2.4 Factoring Quadratic Epressions Goal Use the area model and Distributive Property to rewrite an epression that is in epanded form into an equivalent epression in factored form The area of a rectangle is used to write the factored form for the quadratic epression. Students use the epanded form of quadratic epressions to subdivide a rectangle into parts and to label each part with an appropriate epression for its area. These epressions provide clues on how to split the middle term, b, of the epression 2 + b+c to find the dimensions of the rectangle or the two linear factors for the factored form using the Distributive Property. We do not epect students to be eperts in factoring. More factoring will be done in Say It With Symbols. We do want students to develop understanding of the Distributive Property and its connection to the area of a rectangle. Launch 2.4 You can use the eample in the student book to talk about combining like terms if you did not do it in the summary of the last problem. In the last problem, Carminda claimed she did not need the rectangular model to multiply (+3) by (+2). You may want to review these steps with the students. Suggested Question Could you reverse Carminda s method to write the epression 2 + 8+2 in factored form? (This is more difficult, but some students may notice that you have to split the 8 to get the first step in reversing Carminda s method. Some may notice that you have to split the 8 to get the area of two of the smaller rectangles in a rectangle whose area is represented by 2 + 8+2. Also, the product of the coefficients of must be 2. Therefore, 8=2+6 and 2 + 8+2= 2 + 2+8+2.) Let the class work in pairs. Eplore 2.4 Look for interesting strategies for Questions A, B, and C. The purpose of these questions is to help students see that given the first and last terms of the epanded form 2 + b+c, such as 2 + b+9, there are many possible factorings of 2 + b+c which all depend on which factors of c you choose. The factors of c one chooses then determine the value of b. Starting with the factors of c might be a good strategy for splitting b into two parts. Students can put their work on large poster paper to use during the summary. If students find one pair of factors that work, ask them if other pairs might work. For Question A, whole-number factors that will work are: and 9 and 3 and 3. Students can also use rational number factor pairs such as and 8. There are many factor pairs. Here it suffices to focus on whole-number pairs. Some students may need more guidance on how to factor the epressions. The area model can be used to help students recall the steps needed to multiply two binomials like (+)(+4) or students can use the Distributive Property. This is true with factoring. Students can use the area model to factor; this method is illustrated with the epression 2 + 4+2+8. However, the distributive property is a more useful tool when factoring epression like a 2 + b+c especially when negative coefficients for a, b, and/or c are involved. To factor an epression like 2 + 4+2+8, you can use the metaphor of subparts of the rectangle to get the four epressions needed: 2 + 4+2+8. This then becomes the first step in factoring a quadratic epression in epanded form. We are reversing the process of multiplying two binomials. Do an eample, 2 + 8+5, to help students see how to use the area model to do the factoring. Sketch a rectangle whose area is represented by the epression 2 + 8+5. 2 I N V E S T I G AT I O N 2 Investigation 2 Quadratic Epressions 69
Subdivide it into four parts. Suggested Questions Which part has an area represented by 2? By 5? How can you split 8 into two pieces so that the pieces represent the remaining two subparts? What clues can you use to split 8? (The sum of the coefficients of each part must equal 8 and their product must equal 5.) So what two numbers have a sum of 8 and a product of 5? (3 and 5. So the areas of the two sub parts are 3 and 5. Students may try several different pairs. ) What are the dimensions of the rectangle? (+3 and +5) 3 2 3 2 5 5 5 How can you use the Distributive Property to show that (+3)(+5) is equal to 2 + 8+5? Have some groups put their work on large sheets of paper. Make sure you get all of the different strategies that might occur in A C. In Question D, ask: Is Alyse correct? How do you know? What did she do in step? Step 2? Step 3? Summarize students work in Question D before you have students do Question E. 5 Summarize 2.4 Call on different students to discuss their strategies for factoring epressions. Refer to the large posters of the student work. Many will have drawn diagrams. Have the class ask questions about the work. Be sure to relate the sub areas of the rectangle to the parts of the epanded epression and then relate the factored form to the dimensions and area of the whole rectangle. Collect several different rectangles for Question A. Ask the students to show: A sketch of their rectangle with the area of its subparts appropriately labeled. The dimensions of the rectangle. The two equivalent epressions for the area of the rectangle. You can tell the students that writing an epanded form in factored form is called factoring. In Questions D and E, ask students to eplain their method and why it is correct. For eample, 2 + 7+2= 2 + 3+4+2= (+3)+4(+3)=(+3)(+4). Check for Understanding Describe the steps you used to factor the epanded form of a quadratic. Use the following epressions as eamples: 2 + 7+8 2-3+2 4 2 + 8 (Students should start by looking for factors of the constant term. Then they choose the pair whose sum is equal to the coefficient of the middle term or to b in b. Some may start with finding various sums for b and then choosing the two whose product is equal to the constant term, c. Note: If the coefficient of 2 is not, then finding the coefficient of is more complicated.) Suggested Question How can you check that your answers are equivalent to the original epressions? (Multiply the two factors to see if you get the original epanded form or use a rectangular model to check your work or check the graphs of each equation y=factored form and y=epanded form for each epression. The graphs should be identical.) 70 Frogs, Fleas, and Painted Cubes
