Completing the Square TEACHER NOTES MATH NSPIRED
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- Archibald Bryant
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1 Math Objectives Students will use algebra tiles to build a geometric model of a perfect square quadratic. Students will recognize the characteristics in the algebraic expression of a perfect square quadratic. Students will complete the square in an algebraic expression. Students will relate the patterns for (x + n) 2 to (x n) 2. Student will model with mathematics (CCSS Mathematical Practice). Students will look for regularity in repeated reasoning (CCSS Mathematical Practice). Vocabulary perfect square quadratics coefficient of the x-term constant term About the Lesson Students will begin the lesson by manipulating algebra tiles to build perfect square quadratics and recording the successful expressions. Students will continue the lesson by finding the patterns in the algebraic representation of a perfect square quadratic. Students will identify perfect square quadratic expressions. Students will fill in the missing terms of perfect square quadratics. Students will apply the patterns they found to perfect square quadratics with a negative coefficient in the x-term. TI-Nspire Navigator System Use Screen Capture to examine patterns that emerge. Use Live Presenter to demonstrate how to build perfect square quadratics. Use Quick Polls to assess student understanding throughout the lesson. TI-Nspire Technology Skills: Download a TI-Nspire document Open a document Move between pages Move between windows on a page Grab and drop algebra tiles Tech Tips: Make sure the font size on your TI-Nspire handheld is set to Medium. Lesson Materials: Student Activity Completing_the_Square_ Student.pdf Completing_the_Square_ Student.doc TI-Nspire document Completing_the_Square_Lua.tns Visit for lesson updates and tech tip videos Texas Instruments Incorporated 1 education.ti.com
2 Discussion Points and Possible Answers Tech Tip: To manipulate the algebra tiles, reinforce the idea that the tiles are moved up by grabbing the point on the tile and dragging it to the middle space. The x tile may be placed on the bottom of the x 2 tile by moving the point to the bottom-left corner of the x 2 tile. Click the x tile to rotate it to horizontal. For the larger perfect squares, it is important to move the x 2 tile to the top of the space. In the bottom of the screen, students should click (R)eset to clear all tiles from the mat. Move to page Build perfect square quadratics with lead coefficient 1 by dragging the algebra tiles to the middle window. Record the perfect squares found. Click (R)eset to start over to find a new perfect square. Answer: Completed table is below. Side of Square Perfect Square Quadratic Coefficient of x-term Constant Term x x x + 1 x 2 + 2x x + 2 x 2 + 4x x + 3 x 2 + 6x x + 4 x 2 + 8x Tech Tip: To clear the screen before trying to build another perfect square, click (R)eset and press. Teacher Tip: If students are unfamiliar with using the TI-Nspire technology algebra tiles, a demonstration might be appropriate. Perfect squares can be built without actually creating a square. If the tiles are just dragged into the middle window, the file keeps track of the tiles. Encourage students to put the tiles next to each other to create the square. They will snap together. Make sure to connect the geometric with the algebraic relationship Texas Instruments Incorporated 2 education.ti.com
3 TI-Nspire Navigator Opportunities If students have difficulty, use the Screen Capture or Live Presenter with TI-Nspire Navigator to demonstrate how to build perfect square quadratics. 2. What patterns do you notice for all perfect squares? Answer: Students may notice that the coefficient of the x-term is increasing by 2 and that the constant terms are the perfect squares. a. What relationship exists between the side of the square and the coefficient of the x-term? Answer: The constant term on the side of the square doubles to become the coefficient of the x-term. b. What relationship exists between the side of the square and the constant term? Answer: The constant term on the side of the square is squared to become the constant term of the perfect square quadratic. c. What relationship exists between the coefficient of the x-term and the constant term? Answer: The coefficient of the x-term is halved and then squared to become the constant term. d. Why is this called completing the square? Sample answer: Answers will vary but should include some discussion of the geometric model of building a square and/or the algebraic expression of (x + n) 2. Teacher Tip: If you have TI-Nspire CAS handhelds, students can use Menu > Algebra > Expand to find the answers here. 3. Expand the following: a. (x)(x) Answer: x Texas Instruments Incorporated 3 education.ti.com
4 b. (x + 1)(x + 1) Answer: x 2 + 2x + 1 c. (x + 2)(x + 2) Answer: x 2 + 4x + 4 d. (x + 3)(x + 3) Answer: x 2 + 6x + 9 e. (x + n)(x + n) Answer: x 2 + 2xn + n 2 TI-Nspire Navigator Opportunities You could do a Quick Poll to ensure that students understand perfect squares. For example, have students expand (x + 7)(x + 7). 4. Use either method to find (x + 5) 2. Answer: Students may answer using the pattern of doubling the constant term in the binomial for the coefficient of the x-term and squaring the constant term in the binomial to get the constant term in the perfect square quadratic: x x State whether the following are perfect square quadratics. Explain why or why not. a. x 2 + 3x + 9 Answer: Not perfect; half of 3, squared is not Texas Instruments Incorporated 4 education.ti.com
5 b. x x + 49 Answer: Perfect square; half of 14, squared is 49. c. x x Answer: Perfect square; half of 24, squared is 144. d. x 2 + 6x + 36 Answer: Not perfect; half of 6, squared is not Fill in the missing terms to make the following perfect square quadratics. a. x x + Answer: 64 b. x Answer: 18x c. x x + Answer: 121 d. x Answer: 20x 2012 Texas Instruments Incorporated 5 education.ti.com
6 e. x 2 + 3x + Answer: 9 4 or In your own words, explain how to complete the square algebraically. Sample answer: Answers will vary but students should mention the relationship between the coefficient of the x-term and the constant term. 8. Expand the following: a. (x)(x) Answer: x 2 b. (x 1)(x 1) Answer: x 2 2x + 1 c. (x 2)(x 2) Answer: x 2 4x + 4 d. (x 3)(x 3) Answer: x 2 6x + 9 e. (x n)(x n) Answer: x 2 2xn + n Texas Instruments Incorporated 6 education.ti.com
7 9. Do the negative values in question 8 change the pattern of perfect square quadratics? Explain. Answer: The negatives do not change the pattern. The coefficient of the x-term is still double the constant term in the binomial, and the constant term is the square of the constant term in the binomial. 10. Fill in the missing terms to make the following perfect square quadratics. a. x Answer: 34x b. x 2 26x + Answer: 169 c. x 2 36x + Answer: 324 d. x Answer: 30x e. x 2 5x + Answer: 25 4 or 6.25 TI-Nspire Navigator Opportunities Use Quick Polls throughout the lesson to assess student understanding. Wrap Up Upon completion of the discussion, the teacher should ensure that students understand: The patterns present in perfect square quadratics. How to recognize a perfect square quadratic expression. How to complete the square in an algebraic expression Texas Instruments Incorporated 7 education.ti.com
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