Transformation 2013 PBL Planning Form Guide PBL Title: Quadratics Galore Teacher(s): Bonnie McClung School: Transformation 2013 T-STEM Center Subject: Algebra 1 Abstract: As students move into quadratic equations, there are much more vocabulary and techniques to be learned involving solving equations. This PBL unit is designed to make the experience meaningful and present students with various ways to use quadratic functions. It also relates the new concepts with previously learned concepts.

MEETING THE NEEDS OF STEM EDUCATION THROUGH PBL UNITS

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Begin with the End in Mind 

Does this PBL meet the criteria for STEM student needs (21st century skills, TEKS, TAKS)?

Section 1 Summarize the theme or “big ideas” for this problem based learning unit. Students will learn vocabulary associated with quadratic functions and be able to solve quadratics by factoring or graphing. They will also learn how transformations change the graph of functions. Section 2 Identify the TEKS/SEs that students will learn in the PBL (two or three).

(A.1) Foundations for functions. The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to: (A) describe independent and dependent quantities in functional relationships; (A.2) Foundations for functions. The student uses the properties and attributes of functions. The student is expected to: (A) identify and sketch the general forms of linear (y = x) and quadratic (y = x2) parent functions; (B) identify mathematical domains and ranges and determine reasonable domain and range values for given situations, both continuous and discrete (A.3) Foundations for functions. The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. The student is expected to: (A) use symbols to represent unknowns and variables (A.4) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations.

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The student is expected to: (A) find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations; (C) connect equation notation with function notation, such as y = x + 1 and f(x) = x + 1. (A.9) Quadratic and other nonlinear functions. The student understands that the graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions. The student is expected to: (A) determine the domain and range for quadratic functions in given situations; (B) investigate, describe, and predict the effects of changes in a on the graph of y = ax2 + c; (C) investigate, describe, and predict the effects of changes in c on the graph of y = ax2 + c; and (D) analyze graphs of quadratic functions and draw conclusions. (A.10) Quadratic and other nonlinear functions. The student understands there is more than one way to solve a quadratic equation and solves them using appropriate methods. The student is expected to: (A) solve quadratic equations using concrete models, tables, graphs, and algebraic methods; and (B) make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph of the function. Section 3 Identify key performance indicators students will develop in this PBL.       

Describe dependent and independent quantities in functional relationships. Sketch parent graphs and transformations with or without a calculator. Determine mathematical domains and ranges. Analyze graphs of quadratic functions and make conclusions. Describe functional relationships in a variety of ways. Demonstrate an understanding of the properties and attributes of functions. Demonstrate an understanding of quadratic and other nonlinear functions.

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Demonstrate an understanding of the mathematical processes and tools used in problem solving.

Section 4 Identify the 21st century skills that students will practice in this PBL (one or two). Creativity, critical thinking, problem solving, flexibility, team work Section 5 Identify STEM career connections and real world applications if content learned in this PBL.

Careers: Engineering, Athletics, Firefighting Applications: Design, ball trajectories, water trajectories

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The PBL You are on the writing staff of a local company that is in the process of developing crossword puzzles that incorporate mathematical terms and the real-world applications of those terms. Your team has been assigned the task of writing the crossword puzzle targeting quadratic functions. The “across” words must be the key vocabulary targeting quadratics, and the “down” words must be the applications of quadratics. In order for your team to succeed at the assignment, you must learn all about quadratic functions and their applications. The puzzle book goes to the printer in seven days, so you don’t have much time. Let’s get started!

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Map the PBL

Performance Indicators (Refer to Step I, Section 3)

1. Describe dependent and independent quantities in functional relationships 2. Sketch parent graphs and transformations with or without a calculator 3. Determine mathematical domains and ranges 4. Analyze graphs of quadratic functions and make conclusions 5. Describe functional relationships in a variety of ways 6. Demonstrate an understanding of the properties and attributes of functions 7. Demonstrate an understanding of quadratic and other nonlinear functions 8. Demonstrate an understanding of the mathematical processes and tools used in problem solving 9. Multiplication of binomials

