Dear Mathematics Teacher,

What is mathematics and why do we teach it? This question drives the work of the math coaches at ISA. We love mathematics and want students to have the opportunity to begin to have a similar emotion. We hope this unit will bring some new excitement to students.

This unit was originally written seven years ago and has gone through several iterations. Now it has been redesigned to align with the Common Core State Standards. Essential to this work is an inquiry approach to teaching mathematics where students are given multiple opportunities to reason, discover and create. Problem solving is the catalyst to the inquiry process so as you look closely at this unit you will see students constantly put in problem solving situations where they are asked to think for themselves and with their classmates.

The first four Common Core Standards of Practice are central to this unit. Through the constant use of problematic situations students are being asked to develop perseverance and independent thought, to reason abstractly and quantitatively, and to critique the reasoning of others. Throughout the Teacher Guides in this unit we’ve highlighted some places where the Mathematical Practices are expressed. The Mathematical Practices will be denoted with MP followed by a number indicating which specific Mathematical Practice is being expressed. As an example MP2 will refer to Mathematical Practice 2: Reason abstractly and quantitatively.

Mathematical modeling is present throughout the unit as students are asked to analyze different real world situations and represent them mathematically. Students are also asked to create models including the final project which is to create a fair game based on the principles of probability.

The other four Standards of Practice are also present in this unit. Two of them are central to the inquiry approach. You will see these two statements in the last two standards.

Mathematically proficient students look closely to discern a pattern or structure. Mathematically proficient students notice if calculations are repeated, and look for general methods

and shortcuts.

We believe, as do many mathematicians, that mathematics is the science of patterns. This underlying principle is present in all the work we do with teachers and students. In this unit you will see that students are often asked to discern a pattern within a particular situation. This leads students to make conjectures and possibly generalizations that are both conceptual and procedural.

Thank you for looking at this unit ad we welcome feedback and comments.

Sincerely,The ISA math coaches

Becoming a Wonderful Mathematics Teacher

“… a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting their problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking.” – George Polya (1945)

IntroductionMathematics is a subject most people studied in school for 11 or 12 years but many people never came to appreciate or understand it. Why is that? Is it the discipline itself or the way it has been presented to people? Can mathematics be a subject that most students can begin to make sense of and enjoy if they are given meaningful opportunities?

The Common Core is making new demands on the teachers of mathematics throughout the country. The demands impact both the content teachers are being asked to teach and the standards of practice they employ in the classroom. An inquiry-based approach offers teachers a means of giving students an opportunity to engage with the new content while also developing as mathematical thinkers.

What is inquiry-based instruction in mathematics?Inquiry-based instruction has a long history in both education and mathematics education. In inquiry-based instruction, mathematics is viewed as a humanistic discipline where students construct meaning and understanding within a community of learners (Borasi, 1992). It is a multifaceted approach to learning. Students are encouraged to wonder about mathematical ideas, raise questions, make observations, gather data, consider possible relationships and patterns within the data, make conjectures, test one’s conjectures, and finally generalize a discovery supported by evidence (Borasi, 1992; NRC, 1996; Suchman, 1968; Wells, 1999). Generating an idea or concept and arguing for its authenticity is an essential aspect of inquiry and tells a teacher what a student knows about mathematics (Koehler & Grouws, 1992; Lampert, 1990).

The Common Core demands that teachers move away from math being viewed as a set of procedures students learn to replicate and show on an exam to the presentation of mathematics as a coherent discipline that one begins to understand over time so it can be used within multiple problematic situations.

Using an inquiry/problem-based approach can help students develop both conceptual and procedural understanding while learning to reason quantitatively and abstractly.

Using an inquiry/problem-based approach can help students move from a concrete way of thinking to an abstract way of thinking. This can take the form of moving from an arithmetic way of thinking to an algebraic way of thinking.

Using an inquiry/problem based approach helps students develop independence and perseverance. Students learn to take risks, rethink strategies and make sense of process and solution.

8 Aspects of a Wonderful Mathematics TeacherIn order to develop students who make sense of the discipline of mathematics they must experience high school mathematics courses that enable them to develop as mathematical thinkers. Teachers of mathematics have before them an opportunity to enter into the wonder-filled world of mathematics with their students in such a way that their students leave thinking differently. Our work focuses on helping you develop in the 8 aspects of what makes for a wonderful math teacher: Learners, Artists, Decision Makers, Questioners, Modelers of Mathematical Thinking, Provocateurs, Coaches, and Reflectors.

Learner –Teaching is always a learning experience. The learning takes different forms. A mathematics teacher is always learning about his/her discipline and about different ways of presenting this wonderful discipline to his/her students. Often what teachers are struggling with in the mathematics classroom goes beyond mathematics and into what the students bring with them to the classroom. From life and past math experiences students come to classrooms as complex human beings that cannot be summarized by one dimensional measures or encapsulated by a set of behaviors. This implies that students’ understandings and misunderstandings of mathematics have more nuance than is present on the surface and requires deeper inspection. Thus, teachers need to see themselves as always learning about how their students think and make sense of mathematics.

Artist – Teaching is an art. It takes great imagination and creativity. Mathematics teachers need to find their connection and passion for their discipline and find out how to express it within their teaching. Teachers need to ask, “Why do I love mathematics and how do I bring some of that emotion to my students?” Becoming an artist in the classroom takes time. In the process of doing this teachers move along a continuum of creation. Often, teachers start out through imitation. They use the material of others as they try to get their footing. As teachers develop as artist they move between modifying/tweaking material and creating their own. As they develop their craft they begin to create interesting experiences for students that will lead to conceptual and/or procedural understanding. The teacher as artist is always looking for opportunities to engage students with meaningful problems that bring out the students’ creativity and imagination.

Decision Maker – Mathematics teachers are always making decisions. These decisions are about curriculum, selecting problems to use with their students, and possible next steps or activities for supporting individual students. Decisions also happen within lessons. Teachers are always making decisions about stepping back or entering in as students grapple with a problem. If they join a student in his/her struggle teachers need to decide if asking a question would be best or if the student needs some explanation. The big question is always, “What will be the best thing I do for this (these) student(s) in this moment so that their mathematical thinking is being enhanced and not stifled?”

Questioner – Questions are central to every aspect of a math teacher’s work. When creating units and their accompanying lessons teachers should always ask, “What are the questions that guide this unit and accompanying lessons? Why would these questions help students to deepen their mathematical understanding? Have I created situations within the unit where my students can ask questions?” Questions are also central to the classroom facilitation. The right questions can motivate students to engage in an activity or further engage in solving a complex problem. Asking the right question is part of the art of teaching and teachers need to see themselves as questioners of their own questions.

Modeler of Mathematical Thinking – Modeling has always been seen as central to a teacher’s job in the classroom. A problem with the modeling approach that has been used in the past is that the teacher

modeling focused on handing students a recipe to follow and then it was the students’ job to imitate. Making the focus of modeling procedures or recipes did not enhance mathematical thinking, but rather stifled it. Modeling needs to have as its purpose the strengthening of student mathematical thinking. Teachers can do this by modeling their own thinking process. They share with the students the questions they ask themselves while they are engaged in the problem solving process. A discussion about the question and process should follow where students then can decide how they want to integrate the teacher’s way of thinking into their own way of thinking.

Provocateur – Students need to be put into situations that take them out of their comfort zone and challenges them to expand their thinking and take risks. These types of experiences help students develop a comfort with going to new places in their thinking, and develop perseverance in solving problems. By provoking student thinking, with problems that can be thought about in multiple ways and having multiple solutions, a teacher tries to get his/her students to go deeper into mathematics with more nuance and breadth.

Coach – Students come into the high school math classroom with many experiences (often bad) that affect how they feel about mathematics. Thus teachers have to see themselves as working to change students’ productive disposition in the math classroom. Mathematics teachers are both motivators and critics (raise important questions to students when they go off a path.) They have to see themselves as deeply engaged with each student and figuring out ways to support him/her.

Reflector – Successful mathematics teachers are reflective teachers. They reflect on both their teaching and their students’ thinking and learning. Reflection takes place while the lesson is going on and after class as means of informing instruction. Reflection is essential for growth and so all math teachers and teams of math teachers need to create a set of structures where teachers have the opportunity to reflect and get feedback from their colleagues.

A special thank you to Sapphira Hendrix of Brooklyn Prep HSfor spending some time with us this summer going revising this unit. 

Structure for Unit 6: Quadratics & Other Functions

Essential Questions: What makes quadratic functions unique?

Interim Assessments/Performance Tasks Phoebe’s Rocket (Lesson 17) The Gateway Arch (Lesson 18) Old Pizza Menu (Lesson 19) Throwing Paper (Lesson 20) Angry Birds (Lesson 22)

Final Assessment:

What will students understand and be able to do at the end of the unit? Students will understand the way parabolas transform based on “a”, “b”, and “c” Students will be able to distinguish different functions based on tables, graphs, and equations Students will understand how to find the vertex and axis of symmetry Students will be able to sketch a graph given a quadratic equation Students will be able to find the roots of a quadratic equation using multiple approaches Students will understand how to factor using multiple approaches Students will understand how to solve quadratic equations using multiple approaches

What enduring understanding will students have? Each function has a unique set of patterns Students will understand that quadratic equations can be used to model many events in the world Students will understand how to think about quadratics using different representations

Bookmarks to Lessons in this Unit

Calculator Toolkit

Lesson 1: How are four different functions similar and how are they different?

Lesson 2: Connecting table, and graph of a quadratic

Lesson 3: Transformations of the quadratic function

Lesson 4: Symmetry in Quadratic Functions

Lesson 5: Connecting Axis of Symmetry to the Vertex and Equation

Lesson 6: How can we calculate the x- intercepts of the quadratic equation?

Lesson 7: Doing and undoing in Algebra

Lesson 8-9: Modeling Quadratics

Lesson 10: Strategizing the distributive property and factoring quadratic expressions by looking for patterns.

Lesson 11: Factoring when a ≠1

Lesson 12: Factoring Polynomials

Lesson 13: Determining the x-intercept(s) by factoring.

Lesson 14: Factoring Strategies/Equivalent Expressions

Lesson 15: Strategizing and demonstrating flexibility in determining the x-intercept(s) algebraicallyLesson 16: Transformations re-visited

Lesson 17: Phoebe Small’s Rocket Problem

Lesson 18: Gateway Arch Problem

Lesson 19: An Old Pizza Menu

Lesson 20: Throwing Paper (with Permission) in the Classroom

Lesson 21: Quadratic Word Problems

Lesson 22:Preformance Task - Angry Birds

Lesson 23: Square Root Function

Lesson 24: Comparing f(x)=x2 and y=x3

Lesson 25: Comparing y = x3 and y= 3√x

Appendices

Appendix A: Factoring Lessons (without Algebra Tiles)Appendix B: Calculator Toolkit

Entering a List of Data

Standards Addressed:F-IF.1. Understand that a function from 54one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

F-IF.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

F-IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F-IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

F-BF.1. Write a function that describes a relationship between two quantities.

F-BF.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Decision Maker: If you need graph paper for any of the lessons that follow please feel free to copy and paste to where it is needed.

Lesson BookmarksQuadratics Unit

Lesson 1Teacher Guide

Lesson 1: How are four different functions similar and how are they different?

Within this lesson we want to equip students with a new tool for investigating different kinds of relationships. The first difference is a tool that will allow students to identify the different types of functions they are working with. This will be especially important when working with higher order polynomials later.)

Opening Activity:

Learner: You want students to recognize that each table represents a different function they have worked with before and, therefore, they should be able to name all them except for the quadratic. What are the patterns they are seeing within the columns of the table? Linear (Table 1) displays a constant rate of change. The quadratic relationship (Table 2) displays that it has neither a constant difference nor constant ratio. The exponential (Table 3) displays a constant ratio of growth. The absolute value (Table 4) has characteristics of linearity and only representing magnitudes.

Look at the following four tables.

TABLE 1

x coordinate y coordinate1 32 63 94 125 156 18

TABLE 2

x coordinate y coordinate-3 9-2 4-1 10 01 12 43 9

TABLE 3

x coordinate y coordinate1 32 93 274 815 243

TABLE 4

x coordinate y coordinate-3 3-2 2-1 10 01 12 23 3

A. How are these tables similar? How are they different?B. Describe the patterns within each table.

Activity 2:

Learner: When students look at the difference of the quadratic they should be able to comment that there is no constant rate of change in the first difference. This should make quadratics stand out from the other functions they have worked with thus far. However, there is an opportunity for surprise when they see that a constant rate of change is present within the second difference. One big idea we can take away from this experience is that patterns don’t always make themselves immediately obvious and that we sometimes need to do further investigation to find the pattern.

Now we are going to compare linear and this new unidentified relationship. Looking at Table 1 and Table 2:

A. What do we know about the differences for Table 1?B. Now create differences for Table 2. What do you observe? C. Now find the difference of the values (differences) created from question B. What do you observe?

Questioner: An important question to ask during the subsequent discussion is, “Will this be a characteristic of all quadratic functions?” Do not feel pressured to answer this question at this moment, however.

Decision Maker: It would be interesting to take a survey of the students’ opinions on this question based on their experience in working with patterns and record the results somewhere in the room. This will enable you to continue to come back to this question in a tangible way as the unit develops.

D. What do you think your observations mean?

Learner/Provocateur: What sense do your students make out of their observations? It’s important that students see the concept of rate of change being present in relationships other than linearity. With quadratic functions it takes on a different form but is still present.

Decision Maker: It would be interesting to take a survey of the students’ opinions on this question based on their experience in working with patterns and record the results somewhere in the room. This will enable you to continue to come back to this question in a tangible way as the unit develops.

Student tables should look like the following once A, B and C are complete. You can use the difference tables as a scaffold for students.

Table 1 (Linear) Table 2 (Quadratic)

x y

1 3Differenc

e

> 32 6

> 33 9

> 34 12

> 35 15

> 36 18

E. Does Table 2 represent a function? Explain.F. Looking at Table 2, how could this be represented as an equation?

Learner/Decision Maker: As a part of the discussion for question F, how do you want to introduce the equation for Table 2 (y = x2)? Did the students agree that Table 2 represents a function? Do you also want to generalize it as f(x) = x2? If students aren’t making the connection you may have them compare 12, 22, 32 to Table 2 and see if that helps.

Learner/Decision Maker: Do students know what that y=x2is called a quadratic? Do you want to introduce the name of this function here?

Provocateur: What about the relationship in Table 2 do you think causes it’s 2nd difference to turn out as it did? Is there something similar going on with Table 1?

x y 1st

-3 9Difference 2nd

> 5Difference

-2 4 > 2

> 3-1 1 > 2

> 10 0 > 2

> -11 1 > 2

> -32 4 > 2

> -53 9

Decision Maker: For those without access to graphing calculators the graph has been provided on the next page. Do you want to embed this graph in the student’s version?

Activity 3:Look at the following table.

Table 5x coordinate y coordinate-2 24-1 150 81 32 03 -14 05 36 87 158 24

A. What do you observe? Do you see a pattern? If so, describe it.B. Compare Table 2, Table 5, and Table 4 to one another. What characteristics are the same between

these tables?C. What questions arise from your observations?

Coach: Have class discuss their observations. Students should see the symmetry and the minimum value of y associated with that symmetry. Question B will set the students up to make a connection back to absolute value as well.

D. Predict the next x and y coordinates in the table in both directions.E. If we plot the points from this table what do you think we would see?

Decision Maker: Now display the graph of the Table 5 quadratic for everyone to see. Obviously you are free to do this however you think is easiest. If you would like to use Desmos (www.desmos.com) please know that Desmos doesn’t have the capabilities to create quadratic regressions. So we would recommend creating the table in advance and plot the equation in plot 2 (y = x2 – 6x + 8).)

Artist/Decision Maker: It would be advantageous and strongly recommend that there is a Calculator Tool Kit where each entry into the Tool Kit captures the button strokes for a particular calculator process, so that students can reference it each time they need it. We will be using the STAT plot again in this unit as well as in the upcoming statistics unit where students will be creating regressions. Students need to be comfortable with their calculators.

F. Go to your TI calculator and plot these points.

Use Stat and Edit. Let L1 be your x value and L2 be the y value for each pair. Use coordinates from Table 5 above.

To Graph: Press 2nd Y= or Stat Plot, Choose 1, Turn on Plot 1 and make sure it uses L1 and L2 from the list you created. Then Press GRAPH. Your points should be displayed.

