brittlindahl.weebly.com€¦ · Web viewCapstone Unit Project. Unit . Introduction/Rationale:...

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Capstone Unit Project

Unit Introduction/Rationale:

OverviewFor my student teaching in the Spring of 2013, I will be teaching an Algebra I course.

Therefore I chose an Algebra I lesson for this 3-day unit. I tried to choose a topic that will be covered later in year, which is why I chose Quadratic Equations. In this chapter, students have already learned how to identify Quadratic equation, finding the direction of the parabola, and finding the vertex. Also, the students have learned the characteristics of Quadratic equation, including finding the zeros of an equation by looking at a graph. Earlier in the year, students also learned how to factor polynomials. Therefore, prior to starting this three-day unit, students should have an understanding of this prior knowledge.

Day one of this unit will cover graphing quadratic equations on a coordinate plane. Students already have prior knowledge about how to find many of the components of the graphs, such as the axis of symmetry and the vertex. This lesson will teach students how to find the y-intercept as well as two other points on the same side of the axis of symmetry. Then students will have to use all this information to form a graph.

Day two of this unit will cover solving quadratic equations by graphing. In this lesson, students will learn how to solve by graphing on a coordinate plane, as well as on their calculator. This day will include a calculator activity about how to find the roots, zeros, and x-intercepts on their calculators.

Day three of this unit will cover solving quadratic equations by factoring. In this lesson, students will use their prior knowledge of how to factor polynomials to find the solutions to a quadratic equation.

My lessons will involve using the SMART board. This allows for visual representation throughout the lesson to aid student understanding. Also, on day 2, we will be using the TI-83/84 calculators to do an activity about graphing quadratic equations.

Objective/Purpose

Prior to this point, students have learned mainly about linear equations and inequalities. This unit is focused on students getting a greater understanding of quadratic equations. Specifically, students should be able to graph and solve quadratic equations. Similar to linear equations, quadratic equations have a lot of real life applications. Therefore, understanding quadratic equations is very important for future mathematics courses.

My main mathematical objectives for this lesson are about graphing quadratic equations, solving quadratic equations using a graph, and solving quadratic equations by factoring. I want my students to learn how to graph a quadratic equation on an xy-coordinate plane. Next, I would like my students to be able to use the graph of a quadratic equation to find the values of x.

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Finally, my students should be able to be able to factor a quadratic equation and then use the factored form to solve for x.

Throughout my lesson, I would like students to learn how to work respectively with others. There are many opportunities for group and partner work throughout the lesson. I would like these activities to be learning experiences, not only mathematically, but also socially. Students should use these opportunities to learn how to work with and support others.

Meeting the Needs of Students

The varying academic abilities in the classroom helped me plan my lesson in order to meet the needs of all levels of students. Making certain accommodations in the classroom can benefit all students, not just ELL students or students with IEPs.

Throughout the lessons, I will be using a SMARTboard for the lecture and examples. This allows for visual representation throughout the lesson. I will also be writing down everything on the SMARTboard that students need to know and take note of during the lesson. I will make sure to verbally as well as visually make sure students know what aspects are important to know for each topic.

My lessons will also include worksheets in order to make notes and following along easier for all students. These worksheets will include space for notes, and some guided notes. They will also include all the examples I plan to complete during class. Therefore students can have everything they need to following along during the lesson. One of the most important aspects of the worksheets I feel is including all the examples. Since this lesson includes graphing, I feel it is important to provide students with the equations as well as the blank coordinate planes. This allows for the most accuracy when students do the examples and minimizes possible errors that could be made when following along.

During my lessons, I also have partner and group work. I feel this is very beneficial for all students because they are not only learning from me, but also learning from each other. Also, students will work and talk to students that they wouldn’t normally talk to outside of the classroom. I feel this allows for classroom unity that will make the classroom more comfortable and safe for everyone.

Day two of this unit will include a calculator activity. This allows for another visual way to solve quadratic equations. This representation will help students who are more hands-on learners to have another learning experience.

These accommodations will meet the needs of the ELL students as well as the students with IEPs who struggle with reading. It allows for less oral learning and more visual and hands-on learning. Therefore, it will be more effective for these types of students.