Taught before the project

Taught during the project

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10. Factor trinomials

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11. Find solutions to quadratic functions, relate zeros, roots, solutions, and x-intercepts to graph of quadratic functions 12. Analyze graphs and draw conclusions

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13. Understand vocabulary associated with quadratic functions and use in a creative manner

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5E Lesson PBL Title: Quadratics Galore TEKS/TAKS objectives: A1.A; A2.A,B; A3.A; A4.A,C; A9.A,B,C,D; A10.A,B

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their applications. The puzzle book goes to the printer in seven days, so you don’t have much time. Let’s get started!” Have the groups fill in a “Know/Need to Know” Chart to help them get started (see below) and paste the chart into their math journals. The students will continue to add to this chart as the unit progresses. Engage Activity Products and Artifacts Commonalities between pictures, discussion of articles, Know/Need to Know Chart, math journals Engage Activity Materials/Equipment Computer, Internet access, projector or overhead, pictures/transparencies of aforementioned objects, chart paper, markers, articles, “Know/Need to Know” Chart Engage Activity Resources http://regentsprep.org/Regents/math/math-topic.cfm?TopicCode=polymult (optional) http://www.freeimages.co.uk/ (free photos) http://plus.maths.org/issue29/features/quadratic/index.html http://plus.maths.org/issue30/features/quadratic/index-gifd.html

Explore #1 Activity Present the Explore #1 Activity to the students (see below). Provide students with small dry erase boards, markers, and erasers. Have students work problems #1-#3 in their groups. Monitor progress as the students work and answer any questions along the way. When the students reach problem #4, demonstrate that the students must set the equation equal to zero and factor the equation to arrive at the solution. Walk them through the steps necessary to arrive at the solutions. Upon completion of the manual method, demonstrate the graphing method using the graphing calculator. Walk them through finding points that they can easily graph. Show students how to calculate the solutions of the quadratic function using the graphing calculator. Identify additional vocabulary as you talk with them and add the terms to the word wall (line of symmetry, roots, zeros, xintercepts, vertex, minimum, maximum, etc.). Upon completion of the activity, have students add to their Know/Need to Know Charts and turn in their completed activity.

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Explore Activity Products and Artifacts Explore #1 Activity, Know/Need to Know Chart additions Explore Activity Materials/Equipment Explore #1 Activity, graphing calculators, word wall Explore Activity Resources None

Explain #1 Activity Debrief with the students regarding the Explore #1 Activity. Refresh their knowledge of the vocabulary and key concepts (including graphing calculator steps). Discuss with the students additional examples of greatest common factor (GCF). Introduce the students to factoring additional types of polynomials. You may choose to use the Sample Activity below or the following web link: http://regentsprep.org/Regents/Math/mathtopic.cfm?TopicCode=FacEq. Have the groups work together to practice factoring and provide them with a teacher-generated quiz to ensure understanding. Upon completion of the practice and quiz, offer an additional activity where students can discuss the mathematics involved in factoring. Have them work in groups to complete a series of sample problems and post their answers around the classroom. Have students walk around to compare strategies and answers. Sample Problems: Given each of the following trinomials, match each with its equal function in factored form and its solutions: f(x) = x2 + 3x – 10 A. f(x) = (x - 2)(x - 5) 1. x = 2, x = -5 2 B. f(x) = (x + 1)(x + 10) 2. x = 2, x = 5 f(x) = x – 7x + 10 C. f(x) = (x – 2)(x + 5) 3. x = -1, x = 10 f(x) = x2 – 9x – 10 f(x) = x2 + 11x + 10 D. f(x) = (x + 1)( x – 10) 4. x = -1, x = -10 Given each of the following trinomials, match each with its equal function in factored form: 6x4 + 18x2 A. 8x3(3 + 2x3) 2 18x – 12x B. 10x3(2 + 3x2) 24x3 +16x6 C. 6x2(x2 + 3) 3 5 D. 6x(3x – 2) 20x + 30x