Note: The window on the calculator must be adjusted to the following, press WINDOW and change:

G. After observing the table and the graph for Table 5. Would you classify this relationship as a function? Why?

Learner: How did students make their decision about whether or not they think Table 5 is a function? What understandings and misunderstandings are represented in their thinking?

Xmin -10Xmax

10

Xscl 1Ymin -10Ymax 30Yscl 1Xres 1

https://www.desmos.com/calculator/inysedbvs9

Activity 4: Reflection: Describe the new relationship, presented in Table 2 and Table 5. How is it similar to other functions? What makes this new relationship unique?

Quadratics UnitLesson 1

Student Version

Opening Activity: Look at the following four tables.

TABLE 1 A. How are these tables similar? How are they different?

x coordinate y coordinate1 3 2 63 94 125 156 18

TABLE 2

x coordinate y coordinate-3 9-2 4-1 10 01 12 43 9

TABLE 3 B. Describe the patterns within each table.

x coordinate y coordinate1 32 93 274 815 243

TABLE 4

x coordinate y coordinate-3 3-2 2-1 10 01 12 23 3

Activity 2:Now we are going to compare linear and this new unidentified relationship. Looking at Table 1 and Table 2:

A. What do we know about the differences for Table 1?Table 1 (Linear)

x y

1 3Difference

>2 6

>3 9

>4 12

>5 15

>6 18

B. Now create differences for Table 2. What do you observe?

C. Now find the difference of the values (differences) created from question B. What do you observe?

D. What do you think your observations mean?

E. Does Table 2 represent a function? Explain.

F. Looking at Table 2, how could this be represented as an equation?

Activity 3:Look at the following table.

Table 5x coordinate y coordinate

-2 24-1 150 81 32 03 -14 05 36 87 158 24

A. What do you observe? Do you see a pattern? If so, describe it.

B. Compare Table 2, Table 5, and Table 4 to one another. What characteristics are the same between these tables?

C. What questions arise from your observations?

D. Predict the next x and y coordinates in the table in both directions.

E. If we plot the points from this table what do you think we would see?

F. Go to your TI calculator and plot these points.

Use Stat and Edit. Let L1 be your x value and L2 be the y value for each pair. Use coordinates from Table 5 above.

To Graph: Press 2nd Y= or Stat Plot, Choose 1, Turn on Plot 1 and make sure it uses L1 and L2 from the list you created. Then Press GRAPH. Your points should be displayed.

Note: The window on the calculator must be adjusted to the following, press WINDOW and change:

G. After observing the table and the graph for Table 5. Would you classify this relationship as a function? Why?

Activity 4: Reflection: Describe the new relationship, presented in Table 2 and Table 5. How is it similar to other functions? What makes this new relationship unique?

Xmin -10Xmax

10

Xscl 1Ymin -10Ymax 30Yscl 1Xres 1

Lesson BookmarksQuadratics Unit

Lesson 2Teacher Guide

Lesson 2: Connecting table, and graph of a quadratic

Focus Question(s):How do the features of a quadratic show up in a table, and graph?

Opening Activity: Use f(x) = x2 to create the table and plot the points from your table on the graph below. What connections do you see between the table and the graph?

Decision Maker: If students didn’t connect y= x2 or f(x) = x2 to the word quadratic yesterday, do you want to do that today? What are the important features you want student connecting with the word quadratic at this time? How do you want to capture this?

Here’s the Desmos graph for all the functions in this lesson: https://www.desmos.com/calculator/fapbu5vhqq You can use this graph to create any screen captures.

f(x) = x2

x f(x)

-3

-2

-1

0

1

2

3

Activity 2:How do you think could you use the table for f(x) to create the table for g(x)? Why does this work?

Coach: If students struggle with getting started in Activity 2, you could ask the students to compare the function in Activity 1 and consider how it’s different from the function in Activity 2? Once they have observed that g(x) is the negative of f(x), you could ask what impact do you this observation has on the table? How would you account for this different in the values of the table?

If the approach above isn’t effective, you could follow up with asking the student to create the table for g(x) and compare the values of g(x) to f(x) then discuss what about the equation caused this difference.

Provocateur: What based on your comparison of g(x) to f(x) what hypothesis do you have about the way the sign on x2 effects the table and the graph? Do you think this will always be true? Why or why not?

g(x) = -(x2)

x g(x)

-3

-2

-1

0

1

2

3

Activity 3: How do you think could you use the table for f(x) to create the table for h(x)? Why does this work?

Coach: If students struggle with getting started in Activity 3, similar to the line of questioning for Activity 2 you could ask the students to compare the h(x) to f(x) and consider they are different? Once they have observed that h(x) has a 2 added to x2, you could ask what impact do you this observation has on the table? How would you account for this different in the values of the table?

If the approach above isn’t effective, you could follow up with asking the student to create the table for h(x) and compare the values of h(x) to f(x) then discuss what about the equation caused this difference.

Provocateur: What based on your comparison of h(x) to f(x) what hypothesis do you have about the affect that adding a number to x2has on the graph? Do you think this will always be true? Why or why not?

h(x) = x2 + 2

x h(x)

-3

-2

-1

0

1

2

3

Activity 4: Calculate the values for k(x) in the table below and graph k(x). What connection do you see between the values in the table for k(x) and f(x)? What connection do you see between the graph of k(x) and f(x)? Do you see a way you could have used the table or graph for f(x) to create the table or graph for k(x)?

Decision Maker: Do you want to use the same line of questioning in Activity 4 that was used in Activity 2 and 3? If so: How do you think could you use the table for f(x) to create the table for h(x)? Why does this work?

k(x) = (x + 2)2

x k(x)

-4

-3

-2

-1

0

1

Closing Activity:

Decision Maker/Reflector: How do you want to facilitate the share out for this closing activity? Please read through the questions and the included notes about each question and then reflect on what opportunities these questions provide for you and your students and decide which ones you want to capitalize on during the share out.

How would you describe the way f(x), g(x), h(x), and k(x) are the same? Think about the graphs, tables, and equations. Feel free to use sketches or drawings to illustrate your descriptions.

Learner/Decision Maker: What features do the students identify? What language do they use to describe these features? What understandings and misunderstandings do you detect in their thinking?

The hope is that students observe some of the following about quadratics and it doesn’t need to be in mathematical language or exactly in the terms that have been used below. This closing activity can act as the beginning of a discussion where the student’s observations become more formalized and you provide them with the mathematical language that goes along with their observational language. Introducing vertex as a part of this discussion will be important for the work in the upcoming lessons.

They are symmetric; if folded in half the curves will line up, they look like the reflect around some center piece

They have a vertex; turning point The shape is similar; they all look like the letter U or some version of a U There’s a portion of the table/graph where the values of the function (or the y values) are

increasing and a portion where they are decreasing

How would you describe the way f(x), g(x), h(x), and k(x) are the different? How did those differences show up? What do you think has caused those differences?

Learner/Decision Maker: What features do the students identify? What language do they use to describe these features? What understandings and misunderstandings do you detect in their thinking?

Some face up, some face down The vertex is sometimes on the y-axis only, sometimes on both, sometimes only on the x-axis g(x) is getting bigger for x values that f(x), h(x), and k(x) are all getting smaller

Why do you think someone might refer to f(x) as the parent function and g(x), h(x), and k(x) as the children functions?

Coach: The language of parent and child(ren) functions is useful in helping students see that g(x), h(x), k(x) aren’t completely new and are related to f(x) in some way. This connection will be more explicitly explored in Lesson 3.

Key features of a quadratic so far

What are the important elements of a quadratic that we’ve worked with so far?

Decision Maker: Throughout the course of this unit the class will have the opportunity to update this sheet with the new understandings they have about quadratics. Other than the vertex are there other observations you or the students want to record here?

VertexTurning PointMaximum (-2, 1)

(2, -1)VertexTurning PointMinimum

Quadratics UnitLesson 2

Student Version

Opening Activity: Use f(x) = x2 to create the table and plot the points from your table on the graph below. What connections do you see between the table and the graph?

f(x) = x2

x f(x)

-3

-2

-1

0

1

2

3

Activity 2:How do you think could you use the table for f(x) to create the table for g(x)? Why does this work?

g(x) = -(x2)

x g(x)

-3

-2

-1

0

1

2

3

Activity 3: How do you think could you use the table for f(x) to create the table for h(x)? Why does this work?

h(x) = x2 + 2

x h(x)

-3

-2

-1

0

1

2

3

Activity 4: Calculate the values for k(x) in the table below and graph k(x). What connection do you see between the values in the table for k(x) and f(x)? What connection do you see between the graph of k(x) and f(x)? Do you see a way you could have used the table or graph for f(x) to create the table or graph for k(x)?

k(x) = (x + 2)2

x k(x)

-4

-3

-2

-1

0

1

Activity 5:How would you describe the way f(x), g(x), h(x), and k(x) are the same? Think about the graphs, tables, and equations. Feel free to use sketches or drawings to illustrate your descriptions.

How would you describe the way f(x), g(x), h(x), and k(x) are the different? How did those differences show up? What do you think has caused those differences?

Why do you think someone might refer to f(x) as the parent function and g(x), h(x), and k(x) as the children functions?

Key features of a quadratic so far

What are the important elements of a quadratic that we’ve worked with so far?

VertexTurning PointMaximum (-2, 1)

(2, -1)VertexTurning PointMinimum

Lesson BookmarksQuadratic Unit

Lesson 3Teacher Guide

Lesson 3: Transformations of the quadratic function

Teacher GuideFocus Question(s):What happens when parts of the equation of a quadratic function are changed?

Coach: Remind students of the language of parent and child function to help students connect that x2 + 5 isn’t a new quadratic, but is a transformed quadratic from x2.

Coach/Learner: What connections did the students make in working with quadratics in previous experiences?

Decision Maker: This Opening Activity could go fairly quickly. How do you want to facilitate it? In what ways do you want the students interacting and clarifying one another’s ideas?

Opening Activity:

f(x) = x2

Create a table for f(x) with a domain of -3 to 3.

Which type of function would this equation and its accompanying table represent? Why?

Coach: Give this new function its proper name.

Activity 2:The general form of a quadratic equation can be written as p(x) = ax2 + bx + c where the letters a, b, and c represent numbers (called coefficients) in a specific equation.

Write down the values of a, b and c for the following equations.1. q(x) = 2x2 +3x + 5 a = ____, b = ____, c = ____2. r(x) = x2 -5x + 8 a = ____, b = ____, c = ____3. s(x) = -x2 – 12 a = ____, b = ____, c = ____4. t(x) = 7x2 -8x a = ____, b = ____, c = ____

The general form of the quadratic equation is y = ax2 + bx + c. When b = 0 and c= 0, it becomes the parent function y = ax2. Look at the following equations in this form.

Decision Maker/Artist: Students will likely struggle to identify b in 3 and c in 4, because in both cases they are 0.

Activity 3:Decision Maker/Artist: It might be helpful to use Desmos or other graphing calculator program projected on a screen so the whole class can see the image.

Coach: Encourage students to make sketches of the graphs to give them a visual they can always refer back to.

What do you think a, the coefficient of x2, has on the graph? Why? Use sketches in your reasoning.

A. f(x) = x2

B. g(x) = 4x2

C. h(x) = 2x2

D. j(x) = .02x2

E. k(x) = -.02x2

F. m(x) = -2x2

G. n(x) = -4x2

What statement can you make about a in f(x) = ax2 + bx + c?Did your observations match up with what you thought would happen? Explain why a has the impact it does.

Coach: During the course of this activity make the connection that a vertex can either be a maximum or a minimum. For which functions is the vertex a maximum? For which function is the vertex a minimum?

Coach: There should be a full class discussion about the effect of a on the graph. This could be related to the transformation work with absolute value. Sketches would be useful for students as they begin their vocabulary of how the transformations affect the parent.

Decision Maker: Does your setting have graphing calculators? Using a graphing calculators or Desmos is best for this activity. It’s important for the students to go through the process.

Activity 4:

A. We’ve become familiar with three different types of functions: Linear, Exponential, Absolute Value

Type of functiona(x) = 3x + 2 Linearb(x) = 3x + 2 Exponentialc(x) = |x| + 2 Absolute Value

Decision Maker: The graphs of 3x and 3x+2 and 3x and 3x+2 and │x│ and │x│+ 2 provided here.

a(x)=3x + 2y =3x

b(x)=3x + 2

y =3x

What does the 2 in each of these functions tell you? Do you think this will hold true for quadratic functions? Why?

Coach: Have a discussion about y-intercepts. They should also talk about why they think it will or won’t.

B. Test your conjecture: Create and graph 3 transformed functions from f(x) = x2 so that they each have a different value for c.

Does this confirm your conjecture? Can you explain why?

g(x) = x2 + ____h(x) = x2 + ____k(x) = x2 + ____

C. What statement can you make about c in f(x) = ax2 + bx + c?

y = |x|

c(x) = |x|+2

Quadratic UnitLesson 3

Student Version

Opening Activity:

f(x) = x2

Create a table for f(x) with a domain of -3 to 3.

Which type of function would this equation and its accompanying table represent? Why?

Activity 2:The general form of a quadratic equation can be written as p(x) = ax2 + bx + c where the letters a, b, and c represent numbers (called coefficients) in a specific equation.

Write down the values of a, b and c for the following equations.

1. q(x) = 2x2 +3x + 5 a = ____, b = ____, c = ____

2. r(x) = x2 -5x + 8 a = ____, b = ____, c = ____

3. s(x) = -x2 – 12 a = ____, b = ____, c = ____

4. t(x) = 7x2 -8x a = ____, b = ____, c = ____

Activity 3:What do you think a, the coefficient of x2, has on the graph? Why? Use sketches in your reasoning.

A. f(x) = x2

B. g(x) = 4x2

C. h(x) = 2x2

D. j(x) = .02x2

E. k(x) = -.02x2

F. m(x) = -2x2

G. n(x) = -4x2

What statement can you make about a in f(x) = ax2 + bx + c?

Did your observations match up with what you thought would happen? Explain why a has the impact it does.

Activity 4:

D. We’ve become familiar with three different types of functions: Linear, Exponential, Absolute Value

Type of functiona(x) = 3x + 2 Linearb(x) = 3x + 2 Exponentialc(x) = |x| + 2 Absolute Value

What does the 2 in each of these functions tell you? Do you think this will hold true for quadratic functions? Why?

E. Test your conjecture: Create and graph 3 transformed functions from f(x) = x2 so that they each have a different value for c.

g(x) = x2 + ____

h(x) = x2 + ____

k(x) = x2 + ____

Does this confirm your conjecture? Explain why.

What statement can you make about c in f(x) = ax2 + bx + c?

Lesson BookmarksQuadratic Unit

Lesson 4Teacher Guide

Lesson 4: Symmetry in Quadratic Functions

Teacher Guide

Focus Question(s): How are symmetry, shape, and vertex (minimum, maximum) connected?How are the equation and the vertex connected?

Opening Activity:Decision Maker: When do you want to have a discussion about the experience of classifying these different symmetries? With this activity we are wanting to develop a baseline for symmetry, so the focus is on the ways in which symmetry is expressed and not on specific types of symmetry that might are present. Certainly you want to be open to the understandings the students bring to the conversation as such there’s nothing lost if the discussion doesn’t get to the differences between point symmetry and line symmetry.

Learner: What understandings of symmetry do you students express through this activity?

A. Looking at the images, would you describe each of the images as symmetric?

If you think it is symmetric, why do you think it is and how would you describe the symmetry?

If you don’t think it’s an example of symmetry, how would you support your position?

A. B.

C. D.

Activity 2:

Coach: Here students are beginning to develop their experience with symmetry in quadratic functions. It is important that students see the symmetry within quadratic tables. This leads to an understanding of why the graph is what it is and the axis of symmetry makes sense.

Look at the following tables

Table 5

f(x) = x2 – 6x + 8x coordinate f(x)

-2 24-1 150 81 32 03 -14 05 36 87 158 24

B. Quadratic functions have been described as being symmetric. Look at the table and find where symmetry exists in each of the tables.

Learner/Decision Maker/Coach: How did your students describe the way in which they saw the symmetry?

C. Graph the points from the table for Table 6. Describe the symmetry you see in the visual representation (the graph)?

Decision Maker: Do you want to provide graph paper where the x-axis and the y-axis have been labeled?

