Semester Unit PlanAlgebra I Text

Chapter 8: QuadraticsBy: Kaitlin BurkeNatalie Jenkins

Jaclyn WolfeHannah Worman

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Table of Contents

Rationale…………………………………………………………………………………………………………….3

Unit Calendar……………………………………………………………………………………………………...4

Mini Lesson #1……………………………………………………………………………………………………7

Full Lesson #2 (Jackie)………………………………………………………………………………………..8

Mini Lesson #3………………………………………………………………………………………………….12

Mini Lesson #4………………………………………………………………………………………………….13

Mini Lesson #5 (Quiz)………………………………………………………………………………………..14

Full Lesson #6 (Hannah)……………………………………………………………………………………15

Mini Lesson #7………………………………………………………………………………………………….19

Full Lesson #8 (Natalie)…………………………………………………………………………………….20

Mini Lesson #9………………………………………………………………………………………………….25

Mini Lesson #10………………………………………………………………………………………………..26

Mini Lesson #11………………………………………………………………………………………………...27

Full Lesson #12 (Katie)………………………………………………………………………………………28

Mini Lesson #13…………………………………………………………………………………………………

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Mini Lesson #14…………………………………………………………………………………………………

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Assessment Pieces……………………………………………………………………………………………..37

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Rationale

The overall goals of this unit involve using different methods to solve quadratic equations. Students will use factoring, completing the square and the quadratic formula to solve quadratic equations algebraically. Students will solve real life applications of quadratics, including optimization by finding the vertex of parabolas. They will be able to write quadratic equations based on graphs and then be able to draw a graph if they are given a quadratic equation. Students will be able to graph linear and quadratic inequalities of two variables and find solutions of the systems of inequalities.

The students are familiar with polynomials and are able to factor monic quadratic equations and complete the square. They know how to graph linear equations and make a table of data points. Students are able to graph one variable inequalities on a number line.

We understand that our students show potential but are not motivated to do their homework so we made our lessons as engaging as possible through different types of activities with different types of technology. When homework is assigned, we give students the opportunity to ask questions the next day and go over the solutions to check for understanding. Some of our lessons included exit slips to give students the opportunity to share their opinions about the lesson and things that they struggled with to give us an idea of how to improve student understanding. We include several types of assessment besides homework and tests grades. During the class periods we walk around to assess student understanding while they work on problems. Exit slips provide us with a good idea of where the students are at after the lesson has been taught. Two of our students struggle with reading and are allowed extended testing time so we allow these students to come back during lunch or at the end day to finish their tests.

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Unit Calendar

Date Lesson Plan By:

Brief Description of Content and Lesson

Technology, Special Activities, Manipulatives, Problem-Based, Instructional Strategies

May 5 (Day 1)

Jackie 8.1 and start 8.2The day will start with a fun video on factoring quadratic equations. Students will work on factoring and completing the square to solve quadratic equations.

Projector and laptop for video

May 6 (Day 2)

Jackie (Full)

8.2- Quadratic FormulaToday the students will be deriving the quadratic formula by using the completing the square method. They will practice using the quadratic formula to solve quadratic equations.

May 7 (Day 3)

Jackie 8.4- Factoring Nonmonic QuadraticsThe class will be learning a method of factoring nonmonic quadratic equations by making the look monic. They will practice using this method of factoring and comparing it to the quadratic formula

May 8 (Day 4)

Hannah 8.3- “Going the Other Way”, Review for QuizStudents will learn how to find quadratic equations given the roots. They will discuss how there are unlimited quadratic equations

Graphing calculator that teacher can project

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with the same roots. They will also review for tomorrow’s quiz by doing review problems.

May 9 (Half-Day) (Day 5)

All Quiz 8.1-8.4Students will be taking a quiz that assesses their ability to solve quadratic equations by factoring, using the quadratic formula, and being able to construct a quadratic equation if they are given the roots.

May 12 (Day 6)

Hannah (Full)

8.5 Introducing OptimizationToday, students will work on a problem that involves maximizing the area given a certain amount of fence. Then they will be asked to model the situation with a quadratic equation. They will discover that the maximum occurs at the vertex of the parabola.

Problem-basedgraphing calculator for each group

May 13 (Day 7)

Hannah 8.6- OptimizationToday, students will work more with optimizing quadratics. They will complete the square in order to find the vertex of the parabola. They will discuss how the average of the roots is the x-coordinate of the vertex. Students will work through multiple optimization word problems.

Graphing calculators

May 14 (Day 8)

Natalie (Full)

8.7- Graphing Quadratic Functions

Students will get an introduction to parabolas, and learn how to graph the shape based on vertex form equations. They will also learn how manipulations to the equation change the graph.

iPads or laptops

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May 15 (Day 9)

Natalie 8.7- Graphing Quadratic Functions Day 2, 8.8- Jiffy Graphs

Students will have half of the class to review material from section 8.7.  Then they will explore how they can use given material to find axis of symmetry and make quick sketches of parabolas.

Graphing Calculators

May 16 (Day 10)

Natalie 8.10- Solving with Graphing

Students will use graphing calculators to relate intersection points to solutions of a system of equations.  They will use their calculators to find these solutions.