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Pose the question “In the first four problems, what do you observe about the sign of the constant term and the signs of the factors?” When the constant term is negative, the factors have opposite signs. When the constant term is positive, the factors have the same sign. Upon completion of this activity, have teams create five trinomials to be factored and solved and five binomials to be factored and solved (and the factored forms and solutions as an answer key). Teams will write their problems on large pieces of chart paper and hang them up around the classroom. Groups will rotate around the classroom working through all of the problems. Debrief the activity with the students upon completion and clear up any misunderstandings. Make sure that you continue to introduce and reintroduce key quadratic vocabulary during the discussion and add to the word wall. Have students record key concepts in their journals and describe what they are learning. Additionally, have students continue to record vocabulary terms and definitions in their journals. Explain Activity Products and Artifacts Explain #1 Sample Activity, teacher-generated quiz, sample problems, team generated problems and factored forms, word wall, journal entries, Know/Need to Know chart additions Explain Activity Materials/Equipment Handouts, quiz, chart paper, markers, journals, pencils Explain Activity Resources http://regentsprep.org/Regents/Math/math-topic.cfm?TopicCode=FacEq (optional)

Explore #2 Activity We have seen how quadratic equations can be used to model real-life experiences, and we have also seen how linear equations are important. Ask the students to share some characteristic they have in common with one or both of their parents (eye color, height, hair color, dimple, sense of humor, etc.). Explain that they are part of a family and, as such, they share certain characteristics of that family. Follow up the discussion with a comparison to families of functions and parent functions. Provide examples of parent functions and have students sketch the graph of each in their math journals. They are welcome to use the graphing calculator if they need assistance.

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y = x or f(x) = x y = x2 or f(x) = x2 y = |x| or f(x) = |x| y=

x or f(x) =

x

Our emphasis has been on quadratic equations in this PBL but I wanted to review some of the other functions we have studied as well. Have the students get with a partner to work on the following problem, recording it in their journals: The function f(x) = -0.06x2 + 0.6x + 10.26 can be used to model the height in meters of an arch support for a bridge, where the x-axis represents the water level and x represents the distance in meters from where the arch support enters the water. Can a sailboat that is 14 meters tall pass under the bridge? (Holt Algebra I, Holt, Rinehart and Winston, 2007).  Enter the function in Y1 in your calculators.  Enter a friendly viewing window (students should be able to do this by now but the teacher may need to provide guidance).  Calculate the maximum (the highest point on the graph).  Solution: Since this point is 11.76 m above the water, it is not possible for the sailboat to pass under the bridge. Debrief and see if students have any questions. Next, have the students work a problem independently and follow with a discussion of the solutions. Students will record all work in their journals. A golf ball is hit and it has to go over water. The path of the ball can be modeled by the quadratic function f(t) = -16t2 + 80t , where t is time in seconds and the beginning height of the ball is zero feet. Follow the path of the ball and answer the following questions.  What is the maximum height of the ball?  How long did it take the golf ball to reach that maximum height?  If the ball must remain in the air at least 4.7 seconds to go over the water, did the golfer succeed? Explain your answer in the context of the problem. Explore Activity Products and Artifacts Journal entries, Know/Need to Know chart additions Explore Activity Materials/Equipment Journals, pencils, above problems, graph paper, graphing calculators

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Explore Activity Resources Holt Algebra I, Holt, Rinehart and Winston, 2007

Explain #2 Activity After debriefing the Explore #2 with students, draw two parabolas on a transparency, one that opens up and the other that opens down. Label the vertex and the axis of symmetry. Using f(x) = ax2 + bx + c as the model, ask the students which variable they think controls whether the graph opens up or down (let them use dry erase boards to check their responses). Give the students some problems to graph using their calculators and ask them the same question after they graph them. f(x) = -3x2 + 7 f(x) = 4x2 – 12 -2x2 + y = - 15 5x2 + y = 7 **they will have to solve for y in the last two before graphing. Next, have students graph the following quadratic equations and identify the maximum or minimum value. Show the students how to use TRACE and CALC: 3 or 4 to identify the maximum or minimum value. You might want to check and make sure they understand the vocabulary of maximum and minimum. y = x2 – 6x + 10 y = = -x2 + 6 y = x2 – 2x – 3 Debrief and follow with a discussion involving domain and range. Ask the students to identify the independent variable (x). Is there any value they cannot replace x with in the equation? This leads the students to understand that the domain is all real numbers. What is the y value called? (dependent variable, range). In the first problem, the minimum value is 1. Students should know that the point (3, 1) is the vertex of the parabola but the minimum value is 1. The entire graph is seen above that y-value. Therefore the range is y ≥ 1. Have students use dry erase boards to identify the maximum or minimum point of the other two problems. Then state the domain and range of each. Debrief for any questions. Add to the word wall as additional terms surface in the discussion. Explain Activity Products and Artifacts Graphs and solutions to above problems ©2008 Transformation 2013