Note: If using graphing calculator, the window on the calculator must be adjusted to the following:

Xmin -10Xmax 10Xscl 1Ymin -20Ymax 20Yscl 1Xres 1

Table 6

g(x) = -x2 + 8x - 6x coordinate g(x)

-1 -150 -61 12 63 94 105 96 67 18 -69 -15

Learner: How do your students describe the way symmetry shows up in the graph? Do students talk about folding the paper, or using a mirror? Do the students talk about one side of the parabola lining up with the other after the fold or from using a mirror? Did anyone use the ideas of symmetry in when graphing the points in tables 5 and 6? All these ideas lead towards their understanding of the axis of symmetry.

Coach/Artist: Depending on how this discussion goes and what the students share this may be an appropriate time to establish the presence of the axis of symmetry.

f(x) = x2 – 6x + 8

g(x) = -x2 + 8x - 6g(x) Vertex: (4, 10)

f(x) Vertex: (3, -1)

D. Graph the points from Table 5 on the same axes as Table 6. Compare the way symmetry presents itself when the two functions have been graphed.

E. Do you think that for every parabola there is a line that we can draw so that if you fold the parabola along that line one side will land on top of the other?

Coach: If students are having difficulty engaging with this question, have them sketch a few other parabolas or look back on the parabolas they made in previous lessons to further explore this question. As an additional differentiation move you could sketch a few different parabola or have a few graphed parabolas prepared in advance.

Decision Maker: Here’s an opportunity for a discussion about symmetry.

Learner: In the next activity students are going to find more specifically the connection between the equation, the vertex, and the axis of symmetry.

F. Look at the graph you created. Find the axis of symmetry. Draw the axis of symmetry on your graph for f(x) and g(x).

G. Where does the axis of symmetry cross the parabola? What is significant about the point where it crosses the parabola?

Learner: Do the students make the connection that the axis of symmetry is passing through the vertex of the parabola?

Decision Maker: How do you want to facilitate the discuss that follows this question? As a part of this discussion we want explore why does it make sense that the axis of symmetry passes through the vertex?

H. How might we find this point if we were only given the equations? Or what do you think may be the relationship between this point and the equation? In this case f(x) = x2 – 6x + 8 and g(x) = -x2 + 8x – 6.

Coach: The purpose of this question is to get students wondering about this connection and to setup the large question for the following activities. Students can create different hypotheses about this connection. These hypotheses can be record in the class in some way and revisited after the coming activities. This could be on a poster or whatever way makes the most sense for your classroom.

f(x) = x2 – 6x + 8

g(x) = -x2 + 8x - 6

Orange line the axis of symmetry for f(x): x = 3

Purple line the axis of symmetry for g(x): x = 4

Decision Maker: This is an opportunity to bring together all that the students have thought through in the Opening Activity and Activity 2. You’ll need to decide from the information below what you want to give students and what you don’t.

Summary: Making Sense of Symmetry in QuadraticsHow does symmetry in quadratics show up in different ways? Where do you see symmetry in the following examples?

Vertex: (-2, 1)

Vertex: (2, -1)

Axis of Symmetry: x = -2

Axis of Symmetry: x = 2

q(x) = -x2 – 4x – 3 r(x) = x2 – 4x + 3

x r(x)-1 80 31 02 -13 04 35 8

x q(x)-5 -8-4 -3-3 0-2 1-1 00 -31 -8

r(x)

q(x)

Activity 3: Making use of Symmetry

Coach: In these practice exercises, you want students to recognize the symmetry of a parabola as a tool for locating more points. In the second exercise, students must also use the second difference to locate more points.

1) The middle point in the following sets of coordinates is the vertex of a quadratic equation. Fill in the missing x values using your knowledge of symmetry.

a) (-2,4) (0,6) (_____,4) b) (3,6) (7,9) (_____,6) c) (_____,-2) (4,-6) (5,-2) d) (_____,8) (-2,5) (3,8)

2) Look at the following series of points that are solutions to a quadratic equation (and fall on its graph) where (2,24) is the vertex point. Fill in the missing y values using your knowledge of symmetry.

x y-4 -48-3-2 -8-1012 243 224 165 667 -26

Quadratic UnitLesson 4

Student Version

Opening Activity:I. Looking at the images, would you describe each of the images as symmetric? If you think it is symmetric,

why do you think it is and how would you describe the symmetry? If you don’t think it’s an example of symmetry, how would you support your position?

B. B.

C. D.

Activity 2:Look at the following tables

Table 5

f(x) = x2 – 6x + 8x coordinate f(x)

-2 24-1 150 81 32 03 -14 05 36 87 158 24

A. Quadratic functions have been described as being symmetric. Look at the table and find where symmetry exists in each of the tables.

B. Graph the points from the table for Table 6. Describe the symmetry you see in the visual representation (the graph)?

Note: If using graphing calculator, the window on the calculator must be adjusted to the following:

Xmin -10Xmax 10Xscl 1Ymin -20Ymax 20Yscl 1Xres 1

Table 6

g(x) = -x2 + 8x - 6x coordinate g(x)

-1 -150 -61 12 63 94 105 96 67 18 -69 -15

C. Graph the points from Table 5 on the same axes as Table 6. Compare the way symmetry presents itself when the two functions have been graphed.

D. Do you think that for every parabola there is a line that we can draw so that if you fold the parabola along that line one side will land on top of the other?

E. Look at the graph you created. Find the axis of symmetry. Draw the axis of symmetry on your graph for f(x) and g(x).

F. Where does the axis of symmetry cross the parabola? What is significant about the point where it crosses the parabola?

G. How might we find this point if we were only given the equations? Or what do you think may be the relationship between this point and the equation? In this case f(x) = x2 – 6x + 8 and g(x) = -x2 + 8x – 6.

Making Sense of Symmetry in QuadraticsHow does symmetry in quadratics show up in different ways? Where do you see symmetry in the following examples?

Vertex: (-2, 1)

Vertex: (2, -1)

Axis of Symmetry: x = -2

Axis of Symmetry: x = 2

q(x) = -x2 – 4x – 3 r(x) = x2 – 4x + 3

x r(x)-1 80 31 02 -13 04 35 8

x q(x)-5 -8-4 -3-3 0-2 1-1 00 -31 -8

r(x)

q(x)

Activity 3: Making use of Symmetry

1. The middle point in the following sets of coordinates is the vertex of a quadratic equation. Fill in the missing x values using your knowledge of symmetry.

a. (-2,4) (0,6) (_____,4)

b. (3,6) (7,9) (_____,6)

c. (_____,-2) (4,-6) (5,-2)

d. (_____,8) (-2,5) (3,8)

2. Look at the following series of points that are solutions to a quadratic equation (and fall on its graph) where (2,24) is the vertex point. Fill in the missing y values using your knowledge of symmetry.

x y-4 -48-3-2 -8-1012 243 224 165 667 -26

Lesson BookmarksQuadratic Unit

Lesson 5Teacher Guide

Lesson 5: Connecting Axis of Symmetry to the Vertex and Equation

Focus Question(s): How are symmetry, shape, and vertex (minimum, maximum) connected?How are the equation and the vertex connected?

Decision Maker: The activities in this lesson could stand as their own lesson or could be a part of the last lesson.

Opening Activity:

Decision Maker: One way to setup this opening activity could be to put a student’s work from Activity 4 and a summary sheet from the previous lesson. This way those who weren’t present for Activity 4 have some artifacts to help frame their thinking.

A. How is knowing that quadratics are symmetric helpful?

Learner: Every mathematical pattern/relationship has certain features that help us identify them, form expectations about them, and empower us to work more effectively with them. What are the students taking away from their work with symmetry in quadratics?

Decision Maker: Are there certain aspects of you want to emphasize? Do you want to incorporate parts of Activity 4 from the previous lesson in this conversation?

Activity 2:

Artist: It’s especially important with this activity that you read through the notes and reflect on the activity beforehand in order to work out your facilitation ideas. Working through these functions one at a time will probably be the most productive. Keep the focus of finding the relationship between the a value, the b value, and the x-coordinate of the vertex central to the investigation. This activity is not about filling in boxes or simply pulling out pieces of information out of the table and the equation.

Using the equations and the table of values for h(x), find the vertex, b value, a value, and the x coordinate of the vertex. (Recall standard form is ax2 + bx + c)

Note to Teacher: The writing in green is not included in the student version as those are the values the student is being asked to find.

Equation h(x) = x2 + 4x + 1Table

x h(x)-5 6-4 1-3 -2-2 -3-1 -20 11 62 13

Vertex (-2,-3)b value 4a value 1x coordinate of the vertex

-2

B. Describe the relationship between the b value, the a value, and the x-coordinate of the vertex.

Learner/Decision Maker: Give students a couple minutes to struggle with this, but together you’re going to look at a few different functions so it’s okay if they don’t nail it on the first try. Additionally, establishing a pattern takes more than one instance anyhow, so use the initial description the students come up with as a hypothesis for examining the next functions.

Coach: We specifically posed this question as a describing prompt because often times students are more comfortable giving a concrete description of what they are observing as a part of their process. This description could simply be, ‘the x-coordinate is half of b then make it negative’. Though this is not yet accurate for all a values, it does work so far and is a form of generalization though it’s not yet writing as an expression or equation. A description is a perfectly fine place to start especially if it’s the way the student is most comfortable thinking about it.

Provocateur: On a student by student basis you may want to validate their description and challenge the student to try to represent their description symbolically in some way. Their next pass at

generalizing may look something like x−coordinate=−b value2 . Don’t pressure them to only use

symbols at this point. It’s more important that they understand what they are writing and that they feel comfortable using it with the upcoming examples.

Learner: What do you see in your students’ descriptions of the relationship between the b value, the a value, and the x-coordinate of the vertex? If students struggle to articulate a relationship it may be because there is a lot going on in the table. Once the value for a, b, and the x-coordinate have been identified it might be helpful to pull those values out away from the table and ask the students the question again.

On the next page the functions j, k, and m; we’ve ordered these functions in an order that we’d recommend.

Activity 3:

Coach/Learner: Through the course of analyzing the following functions your students may not need to analyze all of them in order to formulate a version of the relationship between the value of a, b, and the x-coordinate.

Equation g(x) = -x2 + 6x + 4 f(x) = x2 – 8x i(x) = x2 – 2x – 2

Tablex g(x)-1 -30 41 92 123 134 125 9

x f(x)7 -76 -125 -154 -163 -152 -121 -7

x i(x)-3 13-2 6-1 10 -21 -32 -23 1

Vertex (3,13) (4,-16) (1,-3)b value 6 -8 -2a value -1 1 1x coordinate of the vertex

3 4 1

Decision Maker/Coach: Could it help to isolate the values we’re trying to find a pattern? Below is a table of the values we’re most interested in.

h(x) g(x) f(x) i(x)b value 4 6 -8 -2 ba value 1 -1 1 1 ax coordinate of the vertex

-2 3 4 1 ?

Decision Maker/Learner/Coach: A student may start by generalizing the relationship between a, b,

and x as b

−2 or −b2 This generalization will work for h(x), k(x), and m(x), but won’t work for j(x). You

could use this as a part of line of questioning or consider with the students how might we incorporate a so that it works for four functions. Your students might decide on one of the following

representations ( b−2 ) ,

a∗b−2 ,

b−2 a , or

−b2a . All of these representations will work for these four

functions but only b

−2 a or −b2 a will work for sure when a is a value other than 1.

You may come away from this discussion with a disagreement about how to write the generalization. That would be wonderful. Students will have the opportunity in the net activity to text out what ever conjecture they decided on.

Note to Teacher: −b2 a is the way the pattern is more typically generalized when it’s connected to the

quadratic formula which is x=−b±√b2−4 ac2 a

. Since b

−2 a and −b2a are equivalent (

b−2 a

=−b2a ) they

both work. You don’t need to force students who are more comfortable with b

−2a to use −b2a , but it

is important for everyone to know that it’s more commonly written as −b2a .

C. If the a value of a quadratic is larger than 1, do you think your generalization will still work?

Activity 4:Decision Maker: Do you want to facilitate a=2 and a=3 as separate parts of activity 4 or do you want to combine them? As mentioned in the previous note this is the opportunity for the students to start testing their conjectures concerning the relationship between a, b, and the x-coordinate of the vertex.

Artist: When do you want to bring the students back together to discuss their findings? Do multiple transitions derail your students or enable them to focus better in shorter bursts?

A. Using the equations and the table of values for each function provided below, determine the vertex, b value, a value, and the x coordinate of the vertex. (Recall standard form is ax2 + bx + c)

Equation j(x) = 2x2 – 8x k(x) = -2x2 + 12x + 4

Tablex j(x)5 104 03 -62 -81 -60 0-1 10

x k(x)-2 -28-1 -100 41 142 203 224 20

Vertex (2,-8) (3,22)b value -8 12a value 2 -2x coordinate of the vertex

2 3

B. Does your description (from question A) of the relationship between the b values, the a values and the x coordinate of the vertex still hold true now that the a value is 2? If not how would you describe the relationship to account for these new values.

C. Using the equations and the table of values for each function provided below, determine the vertex, b value, a value, and the x coordinate of the vertex. (Recall standard form is ax2 + bx + c)

Equation n(x) = 3x2 – 6x p(x) = -3x2 + 12x + 4

Tablex n(x)-3 45-2 24-1 90 01 -32 03 9

x p(x)-2 -32-1 -110 41 132 163 134 4

Vertex (1,-3) (2,16)b value -6 12a value 3 -3x coordinate of the vertex

1 2

D. Does your descriptions (from question A and/or D) of the relationship between the b values, the a values and the x coordinate of the vertex still hold true now that the a value is 3? If not how would you change your description, or representation to fit the values here.

E. How can we take the description of the relationship above and make an equation?

Challenge Question:When a is 4, what’s the relationship between the value of b and the x coordinate of the vertex? (e.g. f(x) = 4x2 + bx + c)

Summary: Connecting Axis of Symmetry to the vertex and equationHow are the vertex and the equation connected?

Decision Maker: Do you want to include some practice here with his new connection that has been developed

Vertex: (-2, 1)

Vertex: (2, -1)

q(x) = -x2 – 4x – 3 r(x) = x2 – 4x + 3

x r(x)-1 80 31 02 -13 04 35 8

x q(x)-5 -8-4 -3-3 0-2 1-1 00 -31 -8

r(x)

q(x)

Quadratic UnitLesson 5

Student Version

Opening Activity:F. How is knowing that quadratics are symmetric helpful?

Activity 2:Using the equations and the table of values for h(x), find the vertex, b value, a value, and the x coordinate of the vertex. (Recall standard form is ax2 + bx + c)

Equation h(x) = x2 + 4x + 1Table

x h(x)-5 6-4 1-3 -2-2 -3-1 -20 11 62 13

Vertexb valuea valuex coordinate of the vertex

A. Describe the relationship between the b value, the a value, and the x-coordinate of the vertex

Activity 3:Using the equations and the table of values for h(x), find the vertex, b value, a value, and the x coordinate of the vertex.

Equation g(x) = -x2 + 6x + 4 f(x) = x2 – 8x i(x) = x2 – 2x – 2

Tablex g(x)-1 -30 41 92 123 134 125 9

x f(x)7 -76 -125 -154 -163 -152 -121 -7

x i(x)-3 13-2 6-1 10 -21 -32 -23 1

Vertexb valuea valuex coordinate of the vertex

h(x) g(x) f(x) i(x)b value ba value ax coordinate of the vertex

If the a value of a quadratic is larger than 1, do you think your generalization will still work?

Activity 4:A. Using the equations and the table of values for each function provided below, determine the vertex, b

value, a value, and the x coordinate of the vertex. (Recall standard form is ax2 + bx + c)

Equation j(x) = 2x2 – 8x k(x) = -2x2 + 12x + 4

Tablex j(x)5 104 03 -62 -81 -60 0-1 10

x k(x)-2 -28-1 -100 41 142 203 224 20

Vertexb valuea valuex coordinate of the vertex

B. Does your description (from question A) of the relationship between the b values, the a values and the x coordinate of the vertex still hold true now that the a value is 2? If not how would you describe the relationship to account for these new values.

C. Using the equations and the table of values for each function provided below, determine the vertex, b value, a value, and the x coordinate of the vertex. (Recall standard form is ax2 + bx + c)

Equation n(x) = 3x2 – 6x p(x) = -3x2 + 12x + 4

Tablex n(x)-3 45-2 24-1 90 01 -32 03 9

x p(x)-2 -32-1 -110 41 132 163 134 4

Vertexb valuea valuex coordinate of the vertex

D. Does your descriptions (from question A and/or D) of the relationship between the b values, the a values and the x coordinate of the vertex still hold true now that the a value is 3? If not how would you change your description, or representation to fit the values here.

