# Unit #3: Investigating Quadratics (9 days + 1 jazz day + 1 summative evaluation day) BIG Ideas:

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3 9 Math's Next Top Model students will determine an appropriate quadratic model using x-intercepts and one point Lesson Included N N QF3.02 determine, through investigation using a variety of Suggestion..Bouncing ball students strategies (e.g., applying properties of quadratic can find the equation using the x-intercepts functions such as the x-intercepts and the vertex; using and 1 point transformations), the equation of the quadratic function that best models a suitable data set graphed on a scatter plot, and compare this equation to the equation of a curve of best fit generated with technology (e.g., graphing software, graphing calculator); 10 Review Day (Jazz Day) 11 Summative Unit Evaluation Grade 11 U/C Unit 3: Investigating Quadratics 3

5 3.1.1 Pair Share Partner A will explain to partner B how you would answer question #1 in column A. Then partner B will explain to partner A how to answer question #1 in column B. The process is repeated for each question number. Partner A Partner B 1. What is the x-intercept for the function? 1. What is the x-intercept for the function? 2. Sketch a linear function with an x-intercept of Sketch a linear function with an x-intercept of Solve the equation x + 6 = Solve the equation x 5 = Given f ( x) = x + 6, evaluate f ( 6). 4. Given f ( x) = x 5, evaluate f (5). 5. Graph the function y = x + 6. State the x-intercept. 4. Graph the function y = x 5. State the x-intercept. Grade 11 U/C Unit 3: Investigating Quadratics 5

6 3.1.2 As a matter of fact ors Match the graphs with their factors. Reminder: Only the x-intercepts need to match (do not worry about the direction and/or shape of the graph). y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) x x + 1 x + 2 x + 3 x + 4 x + 5 x x - 1 x - 2 x - 3 x - 4 x - 5 Grade 11 U/C Unit 3: Investigating Quadratics 6

7 3.1.3 Zero ing In Match the functions with their x-intercepts. 2 y = x = ( 4x + 1)( 2x 3) y = ( 4x + 1)( 2x 1) y = x( 2x + 3) y y = ( 2x + 3)( 2x 3) y = ( x 2 )( x + 2) = ( 4 x 1)( 2x + 1) y = x( 2x 1) y = ( 4 x 3)( 2x + 3) y y = x( 2x 3) ( 4 x 1)( 4 + 3) y = x = x( x 2) y y = ( 2 x 3)( 4x + 3) y = ( 4 x 1)( 2x + 3) y = ( 2x + 1)( 4x 3) y = x( 4x 3) y = x( 2x + 1) y = ( 4x + 1)( 4x 3) y = x( 4x + 1) y = ( 2 x 3)( 2x + 1) ( 2x + 3)( 2 1) y = x = x( x + 2) y = ( 4x + 3)( 2x 1) y y = x( 4x 1) y = x( 4x + 3) Grade 11 U/C Unit 3: Investigating Quadratics 7

8 3.1.4 Quadratic Equation vs. Quadratic Function Complete the following table. The zeros and roots must be the same within a row. Algebraic Model Of Quadratic Function Graphical Model Of Quadratic Function x-intercepts or zeros Equation and Roots Of Equation y y =( )( ) =( )( ) ( x + 4)( 3) y = x y =( )( ) -3 and 5 y =( )( ) Equation: 0 = ( x )( x ) Roots: x = and x = Equation: ( x + 2)( 1) 0 = x Roots: x = and x = Equation: 0 = ( x )( x ) Roots: x = and x = Equation: 0 = ( x )( x ) Roots: x = and x = Equation: 0 = ( )( ) Roots: x = 4 and x = 1 2 Grade 11 U/C Unit 3: Investigating Quadratics 8

10 3.1.5 Teacher Instructions - BLM and BLM Rows 1 4 of BLM y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) y = ( )( ) 1. First player places a paper clip under the factors (one on the left set and one on the right) X x + 1 x + 2 x + 3 x + 4 x + 5 x x - 1 x - 2 x - 3 x - 4 x 5 2. They then try to match with a function above that would cross the x-axis at the same place when defined in the form y = (x + 4)(x -2). It will be important to point out to students that the orientation and shape (stretch factor) of the match is irrelevant at this point. 3. The other player will verify the choice using the graphing calculator [e.g., y = (x + 4)(x -2)]. A possible prompt for this would be Does the function on the calculator cross at the same places as the function selected by your opponent? 4. If the selected function has the same x-intercepts, the player is able to put his/her marker on the game board and record the factors in the defining equation. 5. The other player then moves one of the paper clips to try to gain an unclaimed spot on the board. The paper clip on the right must stay with the group of factors on the right and the paper clip on the left must stay with the group of factors on the left. Reminder: Students have not factored at this point in the course. Grade 11 U/C Unit 3: Investigating Quadratics 10