May 20 (Day 11)

Katie 8.11- Inequalities: Practice with dotted line/ solid line when graphing inequalities. A project using plugging in points (using M&Ms as points) to find the solution to the system of linear inequalities.

SmartBoard, Group work with M&Ms activity

May 21 (Day 12)

Katie (Full)

8.11- Inequalities: Students will use their knowledge of solving equations of linear inequalities and do a Geogebra investigation to discover how to solve systems of quadratic equations

Geogebra, “Red Light Green Light", Problem-Based Learning, SmartBoard, Journaling

May 22 (Day 13)

Katie Review for Test, Jeopardy Game and time for questions, Jeopardy answer key passed out as a study guide

SmartBoard, Jeopardy Powerpoint

May 23 (Day 14)

All Two part assessment, calculator and no calculator, covers entire unit, taken individually

Graphing Calculators

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Day #1 Lesson Title: Getting Started

Goal: To continue practice on factoring and finding the zeros of a polynomial and introduce how to solve nonmonic polynomials.

Objectives: Students will factor polynomials to find the zeros of a function.

Students will make a table of inputs and outputs for a function, sketch its graph, and find the points that cross the x-axis

Students will divide polynomials by the leading coefficient and then factor to find the zeros.

Lesson Summary The class will begin watching a video “Teach Me How to Factor,” which will get them thinking about solving quadratics. They will then do a warm up of problems that involve factoring polynomials and completing the square. The first few can be factored easily and the last couple will have a leading coefficient that is not one. The teacher will introduce the idea of dividing by the leading coefficient then factoring. After those practice problems, the teacher will give the class a worksheet. The worksheet will have some polynomials with leading coefficient one, but most will not. The students will make a table of inputs and outputs for each polynomial and then sketch its graph to find where it crosses the x-axis. At the end of the lesson, the teacher will talk about a formula that can be used to solve quadratic equations derived by completing the square.

HW No Homework

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Name: Jaclyn WolfeDate: Day 2Grade Level: FreshmenCourse: Algebra 1Time Allotted: 50 minutesNumber of Students: 25

I. Goal(s): To divide a nonmonic quadratic equation by its leading coefficient to

make it monic To complete the square and factor a quadratic equation to solve for x To introduce the quadratic formula and how it is derived from completing

the square

II. Objectives: Students will complete the square to solve for the zeros of a quadratic

equation Students will divide by the leading coefficient of a quadratic, complete the

square and then factor as a difference of squares to solve for x. Students will solve equations using the quadratic formula

III. Materials and Resources: The teacher will need the Algebra I textbook Students will need notebooks and a pencil to take notes Students will need the Algebra I textbook to complete their homework

IV. Motivation (4 minutes): The teacher will start out the lesson as a continuation from section 8.2

from the previous day. Ask the students, “Who can tell me what the general equation is for

completing the square when solving for x for a quadratic equation? This is where we left off yesterday.”

Give students a minute or two to flip through their notes and come up with the equation. Have one student write it on the board.

“Great! We will be using this today. Now let’s take a look at this. What types of quadratic equations do we need to use completing the square for?”

Give students a minute to think. Expect answers such as “when we can’t factor.”

“That’s correct. We need to use completing the square when we can’t factor the equation nicely.”

Transition: “Now I am going to use this formula to show that if a general monic quadratic x2+rx+s = 0 has real solutions, we can find them.”

V. Lesson Procedure (35 minutes)

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State theorem 8.1 on page 684 of the textbook and write it on the board. Show that you can write these two solutions in one statement with the “plus or minus” signs. Expect students to ask questions about the plus or minus signs and the square roots. (3 minutes)

“I will show you how we can derive these answers from the form of completing the square we came up with yesterday.”

Go through the proof on page 684 step by step on the board of how to get the two solutions of the general monic quadratic. (7 minutes)

“Okay so we start out with x2+rx+s = 0. Since this is already monic, we don’t have to divide by the coefficient of x2. Next, we go through the steps of completing the square for this equation.”

Write out step 2 of the proof for completing the square Expect students to ask questions about this step. Show them how they can

rewrite – (r2/4) + s as – ((r2- 4s)/4). “Now we factor this result as a difference of squares.” Write the factored

version of step 3. “We now have our two factors. How do we solve for x?” Call on a

student and wait for the answer of “set each factor equal to 0 and solve for x.”

Go through the steps of solving for x on the board to get the two solutions.

Once students have gone through the proof and understand how the solutions were derived, give an example for them to do in their notes.

“Solve the equation x2 – 6x + 7 = 0. Apply theorem 8.1, which we just proved, to find the solutions.”

Give students a few minutes to solve this equation. Expect the error of forgetting the negative sign in front of the 6 and errors with simplifying the solutions. (3 minutes)

Once students have worked for a couple minutes, ask a student do go through the answer on the board.

While this student is going through the answer, correct any mistakes he/she does. “Make sure you are careful with the negative signs. We plug in -6 for the r so we will have –(-6).

Make sure the student includes both answers and writes them both out separately. (3 minutes)

“Great job! Now what if we don’t have a monic equation? Can we use a formula like this to solve for nonmonic equations?”