E. How can we take the description of the relationship above and make an equation?

Challenge Question:When a is 4, what’s the relationship between the value of b and the x coordinate of the vertex? (e.g. f(x) = 4x2 + bx + c)

Summary: Connecting Axis of Symmetry to the vertex and equationHow are the vertex and the equation connected?

Vertex: (-2, 1)

Vertex: (2, -1)

q(x) = -x2 – 4x – 3 r(x) = x2 – 4x + 3

r(x)

q(x)

x q(x)-5 -8-4 -3-3 0-2 1-1 00 -31 -8

x r(x)-1 80 31 02 -13 04 35 8

Lesson BookmarksQuadratic Unit

Lesson 6Teacher Guide

Lesson 6: How can we calculate the x- intercepts of the quadratic equation?

Learner: In this lesson we are going to introduce the need for factoring… as a means of finding the zeroes or roots of a quadratic equation. We want students to be flexible in their use of representations, to understand the connections between each form, and to appreciate the utility of each. In the next few lessons, we introduce a group of students – Fatima, Frank, Felix and Quentin. Each student, like your own, are in a different place in terms of experience and understanding, yet all are questioners and inquirer. As you go through the lesson, point out to students the types of questions these fictional students are asking, and how it leads them to new understandings. Specifically, mathematicians: Look for and make use of structure; Attend to precision; Make sense of problems and persevere in solving them.

Opening Activity

Fatima and Frank have been exploring different representations of quadratic equations.

Fatima says, “I know that values of a, b and c in the equation determine what the graph will look like. For example, I can determine the line of symmetry, vertex and y-intercept of the parabola just by looking at its equation.”

Frank questions, “I wonder if it’s possible to easily determine the x-intercepts.”

Fatima responds, “It might be helpful for us to analyze the table and graph of several quadratic equations to see if we can make sense of their x-intercepts.”

Examine the graphs of the four quadratic equations below.

y = x2 – 6x + 8 y = x2 + 4x + 3 y = -x2 +6x -9 y = -x2 – 2x - 3

EquationNumber of x – interceptsCoordinates of x-

intercept(s)

What observation can be made about the coordinates of any x-intercept?Coach: The goal is to have students see that the y-value will always be zero

Using that information, determine the x-intercept(s) of each quadratic equation given the three tables below. Circle the points.

y = x2 – 6x + 8 y = x2 + 4x + 3 y = -x2 +6x -9 y = -x2 – 2x – 3

Fatima explains, “So to find the x-intercepts of any quadratic function, I just need to examine the graph or table of values. To save time, I’ll just enter the equation into the calculator.”

What was Frank and Fatima’s big discovery? What general statement can be made about the x-intercept of any quadratic function? Coach: The goal is to have students see that the y-value will always be zero

Coach: Here is an opportunity to introduce the term “zeroes” of the function. Other terms they should also be familiar with include “solution” and “roots”. Those will be introduced later.

Practice a few.

a) y = x2 + 4x + 4 b) y = x2 + 8x + 12 c) y = - x2 – 6x – 5 d) y = x2 - 6x - 1

What did we find out about the table and the graph in exercise d? What do you think about that?

Decision maker: You want to have a discussion about this. How do you want to facilitate this discussion Why isn’t it okay to estimate? Here is an opportunity to have a conversation about precision. It is also a good time to remind students of the value of one representation over another.

X Y -3 0-2 -1-1 00 31 8

X Y -3 0-2 -1-1 00 31 8

Activity 2Provocateur: Students will struggle with trying to solve this equation, because it something entirely new to them. It’s really important for students to think about two things: 1) the x-intercept can be determined by looking at the x-value when y = 0; 2) Traditional methods for solving an equation don’t work in this situation, thus prompting the need for another method.

Fatima is curious about equation, y = x2 - 6x + 8.

Frank recalls that there was a third way to determine the line of symmetry and vertex of a quadratic equation – algebraically. He wonders if that might also work for finding the x-intercept.

Use Frank’s idea to set up an equation that we could solve to determine the x-intercept(s) for the quadratic function y = x2 - 6x - 1. (What equation would we set up?)

Try to solve that equation. What happens?

Activity 3Learner/Decision Maker: You need to decide how to structure the conversation about Activity 3. Chart student’s observations, etc. When we encourage students to wonder about mathematical ideas and raise questions about them, we are getting to the heart of inquiry. Use your students’ wonderings to guide them through the next few lessons. Refer to them often to encourage them to do more questioning.

Felix comes by and looks at their work, and says, “You might want to try factoring that.”Fatima and Frank are stumped. They have no idea what Felix is talking about. He shows them the following work:

Can you make sense of what he did?

What did you observe? What confused you? What questions do you have?

Quadratic UnitLesson 6

Student Version

Opening Activity:

Fatima and Frank have been exploring different representations of quadratic equations.

Fatima says, “I know that values of a, b and c in the equation determine what the graph will look like. For example, I can determine the line of symmetry, vertex and y-intercept of the parabola just by looking at its equation.”

Frank questions, “I wonder if it’s possible to easily determine the x-intercepts.”

Fatima responds, “It might be helpful for us to analyze the table and graph of several quadratic equations to see if we can make sense of their x-intercepts.”

Examine the graphs of the four quadratic equations below.

y = x2 – 6x + 8 y = x2 + 4x + 3 y = -x2 +6x -9 y = -x2 – 2x - 3

EquationNumber of x – interceptsCoordinates of x-intercept(s)

What observation can be made about the coordinates of any x-intercept?

Using that information, determine the x-intercept(s) of each quadratic equation given the three tables below. Circle the points.

y = x2 – 6x + 8 y = x2 + 4x + 3 y = -x2 +6x -9 y = -x2 – 2x – 3

Fatima explains, “So to find the x-intercepts of any quadratic function, I just need to examine the graph or table of values. To save time, I’ll just enter the equation into the calculator.”

What was Frank and Fatima’s big discovery? What general statement can be made about the x-intercept of any quadratic function?

Practice a few.

a) y = x2 + 4x + 4 b) y = x2 + 8x + 12 c) y = - x2 – 6x – 5 d) y = x2 - 6x – 1

What did we find out about the table and the graph in exercise d? What do you think about that?

X Y -3 0-2 -1-1 00 31 8

X Y -3 0-2 -1-1 00 31 8

Activity 2:

Fatima is curious about equation, y = x2 - 6x -1.

Frank recalls that there was a third way to determine the line of symmetry and vertex of a quadratic equation – algebraically. He wonders if that might also work for finding the x-intercept.

Use Frank’s idea to set up an equation that we could solve to determine the x-intercept(s) for the quadratic function y = x2 - 6x - 1. (What equation would we set up?)

Try to solve that equation. What happens?

Activity 3:

Felix comes by and looks at their work, and says, “You might want to try factoring that.”Fatima and Frank are stumped. They have no idea what Felix is talking about. He shows them the following work:

Can you make sense of what he did?

What did you observe? What confused you? What questions do you have?

Lesson BookmarksQuadratic Unit

Lesson 7Teacher Guide

Lesson 7: Doing and undoing in Algebra

Coach: In this activity we want students to understand that factoring and the distributive properties “undo” each other. Keep in mind that although we are seeming to go off-track from finding the x-intercept, the ultimate goal in these next few lessons/activities is for students to be able to solve a quadratic equation/find the x-intercept(s) and connect that idea back to the table and equation. You should give students the opportunity to experience several different methods to develop a flexibility in solving quadratic equations. You might find yourself getting away from the main idea; remember to always connect back to the x-intercept.

Decision Maker/Artist: You have the option to use algebra tiles as a way to help students visualize the math. Algebra tiles are a form of physical representation that can be used to facilitate and make sense of the distributive property and factoring of quadratic expressions. When deciding whether or not to use them, keep in mind the different learners in your room – visual, physical, logical. Algebra tile kits can be purchased, or you can create them yourself. The activities that follow incorporate the use of algebra tiles.

Opening Activity. Coach: Students have learned factoring before. It is extremely important that students are proficient in this skill. Something new for students is the notion that factors can be negative integers. Previously, students only thought of positive integers (counting numbers) as the factors of a number. Therefore: Factors of 25: 1, 25, -1, -25, 5, 5, -5, -5. What is it about the structure of a quadratic equation that necessitates both positive and negative factors?

Yesterday we started examining Felix’s work. In these next few lessons, we are going to answer many of your questions.

Fatima and Frank makes it their goal learn how to “factor” y = x2 - 6x - 1.

Felix called his method factoring. You have seen factoring before. What does it mean to factor in mathematics? Give several examples of factoring. Practice: List all of the factors of:

a) 6 b) 24 c) 7 d) -8 e) 25

Activity 2Learner/Decision Maker: Do your students understand the laws of exponents? We do a quick review here.

Learner/Coach: Try not to teach students quick methods, such as “foil”. Let them come up with their own methods. In this lesson, we introduce students to several models to help them make sense of the structure and look for patterns.

Remember from your 8th grade math class that you learned about exponents. Let’s review a few:a) x · x3 = b) x2 · x4 = c) x · x =

Perform the given operation. What mathematical property did you use?

a) 2(2x + 4) b) x (x+ 3) c) x(x2 + 2x + 1) d) (x + 3) (x + 2)

Practice using the distributive property to multiply the following binomials.

1) (x + 1)(x + 3) 2) (x + 3)(x - 2) 3) (x - 3)(x - 2)

Activity 3: Coach: The big idea of the lesson is that the distributive and factoring “undo” each other. You want students to see that in Felix’s work. Make sure students are able to articulate that (x + 1)(x + 3) is an equivalent expression to x2 + 4x + 3. Make it clear to your students that the next few lessons will be spent learning different methods for factoring so that they may answer Felix and Fatima’s initial dilemma – can the x-intercepts of a quadratic equation be determined algebraically?

Let’s re-examine Felix’s work, which he called factoring. How might an understanding of the distributive property help us to make sense of factoring? How will knowing how to factor help Frank and Fatima?

Quadratic UnitLesson 7

Student Version.

Opening Activity: Yesterday we started examining Felix’s work. In these next few lessons, we are going to answer many of your questions.

Fatima and Frank makes it their goal learn how to “factor” y = x2 - 6x - 1.

Felix called his method factoring. You have seen factoring before. What does it mean to factor in mathematics? Give several examples of factoring.

Practice: List all of the factors of:

a) 6 b) 24 c) 7 d) -8 e) 25

Activity 2:Remember from your 8th grade math class that you learned about exponents. Let’s review a few:

a) x · x3 = b) x2 · x4 = c) x · x =

Perform the given operation. What mathematical property did you use?

a) 2(2x + 4) b) x (x+ 3) c) x(x2 + 2x + 1) d) (x + 3) (x + 2)

Practice using the distributive property to multiply the following binomials.

a) (x + 1)(x + 3) b) (x + 3)(x - 2) c) (x - 3)(x - 2)

Activity 3: Let’s re-examine Felix’s work, which he called factoring. How might an understanding of the distributive

property help us to make sense of factoring? How will knowing how to factor help Frank and Fatima?

Lesson Bookmarks

Quadratic UnitLesson 8-9

Teacher Guide

Lesson 8-9: Modeling Quadratics

Note to Teacher: This was originally part of Lesson 8. This can be broken into 2 lessons or taught over 2 days (Opening Activity – Activity 3 and Activity 4 –Activity 8). This allows time for students to work with and understand the use of the manipulatives and how they relate to factoring and the distributive property.

Opening Activity:

Learner/Artist: When you introduce the tiles to students, direct them to pay special attention to the dimensions of the tiles in order to make meaning of the representation.

Coach/Decision Maker: This part of the lesson is somewhat direct-instruction because you are giving students a new tool and need them to make sense of it. This part can be a class discussion, as you ask students to try to make sense of the tool.

Give the tiles to students. Ask them to describe them. (Three different tiles. Each side has different color. Dimensions match, etc)

Coach: Allow students to play with this physical representation. Give them 12 algebra tiles You want students to discover which arrangements of dimensions can and cannot form rectangles.

This is a unit tile. Its dimensions are 1 x 1.

How many different rectangles can be formed using 12 unit tiles? What are their dimensions?

Draw them in the grid below.

Coach: This is a great opportunity to discuss the commutative property and the concept of area. Is a 3 x 4 rectangle the same as a 4 x 3 rectangle (commutative property)? Why or why not? How are all of these representations equivalent? How are they different?

How many different rectangles can be made with 7 tiles? 10 tiles? 24 tiles? 13 tiles?How is this connected to the concept of factoring?Questioner: Ask students why it is important to form the tiles into rectangles. The grid makes it easier for students to see the connection between the area of a rectangle and its dimensions. You want students to conclude that prime numbers will have only one rectangle configuration. This point is extremely important for students to understand as it will make strategizing easier when factoring.

Activity 2:Coach: Have a class-discussion about the ideas learned here about the distributive property.

We use this tile to represent 1x. Label its dimensions.

How can you use these tiles to create a rectangular model for 4x + 2? What are its dimensions, or factors?

Coach: Discuss how the tiles fit together – the dimensions must match up.

Use the distributive property to prove the product of those two factors is 4x + 2.

This model, called an area model, can be drawn to represent the algebra tile model you made for 4x + 2. Try to make sense of this model. Describe how it works. How is it connected to the distributive property?

Coach: When you multiply the length and width of the rectangle, the resulting polynomial represents the area.

Questioner: In other words, is it true that 2(2x + 1) = 4x + 2?

Activity 3:Coach: Have a class-discussion about the ideas learned here about the distributive property.

Finally, this is the third tile you will be working with. What are the lengths of its sides? Label them.

Use the algebra tiles to create a rectangular model for x2 + 3x. What are its dimensions? Use the distributive property to show the product of those two factors is x2 + 3x.

Use this area model to explain the relationship between the factors of x2 + 3x, the distributive property and the area of the rectangle.

Coach: You might want to review with students factoring expressions that result in a monomial-polynomial factor pair.

Practice:Draw a rectangle model for 2x + 10.

What are the two factors? Use the Distributive Property to show the product of those two factors is 2 x+10.

Draw a rectangle model for 3 x2+12 x.

What are the two factors? Use the Distributive Property to show the product of those two factors is 3 x2+12 x

Draw a rectangle model for 2 x2+10 x+8. What are the two factors? Use the Distributive Property to show the product of those two factors is: 2 x2+10 x+8

Activity 4:

Provocateur: When a quadratic expression is factorable, it can be modeled as a rectangle using algebra tiles. In order for the algebra tile model to work, students need to be clear on this idea. It seems, then, that these two expressions cannot be factored.

Use algebra tiles to find the factors of 3x + 2, then x2 + 7. What do you notice?

Activity 5:Coach: In this activity, students practice using each model and should develop a preference for a particular model. You want students to continue to develop the notion that different representations of the same idea are equivalent.

Felix was showing Frank and Fatima how to factor quadratic expressions using algebra tiles. He created several different models. What are the factors (dimensions of the sides) of each expression? What quadratic expression does each represent? Use the distributive property or area model, whichever you prefer, to show you are correct.

Algebra Tile model Area model Distributive Property X +2

X

+1

Activity 6:Coach: This activity is meant to help students visualize the distributive property.

Algebra tiles Area Model Distributive Property

Now you have several models – distributive property, algebra tiles, area model - for multiplying binomials and factoring quadratic expressions. Which do you prefer? What do you think are the advantages of each?

Activity 7:Learner/Coach: Try to encourage students to make use of the area model or the distributive property. While the algebra tiles facilitate understanding of the procedure, using them is not always the most efficient representation, and algebra tiles might not always be available – for example on the CC Exam. You do, however, want students to take note of the relationship between each method. In future lessons, algebra tiles will be used as a visual representation that facilitate learning.

Use any model to find the product of each of the following binomial expressions:

a) (x + 3) (x + 2)b) (x - 3)(x - 2)c) (x + 3) (x – 2)

Activity 8:Provocateur: In exercises c and d the idea of additive inverses comes into play. Students need five -1x tiles, but where do the extra -1 x and +x tiles come from?

Use any model to determine the factors of the following quadratic expressions:

a) x2 + 4x + 4b) x2 + 8x + 12c) x2 – 5x – 6d) x2 – 3x – 4

Artist/Learner: Here is an example of how the algebra tiles facilitate understanding of an otherwise abstract concept.