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Formal Lesson Plan SequenceName: Britt Lindahl

Date: 04/08/12Grade Level: 9

Time Allotted: 50 minutesNumber of Students: 24

I. Goals Students will learn how to graph a Quadratic equation on an xy-coordinate plane.

II. Objectives Students will find the axis of symmetry and vertex of a quadratic equation. Students will learn how to find the y-intercept of a quadratic equation. Find two more points on the same side of the axis of symmetry as the y-intercept. Students will learn how to graph these points as well as the rest of the quadratic equation.

III. Materials Algebra I textbook Notebook, folder, pencil Graph paper, or worksheet with coordinate planes SMARTboard with lecture notes and examples

IV. Motivation1. [10 minutes]Students will start with a warm-up that reviews finding the axis of symmetry

and the vertex of quadratic equations. This will be an independent work activity.2. Then students will be allowed to compare their answers with their partner.3. Finally, we will go over the answers to the warm-up as a class and answer any remaining

questions students may have over the material.4. Students will keep this warm-up for themselves.

Transition: “Today we will being using our knowledge of finding the axis of symmetry and the vertex to graph quadratic equations on a coordinate plane.”

V. Lesson Procedure5. [15 minutes] First, pass out the worksheet of notes for the day. The worksheet includes a

blank space for notes as well as all the examples with coordinate planes that will be done in class so students can complete the problems.

6. All notes and examples will be written on the SMARTboard.7. Begin the lesson and notes by going over the steps needed to graph a quadratic equation. 8. Speak and write on the SMARTboard. “There are four things that need to be found in

order to graph the equation. These four items are the axis of symmetry, vertex, y-intercept, and two points on the same side of symmetry as the y-intercept.”

9. “Next graph the axis of symmetry, the vertex, the y-intercept, and the two other points.”10. “Then, reflect the points across the axis of symmetry.”11. “Finally connect the points with a smooth curve.”12. Then, do a quick review of how to find each of the four part of graphing.

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13. “How do we find the axis of symmetry?” Give wait time, and then call on a student, or multiple students. Students should reply, “To find the axis of symmetry, the formula is x = -b/2a.” Write responses and correct definition on SMARTboard.

14. “Next, how to we find the vertex?” Give wait time, then call on a student. Student should reply, “To find the vertex, use the value you got for the axis of symmetry, and plug this value in for x, and solve for y.” Write responses and correct definition on SMARTboard.

15. “How do we find the y-intercept of the graph?” Students will not be as familiar with, but see what student responses will be. State or restate that, “When finding the y-intercept, find the c value, ax2 + bx +c, where (0, c) is the vertex. Similarly, if you plug in x=0, then the value for y= equals the y-intercept.” Write responses and correct definition on SMARTboard.

16. “Let’s find two other points that are on the same side of the axis of symmetry as the y-intercept, as I previously stated.” Write on SMARTboard.

17. [18 minutes] “Let’s do an example on the worksheet. The first example is y= x2-x-6. What is the axis of symmetry?” Give wait time and call on a student. Ask for the student’s answer and how they arrived at that answer. “What is the vertex?” Give wait time and call on a student. Ask for the student’s answer and how they arrived at that answer. “What is the y-intercept?” Give wait time and call on a student. Ask for the student’s answer and how they arrived at that answer. “Finally, what are two other points on the same side of the axis of symmetry as the y-intercept?” Give wait time and call on a student. Ask for the student’s answer and how they arrived at that answer. “Now plot these points on the graph (plot points on SMARTboard), mirror the points over the axis of symmetry, and connect them with a smooth line.” Complete actions on SMARTboard

18. Repeat this same procedure of wait time, asking for participation, and doing the example on the SMARTboard for the first three examples. If the students are participating and seem to have an understanding of the material, have students come up to the board alone, or with their classmates help, and complete the rest of the examples. One student or two students per example. If students do not seem to understand the material, then continue doing examples as a class for example number 4, and possibly 5-8. Student participation and understanding will determine whether students need more or less scaffolding.

19. At the end of the period, have students do an exit slip in order to assess what they learned today and how well they understood the material. Also homework will be written on the board out of the textbook. Holt McDougal Algebra 1 pg. 609-610 #1-6, 8-13, 15-26

VI. Closure[2 minutes] “Today we expanded on our knowledge of recognizing quadratic equations, finding the axis of symmetry and vertex in order to graph quadratic equations. Today we also learned how to find the y-intercept and two more points on the same side of symmetry. Then we connected all our knowledge together in order to connect the points and draw the graph.” [5 minutes] “I will now pass our your exit slip to complete in the next 5 minutes. Please work individually and when you are done you can turn it into me. After you complete your exit slip you can start your homework for the last few minutes of class.”

