THE CASE OF THE MISSING MIDDLE TERM EXAMPLES Consider the two squres tht re shown.. Wht is the re of the smller squre? Answer 2 2. Wht is the re of the lrger squre? Answer 2 3. If I took pir of scissors nd cut squre from squre, wht re would e remining? Answer 2 2 - Region Region 2 - The Cse of the Missing Middle Term 2003 www.econlerningcenter.com Rev. 06.09.03
- Region - Region - Region 2 Rerrnging Region, these regions cn e ressemled s shown elow. You cn see tht 2 2 is equl to the product of ( )( + ). ( + ) - Region Region 2 - ( + ) 4. Hve students complete these products:. ( + 4)( 4) 2 6. (p 5)(p + 5) p 2 25 c. (4m )(4m + ) 6m 2 d. (3x + 2y)(3x 2y) 9x 2 4y 2 e. ( + )( ) 2 2 ( + 4)( 4) 2 6 The first term is the squre of the first product, nd the lst term is the squre of the lst product. 5. From the exmples ove, hve the students mke comments out the pttern tht is eing developed. 6. Introduce students to the Difference of Squres eqution 2 2 = ( + )( ). Students should memorize the form of difference of two squres. 7. Fctor 2 64 using Difference of Squres. 2 64 () 2 (8) 2 ( 8)( + 8) The Cse of the Missing Middle Term 2003 www.econlerningcenter.com Rev. 06.09.03
8. Fctor m 2 4 using Difference of Squres. m 2 4 (m) 2 (2) 2 (m 2)(m + 2) 9. Fctor 9x 2 00y 2 using Difference of Squres. 9x 2 00y 2 (3x) 2 (0y) 2 (3x 0y)(3x + 0y) 0. Fctor 4x 2 25m 2 using Difference of Squres. 4x 2 25m 2 (2x) 2 (5m) 2 (2x 5m)(2x + 5m). Fctor 4 x 2 9 4 y 2 using Difference of Squres. x 2 4 y 2 4 9 ( x) 2 2 ( y) 2 2 3 ( 2 x 3 2 y)( 2 x + 3 2 y) 2. Fctor x 2 9 y 2 using Difference of Squres. 6 25 x 2 9 y 2 6 25 ( x) 2 3 ( y) 2 4 5 ( 4 x 5 3 y)( 4 x + 5 3 y) 3. Review the FOIL method with students. For exmple, multiply (x + 2)(x + 5). F O I L FIRST OUTER INNER LAST (x + 2)(x + 5) = x 2 + 5x + 2x + 0 = x 2 + 7x + 0 The Cse of the Missing Middle Term 2003 www.econlerningcenter.com Rev. 06.09.03
4. Ask the students to use the FOIL method on (7 2) nd explin why there is only two terms in the finished product when you re multiplying two terms times two terms (two terms)(two terms) which should yield four terms. ANSWER Middle terms cncel. Sometimes the terms of inomil hve common fctors. If so, the GCF (Gretest Common Fctor) should e fctored out efore fctoring the difference of squres. Occsionlly, fctoring the difference of squres needs to e used more thn once in prolem. 5. Fctor 8x 2 8y 2 8x 2 8y 2 using Difference of Squres. 2(4x 2 9y 2 ) 2(2x 3y)(2x + 3y) The Cse of the Missing Middle Term 2003 www.econlerningcenter.com Rev. 06.09.03
Nme: Dte: Clss: THE CASE OF THE MISSING MIDDLE TERM WORKSHEET Stte whether ech inomil cn e fctored using the difference of squres.. x 2 y 2 6. 9x 2 5 2. x 2 + y 2 7..44x 2 0.45y 2 3. x 2 25 4. x 2 4y 4 8. 4 x 2 9 4 5. 9x 2 + y 2 6 9. x 2 + 25 Fctor ech polynomil completely! Check Using FOIL method. 0. 2 9 7. 2x 2 98. x 2 49 8. 2 2 48 2. 4x 2 9y 2 9. 8x 2 8 3. x 2 36y 2 20. x 2 25 6 4. 9y 2 5. 49 4 2 2 6. 2 2 + 8 The Cse of the Missing Middle Term 2003 www.econlerningcenter.com Rev. 06.09.03
THE CASE OF THE MISSING MIDDLE TERM WORKSHEET KEY Stte whether ech inomil cn e fctored using the difference of squres.. x 2 y 2 Yes 6. 9x 2 5 No 2. x 2 + y 2 No 7..44x 2 0.45y 2 Yes 3. x 2 25 Yes 4. x 2 4y 4 Yes 8. 4 x 2 Yes 9 4 5. 9x 2 + y 2 No 6 9. x 2 + No 25 Fctor ech polynomil completely! 0. 2 9 (x 3)(x + 3) 6. 2 2 + 8 2( + 9). x 2 49 (x 7)(x + 7) 7. 2x 2 98 2(x 7)(x + 7) 2. 4x 2 9y 2 (2x 3y)(2x + 3y) 8. 2 2 48 2( + 2)( 2) 3. x 2 36y 2 (x 6y)(x + 6y) 9. 8x 2 8 2(2x + 3)(2x 3) 4. 9y 2 ( 3y)( + 3y) 20. 2 x 25 ( x 5)( x + 5) 6 4 4 5. 49 4 2 2 (7 2)(7 + 2) The Cse of the Missing Middle Term 2003 www.econlerningcenter.com Rev. 06.09.03
Student Nme: Dte: THE CASE OF THE MISSING MIDDLE TERM CHECKLIST. On questions thru 9, did the student stte whether ech inomil could e fctored using the difference of squres correctly?. All nine (40 points). Eight of the nine (35 points) c. Seven of the nine (30 points) d. Six of the nine (25 points) e. Five of the nine (20 points) f. Four of the nine (5 points) g. Three of the nine (0 points) h. Two of the nine (5 points) 2. On question 0, did the student fctor the polynomil completely?. Yes (0 points). No, ut polynomil ws prtilly fctored (5 points) 3. On question, did the student fctor the polynomil completely?. Yes (0 points). No, ut polynomil ws prtilly fctored (5 points) 4. On question 2, did the student fctor the polynomil completely?. Yes (0 points). No, ut polynomil ws prtilly fctored (5 points) 5. On question 3, did the student fctor the polynomil completely?. Yes (0 points). No, ut polynomil ws prtilly fctored (5 points) 6. On question 4, did the student fctor the polynomil completely?. Yes (0 points). No, ut polynomil ws prtilly fctored (5 points) 7. On question 5, did the student fctor the polynomil completely?. Yes (0 points). No, ut polynomil ws prtilly fctored (5 points) The Cse of the Missing Middle Term 2003 www.econlerningcenter.com Rev. 06.09.03
8. On question 6, did the student fctor the polynomil completely?. Yes (0 points). No, ut polynomil ws prtilly fctored (5 points) 9. On question 7, did the student fctor the polynomil completely?. Yes (0 points). No, ut polynomil ws prtilly fctored (5 points) 0. On question 8, did the student fctor the polynomil completely?. Yes (0 points). No, ut polynomil ws prtilly fctored (5 points). On question 9, did the student fctor the polynomil completely?. Yes (0 points). No, ut polynomil ws prtilly fctored (5 points) 2. On question 20, did the student fctor the polynomil completely?. Yes (0 points). No, ut polynomil ws prtilly fctored (5 points) Totl Numer of Points A B C D F 35 points nd ove 20 points nd ove 05 points nd ove 90 points nd ove 89 points nd elow Any score elow C needs remedition! The Cse of the Missing Middle Term 2003 www.econlerningcenter.com Rev. 06.09.03