Coach: Through this writing exercise want students to start strategizing a method to quickly multiply any two binomials. They will formally do that in the next lesson.

Journal Writing: When you multiply two terms by two terms, you get four terms. Why does the final result, when you multiply two binomials, usually result in three terms?

Give an example of how your final result can be two terms.

Quadratic UnitLesson 8- 9

Student Version

Opening Activity:

This is a unit tile. Its dimensions are 1 x 1.

How many different rectangles can be formed using 12 unit tiles? What are their dimensions?

Draw them in the grid below.

How many different rectangles can be made with 7 tiles? 10 tiles? 24 tiles? 13 tiles?

How is this connected to the concept of factoring?

Activity 2: We use this tile to represent 1x. Label its dimensions.

How can you use these tiles to create a rectangular model for 4x + 2? What are its dimensions, or factors?

Use the distributive property to prove the product of those two factors is 4x + 2.

This model, called an area model, can be drawn to represent the algebra tile model you made for 4x + 2. Try to make sense of this model. Describe how it works. How is it connected to the distributive property?

Activity 3:

Finally, this is the third tile you will be working with. What are the lengths of its sides? Label them.

Use the algebra tiles to create a rectangular model for x2 + 3x. What are its dimensions? Use the distributive property to show the product of those two factors is x2 + 3x.

Use this area model to explain the relationship between the factors of x2 + 3x, the distributive property and the area of the rectangle.

Practice:Draw a rectangle model for 2x + 10.

What are the two factors? Use the Distributive Property to show the product of those two factors is 2 x+10.

Draw a rectangle model for 3 x2+12 x.

What are the two factors? Use the Distributive Property to show the product of those two factors is 3 x2+12 x

Draw a rectangle model for 2 x2+10 x+8.

What are the two factors? Use the Distributive Property to show the product of those two factors is: 2 x2+10 x+8

Activity 4: Use algebra tiles to find the factors of 3x + 2, then x2 + 7. What do you notice?

Activity 5:Felix was showing Frank and Fatima how to factor quadratic expressions using algebra tiles. He created several different models. What are the factors (dimensions of the sides) of each expression? What quadratic expression does each represent? Use the distributive property or area model, whichever you prefer, to show you are correct.

Algebra Tile model Area model Distributive Property X +2

X

+1

Activity 6:Algebra tiles Area Model Distributive Property

Now you have several models – distributive property, algebra tiles, area model - for multiplying binomials and factoring quadratic expressions. Which do you prefer? What do you think are the advantages of each?

Activity 7:Use any model to find the product of each of the following binomial expressions:

d) (x + 3) (x + 2) b) (x - 3)(x - 2) c)(x + 3) (x – 2)

Activity 8:Use any model to determine the factors of the following quadratic expressions:

e) x2 + 4x + 4 b) x2 + 8x + 12 c) x2 – 5x – 6 d)x2 – 3x – 4

Journal Writing: When you multiply two terms by two terms, you get four terms. Why does the final result, when you multiply two binomials, usually result in three terms?

Give an example of how your final result can be two terms.

Lesson BookmarksQuadratic Unit

Lesson 10Teacher Guide

Lesson 10: Strategizing the distributive property and factoring quadratic expressions by looking for patterns.

Coach: You will want students to notice that b = m + n and c = mn. You also want students to recognize the difference between two squares. Have students pay special attention to the contrast between (x+m)2 and (x-m)2

Opening Activity: (groups of 3 students each) Decision Maker/Artist: This activity works best when students are working in groups of 2 or 3. Print out a one copy of the table for each group on plain paper or cardstock, cut out each expression and shuffle them. Ask students to place them into pairs.

Your teacher will give your group several binomials to multiply and several quadratic expressions. Match each binomial factor pair with its quadratic expression.

x2 + 5x + 6 x2 - 6x + 8

x2 + 3x –10x2 -2x - 24

x2 - 9x + 8x2 - 9

x2 + 6x + 5x2 - 100

x2 - 25 x2 + 10x + 25x2 – 10x + 25 x2 + 20x + 100

(x + 3) (x + 2) (x - 8) (x - 1)(x - 2) (x - 4) (x - 3) (x + 3)(x + 5) (x - 2) (x + 5) (x + 1)(x - 6) (x + 4) (x + 10) (x - 10)(x + 5) (x – 5) (x + 5) (x + 5)(x - 5) (x – 5) (x + 10)(x – 10)

Write the binomial product-quadratic expression pairs in the table below:

Binomial Product Quadratic expression

When you matched each quadratic equation with the binomials you were finding the factors of the equation. But how exactly do we factor a quadratic equation?

Activity 2: Coach: Give students an opportunity to create their own groupings. Push students to look for patterns in the structures. Ultimately you are looking for three different groups: difference of two squares, perfect square binomials, other binomials. Students might not know how to call them, and you can help them to describe them. They will be visited in more detail in future lessons.

Observe the data you collected in the table above. Create different groupings of binomial product-quadratic expression pairs. Why did you create these groupings?

Learner: How are your students developing their metacognition? Are they able to articulate their thinking/reasoning?

Journal Write/Class Discussion: What process, clues or strategies did you develop/notice while working on this activity?

Activity 3: Look at the group containing (x + 2)(x + 3) = x2 + 5x + 6. How would you describe all of the binomial product-quadratic expression pairs in this group?

What are some patterns you notice about them?

Activity 4: Coach: Help students see the structure of the quadratic expressions they are factoring. In all of these, a = 1, c≠ 0. You might want to consider giving students opportunities with factoring by using the greatest common factor.

Use the groups you constructed to create a method to find two binomial factors of a quadratic expression.

For example, how would you go about factoring x2 + 4x + 3?

Your method must work for all situations. If it does not, you need to re-evaluate your procedure. Use the Distributive Property to check your results.

Practice: Factor each quadratic expression

a) x2 + 7x + 10 b) x2 - 4 c) x2 - 8x + 16 d) 16 – x2

Journal Write: Write a procedure for determining factors of any quadratic function algebraically. Why do we say we are undoing the distributive property when we factor algebraically?

Provocateur: We want students to understand that this method of factoring doesn’t always work.With this newly acquired knowledge, try to help Frank and Fatima factor the expression y = x2 - 6x -1.

Quadratic UnitLesson 10

Student Version

Opening Activity: (groups of 3 students each) Your teacher will give your group several binomials to multiply and several quadratic expressions. Match each binomial factor pair with its quadratic expression.Write the binomial product-quadratic expression pairs in the table below:

Binomial Product Quadratic expression

When you matched each quadratic equation with the binomials you were finding the factors of the equation. But how exactly do we factor a quadratic equation?

Activity 2: Observe the data you collected in the table above. Create different groupings of binomial product-quadratic expression pairs. Why did you create these groupings?

Journal Writing: What process, clues or strategies did you develop/notice while working on this activity?

Activity 3: Look at the group containing (x + 2)(x + 3) = x2 + 5x + 6. How would you describe all of the binomial product-quadratic expression pairs in this group?

What are some patterns you notice about them?

Activity 4: Use the groups you constructed to create a method to find two binomial factors of a quadratic expression.For example, how would you go about factoring x2 + 4x + 3?Your method must work for all situations. If it does not, you need to re-evaluate your procedure. Use the Distributive Property to check your results.

Practice: Factor each quadratic expression

b) x2 + 7x + 10 b) x2 – 4 c) x2 - 8x + 16 d) 16 – x2

Journal Writing: Write a procedure for determining factors of any quadratic function algebraically. Why do we say we are undoing the distributive property when we factor algebraically?

With this newly acquired knowledge, try to help Frank and Fatima factor the expression y = x2 - 6x -1.

Lesson BookmarksQuadratic Unit

Lesson 11Teacher Guide

Lesson 11: Factoring when a ≠1

Opening Activity:Decision Maker: This simple opening activity may be included or not. It is an opportunity to see what your students know about the structure of a quadratic expression, and gives them practice with factoring and the distributive property. Coach: Remind students of the structure of the quadratic expressions they are factoring. In all of these, a = 1, c≠ 0. Learner: Are students creating binomials of the form (1x + m)(1x + n)? Do they understand the structure and terminology?Artist: This is a bare bone structure of the game. What can you do to flesh it out?

To play this game, you will need a partner, a pencil, and a recording sheet. One partner uses the distributive property to create a quadratic expression, the other partner tries to factor it. Then switch.

Activity 2:Coach: Notice that in these examples, “a” is a prime number. When “a” is not a prime number, there are many possibilities and it becomes less efficient to use this method of factoring. We need a better way.

Frank gives Fatima the following problem:

5x2 + 11x + 3 She struggles to factor the quadratic expression. I give up! What factors did you use? (5x + 1)(x + 2)

Felix says, that’s interesting, in all of the problems we have worked on before, the value of the coefficient a was equal to 1. I guess when a is not equal to 1 factoring works differently.

Work through these expressions and see if you can notice patterns that can help you think about factoring.

1) (2x + 5)(x + 3) 2) (7x + 2) (x + 6) 3) (x + 5)(3x + 2)

What patterns do you see that might help you with the factoring? Try using your ideas to factor:

1) 3x2 + 13x + 4 2) 2x2 + 3x +1 3) 3x2 + 5x + 2 4) 5x2 + 9x + 4

Note to Teacher: As you can see, we can go on and on offering students different experiences with factoring. Let’s stop for now. Let students know that we could continue looking for new ways to factor quadratic expressions, but we’re going to stop for now. Students who are interested can continue to work on this on the side.

We are now going to re-visit the idea of solving for the x-intercept(s) algebraically.

Provocateur: For students who want to grapple with more ideas, offer them the following:

Activity 3:

Quentin comes along and offers up this quandary:

Try factoring 4x2 + 10x + 6 and 4x2 + 11x + 6.

What are some issues that come up? Why is this a problem?

Quadratic UnitLesson 11

Student Version

Opening Activity:

To play this game, you will need a partner, a pencil, and a recording sheet. One partner uses the distributive property to create a quadratic expression, the other partner tries to factor it. Then switch.

Activity 2:Frank gives Fatima the following problem:

5x2 + 11x + 3 She struggles to factor the quadratic expression. I give up! What factors did you use? (5x + 1)(x + 2)

Felix says, that’s interesting, in all of the problems we have worked on before, the value of the coefficient a was equal to 1. I guess when a is not equal to 1 factoring works differently.

Work through these expressions and see if you can notice patterns that can help you think about factoring.

2) (2x + 5)(x + 3) 2) (7x + 2) (x + 6) 3) (x + 5)(3x + 2)

What patterns do you see that might help you with the factoring? Try using your ideas to factor:

2) 3x2 + 13x + 4 2) 2x2 + 3x +1 3) 3x2 + 5x + 2 4) 5x2 + 9x + 4

Activity 3:

Quentin comes along and offers up this quandary:

Try factoring 4x2 + 10x + 6 and 4x2 + 11x + 6.

What are some issues that come up? Why is this a problem?

Lesson BookmarksQuadratic Unit

Lesson 12Teacher Guide

Lesson 12: Factoring Polynomials

Opening Activity:We have explored how to factor quadratics in the form a x2+bx+c=0 ¿ when a = 1 and when a ≠ 1. Today we are going to explore how to factor polynomials.

Observe the following polynomials and their factors:

Polynomial Factorsa. x2+5x+6 (x+2)( x+3)b. x4+8 x2+16 ( x2+4 ) ( x2+4 )c. x4+2 x2−3 (x2+3)(x2−1)d. x2−6 x+9 (x−3)(x−3)e. x6−6 x3+9 (x3−3)(x3−3)

Coach: Notice that example c is not completely factored. This example will be revisited in Activity 2

A. How are these polynomials similar to the ones we’ve worked with before? How are they different?Coach: We want students to notice that the “c” value is always factored. Students should see that the degrees of examples b, c, and e are different.

B. What patterns do you see between the polynomial and its factored form?Coach: Students should notice that the degree of the factors is the same as the degree of the middle term. The degree of the factors is also half of the leading degree of the polynomial.

Scaffold: You can remind students that x is also equivalent to x1.

Activity 2:Let’s revisit Example C from the Opening Activity.x4+2 x2−3 = (x2+3)(x2−1)

A. Is this polynomial completely factored? Explain your reasoning.

Coach: Sometimes students will stop factoring once they have 2 factors. We want students to examine the factors and determine if a polynomial can be broken down more. Students should get in the habit of factoring completely. As we can see (x2−1) is the difference of 2 squares and can be factored.

Now that you’ve learned all about factoring, let’s practice a few: Factor completely

Coach: Examples included range in difficulty. You can determine how many examples you would like to use. These examples cover Lessons 8 – 12. You can add your own examples as well. I

Coach: Some examples include a negative coefficient.

Learner: Do students factor out a negative coefficient? What happens when they do not?

2 x3−50x−16 t 2+32 t+48

2 a2+6a+185 x2−5

3 t3+18 t 2−48t4 n−n3

x2−8 x+15−2 x3−2x2+112 x

2 x2+9 x+4−100+99 v+v2

Activity 3:Quentin is given the following problem:

Artist: The following question is from the Alegbra I Common Core Regents exam (Jun 2014, #31). Note how we are able to make this into a more interesting question

He comes up with ( x4+7 ) (x−1 ). When he checked his answer he got x5−x4+7 x−7. Can you teach him how to factor this example? (Hint: include any patterns you look for when factoring yourself)

Quadratic UnitLesson 12

Student Version

Opening Activity:We have explored how to factor quadratics in the form a x2+bx+c=0 ¿ when a = 1 and when a ≠ 1. Today we are going to explore how to factor polynomials.

Observe the following polynomials and their factors:

Polynomial Factorsf. x2+5 x+6 (x+2)( x+3)

g. x4+8 x2+16 ( x2+4 ) ( x2+4 )h. x4+2 x2−3 (x2+3)(x2−1)i. x2−6 x+9 (x−3)(x−3)j. x6−6 x3+9 (x3−3)(x3−3)

C. How are these polynomials similar to the ones we’ve worked with before? How are they different?

D. What patterns do you see between the polynomial and its factored form?

Activity 2:Let’s revisit Example C from the Opening Activity.x4+2 x2−3 = (x2+3)(x2−1)

B. Is this polynomial completely factored? Explain your reasoning.

Now that you’ve learned all about factoring, let’s practice a few: Factor completelya) 2 x3−50x b)−16 t 2+32 t+48 c) 2 a2+6a+18

d) 5 x2−5 e) 3 t3+18 t 2−48t f) 4 n−n3

g) x2−8 x+15 h) −2 x3−2 x2+112 x i)2 x2+9 x+4

j) −100+99 v+v2

Activity 3:Quentin is given the following problem:

He comes up with ( x4+7 ) (x−1 ). When he checked his answer he got x5−x4+7 x−7. Can you teach him how to factor this example? (Hint: include any patterns you look for when factoring yourself)

Lesson BookmarksQuadratic Unit

Lesson 13Teacher Guide

Lesson 13: Determining the x-intercept(s) by factoring.

Opening Activity:Use your graphing calculator to determine the x-intercept(s) of the following quadratic equations; then determine the factors algebraically.

Quadratic Equation x-intercept(s) determined graphically or by table

Factors

y = x2 + 10x + 21y = x2 + x - 6y = x2 - 8x + 15

What do you notice is the relationship between the x-intercepts and the factors of a quadratic expression? Activity 2Coach: You want students to make sense of a few things in this activity: the idea that x-intercept occurs at y = 0; a quadratic expression and its factors are equivalent; Zero-property of multiplication; the x-intercepts are solutions of the quadratic equation when y = 0.

Questioner: Let students struggle to make meaning of Step 4. Why is it okay to make the assumption that 0 = x + 1 or 0 = x + 3?

Let’s re-examine Felix’s work.

Try to reason through Felix’s algebraic solution.Modeler of Mathematical Thinking: This chart is a helpful way for students to examine a process and try to make sense of it. Looking at the structure of a sample algebraic solution to a problem is also a good way for students to develop their quantitative reasoning skills. Felix methodically worked through the problem and each step involved a series of logic and reasoning.

Why was it okay for Felix to create the two equations in Step 4?Coach: Discuss the zero-property of multiplication with students. You might need to give them several prompts to help them arrive at the idea that if the product of two numbers is zero, then one or both of the factors must be zero. ( )(8) = 0; (-10)( ) = 0; etc.