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VII. Extension[10 minutes] If time allows, an extra worksheet will be handed out. On this worksheet, students will have blank coordinate planes. Students are to create their own quadratic equation, find the four parts needed to graph that equation, and then graph the equation. Students can work with a partner and together create 4 equations and graphs.After they complete this activity, students can start their homework assignment.

VIII. AssessmentStudents will do a warm-up as well as an exit slip. A warm-up will assess their starting point before the lesson. It will review and highlight prior knowledge needed for the lesson. An exit slip assesses what the students learned in the lesson, how well they learned it, as well as what areas the students still need to work on. This information can help guide the lesson for the next day. If the students seem to have a good grasp of the concept, then as a teacher, you know you can move onto a new topic. If the students seem to still be struggling with the concept, it may mean the students need more work on that topic the next day. Students will also have homework each night on the material covered in class. The next day, I will ask student to ask question on the homework. What problems the students ask question about will also guide what topics the student’s grasped verse what topics the students did not grasp.

IX. Standards

Common Core StandardsCCSS.Math.Content.HSF-IF.C.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.

In this lesson, students will be given an equation and have to use their knowledge and procedure given to graph the equation on a coordinate plane. They will be expected to find the vertex, y-intercept, and two other points on the graph.

Standards for Mathematical Practice1. Make sense of problems and persevere in solving them

Given an equation, students should analyze the equation in order to find the necessary components to graph the equation. Students will have to solve for the axis of symmetry, vertex, y-intercept, and two other points. This takes knowledge of how to use the equation to find these points. Students should also be able to look at their points and graph and answer the question, “Does this make sense given this equation?” If the answer is yes, then they students most likely graphed it correctly. Although if the answer is no, then the student should check his or her work and see what doesn’t make sense.

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Warm-upFind the axis of symmetry

1. y = 4x2 – 7

2. y = x2 – 3x + 1

3. y = -2x2 + 4x + 3

4. y = -2x2 + 3x – 1

Find the vertex.

5. y = x2 + 4x + 5

6. y = 3x2 +2

7. y = 2x2 + 2x – 8

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Warm-upSOLUTIONS

Find the axis of symmetry

1. y = 4x2 – 7

x=0

2. y = x2 – 3x + 1

x=3/2

3. y = -2x2 + 4x + 3

x=1

4. y = -2x2 + 3x – 1

x=3/4

Find the vertex.

5. y = x2 + 4x + 5

(-2, 1)

6. y = 3x2 +2

(0, 2)

7. y = 2x2 + 2x – 8

(-1/2, -17/2)

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Graphing Quadratic Equations Notes

Notes:

Example:y = x2-x-6

Axis of symmetry:Vertex:y-intercept:Two other points:

Example:y =x2+6x+8


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Example:y = x2+3x-10


Example:y = x2+8x+12


Example:y = x2-8x-20


Example:y = x2-9x+18


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Example:y = -x2-11x-28


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In-class example Solutions

Example:y = x2-x-6

Axis of symmetry: x=1/2Vertex: (1/2, -25/4)y-intercept: y=-6 or (0, -6)Two other points: (-1, -4) (-2, 0)

Example:y =x2+6x+8

Axis of symmetry: x=-3Vertex: (-3, -1)y-intercept: y=8 or (0, 8)Two other points: (-2, 0) (-1, 3)


Example:y = x2+8x+12

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Axis of symmetry: x=-3/2Vertex: (-3/2, -49/4)y-intercept: y=-10 or (0, -10)Two other points: (1, -6) (2, 0)

Axis of symmetry: x=-4Vertex: (-4, -4)y-intercept: y=12 or (0, 12)Two other points: (-3, -3) (-2, 0)

Example:y = x2-8x-20

Axis of symmetry: x=4Vertex: (4, -36)

Example:y = x2-9x+18

Axis of symmetry: x=9/2Vertex: (9/2, -9/4)

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y-intercept: y=-20 or (0, -20)Two other points: (-1, -11) (-2, 0)

y-intercept: y=18 or (0, 18)Two other points: (1, 10) (2, 4)