11 Unit 3 : Day 2 : The Simple Life Description/Learning Goals Minds On: 10 Students will represent situations using quadratic expressions. Action: 45 Students expand and simplify quadratic expressions using computer algebra systems in a game scenario. Consolidate:20 Total=75 min Minds On Pairs Exploration Each pair of students will be given a bag of algebra tiles that have been prepackaged to form a factorable quadratic expression (ex. x 2 + 6x + 8; x 2 + 5x + 6, x 2 + 7x + 10). Students must arrange the tiles into a rectangle. Students will fill in BLM once they have created their rectangle. Assessment Opportunities MCF 3M Materials BLM BLM Algebra tiles Optional: Computer algebra systems Once students have created their rectangles, introduce the names of the tiles (i.e. which tiles are the x-tiles, the x 2 tiles and the unit tiles) and instruct students to continue with BLM. Action! Individual Exploration Students will explore various tools for expanding binomial expressions. The teacher will need to model the chart method for expanding binomials. Example: Expand and simplify: (x + 4)(x -3) Virtual manipulatives can be found at en/nav/topic_t_2.ht ml or in the Sketches folder in GSP. (x + 4)(x -3) = x 2 + 4x 3x 12 = x 2 + x - 12 x +4 x x 2 4x -3-3x -12 Consolidate Debrief Pairs A Simple Game Students will use their expanding skills to complete the game. The object of the game is to piece together the original game board. Mathematical Process of Lesson Focus: Connecting (Students will connect various representations of binomial multiplication). Individual Journal Students will complete a journal entry response to the following question: Dora was absent from today s lesson. She calls you on the phone and asks you to explain how to multiply two binomials. Write your explanation. Make enough copies for each pair of students for BLM The game board should be cut up into individual squares. Skill Drill Home Activity or Further Classroom Consolidation Appropriate selection of practice questions should be assigned for homework. Grade 11 U/C Unit 3: Investigating Quadratics 11

12 3.2.1 "Wrecked"- tangles Arrange the tiles in your bag into a rectangle. Draw the arrangement of the tiles in the space provided below. Record the length, width and area of your rectangle. Length Width Area Trade your bag with another group and repeat the process. Record the length, width and area of your rectangle. Length Width Area Grade 11 U/C Unit 3: Investigating Quadratics 12

13 3.2.2 Expanding our Math Horizons Using the Algebra Tiles Expand and simplify the following expressions. Hypothesis Expression x ( x +1) 3 ( x +1) ( x + 3 )( x +1) x ( x + 4) 4 ( x + 4) ( x + 4) 2 Explain how to expand ( + 3 )( x +1) Expanded Form x without using the tiles. Test your hypothesis x + 4 x + 5. Check with the algebra tiles. Expand ( )( ) Using the Chart Method Expand and simplify ( x 4 )( x + 3). Expand and simplify ( + 6)( x 8) x. Grade 11 U/C Unit 3: Investigating Quadratics 13

14 3.2.2 Expanding our (CAS Version) Using the Computer Algebra System Expand and simplify the following expressions. Expression Expanded Form x x +1 Hypothesis ( ) 3 ( x +1) ( x + 3 )( x +1) 2 x ( x +1) 5( x 4) ( 2 x + 5)( x + 1) x ( x + 4) 4 ( x + 4) ( x + 4 )( x + 4) Explain how to expand ( + 3 )( x +1) x without using your calculator. Test your hypothesis 2x + 1 x 5. Check using your calculator. Expand ( )( ) Grade 11 U/C Unit 3: Investigating Quadratics 14