2.4 Factoring Quadratic Epressions At a Glance PACING 2 days Mathematical Goal Use the area model and Distributive Property to rewrite an epression that is in epanded form into an equivalent epression in factored form Launch In the last problem, Carminda claimed she did not need the rectangular model to multiply (+3) by (+2). You may want to review these steps with the students. Could you reverse Carminda s method to write the epression 2 + 8+5 in factored form? Let the class work in pairs. Eplore Look for interesting strategies for Questions A, B, and C. Students could put their work on large poster paper to use during the summary. If students find one pair of factors that works, ask them if other factor pairs might work. Some students may need more guidance on how to factor the epressions. We are reversing the process of multiplying two binomials. Do an eample, 2 + 8+5, to help students see how to use the area model to do the factoring. Sketch a rectangle whose area is represented by the epression 2 + 8+5. Subdivide it into four parts. Which part has an area represented by 2? By 5? How can you split 8 into two pieces so that the pieces represent the remaining two subparts? What clues can you use to split 8? What two numbers have a sum of 8 and a product of 5? What are the dimensions of the rectangle? How can you use the Distributive Property to show that (+3)(+5) is equal to 2 + 8+5? Have some groups put their work on large sheets of paper. Make sure you get all of the different strategies that might occur in Questions A, B, and C. In Question D, ask: Is Alyse correct? How do you know? What did she do in step? Step 2? Step 3? Summarize Question D before you have students do Question E. Materials Poster Paper (optional) Investigation 2 Quadratic Epressions 7
Summarize Call on different students to discuss their strategies for factoring epressions. Refer to the large posters of the student work. Relate the sub areas of the rectangle to the parts of the epanded epression and then relate the factored form to the dimensions and area of the whole rectangle. In Questions D and E, ask students to eplain their method and why it is correct. For eample, 2 + 7+2= 2 + 3+4+2= (+3)+4(+3)=(+3)(+4). Describe the steps you used to factor the epanded form of a quadratic. How can you check that your answers are equivalent to the original epressions? Materials Student notebooks ACE Assignment Guide for Problem 2.4 Core 29 39 Other Connections 56, Etensions 62 64; unassigned choices from previous problems Adapted For suggestions about adapting ACE eercises, see the CMP Special Needs Handbook. Connecting to Prior Units 56: Moving Straight Ahead Answers to Problem 2.4 A.. Possible drawing: 3 3 2 9 3 3 Since the lengths of the two segments marked with question marks seem to be the same length, students may argue that 3 and 3 are the only factors of 9 that will work, while others will claim that and 9 are possibilities as well. Either case is acceptable. 2. Possible answers: Factored form: (+3)(+3); Epanded: 2 + 3+3 + 9 or 2 + 6+9; Factored form: (+)(+9); Epanded: 2 + +9+9 or 2 + 0+9. 2 9 Note: There are infinitely more factor pairs that will work for 9 such as and 8 2 or and 36. For this problem we can focus 4 on the whole-number factors. B.. If b=9, then 2 + 9+8 in factored form is (+)(+8). If b=6, then 2 + 6+8 in factored form is (+2)(+4). If b=-9, then 2-9+8 in factored form is (-)(-8). 9 9 72 Frogs, Fleas, and Painted Cubes
If b=-6, then 2-6+8 in factored form is (-2)(-4). Note: Not every b value that students pick will result in a quadratic epression that is factorable at this time by students. For eample a b value of 0 gives an epression of 2 + 0+8. This epression cannot be factored by students unless they know the quadratic formula which they may not learn until high school. In this problem, we want students to realize that they are looking at the factor pairs of 8 and finding a pair whose sum is the value for b that will make 2 + b+8 an epression that can be factored. Restricting the factor pairs of 8 to integers, we have ( and 8), (- 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 student eplanation may include the use of an area model such as the one shown below: The factor pair consisting of 2 and 4 yields a b value of 6 and the factored form of (+2)(+4) for 2 + 6+8. 2. Students will probably find that not all of their classmates found the same epressions. However the epressions that students will have will be either (+2)(+4), (+)(+8), (-)(-8) or (-2)(-4). C.. r=4 4 4 2. r=3, s=8 (or r=8, s=3) 3. r= and s=24 or r=24 and s= 4. A common strategy is to compare the c term of the 2 + b+c form to the 2 8 2 2 constant terms in the factors; r and s must be factors of c. Meanwhile, the sum of r and s must make b. D. Yes; Students can make an area model to check that 2 + 0+6=(+2)(+8). They may also reason that Alyse applied the Distributive Property. In step, she separated 0 into 2+8 and in step 2 she used the Distributive Property to rewrite 2 + 2 as (+2) and 8+6 as 8(+2). In step 3, she used the Distributive Property to see that (+2) is multiplied by both and 8 so she can factor the (+2) term out and multiply it by the remaining (+8). E.. Using the Distributive Property: 2 + 5+2+0= (+5)+2(+5)= (+5)(+2). Note: Students may also write this as (+2)(+5) which is equivalent because of the Commutative Property of Multiplication. In the answers for Question E, parts () and (4) the common factor has been pulled out in front since it seems the most natural. For part (), the factor of +5 was pulled out in front to obtain (+5)(+2). 2. Using the Distributive Property: 2 + +0= 2 + 0++ 0= (+0)+(+0)=(+0)(+). 3. Using the Distributive Property: 2 + 3-0= 2 + 5-2-0= (+5)-2(+5)=(+5)(-2). 4. Using the Distributive Property: 2 + 6+5= 2 + 5++5= (+5)+(+5)=(+5)(+). 5. Using the Distributive Property: 2-8+5= 2-3-5+5= (-3)-5(-3)=(-5)(-3). 6. Using the Distributive Property: 2-2 + 36 = 2-6+-6 + 36 = (-6)+-6(-6)=(-6)(-6). Note: Students may need help seeing that -5 is a factor of -5 and 5 in part (5) or that -2 is a factor of -2 and -0 in part (3). Factoring out a negative quantity is represented somewhat differently in the part (6) solution. I N V E S T I G AT I O N 2 Investigation 2 Quadratic Epressions 73
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