Axis of symmetry: x=-5/2Vertex: (-5/2, -81/4)y-intercept: y=-14 or (0, -14)Two other points: (1, 8) (2, 0)

Example:y = -x2-11x-28

Axis of symmetry: x=-11/2Vertex: (-11/2, 9/4)y-intercept: y=-28 or (0, -28)Two other points: (-2, -10) (-3, -4)

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Exit Slip

y = x2-14x+49


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Exit SlipSolution

y = x2-14x+49

Axis of symmetry: x=7Vertex: (7, 0)y-intercept: y=49 or (0, 49)Two other points: (4, 9) (5, 4)

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Extension Activity

y =


y =


y =


y =


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Name: Britt LindahlDate: 04/09/12Grade Level: 9


I. Goals Given a graph of a quadratic equation, students should be able to use the graph to find the

values of x.

II. Objectives Students should be able to graph a quadratic equation. Students should be able to find the zeros on the graph. Students should be able to recognize that the zeros are the solution to the quadratic

equation.

III. Materials Algebra I textbook Notebook, folder, pencil Graph paper, or worksheet with coordinate planes SMARTboard with lecture notes and examples TI-83/84 Calculators (24 total)

IV. Motivation1. [10 minutes] Students will start with a warm-up that reviews graph quadratic equations

from the day before. 2. Then students will be allowed to compare their answers with their partner.3. Finally, we will go over the answers to the warm-up as a class and answer any remaining


Transition: “Yesterday we learned how to graph quadratic equations. Today we will take the knowledge and use it to solve quadratic equations by graphing.”

V. Lesson Procedure5. [13 minutes] First pass out notes sheet for the day. This sheet includes a spot of notes

and the examples we will be doing in class for students to do on their paper as the problems are done on the board.

6. “First let’s review what we need in order to graph a quadratic equation. What are the four things we need to find to graph a quadratic equation?” Give wait time and call on one or more students. Write the four things on the SMART board. The four things are axis of symmetry, vertex, y-intercept, and two more points.

7. Do an example of solving using a graph. “Similar to yesterday, when given an equation the first thing you want to do is graph the equation using the four things we talking about yesterday. Example one is y = x2-4x-5. First what is the axis of symmetry?” Give wait time and call on a student. Ask for the student’s answer and how they arrived at that

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answer. “What is the vertex?” Give wait time and call on a student. Ask for the student’s answer and how they arrived at that answer. “What is the y-intercept?” Give wait time and call on a student. Ask for the student’s answer and how they arrived at that answer. “Finally, what are two other points on the same side of the axis of symmetry as the y-intercept?” Give wait time and call on a student. Ask for the student’s answer and how they arrived at that answer. “Now plot these points on the graph, mirror the points over the axis of symmetry, and connect them with a smooth line.”

8. Once the graph is complete, look on the graph along the x-axis, “How many times does the graph cross the x-axis?” Give wait time and call on a student. “Where does the graph cross the x-axis?” Give wait time and call on a student. “Yes, the graph crosses the x-axis twice at x=-1 and x=5. These are our two solutions.”

9. Repeat this same procedure of wait time, asking for participation, and doing the example on the SMARTboard for two examples. If the students are participating and seem to have an understanding of the material, have students come up to the board alone, or with their classmates help, complete the rest of the examples. One student or two students per example. If students do not seem to understand the material, then continue doing examples as a class. Student participation and understanding will determine whether students need more or less scaffolding.

10. [20 minutes] At the end of the period, students will do a calculator activity. There are two activities. The first activity is how to use a table on a TI-83/84 calculator to find the solutions to a quadratic equation. The second activity is how to use a table and a graph on a TI-83/84 calculator to find the solutions to a quadratic equation.

11. First pass out worksheet with for the calculator activity. “Yesterday and today we have been graphing quadratic equations by hand. Now we are going to learn how to graph them on our calculator. The calculator can help us graph the equation and find the zeros very easily.”

12. “You will work with your partner on this activity for the remainder of class. Each student is expected to turn in an assignment at the end of class. Both partners are expected to put in equal effort and participation.”