Expect students to say they can just divide by the leading coefficient. “We could do that but we might end up with some messy fractions. There

is another formula we can use called the Quadratic Formula.” Write down the general nonmonic equation on the board: ax2 + bx + c =

0. “I’m going to use this equation now for nonmonic equations. I’ll prove

that we can still find solutions for these types of quadratic equations”

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Go through the proof on page 686 of the textbook step by step. Be sure to show the similarities between this proof and the previous proof of completing the square. (5 minutes)

“This is essentially the same process, but this time we just need to replace r and s with a, b, and c. When we divide by a to make this equation monic for our proof, r becomes b/a and s becomes c/a. We simply plug these into our solutions we just found and come up with a new formula.”

Replace r and s with b/a and c/a respectively to give the solutions for x. Make sure the final solution includes the plus and minus signs.

“We just derived the Quadratic Formula. I will state the theorem for you to write down in your notes for the future.”

Write out theorem 8.2 on the board. Give students time to write this in their notes. (2 minutes)

“Now let’s use this to solve an equation.” Write down the equation 2x2+7x-15=0 and make sure the students are writing this in their notes.

“What are our a, b and c values in this equation?” Call on a student to answer the question. Ask the rest of the class if they agree.

“Good. Our a is 2, b is 7, and c is -15. Now I want you to go through the rest of the problem on your own. You know the values for a, b and c, and you just need to plug them into the quadratic formula now. (2 minutes)

Give students a couple of minutes to finish the problem. Walk around to see how students are doing and answer any questions they have. Expect errors in simplifying and forgetting negative signs. (3 minutes)

When most students are done, go through the problem step by step on the board for students to check their answers. Expect a couple questions on how to simplify the solutions. Make sure the students included both solutions for x. (3 minutes)

VI. Closure (7 minutes): The teacher will provide a quadratic equation that has no real roots: 3x2-

5x+7=0 Ask the students what will happen if they use the quadratic formula.

Have the students guess what will happen, and then let them try to use the quadratic formula. They will get a negative under the square root sign. (4 minutes)

End the lesson with a discussion of writing the roots of the form a+bi and a-bi. (2 minutes)

“When we get a negative under the square root, we still write the two solutions for x. We know that i = (-1)2, so we factor out an i from under the square root.”

VII. Extension: Students may begin working on their homework from the textbook. Homework is pages 688-689: 1-8, 10, 12-16, 18, 19, 27

VIII. Assessment Summary

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The teacher will be asking questions throughout the lesson to check for understanding

The teacher will be walking around and watching students solve problems throughout the lesson.

The students will complete a homework assignment from the textbook that will be collected the next day in class.

Homework is on pages 688-689, numbers 1-8, 10, 12-16, 18, 19, 27

IX. Standard(s): CCSS.Math.Practice.MP1 Make sense of problems and persevere in

solving them. CCSS.Math.Content.HSA-REI.B.4a Use the method of completing the

square to transform any quadratic equation in x into an equation of the form (x-p)2 = q that has the same solutions. Derive the quadratic formula from this form.

CCSS.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 and a-bi for real numbers a and b.

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Day #3 Lesson Title: Factoring Nonmonic Quadratics

Goal: To factor nonmonic quadratic equations using different methods

Objectives: Students will factor nonmonic quadratic equations by scaling them so they look monic.

Students will compare this method of solving for x with the quadratic formula method.

Lesson Summary The lesson will begin by going over the assigned homework so students have the chance to ask any questions. After the homework discussion, the lesson will begin with episode 37 on page 695 of the textbook. Students will go through Sasha’s method of factoring and then check the answer by multiplying it out. The teacher will explain how you can make a nonmonic quadratic seem monic and be factored. The teacher will then go through another example that doesn’t have a perfect square as the leading coefficient to test this method. The lesson will end by comparing this method of factoring to the quadratic formula and then coming up with an algorithm for the method of factoring in this section.

HW Pages 698-699, numbers 1, 2, 5-11

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Day #: 4 Lesson Title: Going the Other Way

Goal: Construct a quadratic equation given the equation’s two roots

Objectives: 1. Students will be able to multiply the roots together to find a quadratic equation

2. Students will be able to give multiple equations by using multiples of the original equation they found

Lesson Summary (one paragraph maximum)

Class will start by reviewing the homework from yesterday and allowing students to ask any questions that they may have about it. Introduce students to a “brain teaser:” I had a quadratic equation, solved it, and got rots of 5 and 7. What is my equation? Students will work on this together to find the equation. Then there will be a class discussion on how they found their equations. The teacher will then ask the class if there are other equations that would have the same roots. The teacher will show a graph of multiple parabolas with the same roots. After that, the teacher will review the quadratic formula with a couple of practice problems to help prepare the students for the quiz the next day.

HW Practice problems for the quiz: page 701 #1-13 (Review for the quiz)

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Day #5 Quiz

Goals: To assess the students’ knowledge of applying the quadratic formula to find the zeros of a quadratic equation

To assess the students’ ability of factoring nonmonic quadratics and to be able to “go the other way” to form quadratic equations

Objectives: Students will factor nonmonic quadratic equations

Students will state the quadratic formula and apply it to solve a quadratic equation

Students will construct a quadratic equation with two given roots.

Lesson SummaryThe quiz will assess the students on the first four days of the unit. They will be asked to factor nonmonic quadratic equations and use the quadratic formula to find the zeros. They will also be given the values of the roots and asked to write a quadratic equation. They will then be given one or two quadratic equations and asked to use any method to solve them. The quiz will be relatively short because this is a half-day.