Activity 3

Try using the zero-property to find the x-intercept(s) when given the following factors:

a) (x + 3)(x + 7) = 0 b) (x + 3)(x – 2) = 0 c) (2x – 4)(x – 3) = 0

Activity 4

For each quadratic equation given below, determine the solutions algebraically.

a) x2 + 5x +6 = y b) x2 + 3x – 10 = y c) x2 – 100 = y

Try this problem and justify your answer.

Challenge

Coach: This problem, taken from the June 2014 Algebra I Common Core Regents exam, combines all the skills they just practiced in these last few lessons. What is a trinomial? What does it mean to solve for x?

Learner: This is a good opportunity to see what your students understand about function notation and how to combine like terms.

Decision maker/Artist: To support and structure student reasoning, you might want to ask students to consolidate their thinking into a chart similar to the table used in Activity 3.

Write an equation that defines m(x) as a trinomial where m(x) = (3x – 1)(3 – x) + 4x2 + 19.Solve for x where m(x) = 0.

Activity 5: Challenge

Coach: The purpose of this activity is to introduce students to a polynomial equation that has 3 roots. Discuss the relationship between the number of DISTINCT factors and how they determine the number of X-intercepts.

1. Felix wondered what would happen if you multiplied (x+1)(x+2)(x+3). He knows when you multiply (x+1)(x+2) you get x2+3x+2.

x2+3x+2 has 2 zeroes and 2 factors.

How many zeroes and how many factors does (x+1)(x+2)(x+3) have? List the zeroes and the factors.

How many times will this graph cross the x-axis?

Graph (x+1)(x+2)(x+3) for −4 ≤ x ≤1

What do you observe about the zeroes, factors, and graph of (x+1)(x+2)(x+3)?

2. Complete the following table

Function x-intercept(s) determined graphically

Distinct Factors

f ( x )=x2−2 x−15g ( x )=(x+3)(x2−11 x+24)

h ( x )=( x−3 )2

*** j ( x )=(x+2)(x2−4 x+4)

Coach: Example j(x) helps emphasize that the number of x-intercepts is determined by the number of DISTINCT factors. In this case it is factored to (x+2)( x−2)(x−2) where the distinct factors are (x+2) and (x−2)

Coach: This is a good opportunity to discuss the fundamental theorem of algebra

Decision Maker: Use desmos.com/calculator display on the Smartboard (if available) to discuss student findings and have a discussion relating zeroes, factors and the graph.

Practice:Coach: The following problems were taken from past Algebra Common Core Regents exams

1) June 2015, #12

2) August 2015, #4

Quadratic UnitLesson 13

Student Version

Opening Activity:Use your graphing calculator to determine the x-intercept(s) of the following quadratic equations; then determine the factors algebraically.

Quadratic Equation x-intercept(s) determined graphically or by table

Factors

y = x2 + 10x + 21

y = x2 + x - 6

y = x2 - 8x + 15

What do you notice is the relationship between the x-intercepts and the factors of a quadratic expression?

Activity 2:Let’s re-examine Felix’s work.

Try to reason through Felix’s algebraic solution.

Why was it okay for Felix to create the two equations in Step 4?

Activity 3:Try using the zero-property to find the x-intercept(s) when given the following factors:

b) (x + 3)(x + 7) = 0 b) (x + 3)(x – 2) = 0 c) (2x – 4)(x – 3) = 0

Activity 4:For each quadratic equation given below, determine the solutions algebraically.

b) x2 + 5x +6 = y b) x2 + 3x – 10 = y c) x2 – 100 = y

Try this problem and justify your answer.

ChallengeWrite an equation that defines m(x) as a trinomial where m(x) = (3x – 1)(3 – x) + 4x2 + 19.Solve for x where m(x) = 0.

Activity 5: Challenge3. Felix wondered what would happen if you multiplied (x+1)(x+2)(x+3). He knows when you multiply

(x+1)(x+2) you get x2+3x+2.

x2+3x+2 has 2 zeroes and 2 factors.

How many zeroes and how many factors does (x+1)(x+2)(x+3) have? List the zeroes and the factors.

How many times will (x+1)(x+2)(x+3) cross the x-axis?

Graph (x+1)(x+2)(x+3) for −4 ≤ x ≤1

What do you observe about the zeroes, factors, and graph of (x+1)(x+2)(x+3)?

4. Complete the following table

Function x-intercept(s) determined graphically

Distinct Factors

f ( x )=x2−2 x−15

g ( x )=(x+3)(x2−11 x+24)

h ( x )=( x−3 )2

*** j ( x )=(x+2)(x2−4 x+4)

Practice:

Lesson BookmarksQuadratic Unit

Lesson 14Teacher Guide

Lesson 14: Factoring Strategies/Equivalent Expressions

Coach: In this lesson, we want students to discover that the old methods for factoring don’t always work and that they need to strategize. Looking at the structure of the quadratic expression helps us to develop a more effective strategy. We introduce students to the method of completing the square.

Decision Maker: The method of completing the square involves solutions containing radical terms. You might want to consider a review of radical expressions which includes rational and irrational expressions as well as the case of a negative radicand (which students will be encounter in Lesson 13).

Opening Activity:Coach: It is a leap for students to realize that they just need to solve the equation (no factoring needed here because it’s already factored).

Learner/Coach: What do students do with this? Do some of your more arithmetic thinkers guess-and-check (or use some other form) and determine that x=5 or x = 1? Do those students only find 1 value for x? Remind students that there could also be 2 x-intercepts.

Learner/Decision Maker: Are students comfortable working with algebraic radical expressions? You might consider a mini-activity reviewing the following examples: √ x2, √(x−3)2

Determine the x-intercept(s) of the following quadratic equation.

(x - 3)2 = 4

Activity 2: Coach: We are now forming squares instead of rectangles. Length and width are now equal. Factors have to be the same.

Questioner: We want students to think about why this method is called completing the square. Have them focus on the different squares that are being formed and how they can be strategic about determining how many unit tiles they’ll need to complete the square. What is so special about a square? What does a square look like? What is the usefulness of this method?Learner: Do students understand the power and utility of the additive inverse? Do they understand that x2 – 6x – 1 and (x+3)2 – 10 are equivalent expressions?

Coach: Talk about associative property and laws of equality that they learned in Unit 2, and how they help make this method work.

Decision Maker: Algebra tiles make more sense here because students are making the squares.

Felix likes factoring using physical models. He is fascinated by Frank and Fatima’s original problem: What are the x-intercepts of y = x2 – 6x – 1?

How many of each tile will he need to model this problem?

He creates the following model and calls his strategy completing the square.

He declares x2 – 6x – 1 = (x+3)(x+3)

Is Felix correct? Is it true that x2 – 6x – 1 = (x+3)(x+3)? Look through Felix’s work.

Can you describe his strategy? Why does Felix call this strategy completing the square?

Questioner: This should be a class discussion, as students will need guidance. Ask students what needs to be done to maintain equivalence. Decision Maker: It might be helpful to show this diagram to students.

What mistake did Felix make? (Hint: Where did those nine +1 tiles come from, and what happened to the -1?)

How would you fix his mistake?

Activity 3

Coach: The purpose of this activity is to show that the new method works, and to show the equivalence of each method.

Rewrite the expression x2 + 4x + 3 by factoring and then by completing the square using a physical model as an aid.

Factoring Completing the Square

For each of the following expressions, use completing the square to create an equivalent expression.a) x2 – 8x + 5

b) x2 + 12 + 4

Activity 4

Rewrite the following perfect square binomial expressions in standard form. PERFECT SQUARE FACTORS STANDARD FORM

( x+1 )2 (x+1)(x+1) x2+2 x+1( x+2 )2

( x+3 )2

( x+4 )2

( x+20 )2

Write about any patterns you notice.

How would you recognize a trinomial that can be written as a perfect square binomial?

Can x2 + 4x + 8 be written as a perfect square binomial?

Can x2 +10x + 25 be written as a perfect square binomial?Coach: We want students to generalize that (x + n)2 = x2 + 2n + n2 [Another way to think of it is

n=( b2 )

2

] This is extremely important help students to not only comprehend why completing the

square works, but also to make sense of the derivation of the quadratic formula in the next lesson.

Activity 5Coach: Point out to students that here you are doing and undoing. Doubling and taking half are inverses; squaring and taking the square root are inverses.Now work backwards. Rewrite the following quadratic expressions as perfect squares.

x2 + 2x + 1 x2 + 12x + 36 x2 - 6x + 9

What is a way to easily factor any perfect square quadratic expressions?

Practice:

1) x2+100 x+2500

2) x2−3 x+ 94

Activity 6

Let’s go back to Felix’s original problem of x2 – 6x – 1. Examine the model. Explain the connection between using this physical model to complete the square and the method of completing the square algebraically.

Activity 7How do you use Felix’s completing the square strategy to solve the equation?Try to make sense of Felix’s work towards a solution using the completing the square method.

Practice

Here are six quadratic equations. Two of them are already in a form that easily factor into a perfect square binomial expression (the others need completing the square). Which two are they?

Now find the zeroes of the following equations:

a) x2 + 8x + 16 = 0b) x2 - 6x + 9 = 0c) x2 - 8x + 15 = 0d) x2 + 6x - 7 = 0e) x2 - 2x – 1 = 0f) x2 - 2x = 12

ChallengeDecision Maker: It is worth a discussion with students why these are considered challenging. (b is prime,

which means b2 will be a fraction.

g) x2 - 5x – 24 = 0h) x2 - 3x + = 4

Quadratic UnitLesson 14

Student Version

Opening Activity:Determine the x-intercept(s) of the following quadratic equation.

(x - 3)2 = 4

Activity 2:

Felix likes factoring using physical models. He is fascinated by Frank and Fatima’s original problem: What are the x-intercepts of y = x2 – 6x – 1?

How many of each tile will he need to model this problem?

He creates the following model and calls his strategy completing the square.

He declares x2 – 6x – 1 = (x+3)(x+3)

Is Felix correct? Is it true that x2 – 6x – 1 = (x+3)(x+3)? Look through Felix’s work.

Can you describe his strategy? Why does Felix call this strategy completing the square? What mistake did Felix make? (Hint: Where did those nine +1 tiles come from, and what happened to the -1?)

How would you fix his mistake?

Activity 3:

Rewrite the expression x2 + 4x + 3 by factoring and then by completing the square using a physical model as an aid.

Factoring Completing the Square

For each of the following expressions, use completing the square to create an equivalent expression.c) x2 – 8x + 5 b) x2 + 12 + 4

Activity 4

Rewrite the following perfect square binomial expressions in standard form. PERFECT SQUARE FACTORS STANDARD FORM

( x+1 )2 (x+1)(x+1) x2+2 x+1( x+2 )2

( x+3 )2

( x+4 )2

( x+20 )2

Write about any patterns you notice.

How would you recognize a trinomial that can be written as a perfect square binomial?

Can x2 + 4x + 8 be written as a perfect square binomial?

Can x2 +10x + 25 be written as a perfect square binomial?

Activity 5:Now work backwards. Rewrite the following quadratic expressions as perfect squares.

x2 + 2x + 1 x2 + 12x + 36 x2 - 6x + 9

What is a way to easily factor any perfect square quadratic expressions?

Practice:

a) x2+100 x+2500 b) x2−3 x+ 94

Activity 6:Let’s go back to Felix’s original problem of x2 – 6x – 1. Examine the model. Explain the connection between using this physical model to complete the square and the method of completing the square algebraically.

Activity 7:How do you use Felix’s completing the square strategy to solve the equation?Try to make sense of Felix’s work towards a solution using the completing the square method.

Practice:Here are six quadratic equations. Two of them are already in a form that easily factor into a perfect square binomial expression (the others need completing the square). Which two are they?

Now find the zeroes of the following equations:i) x2 + 8x + 16 = 0 b) x2 - 6x + 9 = 0 c) x2 - 8x + 15 = 0

d) x2 + 6x - 7 = 0 e) x2 - 2x – 1 = 0 f) x2 - 2x = 12

Challengeg) x2 - 5x – 24 = 0 h) x2 - 3x + = 4

Lesson BookmarksQuadratic Unit

Lesson 15Teacher Guide

Lesson 15: Strategizing and demonstrating flexibility in determining the x-intercept(s) algebraically.

Opening ActivityCoach: Students might have a preferred method, but through practice should become flexible with using each method. Remind students to look at the structure of the equation to determine the best strategy. You also want to have a discussion about strategy and efficiency. You could use completing the square for all problems, but is it always the most efficient? Questioner: Which of these problems makes more sense to solve by completing the square? Which makes more sense to solve by factoring? How could you tell early in the solving process which strategy to use?Coach: It is important for students to acknowledge that either method will give the same result. This will increase their flexibility, and also proves the method works.

You have investigated different methods for solving quadratic functions. Here are several quadratic equations. Use any algebraic method you prefer to determine the x-intercept(s).

a) x2 - 4x + 15 = y b) x2 - 2x + 15 = y c) x2 + 20x – 40 = y d) x2 - 3x + 2 = y

Which method do you prefer? Is there a method that you were able to use for all four examples?

How can you recognize which method to use with a particular type of equation?

Activity 2Coach: It is important for students to acknowledge that either method will give the same result. This will increase their flexibility, and also proves the method works.

Quentin comes by and says, “You guys are wasting your time. I have a method that always works!” Fatima, Frank and Felix are flabbergasted. Quentin continues, “First you guys try solving the equation x2 + 6x + 8 = y by factoring, then by completing the square. Then, I’ll show you guys my fool-proof way.”

Solve x2 + 6x + 8 = y by factoring, then by completing the square.

Factoring Completing the Square

What do you observe?

Quentin presents them with the following equation: x=−b±√b2−4 ac2 a

He calls it the quadratic formula.

Quentin’s formula contains several variables. How will you know the value of the variables?

a = _____ b = _____ c = _____

Try using the quadratic formula to solve x2 + 6x + 8 = 0. How do we know Quentin’s formula is correct?

Activity 3Coach: The quadratic formula really is a fool-proof method, but it’s not always the most efficient, it requires students to memorize the formula and it requires a lot of calculations that require precision.

Solve each quadratic equation by factoring, completing the square and by using the quadratic formula. [Note: It may not be possible to use all three methods.]

Equation Solved by Factoring Solved by Completing the Square

Solved by using the Quadratic Formula

x2 + 2x - 3 = 0x2 - 4 = 0x2 + 6x + 9 = 03x2 + 12x - 8 = 0

Does the quadratic formula always work? Is it always the most efficient method? When might you want to use the quadratic formula? Why does Quentin call the quadratic formula a fool-proof strategy?

Coach: Once again, we want to focus students’ attention on structure and how it helps with strategizing and efficiency.

Journal writing: When is completing the square the most efficient strategy to use for solving a quadratic equation? When is the quadratic formula best? When is factoring the most efficient method to us for solving a quadratic equation?Here are several equations to help you think about these questions: y = x2 + 4x + 1 y = x2 - 25 y = 6x2 + 4x + 5y = x2 + 6x - 7

Activity 4: Derivation of the quadratic formulaDecision Maker/Provocateur: How are you going to present this to students? One suggestion is to show them the derivation for a few minutes and ask them to try to make sense of any part of it.

Drawing student’s attention to the term x=±√b2−4 ac2a

− b2a

you can ask them if any part of it looks

familiar. That can lead into a deeper discussion in which you guide them through the derivation.

Fatima was fervent. “Quentin, that is really fascinating! Can you show me where that formula is derived from and why it works?”

Quentin begins, “Recall the general form of a quadratic equation is y = ax2 + bx + c ….”

Quentin makes the following calculations. Can you make sense of what he did?

y=a x2+bx+c

a x2+bx+c=0a x2+bx=−c a x2

a+ b

ax=−c

ax2+ b

ax=−c

a x2+ ba

x+( b2 a )

2

=−ca

+( b2 a )

2

(x+ b2a )

2

=−ca

+( b2a )

2

(x+ b2a )

2

=−ca

+ b2

4 a2 (x+ b2a )

2

=4 a4 a

° −ca

+ b2

4 a2(x+ b2a )

2

=−4 ac4a2 + b2

4 a2(x+ b2a )

2

=b2−4 ac4a2

√(x+ b2 a )

2

=±√ b2−4ac4 a2 (x+ b

2a )=±√ b2−4 ac4 a2

x+ b2a

=±√b2−4 ac√4 a2 (x+ b

2a )=±√b2−4ac2a x=±√b2−4 ac

2a− b

2 a

x=±√b2−4 ac−b2a

x=−b±√b2−4ac2 a

Coach: Although not all students will be able to make sense of the derivation, it is an important experience for them to engage with. Here the derivation is broken into several parts; most importantly: completing the square, recognizing the equation for line of symmetry (and you can point out the discriminant, which comes up in the next activity), and algebraic manipulation to simplify the formula. Provocateur: How can I use the properties of equality and equivalence to help me to manipulate equations into a form that I can work with?Questioner: This is a literal equation. We explored these in Unit 2. Why do we want to get x alone? [Remind students we are ultimately looking for the x-value that represents the x-intercept(s)]

Activity 5Coach: You want to discuss with students two important features of the quadratic formula – they are

already familiar with first term,x=−b2a , which determines the axis of symmetry; the second term

contains the formula for the discriminant, b2 – 4ac. What does the discriminant allow us to predict/determine? As you know a positive discriminant yields a graph with two x-intercepts, a disciminant of zero results in a graph which has only one x-intercept (which is also the vertex), a negative discriminant indicates that there are no real x-intercepts

We call b2 – 4ac the discriminant. It plays a special role in the quadratic formula. In this activity, we will investigate its role.