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Explain Activity Materials/Equipment Graphing calculators, pencils, word wall, transparencies, overhead projector Explain Activity Resources None

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Explore/Explain Activity Resources http://www.ronblond.com/M11/QFA.sf.APPLET/index.html

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Evaluate Activity Provide students with the crossword puzzles from each group and give them time to work to solve the puzzles. Students will provide each group with feedback regarding the difficulty of the puzzles and things that they might change. Evaluate Activity Products and Artifacts Completed crossword puzzle packets Evaluate Activity Materials/Equipment Puzzle packets, pencils, and rubric Evaluate Activity Resources Rubric

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Topic: What I Know

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What I Need to Know

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Explore #1 Activity Use the mathematical model, d = 1/2at2 + v to solve the following problem: d = distance traveled in meters (m) a = acceleration rate in meters per seconds squared (m/s2) t = time in seconds (s) v = beginning velocity in meters per second (m/s) â&#x20AC;&#x153;A car is stopped at a red light. The light changes and the car accelerates at a rate of 6 m/s2. Meanwhile, an SUV is in another lane traveling at a speed of 21 m/s and continues through the light that has now turned green. His distance can be modeled by d= 21t, where d is the distance in meters, since it is traveling at a constant speed.â&#x20AC;? 1. Write an equation for the distance the car travels in time, t.

2. Write the mathematical model, d = 1/2at2 + v, in function notation where distance is a function of time.

3. Identify the independent and dependent variables in the function above in #2.

4. When will the SUV and the car be side by side (equations equal to each other)?

5. Graph the function and identify the solutions. Also, identify three additional terms that describe the solutions of a quadratic function.

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Explain #1 Sample Activity Find the Greatest Common Factor (GCF) of each of the following quadratic equations. 1. 2. 3. 4. 5.

2y3 + 6y2 27m3 + 18m4 15a4 – 25a2 100x4 – 25x2 18c2 + 24c

Factor each of the following polynomials using the GCF method. 6. 5a3 + 15a2 + 20a 7. 12w2 + 4w + 2 8. 25p5 – 10p3 – 5p2 9. 4m5 + 8m4 + 12m3 + 6m2 10. 3n5 + 9n3 + 12n2 + 15n For each problem below, find one pair of numbers that will satisfy both equations. 11. ____ + ____ = 14 12. ____ + ____ = 10 13. ____ + ____ = -3 14. ____ + ____ = -4 15. ____ + ____ = 1 16. ____ + ____ = -7 17. ____ + ____ = 12 18. ____ + ____ = 8 19. ____ + ____ = -1 20. ____ + ____ = 14

and and and and and and and and and and

____ x ____ = 40 ____ x ____ = 25 ____ x ____ = -28 ____ x ____ = 4 ____ x ____ = -30 ____ x ____ = 12 ____ x ____ = 20 ____ x ____ = 15 ____ x ____ = -42 ____ x ____ = 48

Find the factors for of the following trinomials. Then set each of the factors equal to zero and solve the quadratic function. 21. c2 – 5c – 14 22. y2 + 14y + 49 23. x2 + 7x + 10 24. k2 – 4k + 3 25. x2 – 3x – 28 26. a2 – 7a + 6 27. x2 – x – 2 28. m2 – 2m – 48 29. t2 + 5t + 6 30. w2 + 3w – 4

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Pick any 5 of the above problems and use your graphing calculator to solve the equations by graphing. 31.

32.

33.

34.

35.