HW No homework

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Name: Hannah WormanClass: Algebra 1Time Allotted: 50 minutesNumber of Students: 25Goals

To introduce the idea of optimizationObjectives

1. Students will be able to set up an expressions to model a situation2. Students will be able to use their knowledge of quadratics to optimize

some quadratic functionsMaterials

1. Worksheet2. Graphing Calculator to project for students to see3. Graphing calculators for students

Motivation (3 minutes)“Have you ever heard of the idea of getting the most for your money? Ask students about some situations that they have tried to get the most for their money. Well, today we are going to learn how to find the best (or most) option for some different situations. Let’s start with a problem that you might have seen before.”

Lesson Procedure(15 minutes)

1. Hand out worksheet and instruct students to begin working on problem number 1 in groups of 4 students.

2. As students finish up with number 1, make sure that all groups got the correct answer of 50 ft x 50 ft.

3. Direct students to start on problem 2 in their groups. Circulate through the room and see how students are doing. If they are struggling, suggest they start with the formula for the area of a rectangle. If they are still stuck, ask “Would introducing a variable be helpful?” Make sure that groups are able to write an expression for the area.

4. Instruct groups to move on to number 3 and 4 as they finish with problem number 2.

5. As groups begin to finish earlier than other groups, see if they can find a pattern to their solutions to the 4 problems. (The total amount of fence used for the vertical sides is the same as the total for the horizontal sides.)

(10 minutes)6. When most groups are finished, bring the class together for a discussion.

Go over each problem 2, 3, 4 and ask a couple of groups to share on the board how they solved the problems. (Ask groups with different solution methods.)

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7. After students are done presenting, ask students to write an expression for the area using variables. (“What if you labeled one of the sides as x? How can you express the other sides? How can you write an expression for the area of the pen?”)

8. Direct students to work in their groups again to write these expressions.(7 minutes)

9. After most groups have finished writing an expression for area, ask students to graph their expressions using their graphing calculators. Finding the best window might be difficult so warn students that they will have to mess around with the window so they can see the parabola correctly. Circulate to make sure each group have been able to graph their equations.

10. Have students answer the following questions with their groups (project questions or write them on the board):

a. What does the graph look like?b. Draw a sketch of the graph.c. Where on the graph is the maximum area?d. After graphing a couple of the equations for the different

problems, can you make any conclusions on how the maximums you found relate to the graphs?

(3 minutes)11. After students have started to make some conclusions in their groups,

pull the class back together to talk about what they found.12. Ask students to make share their observations/conclusions about the

graph. (Looking for some ideas about where the maximum is and how that relates to the vertex).

(8 minutes)13. Write the quadratic function f ( x )=x2−6x+8 on the board and ask

students to find the minimum value. Allow students to work in groups.14. After most groups have found the minimum, ask students to graph the

equation on their calculator to check their answer. 15. Ask students to complete the square to write f ( x )=x2−6x+8 as a perfect

square. Go through it as a class to make sure that everyone remembers how to do it.

16. Open a discussion about how the equation looks after they completed the square and how it relates to the vertex. (Just basic. Tomorrow, we will talk more in depth about how to complete the square to find the vertex)

Closure (4 minutes)17. “Today, we talked about finding the maximizing and minimizing

quadratic functions. We found that these occur at the vertex of the

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parabola. Tomorrow, we are going to talk more about optimizing these quadratic functions and finding the maximums or minimums.”

18. Ask students to write a quick summary of what it means to optimize a quadratic equation and how they find the optimal point.

ExtensionIf there is extra time, have students work on problems 10-12 on page 704 in the textbook.

Assessment Summary The teacher will walk around while students are working in groups in

order to gauge how students are doing while graphing the equations. The exit slip will allow the teacher to gauge if students understood what

it meant to optimize a quadratic equation. Since we are talking more about optimization tomorrow, any misconceptions can be addressed.

Standards CCSS.Math.Content.HSA-CED.A.1 Create equations and inequalities in one

variable and use them to solve problems. CCSS.Math.Content.HSA-SSE.B.3b Complete the square in a quadratic

expression to reveal the maximum or minimum value of the function it defines.

Build new mathematical knowledge through problem solving Apply and adapt a variety of appropriate strategies to solve problems

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Maximizing Area: Pet PensName___________________________

1. You have 200 feet of fencing to use for building a rectangular dog pen. Find the dimensions of the pen having the greatest possible area.

2. Suppose you build a pen such that it borders an apartment building. You only need to build three walls. Find the dimensions of the pen having the greatest possible area.

3. Suppose you have three pets, a dog, a cat, and a monkey. You want to divide the pen into three smaller rectangular sections. There will be one pen for each animal. The pens will be side by side as shown below. You still only have 200 feet of fencing. You will have to use some of the fencing to build the dividers between the pens. You want to maximize the area of the entire pen. What should the dimensions be?

4. Suppose you build a pen to separate five monkeys. You have 600 feet of fencing. What dimensions give the maximum area for the entire pen?

