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Teacher Notes: Factoring Differences of Two Perfect Squares Using algebra tiles model what a perfect square number looks like. 3 2 1 1 2 3 (1 tile) (4 tiles) (9 tiles) Have students conduct a pair rally. Using one set of student notes have students list as many perfect square numbers as they can in 20 seconds. One student will start with 1 give the paper to the other student to write 4 back to write 9 etc. as many as they can do in 20 seconds. Tell students this is without talking because other teams could steal their answers. After the 20 seconds are up have all students stand up. As you call out (or write on the board) a perfect square number they don’t have down, students sit down with their partner. It is interesting to find out how many students remember the perfect square numbers. (Awards could be given to winners, the last standing.) The more perfect square numbers a student can recognize the easier many areas of math will be. Perfect Square Numbers: {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225} Ask the following questions every time: Can anything be factored out? Is this a difference of two perfect squares? Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. x2 – 81 1. 2. x -9 x x2 -9x +9 +9x -81 Sum: 0x 25x2 – 16 (1x)(81x) (3x)(27x) (+9x)(-9x) 3. (5x + 4)(5x – 4) 5. 8. 2x2 – 98 Work on students understanding and using the shortcut method. Product: - 81x2 Or 100a2 – 1 (x + 9)(x – 9) 4. (10a + 1)(10a – 1) 6. 17 – 68a2 16x2 – 36y2 (4x + 6y)(4x – 6y) 7. -49x2 + 64y2 2(x2 – 49) 17(1 – 4a2) 64y2 – 49x2 2(x + 7)(x – 7) 17(1 + 2a)(1 – 2a) (8y + 7x)(8y – 8y) 36x2 + 49 prime 9. 4 16a2 – 9 2 2 (4a + )(4a + ) 3 3 10. (x + 2)2 – 9 [(x+2) + 3][(x+2) – 3] (x + 5)(x – 1) Teacher Notes: Factoring Perfect Square Trinomials 1. (a + b)2 (a + b)(a + b) a +b a a2 +ab +b +ab + b2 (a2 + 2ab +b2) 2. Ask students if they can see the pattern. Remember as they work some students will pick up the pattern quickly and will be able to use it, while other students will never see the pattern. The will continue to foil or box to fine the product. Perfect square trinomials: (a – b)2 (a – b)(a – b) a -b a a2 -ab -b -ab + b2 (a2 – 2ab +b2) (a + b)2 = a2 + 2ab +b2 (a – b)2 = a2 – 2ab +b2 When testing a trinomial ask the following questions: Is there a GCF for all three terms? Is the first term a positive perfect square number? Is the last term a positive perfect square number? Is the middle term the product of the square root of the first and last term? Is the second term positive or negative? Determine whether each trinomial is a perfect squarer trinomial. If so, factor it. 3. x2 – 8x + 16 4. yes; (x – 4)2 5. x2 +18x + 81 yes; (7x – 2)2 6. 16x2 – 56xy + 49y2 8. yes; (4x – 7y)2 9. 64x2 – 72x + 81 no 4x2 – 28x + 49 yes; (2x – 7)2 yes; (x + 9)2 7. x2 + 50x + 225 2x2 – 10x + 25 no 10. 1 2 x + 3x + 9 4 yes; ( 1 x + 3)2 2