13. At the end of the period, have students do an exit slip in order to assess what they learned today and how well they understood the material. Also homework will be written on the board out of the textbook. Holt McDougal Algebra pg. 625 #2-13, 15-23, 25, 29

VI. Closure[2 minutes] “Continuing from yesterday when we learned how to graph quadratic equations. Today we used our knowledge of graphing in order to use graphs to solve quadratic equations. Then furthering our knowledge, we learned how to use a table and graph an equation on our calculator to solve a quadratic equation. Graphing is one tool that we can use to solve quadratic equations.”[5 minutes] “I will now pass our your exit slip to complete in the next 5 minutes. Please work individually and when you are done you can turn it into me. After you complete your exit slip you can start your homework for the last few minutes of class.”

VII. Extension

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[10 minutes] If extra time is left, then students will do an extra challenge worksheet. It includes using the calculator to solve for the solutions to difficult equations. This worksheet will give students more help with graphing equations on their calculator.


IX. Standards

Common Core StandardsCCSS.Math.Content.HSA-REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Students will be required to take a quadratic equation, graph the equation, and then use the graph to find the solutions. After the graph gives them the solutions, they will need to check these solutions. In order to check their solutions, they must plug the solutions back into the equation and make sure the y=0.

Standards for Mathematical Practice5. Use appropriate tools strategically.

In this lesson, students will learn how to use their calculator to graph and solve quadratic equations. They will use the calculator to analyze the quadratic equation and look for the solutions. This is allowing them to know that they have many tools to use in order to solve these types of equations. They will also learn how to take the answers the calculator gives them, and check these answers by hand.

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Warm-up

y = x2+8x+16


y = -x2+12x-36


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Warm-upSolutions

y = x2+8x+16

Axis of symmetry: x=-4Vertex: (-4, 0)y-intercept: y=16 or (0, 16)Two other points: (-2, 4) (-1, -9)

y = -x2+12x-36

Axis of symmetry: x=6Vertex: (6, 0)y-intercept: y=-36 or (0, 36)Two other points: (4, -4) (3, -9)

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Solving Quadratic Equations by Graphing Notes

Notes:

y = x2-4x+5

Axis of symmetry:Vertex:y-intercept:Two other points:Solutions:

y = -x2-4x-4


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y = x2-6x+9


y =-2x2+2x+4


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Solving Quadratic Equations by Graphing NotesSOLUTIONS

y = x2-4x-5

Axis of symmetry: x=2Vertex: (2, -9)y-intercept: y=-5 or (0, -5)Two other points: (1, -8) (-2, 7)Solutions: x=-1 and x= 5

y = -x2-4x-4

Axis of symmetry: x=-2Vertex: (-2, 0)y-intercept: y=-4 or (0, -4)Two other points: (-1, -1) (-4, -4)Solutions: x=-2

y = x2-6x+9

Axis of symmetry: x=3Vertex: (3, 0)y-intercept: (0, 9)Two other points: (1, 4) (2, 1)Solutions: x=3

y =-2x2+2x+4

Axis of symmetry: x=1/2Vertex: (1/2, 9/2)y-intercept: y=4 or (0, 4)Two other points: (-2, -8) (-3, -20)Solutions: x=-1 and x=2

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Graphing Quadratic Equation on the Calculator

TI-83/84 Calculator Activity

Activity 1:

Solve by using a table:

5x2 + 8x – 4=0

Steps: Enter the equation in Y1 Press 2ND GRAPH to use the TABLE function Scroll through the values by using the up and down arrows. Find the values of 0 in the

Y1 column. The x-value when the Y1=0 are the zeros of the equation. (You should only find one zero at x=-2)

To check this answer plug the value x=-2 back into your equation and the answer should be y=0.

Solve each equation on your own by using a table:

1. x2-4x-5=0

2. x2-x-6=0

3. 2x2+x-1=0

4. 5x2-6x-8=0

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Activity 2:

Solve by using a table and a graph:

5x2+x-8.4=0

Steps: Enter the equation into Y1 To view both the table and the graph at the same time, set your calculator to the Graph-

Table mode. To do this, press MODE and scroll down to the 8th row and go over to the third item called “G-T” and press ENTER.

Press GRAPH. You should see the graph and the table. (If you do not see a full graph, make sure your window is set to the standard setting. Press ZOOM, then scroll down to ZStandard.)

To get a closer view of the graph, press ZOOM and select ZDecimal Press TRACE and use the left and right arrow keys to scroll to find the zeros. (You

should find two zeros at x=-1.4 and x=1.2) To check your solution, plug x=-1.4 and x=1.2 back into the equation and the result

should be y=0.

Solve each equation by using a table and a graph.

