8x+15=x 2 - 3x-5x+15=x(x-3)-5(x-3)=(x-5)(x-3).6. Using the Distributive Property:x 2 - 12x + 36 = x 2 - 6x+-6x + 36 =x(x-6)+-6(x-6)=(x-6)(x-6).Note: Students may need help seeing that-5 is a factor of -5x and 15 in part (5) orthat -2 is a factor of -2x and -10 inpart (3). Factoring out a negative quantity isrepresented somewhat differently in thepart (6) solution.I N V E S T I G AT I O N 2Investigation 2 Quadratic Expressions 73
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2.4 Factoring Quadratic Expressions - Mona Shores Blogs

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