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Day #: 7 Lesson Title: Optimization

Goal: Use knowledge of quadratics to optimize some quadratic functions

Objectives: 1. Students will be able to complete the square to find the vertex2. Students will examine the graphs of quadratics to find the

vertex3. Students will apply their knowledge of a vertex to optimize a

quadratic equationLesson Summary (one paragraph maximum)

Students will continue to explore optimizing quadratics. We will begin by doing a warm-up of a couple completing the square problems. After that, the class will discuss how the minimum or maximum value of each is when the (something)2 is equal to zero. (Two examples will be used: one with a minimum and one with a maximum). Students will compare graphs and the expressions in vertex form. Students will be introduced to the idea that the average of the roots is the x-coordinate of the maximum or minimum. Towards the end of class, students will work in groups on several optimization word problems.

HW No Homework Assigned

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Name: Natalie JenkinsDate: Day 8Grade Level: FreshmenCourse: Algebra 1Time Allotted: 50 minutesNumber of Students: 25

I. Goal(s): To represent quadratic equations graphically using either y=a x2 or vertex

form of a parabola. To draw lines of symmetry in parabolas to create accurate sketches. To write equations of parabolas given points on the line or sketches of the

graph.

II. Objective(s): The students will graph quadratic equations in the form of y=a x2 to observe

the relation the leading coefficient has on the graph of a parabola. Students will write equations in vertex form and solve for other points on the

line using their equation. Given two points, students will solve for equations of parabolas using vertex

form. Students will draw lines of symmetry for parabolas to generalize a rule and

help sketch graphs quickly.

III. Materials and Resources Teacher will need the Algebra 1 textbook. Teacher will need a laptop or iPad hooked up to a projector to display a

video. Each student will need an access to an iPad or laptop computer. Each student will need a pencil and paper to take notes. The teacher will need a stack of graph paper to pass out to students if time

permits.

IV. Motivation (10 minutes) “So far our class has spent a lot of time talking about and working with

quadratic equations, right? But there’s still one huge part of mathematics that we haven’t touched on regarding quadratics, and that’s graphing. First, we learned about linear equations and then we saw those graphs create lines. So hopefully you guys are now curious about what the graphs of these quadratic equations with look like. Today we’re going to explore these types of graphs—what shapes they make, and certain properties about them.”

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“First, let’s watch this video of the popular game Angry Birds. Pay attention to the path of the birds’ flight”

Show video of Angry Birds projection: http://www.youtube.com/watch?v=Yk7xr4FT54I

After viewing the video, ask students what kinds of shapes they saw. Expect many students to draw the shape with their fingers, since they don’t have a word to describe it (2 minutes).

Once students start to describe the general shape of a parabola, ask questions to guide them into some properties of parabolas, such as:

o “Did all three birds have the exact same shape to their flight? What was different about them? What was the same?”. Let students answer for three minutes.

Transition: “So you guys had a really great discussion about the shapes of the birds’ flights. Even though you probably didn’t know it, you guys were discussing parabolas, which are the general shape for the quadratic equations we’ve been working with! Angry Birds is just one cool example of parabolas, but we really do see parabolas everywhere in real life. Let’s learn more about them.”

V. Procedure (35 minutes) Ask every student to turn on their iPad or laptop and navigate to the website:

http://www.mathopenref.com/quadraticexplorer.html Once every student has it running, they should observe sliders on the right

(these will control the a,b,c variables of the parabola). Ask students to click on the bottom right “Snap to integers” and then put in the following for a, b, and c respectively—1, 0, 0.

“The bottom left hand corner of the graph should have a very familiar equation written down. Under these conditions, what does your equation say?” (Students may freely call out y=x2; if not, choose one student to answer).

“Great, this is our most basic quadratic equation, and as I said before, it’s shape is a parabola.”

“Now I know this is a brand new shape for you guys, so I want to give you a couple minutes to explore it. Without changing any of the sliders, write down in your notes some observations you can make about the parabola, anything you notice. “ (4 minutes)

o “If you’re having trouble getting started, pick a few points on the parabola and see if you can notice anything about them. I suggest trying x=-1 and x=1.”

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Record students’ thoughts on the board. Every point is important but specifically make sure to hit ideas about the vertex and symmetry. (2 minutes)

“Now, does this look exactly like the path we saw the Angry Birds take?” (Students should respond no). “Right, so there must be other ways to draw a parabola. I’m going to give you the next five minutes to explore some shapes of parabolas using the slider ONLY for a. Write down observations you make in your notes, and we will come together as a class to make rules after we’re finished. I’ll be walking around to answer questions as you explore.”

After five minutes, come together as a class and talk about properties of parabolas. The students should make most of (if not all) the important observations; the teacher will simply apply mathematical terms to these.

o (i.e) A student might say “There’s a point at the bottom (or top) where the graph changes directions.” Teacher: “Great! This point is called the vertex. The vertex is the maximum or minimum point of the parabola.”

o The parabola gets wider a is between -1 and 1. If a>1 or a<-1 the graph gets more narrow.

o A student might say: “The parabola is the same on the right as the left.” Teacher: “Good observation. We say the parabola has a line of symmetry, in this case at x=0, since it is the same when reflected over the y-axis.”

o End this section describing the axis of symmetry if possible. If it comes up earlier, revisit the idea after you talk about other properties with the class. This discussion should be about 10 minutes.

Tell students they may now put back the iPads and computers, the rest of the lesson we will do without technology.