5. 2x2-x-3=0

6. 5x2+13x+6=0

7. 10x2-3x-4

8. x2-2x-0.96=0

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Graphing Quadratic Equation on the CalculatorSOLUTIONS

1. x=-1 and x=52. x=-2 and x=33. x=-1 and x=1/24. x=-0.8 and x=25. x=-1 and x=1.56. x=-2 and x=-0.67. x=-0.5 and x=0.88. x=-0.4 and x=2.4

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Exit Slip

y = x2+7x+6


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Exit SlipSOLUTION

y = x2+7x+6

Axis of symmetry: x=-7/2Vertex: (-7/2, 171/4) 171/4 = 42.75y-intercept: y=6 or (0, 6)Two other points:Solutions: x=-1, x=-6

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Extension Activity

Use a graphing calculator to find the solutions of each quadratic equation.

1. 5

16x+ 1

4x2=3

5

2. 1200 x2−650 x−100=−200 x−175

3. 15x+ 3

4x2= 7

12

4. 400 x2−100=−300 x+456

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Extension ActivitySolutions

Use a graphing calculator to find the solutions of each quadratic equation.

1. 5

16x+ 1

4x2=3

5

x=-2.3 or x=-1

2. 1200 x2−650 x−100=−200 x−175

no real solutions

3. 15x+ 3

4x2= 7

12

x=-1 or x=.75

4. 400 x2−100=−300 x+456

x=-1.6 or x=0.86

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Name: Britt LindahlDate: 04/10/12Grade Level: 9


I. Goals Students will factor a quadratic equation and then use the factored form to solve the

equation.

II. Objectives Students will use the Zero Product Property to solve the equation. Students will be able to factor a quadratic equation. Students will be able to use the factored form to solve for the x values of the quadratic

equation.

III. Materials Algebra I textbook Notebook, folder, pencil Graph paper, or worksheet with coordinate planes SMARTboard with lecture notes and examples

IV. Motivation1. [10 minutes] Students will start with a warm-up that reviews how to factor a polynomial.

This will be an independent work activity.2. Then students will be allowed to compare their answers with their partner.3. Finally, we will go over the answers to the warm-up as a class and answer any remaining


Transition: “We have been working with quadratic equations and yesterday we learned how to solve a quadratic equation by graphing. Today we are going to use our prior knowledge about factoring polynomials to solve quadratic equations by factoring.”

V. Lesson Procedure5. [20 minutes] All notes, speech, and solutions will be written on the SMART board 6. “In the warm-up, we reviewed how to factor polynomials. Once a polynomial is

factored, we can use the Zero Product Property to solve for x.”7. “The Zero Product Property says ‘For all real numbers a and b, if the product of two

quantities equals zero. At least one of the quantities equals zero.’ For example is we have ab=0 then a=0 or b=0.”

8. Do first example on worksheet. “Given (x-3)(x+7)=0, which his already in factored form. When we apply the Zero Product Property. This means that x-3=0 or x+7=0. Then if we solve for x, we find that x=3 or x=-7.”

9. “Then we should check our answers. If we plug x=3 into the equation, ((3)-3)((3)+7)=(0)(10)=0. Therefore 3 would be a solution. Then check if x=-7 is a solution to the equation. ((-7)-3)((-7)+7)=(-10)(0)=0. Therefore 7 would be a solution to the equation.”

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10. “The second example is (x)(x-5)=0. What is the first thing we would do to solve this?” Give wait time and call on a student. They should say set x=0 and x-5=0. Then ask, “What is the next step to solve this equation?” Give wait time and call on a student. The student should say solve for x. “What are our solutions?” Give wait time and call on a student. Student should reply x=0 or x=5. “If you plug these two solutions back into the original equation, it will show they are solutions.”

11. “This next example is a little more difficult. The next example is x2+7x+10=0. What do we do first to solve this quadratic equation?” Give wait time and call on a student. Student should reply that we should factor the equation. “What is the factored form of the equation?” Give wait time and call on a student. Student should say (x+5)(x+2)=0. “What do we do next to solve this equation?” Give wait time and call on a student. Student should say set x+5=0 and x+2=0. “What is the next and final step to solving our equation?” Give wait time and call on a student. Student should reply solve for x and get x=-5 or x=-2. “If you plug these two solutions back into the original equation, it will show they are solutions.”