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Additional Sample Handouts Handout #1 Solve each of the following quadratic equations by factoring. Show all work. 1. x2 + x – 20 = 0

2. x2 + 11x + 18 = 0

3. x2 – 6x – 27 = 0

4. 2x2 + 7x + 5 = 0

5. 3x2 – 4x – 15 = 0

6. – 3x2 + 16x – 16 = 0

7. 3p2 – p – 4 = 0

8. 3x5 – 12x3 = 0

9. x2 – 49 = 0

10. x2 – 8x + 16 = 0 11-15 Sketch a graph of problems 1-5 clearly marking the zeros on the x-axis. Use your calculator to verify your zeros are the same as you found by factoring. Challenge: What value of “b” would make 2x2 + bx – 12 factorable? Handout #2 With each of the following problems, sketch the parent graph in one color and the other graphs on the same axis in different colors. Make a color key to identify the graphs. 1.

f(x) = x2

g(x) = x2 – 2

r(x) = - x2 + 4

p(x) = 2x2 – 3

2.

f(x) = x

g(x) = x – 2

r(x) = -x + 4

p(x) = 2x – 3

3.

f(x) = |x|

g(x) = |x| - 2

r(x) = - |x | + 4

p(x) = 2 |x| - 3

4.

f(x) = x3

g(x) = x3 – 2

r(x) = - x3 + 4

p(x) = 2x3 – 3

5.

f(x) = √x

g(x) = √x – 2

r(x) = - √x + 4

p(x) = 2√x - 3

6. For each of the above problems, explain in words, the transformations that occurred in each new problem.

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Plan the Assessment Engage Artifact(s)/Product(s): Commonalities between pictures, discussion of articles, Know/Need to Know Chart, math journals

Explore #1 Artifact(s)/Product(s): Explore #1 Activity, Know/Need to Know Chart additions

Explain #1 Artifact(s)/Product(s): Explain #1 Sample Activity, teacher-generated quiz, sample problems, team generated problems and factored forms, word wall, journal entries, Know/Need to Know chart additions Explore #2 Artifact(s)/Product(s): Journal entries, Know/Need to Know chart additions

Explain #2 Artifact(s)/Product(s): Graphs and solutions to above problems

Explore/Explain #3 Artifact(s)/Product(s): Journal entries, teacher-generated worksheets/quiz, Know/Need to Know chart additions

Elaborate Artifact(s)/Product(s): Completed crossword puzzles

Evaluate Artifact(s)/Product(s): Completed crossword puzzle packets

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Rubrics Quality of Idea

Communication

Expert (4)

Competent (3)

Beginner (2)

Novice(1)

Selects a theme that addresses the problem

Selects a theme that mostly addresses the problem

Selects a theme that somewhat addresses the problem

Selects a theme that does not address the problem

Uses vocabulary efficiently and with purpose

Uses vocabulary mostly with purpose

Uses vocabulary with some purpose

Does not use vocabulary with purpose

Product written with careful deliberation

Product written with good deliberation

Product written with some deliberation

Product written with no deliberation

Crossword flows with ease from beginning to end

Crossword flows with ease most of the time

Crossword is somewhat connected but contains some gaps/errors

Crossword does not flow/is not coherent

Vocabulary words used effectively

Vocabulary words used mostly effectively

Vocabulary words used somewhat effectively

Vocabulary words not used effectively

Vocabulary used in a very creative manner

Vocabulary used in a mostly creative manner

Vocabulary words used in a somewhat creative manner

Did not try to be creative

Used 90 â&#x20AC;&#x201C; 100 percent of the vocabulary

Used 80-90 percent of the vocabulary

Used 60-80 percent of the vocabulary

Used less than 50 percent of the vocabulary

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Story Board  Week 1 Activities

 

Week 2 Activities

Day 1 Engage (45 minutes) Explore #1 (45 minutes) Day 6 Elaborate (90 minutes)

Day 2 Explain #1 (90 minutes)

Day 7 Elaborate (90 minutes)

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Day 3 Explore #2 (90 minutes) Day 8 Evaluate (90 minutes)

Day 4 Explain #2 (90 minutes)

Day 9 Evaluate (90 minutes)

Day 5 Explore/Explain #3 (90 minutes)

Day 10

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