15 3.2.3 A Simple Game 2x x - 15 (3x + 1)(2x 3) 2x x + 15 (2x + 3)(x + 5) 5y 2 12y + 4 x 2 + 3x - 13 x x - 45 (x + 15)(x 3) x 2 + 3x - 28 (x + 7)(x 4) 6x 2-9x + 1 3x x x 2 9 5y 2 12y + 4 x 2 + 3x x 2-6x + 1 (2x + 3)(2x 3) (y - 2)(5y 2) (x + 8)(x 5) (3x - 1) 2 x x x 2 + x x 2 + 3x (5x 4)(x + 1) 10x 2 29x + 10 x 2 + 5x + 6 (x + 3)(x + 2) x 2-3x - 40 x 2-9x - 10 (x + 1)(x 10) 9x 2 + 6x + 1 x (5x + 3)(2x 1) 3 + 4x - 15x 2 (3 5x)(1 + 3x) 5x 2 + x - 4 (5x 4)(x + 1) 5x 2 - x - 4 (5x + 4)(x - 1) (2x 5)(5x 2) (x - 8)(x + 5) (3x + 1) 2 (x + 6)(x + 6) x 2 + 6x x 2 - x - 3 (5x - 3)(2x + 1) 25x 2-36 (5x 6)(5x + 6) x x + 40 (x + 8)(x + 5) x 2 + 3x - 10 (5 + x)(x 2) x 2-81 (x + 3)(x 3) x x + 45 (x + 9)(x + 5) 9x 2-1 (3x - 1)(3x + 1) (x 6)(x 6) (5x - 3)(3x + 1) (5x + 1)(5x + 6) (5x + 1)(5x + 1) 6x 2-1 x 2-6 Grade 11 U/C Unit 3: Investigating Quadratics 15

16 Unit 3 : Day 7 : Back to the Future Minds On: 15 Action: 40 Consolidate:20 Total=75 min Minds On Description/Learning Goals Connect the written description and the graphic and algebraic representations of a quadratic function Answer questions arising from real-world applications, given the algebraic representation of a quadratic function (QF3.03) Pairs Think/Pair/Share Activity Pairs will receive 4 cards of graphs and a sheet with the 4 scenarios from BLM In partners, the students will match each graph to a particular scenario. Once students have all finished, they will then have 1 minute to tape their graphs under the appropriate scenario which has been written on chart paper and posted around the room. Whole Class Think/Pair/Share Activity Review placement choices posted under each scenario. Have students share their justification for each posting, starting with those that are different. Grade 11 U/C Materials BLM (one set of graphs cut into individual cards and one scenario sheet for each pair in the class) BLM BLM Tape to post graphs Scenarios written on chart paper Assessment Opportunities Teachers can laminate the sets of 4 cards and the scenario sheets for re-use. Literacy Strategy: Four corners variation Action! Pairs Activity Explain to students that they will determine which equation matches each of the two given graphs and scenarios on BLM Students will work in pairs to complete BLM Points labelled on the graphs in BLM and are the vertices. Consolidate Debrief Concept Practice Mathematical Process Focus: Connecting and representing (Students will connect and represent a variety of graphical, algebraic and written descriptions for real-world scenarios.) Whole Class Discussion Discuss the rationale for determining which equation matches each graph. Take up the answers to BLM 3.7.2, possibly by having students post an answer and explain it on the board. Allow students to share any different strategies that they used to answer the questions. Home Activity or Further Classroom Consolidation Students will complete BLM Draw attention to the use of proper units in solutions, and discuss the domain and range for each scenario. The homework is similar to BLM and uses the other two scenarios from class. Grade 11 U/C Unit 3: Investigating Quadratics 16

17 3.7.1 Back to the Future Scenario The holder places the football on the ground and holds it for the place kicker. The ball is kicked up in the air and lands down field. Graph A four-wheeled cart is held at the bottom of a ramp. It is given a gentle push so that is rolls part of the way up the ramp, slows, stops and then rolls back down the ramp. A motion detector is placed at the top of the ramp to detect the motion of the cart. A student stands facing a motion detector. He quickly walks toward the detector, slows down, stops and then slowly walks away from the detector. He speeds up as he gets farther away from the detector. A diver is on the diving platform at Wonder Mountain in Canada s Wonderland. She jumps up and dives into the water at the base of the mountain. Grade 11 U/C Unit 3: Investigating Quadratics 17

18 3.7.1 Back to the Future (continued) Distance (metres) Distance (metres) Time (seconds) Time (seconds) Distance (metres) Distance (metres) Time (seconds) Time (seconds) Grade 11 U/C Unit 3: Investigating Quadratics 18