“When we moved our slider a, our axis of symmetry always said x=0, right? Do you think our line must always be at x=0? Do you think our vertex always has to be at (0,0)?” (students should say no, that it can move on the coordinate plane)

Now, introduce the second form of an equation: vertex form. Record the equation on the board, and tell students to put it in their notes as well. “Vertex form of a quadratic equation is: y-k=a(x−h)2 with vertex (h,k). It also

has the same shape as the graph of the equation y=ax2. “Now let’s see how using vertex form can help us when graphing quadratic

equations.”

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Write the following problem on the board: “A parabola has vertex (-3,10) and includes the point (2,0). Find an equation for the parabola, and sketch it’s graph.”

o Ask the students if they have any ideas on how to start this problem. Hopefully one student will remember that vertex form uses our vertex (-3,10) so we can plug that in for h and k, respectively. Otherwise, make the observation.

o Ask a student to share what they got when they plugged in the vertex

to the vertex form. y-10=a(x+3)2

o “What other information were we given? The point (2,0). Call on a student if no one responds.

o “We know a point on the line is of the form (x,y) so we can plug these

values in for x and y” 0-10=a(2+3)2. Give the students a few minutes to solve for a and ask for a solution. They should come up with a=-2/5.

o Plugging back into our form, students should get y-10=-2/5( x+3¿¿¿2 )o Give students five minutes to graph their equation, remembering

what they learned about a and using the vertex. Suggest they make a table and try to find some other points. Tell them 3 points is all they need for this graph. Circulate and expect questions since the topic is new.

o Once students all have something down, draw the graph on the board by making a table of values and using the vertex.

o “Is the axis of symmetry x=0 here? What is it?” Students will answer x=-3. They should also notice that it’s the x value of the vertex, but if not guide them to that.

VI. Closure “Great job working today class. I know it was a lot of material, but we will

have more time tomorrow to go over this section before we start the next part. “

Students may get started on their homework with the last few minutes of class. Tell them to raise their hands with any questions they might have.

VII. Extension Pass out a piece of graph paper to each student. Tell them to sketch a parabola on the graph paper. First, they should find a

vertex, and then draw two points symmetrical about the vertex to connect the points. Tell them specifically not to make their vertex (0,0). (3-5 minutes)

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Have every student pass their graph paper to another person in their row. Make sure everyone switches with someone.

Tell students their job is to come up with an equation in vertex form for their friends graph. They must not only include the equation, but also explicitly state what the vertex and axis of symmetry are. (5 minutes)

VIII. Assessment The teacher will circulate the class while the students work to check for

understanding. The teacher will also call on students when necessary and give informal

feedback. The teacher will ask questions to check for understanding while navigating

the lesson. Students should try problems 8-15 for homework and come prepared with

questions tomorrow.

IX. Standards CCSS.Math.Content.HSA-CED.A.1 Create equations and inequalities in one

variable and use them to solve problems. CCSS.Math.Content.HSA-CED.A.4 Rearrange formulas to highlight a quantity

of interest, using the same reasoning as in solving equations. CCSS.Math.Content.HSA-REI.D.10 Understand that the graph of an equation

in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them.

CCSS.Math.Practice.MP2 Reason abstractly and quantitatively.

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Day #: 9 Review of Graphing & A Quick Look At “Sketching” GraphsGoal: Students will strengthen their understanding of vertex form,

lines of symmetry, and parabolas in general. Students will master how to sketch parabolas given various

types of information.

Objectives: Students will sketch parabolas by finding roots and the axis of symmetry.

Students will solve the quadratic equation to find imaginary roots. In doing this, students will still be able to calculate the axis of symmetry and sketch a graph.

Lesson Summary: (one paragraph max)

8.7 was a new a difficult concept, so expect to spend the first 20 minutes of class answering questions on the homework or reviewing any left over material from that section. Warn students that you will be collecting their homework when the discussion is over, so they should ask questions now for an easy 100% homework grade. After collecting homework, move on to section 8.8 and go through episode 41 to show how to calculate roots, axis of symmetry, and sketch the graph. The students will follow this same process to solve problems 1 and 2 on page 721. Next, the teacher will work through the student thought process for episode 42. A brief discussion on imaginary numbers may take place but is not pertinent to the topic. Students should follow along with the episode, asking questions. After finishing, the class should answer question 4 in partners. Finally, an exit slip will be given out at the end of class. It will have one problem covering section 8.7 asking students to make a graph given an equation in vertex form. They should also include the vertex itself, and the axis of symmetry. The last problem on the homework slip will be a journal. It will ask students to write about what they learned in 8.7 and 8.8. Also, it will ask them to state what they are still struggling with, or what they wish they had more time on.

HW: No homework assigned.

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Day #: 10 Solving by Graphing, AgainGoal: Students will learn how to use graphing calculators to solve

systems of equations.

Objectives: Students will find intersection points of two equations on their calculator to solve systems of equations.