The definition of the word discriminant is a characteristic that enables things, people, or classes to be distinguished from one another. Observe the graphs of the quadratic equations and their discriminants in the table below. What characteristic of quadratic equations does the discriminant help us to predict?

Quadratic Equation Sketch of the Graph Value of the discriminanty = x2 + 8x + 16

y = -6x2 + 4x + 2

Quadratic Equation Sketch of the Graph Value of the discriminanty = x2 - 6x + 9

y = 2x2

y = -x2 - 5x - 11

y = x2 + 3x + 5

Coach: Have a class discussion about why knowing if a quadratic equation has 02, 1, or 0 roots is useful.

Examine the quadratic formula: x=−b ±√b2−4ac2a

Why does the discriminant have the ability to determine

how many zeroes a quadratic function has?

Quadratic UnitLesson 15

Student Version

Opening ActivityYou have investigated different methods for solving quadratic functions. Here are several quadratic equations. Use any algebraic method you prefer to determine the x-intercept(s).b) x2 - 4x + 15 = y b) x2 - 2x + 15 = y c) x2 + 20x – 40 = y d) x2 - 3x + 2 = y

Which method do you prefer? Is there a method that you were able to use for all four examples?

How can you recognize which method to use with a particular type of equation?

Activity 2:Quentin comes by and says, “You guys are wasting your time. I have a method that always works!” Fatima, Frank and Felix are flabbergasted. Quentin continues, “First you guys try solving the equation x2 + 6x + 8 = y by factoring, then by completing the square. Then, I’ll show you guys my fool-proof way.”

Solve x2 + 6x + 8 = y by factoring, then by completing the square.

Factoring Completing the Square

What do you observe?

Quentin presents them with the following equation: x=−b±√b2−4 ac2 a

He calls it the quadratic formula.

Quentin’s formula contains several variables. How will you know the value of the variables?

a = _____ b = _____ c = _____

Try using the quadratic formula to solve x2 + 6x + 8 = 0. How do we know Quentin’s formula is correct?

Activity 3:Solve each quadratic equation by factoring, completing the square and by using the quadratic formula. [Note: It may not be possible to use all three methods.]

Equation Solved by Factoring Solved by Completing the Square

Solved by using the Quadratic Formula

x2 + 2x - 3 = 0

x2 - 4 = 0

x2 + 6x + 9 = 0

3x2 + 12x - 8 = 0

Does the quadratic formula always work? Is it always the most efficient method? When might you want to use the quadratic formula? Why does Quentin call the quadratic formula a fool-proof strategy?

Journal writing: When is completing the square the most efficient strategy to use for solving a quadratic equation? When is the quadratic formula best? When is factoring the most efficient method to us for solving a quadratic equation?Here are several equations to help you think about these questions: y = x2 + 4x + 1 y = x2 - 25 y = 6x2 + 4x + 5 y = x2 + 6x - 7

Activity 4: Derivation of the quadratic formulaFatima was fervent. “Quentin, that is really fascinating! Can you show me where that formula is derived from and why it works?”Quentin begins, “Recall the general form of a quadratic equation is y = ax2 + bx + c ….”

Quentin makes the following calculations. Can you make sense of what he did?

y=a x2+bx+c

a x2+bx+c=0a x2+bx=−c a x2

a+ b

ax=−c

ax2+ b

ax=−c

a x2+ ba

x+( b2 a )

2

=−ca

+( b2 a )

2

(x+ b2a )

2

=−ca

+( b2a )

2

(x+ b2a )

2

=−ca

+ b2

4 a2 (x+ b2a )

2

=4 a4 a

° −ca

+ b2

4 a2(x+ b2a )

2

=−4 ac4a2 + b2

4 a2(x+ b2a )

2

=b2−4 ac4a2

√(x+ b2 a )

2

=±√ b2−4ac4 a2 (x+ b

2a )=±√ b2−4 ac4 a2

x+ b2a

=±√b2−4 ac√4 a2 (x+ b

2a )=±√b2−4ac2a x=±√b2−4 ac

2a− b

2 a

x=±√b2−4 ac−b2a

x=−b±√b2−4ac2 a

Activity 5:We call b2 – 4ac the discriminant. It plays a special role in the quadratic formula. In this activity, we will investigate its role.

The definition of the word discriminant is a characteristic that enables things, people, or classes to be distinguished from one another. Observe the graphs of the quadratic equations and their discriminants in the table below. What characteristic of quadratic equations does the discriminant help us to predict?

Quadratic Equation Sketch of the Graph Value of the discriminanty = x2 + 8x + 16

y = -6x2 + 4x + 2

Quadratic Equation Sketch of the Graph Value of the discriminanty = x2 - 6x + 9

y = 2x2

y = -x2 - 5x - 11

y = x2 + 3x + 5

Examine the quadratic formula: x=−b±√b2−4 ac2 a

Why does the discriminant have the ability to determine

how many zeroes a quadratic function has?

Lesson BookmarksQuadratic Unit

Lesson 16Teacher Guide

Lesson 16: Transformations re-visited

Opening activity:Coach: Encourage students to use descriptive phrases such as shifted up/opens wider/vertical stretch (dilation)/reflection across the x-axis.

We experimented with transformations of quadratic functions earlier in the unit. What effect does the coefficient a have on the graph? What effect does the coefficient c have on the graph?

Without using your graphing calculator, describe how the graph of h(x) = -x2 + 5 transforms the parent function, f(x) = x2.

Activity 2

Coach: The purpose of this activity is not only to review transformations in the standard form, which you explored earlier in the unit, but to help students see that the y-intercept, c, and the shifting coefficient, k, are not the same thing. You will explore this in Activity 3.

Observe the graph of y = x2 + 6x – 2.

Label and describe all of its features. Connect the equation to the graph features.

Use completing the square to re-write the quadratic equation y = x2 + 6x – 2 as a perfect square binomial.

Activity 3Coach: In this activity we want students to understand that completing the square (also called vertex form, f(x) = a(x - h)2 + k ) can quickly give us the vertex (-h, k). We also want students to see that in vertex form, the constant added is not the y-intercept. What does that number being added or subtracted indicate? Students should be able to distinguish between the vertex form and the standard form of a quadratic equation.

Examine the following graphs, which are transformations of the parent function, f(x) = (x)2. Equation written in completed-square form

Graph Key features of the graph

g(x) = (x + 3)2 – 11 y-intercept:axis of symmetry:vertex:

h(x) = (x + 2)2 – 3 y-intercept:axis of symmetry:vertex:

j(x) = (x - 3)2 – 2 y-intercept:axis of symmetry:vertex:

k(x) = (x - 1)2 + 1 y-intercept:axis of symmetry:vertex:

What connections can you make between the completed-square form of the function and its

graph?

Without graphing, state the vertex for each of the following quadratic equations.a) y = (x - 5)2 + 3 b) y = 2(x - 6)2 + 10 c) y + 6 = (x + 3)2

Coach: It is an interesting idea that there are an infinite number of different equations with the same vertex, just by changing the coefficient, a. A little discussion about this might help students see the role each coefficient plays in the behavior of the function.[and similarly, an infinite number of quadratic equations that can contain the same two points, which is different from a linear equation.]d) Write a quadratic equation, other than y = (x – 4)2 – 7, that will have a vertex (4, -7).

Activity 4

Note: The window on the calculator must be adjusted to the following:

Xmin -10Xmax 10Xscl 1Ymin -20Ymax 20Yscl 1

Testing “a” in vertex form

We’ve looked at the effect of “a” in standard form, y = ax2 + bx + c.

Examine these three equations, which are written in vertex form, y = a(x - h)2 + k . What is the same about them? What is different about them? What do you think each of the graphs will look like? Make a conjecture.

y=12

( x−1 )2+5

y=2 ( x−1 )2+ 5y=−2 ( x−1 )2+ 5

Now use a graphing calculator to examine the three graphs. Was your conjecture correct? Explain how “a” affects the graph when the equation is written in vertex form.

Activity 5

Note: The window on the calculator must be adjusted to the following:

Xmin -10Xmax 10Xscl 1Ymin -20Ymax 20Yscl 1

Testing h in vertex form.

Examine these three equations. What is the same about them? What is different about them? What do you think each of the graphs will look like? Make a conjecture.

y= (x−1 )2−¿5

y= (x−4 )2−5 y= (x+4 )2−5

Now use a graphing calculator to examine the three graphs. Was your conjecture correct? Explain how “h” affects the graph when the equation is written in vertex form.

Activity 6

Note: The window on the calculator must be adjusted to the following:

Xmin -10Xmax 10Xscl 1Ymin -20Ymax 20Yscl 1

Testing k

Examine these three equations. What is the same about them? What is different about them? What do you think each of the graphs will look like? Make a conjecture.

y= (x−1 )2+10 y= (x−1 )2+0y= (x−1 )2−5

Now use a graphing calculator to examine the three graphs. Was your conjecture correct? Explain how “k” affects the graph when the equation is written in vertex form.

Activity 7:

Sketch the graphs of f ( x )=−2(x+3)2+2and g(x )=5(x+3)2+2. Describe and compare them.

Practice:

1) Thinking of the graph of the parent function, describe, in words, the transformations that would result in the graph of g ( x )=( x+4 )2−5.

2) The graph of a quadratic function f (x)=x2 has been translated 3 units to the right, vertically stretched by a factor of 4 , and moved 2 units up. Write the formula for the function that defines the transformed graph.

3) Write a formula for the function that defines the described transformations of the graph of the quadratic parent function, f (x)=x2.

3 units shift to the right Vertical shrink by a factor of 0.5 Reflection across the x-axis 4 units shift up

4) Describe the effect of g(x) = x2 – 6x – 7 on the graph of the function f(x) = x2.

Learner/Decision Maker: Students who still struggle with completing the square may use a graph or graphing calculator, but still need to be able to describe the transformations.5) The vertex of the parabola represented by f(x) = x2 – 4x + 3 has the coordinates (2, -1). Find the

coordinates of the vertex of the parabola defined by g(x) = f(x – 2). Explain how you arrived at your answer.

Quadratic UnitLesson 16

Student Version

Opening Activity:We experimented with transformations of quadratic functions earlier in the unit. What effect does the coefficient a have on the graph? What effect does the coefficient c have on the graph?

Without using your graphing calculator, describe how the graph of h(x) = -x2 + 5 transforms the parent function, f(x) = x2.

Activity 2:Observe the graph of y = x2 + 6x – 2. Label and describe all of its features. Connect the equation to the graph features.

Use competing the square to re-write the quadratic equation y = x2 + 6x – 2 as a perfect square binomial.

Activity 3:Examine the following graphs, which are transformations of the parent function, f(x) = (x)2.

Equation written in completed-square form

Graph Key features of the graph

g(x) = (x + 3)2 – 11 y-intercept:axis of symmetry:vertex:

h(x) = (x + 2)2 – 3 y-intercept:axis of symmetry:vertex:

j(x) = (x - 3)2 – 2 y-intercept:axis of symmetry:vertex:

k(x) = (x - 1)2 + 1 y-intercept:axis of symmetry:vertex:

What connections can you make between the completed-square form of the function and its graph?

Without graphing, state the vertex for each of the following quadratic equations.a) y = (x - 5)2 + 3 b) y = 2(x - 6)2 + 10 c) y + 6 = (x + 3)2

d) Write a quadratic equation, other than y = (x – 4)2 – 7, that will have a vertex (4, -7).

Activity 4:

Note: The window on the calculator must be adjusted to the following:

Xmin -10Xmax 10Xscl 1Ymin -20Ymax 20Yscl 1

Testing “a” in vertex form

We’ve looked at the effect of “a” in standard form, y = ax2 + bx + c.

Examine these three equations, which are written in vertex form, y = a(x - h)2 + k . What is the same about them? What is different about them? What do you think each of the graphs will look like? Make a conjecture.

y=12

( x−1 )2+5

y=2 ( x−1 )2+ 5y=−2 ( x−1 )2+ 5

Now use a graphing calculator to examine the three graphs. Was your conjecture correct? Explain how “a” affects the graph when the equation is written in vertex form.

Activity 5:

Note: The window on the calculator must be adjusted to the following:

Xmin -10Xmax 10Xscl 1Ymin -20Ymax 20Yscl 1

Testing h in vertex form.

Examine these three equations. What is the same about them? What is different about them? What do you think each of the graphs will look like? Make a conjecture.

y= (x−1 )2−¿5

y= (x−4 )2−5 y= (x+4 )2−5

Now use a graphing calculator to examine the three graphs. Was your conjecture correct? Explain how “h” affects the graph when the equation is written in vertex form.

Activity 6:Note: The window on the calculator must be adjusted to the following:

Xmin -10Xmax 10Xscl 1Ymin -20Ymax 20Yscl 1

Testing k

Examine these three equations. What is the same about them? What is different about them? What do you think each of the graphs will look like? Make a conjecture.

y= (x−1 )2+10 y= (x−1 )2+0y= (x−1 )2−5

Now use a graphing calculator to examine the three graphs. Was your conjecture correct? Explain how “k” affects the graph when the equation is written in vertex form.

Activity 7: Sketch the graphs of f ( x )=−2(x+3)2+2and g(x )=5(x+3)2+2. Describe and compare them.

Practice:

1) Thinking of the graph of the parent function, describe, in words, the transformations that would result in the graph of g ( x )=( x+4 )2−5.

2) The graph of a quadratic function f (x)=x2 has been translated 3 units to the right, vertically stretched by a factor of 4 , and moved 2 units up. Write the formula for the function that defines the transformed graph.

3) Write a formula for the function that defines the described transformations of the graph of the quadratic parent function, f (x)=x2.

3 units shift to the right Vertical shrink by a factor of 0.5 Reflection across the x-axis 4 units shift up

4) Describe the effect of g(x) = x2 – 6x – 7 on the graph of the function f(x) = x2.

5) The vertex of the parabola represented by f(x) = x2 – 4x + 3 has the coordinates (2, -1). Find the coordinates of the vertex of the parabola defined by g(x) = f(x – 2). Explain how you arrived at your answer.

Lesson BookmarksNote to Teacher: The purpose of problems in these next few lessons (lessons 17-21) is to see how students think about quadratic functions in contextual situations. How can they use the concepts and procedures that were developed in the first set of lessons to help them in working with the problems? You may pick and choose which ones you use. Keep in mind that they each address different concepts. Make sure you work through all of the problems you use.

Quadratic UnitLesson 17

Teacher Guide

Phoebe Small’s Rocket Problem

Note to the Teacher: The purpose of this task is to look at:

how students use their understanding of quadratics in a new and unusual situation the contextual significance of the vertex and other features of a quadratic function

Learner: How do students approach the problem? Are they using the graph only, or the table of values? Encourage them to use both.

Coach: This problem can be solved in two ways: using the calculator to perform a quadratic regression, or by reasoning to find the second difference in a table. Either way, a discussion about the minimum number of data points necessary to work through the problems would be helpful.

In groups of two you will work through the following problem.

Phoebe Small is out Sunday driving in her spaceship. As she approaches Mars, she changes her mind, decides that she does not wish to visit that planet, and fires her retro-rocket. The spaceship slows down, and if all goes well, stops for an instant then starts pulling away. While the rocket motor is firing, Phoebe’s distance, from the surface of Mars depends, by a quadratic function, on the number of minutes since she started firing the rocket.

A. Phoebe finds at times 1,2, and 3 minutes, her distances from Mars are 425, 356 and 293 kilometers, respectively. Find the equation expressing distance in terms of time.