12. If the students are participating and seem to have an understanding of the material, have students come up to the board alone, or with their classmates help, complete the rest of the examples. One student or two students per example. If students do not seem to understand the material, then continue doing examples as a class. Student participation and understanding will determine whether students need more or less scaffolding.

13. Do final word problem example as a class. “The height of a diver above the water during a dive can be modeled by h=-16t2+8t+48, where h is height in feet and t is time in seconds. Find the time it takes for the diver to reach the water.” Following same line of question from above examples.

14. Homework will be written on the board out of the textbook. Holt McDougal Algebra pg. 633 #1-31 odds.

15. [12 minutes] “We will be having a quiz tomorrow over graphing quadratic equations, solving by graphing, and solving by factoring. I have a review sheet for you to do with your partner. Work together with your partner and make sure you both know how to solve each question. If you have any questions, I will be walking about to help.”

VI. Closure[3 minutes] “Today we used our prior knowledge of factoring polynomials to help solve quadratic equations. We furthered our knowledge by applying the Zero Product Property, which helped us solve for each possible x in each equation. As I stated before, at the beginning of class tomorrow, we will have a quiz tomorrow over the information we have covered in the past 3 days. This includes graphing quadratic equations, solving quadratic equations by graphing, and solving quadratic equations by factoring. The quiz will be the same format at the review sheet. If you can solve the question on the review sheet, then you will be successful on the quiz tomorrow.”[5 minutes] “I will now pass our your exit slip to complete in the next 5 minutes. Please work individually and when you are done you can turn it into me. After you complete your exit slip you can start your homework for the last few minutes of class.”

VII. Extension

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Since students have a quiz tomorrow, if time remains, I will allow students to work on the review sheet, do their homework, or ask my any questions they may have. I feel this extra time is important since not all students have time outside of class to ask questions before the quiz tomorrow.


IX. Standards

Common Core StandardsCCSS.Math.Content.HSA-REI.B.4b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.CCSS.Math.Content.HSA-SSE.B.3a Factor a quadratic expression to reveal the zeros of the function it defines.

In this lesson, students will be using Zero Product Property as well as factoring to solve quadratic equations. They will have to set each factored equation equal to zero. Then after this, they will solve for the solution(s) of the equation.

Standards for Mathematical Practice7 Look for and make use of structure.

In this lesson, students will be asked to factor a quadratic equation, then to use this factorization to solve for the solutions of x. Factoring involves a lot of looking for and using structure. It is very systematic and procedure. Therefore, students who can look for and make use of structure will be able to factor well and be successful. Being able to look at a quadratic expression or equation and see it has a whole entity, but then break it down is a very important skill.

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Warm-up

Factor each polynomial:

1. x2+4x+3

2. x2+10x+16

3. x2+15x+44

4. 2x2+9x+10

5. 5x2+31x+6

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Warm-upSolutions

Factor each polynomial:

1. x2+4x+3

(x+1)(x+3)

2. x2+10x+16

(x+2)(x+8)

3. x2+15x+44

(x+4)(x+11)

4. 2x2+9x+10

(2x+5)(x+2)

5. 5x2+31x+6

(x+6)(5x+1)

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Solving Quadratic Equations by Factoring Notes

Notes:

Zero Product Property:

Examples:

(x-3)(x+7)=0

(x)(x-5)=0

x2+7x+10=0

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x2+2x-8=0

x2+2x+1=0

-2x2+12x-18=0

The height of a diver above the water during a dive can be modeled by h=-16t2+8t+48, where h is height in feet and t is time in seconds. Find the time it takes for the diver to reach the water.

39

Solving Quadratic Equations by Factoring NotesSOLUTION

Notes:

Zero Product Property:

Examples:

(x-3)(x+7)=0

x=3 or x=-7

(x)(x-5)=0

x=0 or x=5

x2+7x+10=0

x=-5 or x=-2

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x2+2x-8=0

x=-4 or x=2

x2+2x+1=0

x=-1

-2x2+12x-18=0

x=3

The height of a diver above the water during a dive can be modeled by h=-16t2+8t+48, where h is height in feet and t is time in seconds. Find the time it takes for the diver to reach the water.

t=-3/2 or t=2

41

Review Sheet

Solve each quadratic equation by graphing

1. y = x2+x-6

2. y=x2-2x-3

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Solve each quadratic equation by factoring