Students will analyze mathematical cues to determine when graphing is better than algebra

Lesson Summary: (one paragraph max)

The teacher will begin by briefly reviewing what they already know about solving systems of inequalities algebraically. There will be a worksheet as a warm up that the class goes through together. On the warm up, students will be asked to solve several systems of equations algebraically. They should be able to do these problems without much difficulty. The last problem will be arranged so that it doesn’t factor, and students must find another method to solve it. Then the teacher will introduce using graphing calculators to solve systems, and talk through episode 43 with the class. Once they have talked through the steps used in the episode, the students should work on problem 1-3. Once they have finished, they will switch their problems 2 and 3 (unnumbered) with a peer. The peer will have to decide which problem is easier to use algebra for, and which one uses the calculator. Then they will solve both of their peers’ problems. If there is still time remaining, the class will work on exercise 2 under Check Your Understanding. Once the class has completed this activity individually, the class will come together as a whole. The teacher will ask for each problem whether it students solved it algebraically or graphically, and why. The students should leave class with a clear understanding of when to use each method to find a solution.

HW: No homework assigned

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Day #: 11 Lesson Title: Inequalities with Two Variables

Goal: Students will be able to solve inequalities with two variables using the graphing method.

Objectives: Students will be able to solve inequalities with two variables using the coordinate plane.

Lesson Summary (one paragraph maximum)

The lesson will begin with a review of one-variable inequalities by graphing solutions on a number line. Today will focus on strict inequalities versus less than or equal to/greater than or equal to. We will begin as a class using a word problem and discuss the dotted line versus solid line in a linear inequality. Students will then work in groups on an investigation with a system of two inequalities. They will explore where the shaded area will be with guess and check to see if the inequality holds. If it does, the students will place an M&M on the graph. Once they can see the area where the inequality holds, they will shade the area with colored pencil. We will close with an exit slip journal where the students will summarize what they did today and ask any questions they may have.

HW 8.11: 1, 3, 4, 6

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Name: Kaitlin BurkeDate: Day 12Grade Level: FreshmenTime Allotted: 50 minutesNumber of Students: 25

I. Goal: Student will be able to graph linear and quadratic inequalities of two

variables and find solutions of systems of inequalities.II. Objectives:

Students will be able to graph linear inequalities with dotted and solid lines.

Students will be able to graph quadratic inequalities with dotted and solid lines.

Students will be able to solve systems of inequalities (quadratic, linear, and combination).

Students will be able to represent the solution of the system of inequalities on their graph.

III. Materials and Resources: Each student will need a computer with Geogebra. Each student will need a worksheet. Each student will need a pencil and paper. The teacher will need a computer with Geogebra. The teacher will need a projector.

IV. Motivation (10 minutes): We will start with a review of the previous day’s lesson on Geogebra. The teacher will pull up Geogebra on the projector and will solve the

homework problem 8.11: 6. Discuss the problem having students describe how they came to each

conclusion. 6. a) On the same coordinate plane, graph the two lines y=x+1 and

y=-2x+11. Graph on Geogebra. b) Find the value of x such that x+1=-2x+11. What does x represent?

(possible misconception is that x is the y-intercept). Have a student explain. Then highlight the intersection on Geogebra. Say that “x not only represents the intersection. It is the only value that satisfies that equation.”

c) Find two points (x,y) such that y>x+1 and y<-2x+11. (Address dotted line because it is a strict inequality. Ask “what would be different if it had an equal to component?”.) Point out that there are many more points that satisfy the inequality than the one x that satisfies the equality. “Where are these points located on the graph? Can someone come up to the SmartBoard and shade where they think this inequality is always true.”

d) Find two points (x,y) such that y>x+1 and y>-2x+11. “Where are these points located? Can you show me on the graph?”

e) Shade the entire region where y<x+1 and y>-2x+11 are both true. Have a student come up to the board and show their answer.

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Transition: “We are all very talented at finding the solution to systems of linear inequalities. Do you think we can do this with other types of inequalities? Does anyone have any ideas?” Move on to graphing a quadratic inequality. V. Procedure:

Put students in groups of three by counting off. Pass out red and green solo cups to each group. The green solo cup

with be stacked on the red solo cup. Explain that wherever it says “STOP” on the worksheet, the group should switch the cups so that the red cup is showing to indicate that they have stopped and are ready to have their work checked. Once their work has been checked they switch back to the green cup on top and continue. (2 minutes)

Have everyone open Geogebra on their computers (students have prior knowledge of all operations in Geogebra).

Pass out worksheet (attached) and go through directions. Walk around room as students are working and address any questions

that may come up. Expect most questions to be technical about Geogebra.

First stopping point after about 7 minutes. At first stopping point, check students answers and press for further

explanation. Ask questions such as “Why did you decide to use a dotted/solid line?” and “How could you estimate the shaded region without plugging in points?”

Make sure to ask each member of the group questions, do not just let one member answer everything.

Allow students to move on to part 2. (7 minutes) Circulate and address questions. Expect questions about how to

compare the graphs. Ask leading questions such as “What would the shaded area be if the linear inequality was by itself?” and “Look at your tables side-by-side.”

Once groups have started to flip their cups to red, bring the class together.

Bring up the two inequalities on the coordinate plane on Geogebra on the SmartBoard. (10 minutes)

Have a student come up and shade the area that satisfies the quadratic inequality and explain their process. Make sure they explain their choice of a dotted or solid line and how they found the region.

Have a student come up and shade the area (in a different color) that satisfies the linear inequality and explain their process. Again, make sure they explain their choice of a dotted or solid line and how they found the region.