19 3.7.2 Back to the Future Given the two equations 1. 2 d = 0.2t 0.6t d = 5t t The holder places the football on the ground and holds it for the place kicker. The ball is kicked up in the air and lands down field. Equation: A four-wheeled cart is held at the bottom of a ramp. It is given a gentle push so that is rolls part of the way up the ramp, slows, stops and then rolls back down the ramp. A motion detector is placed at the top of the ramp to detect the motion of the cart. Equation: Distance (metres) (2, 20) Distance (metres) (1.5, 0.5) Time (seconds) Time (seconds) 1. What is the height of the football at 0 seconds? 1. How far is the cart from the detector at the start? 2. What is the maximum height of the football? 2. When is the cart closest to the detector? 3. What is the height of the ball after 3 seconds? 3. How far is the cart from the detector at 1 second? 4. When is the football above 15m? 4. How far does the cart travel before it stops and starts going back down the ramp? Grade 11 U/C Unit 3: Investigating Quadratics 19

20 3.7.3 Back to the Future Given the two equations 1. d = 3t 2 + 6t d = 0.08t 1.6t + 9 A student stands facing a motion detector. He quickly walks toward the detector, slows down, stops and then slowly walks away from the detector. He speeds up as he gets farther away from the detector. A diver is on the diving platform at Wonder Mountain in Canada s Wonderland. She jumps up and dives into the water at the base of the mountain. Equation: Equation: Distance (metres) (10, 1) Distance (metres) (1, 48) Time (seconds) Time (seconds) 1. How far is the student from the detector when he starts to walk? 1. How high is the platform above the ground? 2. When is the student closest to the detector? 2. What is the diver s maximum height above the water? 3. What is his distance from the detector after 2 seconds? 3. At what time is she 36m above the water? 4. When is he more than 3m from the detector? 4. When is she less than 21m from the water? Grade 11 U/C Unit 3: Investigating Quadratics 20

22 3.9.1 Connecting the Dots 1. With your partner, determine the maximum number of parabolas that could be drawn through the points given in each of the following graphs. Graph A Graph B Graph C Graph D 2. Complete the following table: Graph Number Of Points Number Of Possible Parabolas A B C D 3. What is the minimum number of points required to define a unique parabola? Grade 11 U/C Unit 3: Investigating Quadratics 22

23 3.9.2 Math s Next Top Model Back to the Wall and Back Here s the data again from the activity we completed in Unit Plot the data on the grid provided. Time (s) Distance from Motion Detector (m) Construct the curve of best fit. 3. Using the intercepts, create an algebraic model for the data and record it in the box below. f(x) = 4. Using your curve of best fit, determine the distance from the wall at 2.75 seconds. 5. Using your algebraic model, determine the distance from the wall at 2.75 seconds. 6. Reflect on the accuracy of your model in determining the distance at 2.75 seconds. Grade 11 U/C Unit 3: Investigating Quadratics 23

24 3.9.2 Math s Next Top Model (continued) 7. Enter the data into your graphing calculator and generate the regression that best fits the data. Record the regression equation below. Round the coefficients to the nearest whole number. g(x) = 8. Using the regression model, determine the distance from the wall at 2.75 seconds. 9. Reflect on the accuracy of the regression model in determining the distance at 2.75 seconds. 10. In order to compare your model to the regression model we need to express them in the same form. Factor the regression model. 11. Compare the factored form of the regression model to your model. What factor do you need to multiply your model by to make it match the regression model? (We will refer to this value as a.) 12. As discussed earlier on we need at least three points to draw a parabola. Define your equation in this form: f(x) = a(x zero)(x zero). f(x) = a(x - )(x - ) 13. Substitute any point from the data into the equation in #12 and solve for a. What do you notice? 14. Pick another point and solve for a. Does the choice of point make a difference? Grade 11 U/C Unit 3: Investigating Quadratics 24

25 3.9.3 The Flight of a Golf Ball The table below gives the height of a golf ball over time. 1. Determine an algebraic model for the data. Time Height (m) (seconds) Verify your model by entering the data into your calculator and graphing your algebraic model. Rate how well your algebraic model fits the data using a scale of 0-1 (1 being a perfect fit). Justify your reasoning. Journal Entry: h(t) = Summarize the process for finding an algebraic model for data that is quadratic. Reflect: Draw a parabola(s) whose equation cannot be determined using the process outlined. Grade 11 U/C Unit 3: Investigating Quadratics 25

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