3. 2x2+6x=0

4. x2+11x-12=0

5. 2x2+3x+1=0

6. x2-11x+24=0

7. x2-3x+10=0

43

Review SheetSolutions

Solve each quadratic equation by graphing

1. y = x2+x-6

x=2 or x=-3

2. y=x2-2x-3

x=-1 or x=3

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Solve each quadratic equation by factoring

3. 2x2+6x=0

x=0 or x=-3

4. x2+11x-12=0

x=-12 or x=1

5. 2x2+3x+1=0

x=-1 or x=-1/2

6. x2-11x+24=0

x=8 or x=3

7. x2-3x+10=0

x=5 or x=-2

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Exit Slip

Solve each quadratic equation by factoring. Check your answer.

1. x2-8x-9=0

2. x2-3x-10=0

3. x2+2x-15=0

4. 3x2+18x+27=0

5. 2x2+6x+18=0

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Exit SlipSolutions

Solve each quadratic equation by factoring. Check your answer.

1. x2-8x-9=0

x=-1 or x=9

2. x2-3x-10=0

x=-2 x=5

3. x2+2x-15=0

x=-5 x=3

4. 3x2+18x+27=0

x=-3

5. 2x2+6x+18=0

x has no real solutions

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Assessment

The end of unit quiz is designed to specifically address all of the objectives stated in each lesson. Therefore if a student can complete the quiz and following the procedures presented in the lessons, then the student will have met the learning objectives.

At the beginning of each day I will start with a warm-up. This warm-up reviews prior knowledge needed for today’s lesson. This will give me a better understanding of student’s prior knowledge related to the day’s lesson. These warm-ups will be completed individually. Then the students will go over it with their partner. Finally, we will go over it as a class. This is to make sure the students have a concrete understanding of the prior knowledge needed for the day’s lesson. Also, at the end of each lesson, students will be given an exit slip. The teacher will collect these before the students leave. The exit slips will be graded in order to collect information about what students learned during the lesson. It will also help the teacher know what concepts need to be reviewed before going onto the next lesson. An exit slip can give information about if a class is ready to move onto harder concepts, or whether they need to be retaught or review a concept. Finally, homework will be assigned almost every day at the end of the lesson. The next day at the beginning of class, students will be allowed to ask questions about the homework. The questions students ask will also help guide me to better understand what concepts to review.

At the end of a chapter, the class will have a test over the entirety of the chapter. Throughout the chapter, students will be given quizzes to assess their understanding. These quizzes will occur ever 2-4 sections, depending on the amount and depth of the material. These quizzes will help students understand what concepts will be covered on the test. I will always work to make sure that every topic that is expected to be on the test will be covered on a quiz. Therefore students know what they are expected to know for the end of chapter test.

The quiz I created for the three sections I covered aligns very well with my objectives. I purposefully used my objectives when creating the quiz questions to make sure every objective was somehow incorporated into at least one question. The quiz will cover graphing quadratic equations, solving using graphing, and solving using factorization. The students must find the axis of symmetry, vertex, y-intercept, and two other points when graphing the quadratic equation. When solving for graphing, students must complete all the steps for graphing, plus then find where the x equals zero on the graph. Finally for solving using factorization, students must factor a quadratic equation, use the Zero Product Property, and then solve for x. All of these concepts were listed in the objectives of each lesson.

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Quadratic Equations Quiz

Graph the equation and fill in the missing details

1. y = x2+10x+25


2. y =9x2-18x+9


3. What are the solution(s) to the graph in #1

4. What are the solution(s) to the graph in #2?

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Solve by factoring:

5. x2+8x+15=0

6.x2-4x+3=0

7. x2-x-12=0

50

Quadratic Equations QuizSOLUTIONS

Graph the equation and fill in the missing details

1. y = x2+5x+4=0

Axis of symmetry: x=-5/2Vertex: (-5/2, -9/4)y-intercept: y=4Two other points:

2. y =9x2-18x+9

Axis of symmetry: x=1Vertex: (1, 0)y-intercept: y=9Two other points:

3. What are the solution(s) to the graph in #1

x=-4 or x=-1

4. What are the solution(s) to the graph in #2?

x=1

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Solve by factoring:

5. x2+8x+15=0

x=-3 or x=-5

6. x2-4x+3=0

x=1 or x=3

7. x2-x-12=0

x=-3 or x=4

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