Next ask for a volunteer to come up to the SmartBoard and explain the solution to the system of inequalities. Have them shade the area in a new color and explain how they arrived at their conclusion.

Summarize the findings of the activity. “Today we found the solution to a system of linear and quadratic inequalities. When we find the solution to the system of equations, it is the POINT at which the two

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equations intersect. For inequalities we are finding the AREA in which they intersect.”

VI. Closure: (10 minutes) Have the students journal “Think of a real life situation in which

solving a system of equations would be useful. Explain how you would use your skills in this situation.”

Each group should discuss their ideas after 5 minutes of journaling. Homework is 8.11: 9, 10, 11.

VII. Assessment: Students will turn in their worksheet and journal at the end of class.

VIII. Extension: Have each group share one of their real world scenarios. Come up with equations for this scenario and solve.

IX. Standards: CCSS.Math.Content.HSA-REI.D.12 Graph the solutions to a linear

inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

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1. Write your name and your group members names on your worksheet.

2. Open Geogebra. 3. Graph the inequality y>(1/2)x+1 on Geogebra.4. Sketch it below.

Inequalities with Two Variables

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5. Fill in this table.

6. Which region of your sketch represents y> x^2+1? Shade that region on your sketch.

STOP32

Point x x^2+1 y Is y>x^2+1?

(1,2)(2, 6)(0, 0)(-3, 3)(-3, -7)

7. Graph y<(1/2)x+4 on the same coordinate plane as the above equation on Geogebra.

8. Sketch the inequality on the graph above.

9. Fill in the table below.

10. Which points satisfy both inequalities? Explain your reasoning.

STOP11. Shade this area.

12. Why does (1,2) not satisfy the quadratic inequality?

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Point x (1/2)x+4 y Is y<(1/2)x+4?

(1,2)(2, 6)(0, 0)(-3, 3)(-3, -7)

13. What would the quadratic inequality look like if (1,2) did satisfy the inequality?

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Day #: 13 Lesson Title: Assessment Review Day

Goal: To help the students review and understand all concepts that will be assessed the next day.

Objectives: Students will review the quadratic formula.Students will review factoring nonmonic equations.Students will review “going the other way”.Students will review optimization.Students will review graphing quadratic functions.Students will review how to solve equations with graphing.Students will review solving inequalities.

Lesson Summary (one paragraph maximum)

To begin the lesson, each group will come up with one (or more) questions that they would like the teacher to answer about the material. Then the class will begin a Jeopardy game. Each group will rotate the person who is allowed to answer with each question. If they know the answer, they should stand. Whoever stands first wins. The categories are Quadratic Formula, Factoring Nomonic Quadratics, “Going the Other Way”, Optimization, Graphing Quadratic Functions, Solving with Graphing, and Inequalities. Each category will have three questions. When the game is over, each student will get a print out of the questions and answers to study. Close with by explaining the format of the assessment and ask for any last minute questions.

HW Study!

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Day #: 14 Test DayGoal: To assess the students overall understanding of quadratics.

Objectives: Students will solve for the roots of expressions by factoring and using the quadratic equation.

Students will find a quadratic equation given its roots. Students will be able to solve systems of inequalities

(quadratic, linear, and combinations). Students will be able to set up a quadratic expression and

find the vertex to solve for the maximum/minimum Students will graph parabolas by hand using the vertex form

of an equation.

Lesson Summary: (one paragraph max)

Day 14 will be a test day that will cover everything learned in the unit so far, with a focus on the material learned after the quiz (day 5). The test will be set up in two sections: calculator and non-calculator. The students will start with the calculator section to solve problems using inequalities and solutions by graphing. Once, they have completed this section, they will turn it in and get the non-calculator section. The non-calculator section will assess the students’ capability to use optimization problems and factor using various methods. The test should last the whole period.

HW: No homework assigned.

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Assessment Pieces

# Learning Objective Assessment Item1 Students will identify how many

solutions a quadratic equation has and solve the equation by factoring and using the quadratic formula.

1. Consider the quadratic equation 4x2+36x+45 = 0 a. How many solutions does this equation have? b. Solve this equation by factoring c. Solve this equation by using the quadratic formula

2Students will be able to set up a quadratic expression and find the vertex to solve for the maximum/minimum.

Last year the yearbook at Central High cost $75 and only 500 were sold. A studentsurvey found that for every $5 reduction in price, 100 more students will buy yearbooks.What price should be charged to maximize the revenue from yearbook sales?

3Students will interpret equations in vertex form to find critical values and graph their equation.

1) Given the equation y+1=(x+3)2

a. Identify the vertexb. Find the y-interceptc. Graph the vertex and the

line of symmetryd. Graph the y-intercept and

it’s reflectione. Graph the parabola

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Students will solve systems of inequalities (quadratic, linear, and combination).

a. Plot the points (2, -2), (1, -5), (5,-2).

b. Use inequalities to describe the region in which these three points lie.

c. Graph the inequalities.d. Shade the region where the system of

inequalities is satisfied.

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5Students will be able to represent the solution of the system of inequalities on their graph.

Sketch the graph of each inequality. You might start by finding some points that make the inequality work, or by finding the boundaries.

a. |x|+|y|≤2b. |x+y|≤2

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