B. Find the y intercept and tell what this number represents in the real world.C. Does your model tell you that Phoebe crashed into the surface of Mars, just touches the surface, or

pulls away? Explain. (Hint: Find the vertex. What does the y value tell you?)D. Sketch the graph, labeling the axes and the vertex, the y intercept and the x intercepts.E. How far was Phoebe from Mars when she fired her retro-rocket?F. Based on your answers to the above questions, in what domain do you this quadratic function will

give reasonable values for time? Therefore, what is the range for the distance?

[Decision Maker: Here is a graph of the Rocket problem for your use. You may want to support your students by showing part or all of this.]

Quadratic UnitLesson 17

Student Version

Phoebe Small’s Rocket Problem

In groups of two you will work through the following problem.

Phoebe Small is out Sunday driving in her spaceship. As she approaches Mars, she changes her mind, decides that she does not wish to visit that planet, and fires her retro-rocket. The spaceship slows down, and if all goes well, stops for an instant then starts pulling away. While the rocket motor is firing, Phoebe’s distance, from the surface of Mars depends, by a quadratic function, on the number of minutes since she started firing the rocket.

G. Phoebe finds at times 1,2, and 3 minutes, her distances from Mars are 425, 356 and 293 kilometers, respectively. Find the equation expressing distance in terms of time.

H. Find the y intercept and tell what this number represents in the real world.

I. Does your model tell you that Phoebe crashed into the surface of Mars, just touches the surface, or pulls away? Explain. (Hint: Find the vertex. What does the y value tell you?)

J. Sketch the graph, labeling the axes and the vertex, the y intercept and the x intercepts.

K. How far was Phoebe from Mars when she fired her retro-rocket?

L. Based on your answers to the above questions, in what domain do you this quadratic function will give reasonable values for time? Therefore, what is the range for the distance?

Lesson BookmarksQuadratic Unit

Lesson 18Teacher Guide

Gateway Arch Problem

Coach: Be aware that the value of c, which should be zero, will be nonzero with these numbers due to round-off error. You need to explain this to your students. When they write it up, they should just set c to zero. Also, be aware that students may approach this in two different ways.

1. By substituting 170 for y and solving for the two values of x, then seeing if the difference between the x values is more than 40 meters.

2. The other way is to find the y value for which the x is 20 meters less than half of the axis of symmetry (162/2 = 81). This corresponds to x = 61 meters. If y is greater than the 170 meters, the plane can get through, if not, it crashes.

Activity: Work with a partner to reason through this problem.

You and your friend decide to visit the city of St. Louis. You encounter an amazing structure, called the Gateway Arch, that makes you think of McDonalds. The two of you talk about its tremendous height and then you find out it has an elevator that takes you to the top of the structure.

Your friend says, “I wonder how high this is?!” The two of you reason through the situation: This looks like a parabola, so if we can find three ordered pairs representing the width and the

height, we can use that information to find its height. We can make a drawing to help determine out the height.

o We measured the width of the arch at the base and found it to be 162 meters. o As we’re waiting for the elevator, we measured the height of the platform and found the

the arch at this point was 4.55 meters above the ground and 1 meter horizontally from the base.

o We were not able to get any other measurements.

A. Can you help us find the height? (Remember, you need three points to make a quadratic equation.)B. We noticed there is going to be an airplane show and are told that a plane with a wing span of 40

meters will fly through the arch 170 meters above the ground. Do you think this will work? Show your work and explain your answer.

Quadratic UnitLesson 18

Student Guide

Gateway Arch Problem Activity: Work with a partner to reason through this problem.

You and your friend decide to visit the city of St. Louis. You encounter an amazing structure, called the Gateway Arch, that makes you think of McDonalds. The two of you talk about its tremendous height and then you find out it has an elevator that takes you to the top of the structure.

Your friend says, “I wonder how high this is?!” The two of you reason through the situation: This looks like a parabola, so if we can find three ordered pairs representing the width and the

height, we can use that information to find its height. We can make a drawing to help determine out the height.

o We measured the width of the arch at the base and found it to be 162 meters. o As we’re waiting for the elevator, we measured the height of the platform and found the

the arch at this point was 4.55 meters above the ground and 1 meter horizontally from the base.

o We were not able to get any other measurements.

C. Can you help us find the height? (Remember, you need three points to make a quadratic equation.)

D. We noticed there is going to be an airplane show and are told that a plane with a wing span of 40 meters will fly through the arch 170 meters above the ground. Do you think this will work? Show your work and explain your answer.

Lesson BookmarksQuadratic Unit

Lesson 19Teacher Guide

An Old Pizza Menu

Note to the Teacher: This lesson should not be done quickly. Students should work in groups of two and be ready to present their findings at various points in the lesson.

Coach: Consider the real-world significance of the parts of the equation and its solution. For example, diameter = 0, the price is 45 cents. This can be explained in real world terms as overhead. Students should find both a positive and negative values of x. We reject the negative value because the idea of a negative diameter is invalid. The discriminant will be negative and therefore indicate there are no x intercepts. In real terms this means that there are no diameters for which the price of pizza would be zero.

An old menu at a Pizza Parlor from the 1950s lists the following prices for plain cheese pizzas:

Small (8” diameter)……………..$0.85Medium (10” diameter)…………$1.15Larger (13” diameter)……...……$1.75

Assume that the price is a quadratic function of the diameter.

1. Therefore which variable is independent and which is dependent? 2. Using your calculator, plot the points and find the equation using quadratic regression. (Note: The

equation is easier to work with if you eliminate the decimal point and you put money in pennies.) Explain the process you go through to do this.

3. If the pizza parlor had made 20 inch pizzas, using your calculator, find the cost. Explain how you did it.4. Suppose that the menu had listed a “Colossal” pizza costing $5.95. What would be its diameter? How

many values did you find? Why do you want to reject one of them?5. Suppose the cost is $3.25. Using the table find the value of the diameter. What do you notice?6. To find this value using the quadratic formula requires some manipulation of the equation.

Step 1: Substitute 325 for the y value in your equation.

Step 2: To use the quadratic formula, y must equal 0. To do this, you must subtract 325 from both sides. Now you are left with 0 = ax2 + bx +c and you can apply the formula. Explain your solution in real world terms.

7. What is the price when the diameter is 0? Does this make sense? How would you interpret this situation in the real world?

8. Find the discriminant. What does it tell you? What does this mean in real world terms?

Quadratic UnitLesson 19

Student VersionAn Old Pizza Menu

An old menu at a Pizza Parlor from the 1950s lists the following prices for plain cheese pizzas:

Small (8” diameter)……………..$0.85Medium (10” diameter)…………$1.15Larger (13” diameter)……...……$1.75

Assume that the price is a quadratic function of the diameter.

9. Therefore which variable is independent and which is dependent?

10. Using your calculator, plot the points and find the equation using quadratic regression. (Note: The equation is easier to work with if you eliminate the decimal point and you put money in pennies.) Explain the process you go through to do this.

11. If the pizza parlor had made 20 inch pizzas, using your calculator, find the cost. Explain how you did it.

12. Suppose that the menu had listed a “Colossal” pizza costing $5.95. What would be its diameter? How many values did you find? Why do you want to reject one of them?

13. Suppose the cost is $3.25. Using the table find the value of the diameter. What do you notice?

14. To find this value using the quadratic formula requires some manipulation of the equation.

Step 1: Substitute 325 for the y value in your equation.

Step 2: To use the quadratic formula, y must equal 0. To do this, you must subtract 325 from both sides. Now you are left with 0 = ax2 + bx +c and you can apply the formula. Explain your solution in real world terms.

15. What is the price when the diameter is 0? Does this make sense? How would you interpret this situation in the real world?

16. Find the discriminant. What does it tell you? What does this mean in real world terms?

Lesson BookmarksQuadratic Unit

Lesson 20Teacher Guide

Throwing Paper (with Permission) in the Classroom

Note to teacher: The following activity will enable students to develop data about projectile motion. This data will be used to make a quadratic mathematical model that will then be used to make some predictions.

Coach: Once again the real-world significance of a point is considered. One of the x intercepts is meaningless, since the first point in the trajectory is the y intercept and not the origin. (The thrower is releasing the paper at some point above the ground.)

Introduction: What happens when you throw a baseball or a football? Sketch this motion.

Physicists know that when an object is thrown, the motion follows a curve that can be represented by a quadratic equation. This is called parabolic motion.

Activity: You will be organized in groups of four with the following task. You’re going to throw a crumpled-up piece of paper from a given point to an area marked on the floor. Your task is to develop three points with x and y coordinates representing the horizontal and vertical distance respectively for the paper at different points in the flight. You will have the use of a tape measure to find these coordinates. Two of your points can be the starting and ending points. Use the colored tape that is seven feet from the ground to help you find the third point.

Each student will have a task.1. The thrower2. The intermediate point sighter3. The end point sighter4. The measurer and note taker

Each group should perform the throws such that you have three successful throws into the marked area. Choose the throw for which you are most certain about your sighting data.

A. Using your calculator and the three points you’ve just located, find the equation that describes this parabolic curve.

B. What was the highest point that your paper ball reached? Explain how you got this.C. Sketch the graph the graph, by finding the vertex, y-intercept and its symmetric point and the x

intercepts.D. What do the x and y intercepts represent in real world terms?

E. Looking at your equation, what is a realistic domain and range for this activity?Quadratic UnitLesson 20

Student Version

Throwing Paper (with Permission) in the Classroom

Introduction: What happens when you throw a baseball or a football? Sketch this motion.

Physicists know that when an object is thrown, the motion follows a curve that can be represented by a quadratic equation. This is called parabolic motion.

Activity: You will be organized in groups of four with the following task. You’re going to throw a crumpled-up piece of paper from a given point to an area marked on the floor. Your task is to develop three points with x and y coordinates representing the horizontal and vertical distance respectively for the paper at different points in the flight. You will have the use of a tape measure to find these coordinates. Two of your points can be the starting and ending points. Use the colored tape that is seven feet from the ground to help you find the third point.

Each student will have a task.1. The thrower2. The intermediate point sighter3. The end point sighter4. The measurer and note taker

Each group should perform the throws such that you have three successful throws into the marked area. Choose the throw for which you are most certain about your sighting data.

A. Using your calculator and the three points you’ve just located, find the equation that describes this parabolic curve.

B. What was the highest point that your paper ball reached? Explain how you got this.

C. Sketch the graph the graph, by finding the vertex, y-intercept and its symmetric point and the x intercepts.

D. What do the x and y intercepts represent in real world terms?

E. Looking at your equation, what is a realistic domain and range for this activity?

Lesson BookmarksQuadratic Unit

Lesson 21Teacher Guide

Lesson 21: Quadratic Word Problems

Coach: Students should spend 1-2 days working on these exercises from the EngageNY module. Pick and choose which ones you think would helpful to students. We do not consider them to be pure problems because they are less open-ended and more routine. Still, since students will be taking standardized tests, they may benefit from practice with these types of exercises.

1) The measure of a side of a square is x units. A new square is formed with each side 6 units longer than the original square’s side. What is the area of the new square?

2) In the accompanying diagram, the width of the inner rectangle is represented by x−3 and its length by x+3. The width of the outer rectangle is represented by 3 x+4 and its length by 3 x−4.

Express the area of the pink shaded region as a polynomial in terms of x.

3) Two mathematicians are neighbors. Each owns a separate rectangular plot of land that shares a boundary and have the same dimensions. They agree that each has an area of 2 x2+3 x+1 square units. One mathematician sells his plot to the other. The other wants to put a fence around the perimeter of his new combined plot of land. How many linear units of fencing will he need? Write your answer as an expression in x .

Note: This question has two correct approaches and two different correct solutions. Can you find them both?

4) Lord Byron is designing a set of square garden plots so some peasant families in his kingdom can grow vegetables. The minimum size for a plot recommended for vegetable gardening is at least 2 m on each side. Lord Byron has enough space around the castle to make bigger plots. He decides that each side

3 x−4

x+3

x−3

will be the minimum (2 m) plus an additional x m. There are 12 families in the kingdom who are interested in growing vegetables in the gardens. If the total area available for the gardens is 300 sq. m, what are the dimensions of each garden?

5) Mischief is a toy poodle that competes with her trainer in the agility course. Within the course, Mischief must leap through a hoop. Mischief’s jump can be modeled by the equation h = -16t2 + 12t, where h is the height of the leap in feet and t is the time since the leap, in seconds. At what values of t does Mischief start and end the jump?

6) The length of a rectangle is 5 in. more than twice a number. The width is 4 in. less than the same number. If the area of the rectangle is 15 in2, find the unknown number.

7) The length of a rectangle 4cm more than 3 times its width. If the area of the rectangle is 15 cm2, find the width.

8) A garden measuring 12m by 16m is to have a pedestrian pathway that is w meters wide installed all the way around it, increasing the total area to 285 sq. m. What is the width, w , of the pathway?

9) Find two consecutive odd integers whose product is 99. (Note: There are two different pairs of consecutive odd integers and only an algebraic solution will be accepted.)

10) A plot of land for sale has a width of xft. and a length that is 8 ft. less than its width. A farmer will only purchase the land if it measures 240 sq. ft. What value for xwill cause the farmer to purchase the land?

Quadratic UnitLesson 21

Student Version

1) The measure of a side of a square is x units. A new square is formed with each side 6 units longer than the original square’s side. What is the area of the new square?

2) In the accompanying diagram, the width of the inner rectangle is represented by x−3 and its length by x+3. The width of the outer rectangle is represented by 3 x+4 and its length by 3 x−4.

Express the area of the pink shaded region as a polynomial in terms of x.

3 x−4

x+3

x−3

3) Two mathematicians are neighbors. Each owns a separate rectangular plot of land that shares a boundary and have the same dimensions. They agree that each has an area of 2 x2+3 x+1 square units. One mathematician sells his plot to the other. The other wants to put a fence around the perimeter of his new combined plot of land. How many linear units of fencing will he need? Write your answer as an expression in x .

Note: This question has two correct approaches and two different correct solutions. Can you find them both?

4) Lord Byron is designing a set of square garden plots so some peasant families in his kingdom can grow vegetables. The minimum size for a plot recommended for vegetable gardening is at least 2 m on each side. Lord Byron has enough space around the castle to make bigger plots. He decides that each side will be the minimum (2 m) plus an additional x m. There are 12 families in the kingdom who are interested in growing vegetables in the gardens. If the total area available for the gardens is 300 sq. m, what are the dimensions of each garden?

5) Mischief is a toy poodle that competes with her trainer in the agility course. Within the course, Mischief must leap through a hoop. Mischief’s jump can be modeled by the equation h = -16t2 + 12t, where h is the height of the leap in feet and t is the time since the leap, in seconds. At what values of t does Mischief start and end the jump?

6) The length of a rectangle is 5 in. more than twice a number. The width is 4 in. less than the same number. If the area of the rectangle is 15 in2, find the unknown number.

7) The length of a rectangle 4cm more than 3 times its width. If the area of the rectangle is 15 cm2, find the width.

8) A garden measuring 12m by 16m is to have a pedestrian pathway that is w meters wide installed all the way around it, increasing the total area to 285 sq. m. What is the width, w , of the pathway?

9) Find two consecutive odd integers whose product is 99. (Note: There are two different pairs of consecutive odd integers and only an algebraic solution will be accepted.)

10) A plot of land for sale has a width of xft. and a length that is 8 ft. less than its width. A farmer will only purchase the land if it measures 240 sq. ft. What value for xwill cause the farmer to purchase the land?

Lesson BookmarksQuadratic Unit

Performance TaskTeacher Version

Note to Teacher: This performance task was contributed by Sapphira Hendrix and Jason Warren, Math teachers at Brookly Preparatory HS.Coach: This performance task allows students to use what they’ve learned about quadratics and quadratic regressions in order to model an Angry Birds game.

Lesson 22: Angry Birds'Take aim and shoot those naughty pigs!'

The pigs have stolen the birds’ eggs. That makes them angry, very angry. They take aim and launch themselves towards the pigs to get their revenge and reclaim their babies. Based on the classic angry birds game you will be guiding the birds to ensure that their aim is good. Each successive level gets more difficult as the information makes the calculation more challenging. Before attempting this game, you should know and understand the basic properties of the quadratic function, findings zeros, the vertex and it might be helpful to be able to solve simultaneous equations.

You have been given the job of designing your own Angry Bird level. Your Angry Bird level will need to show the birds, pigs, and any obstacles that are in the path. The flight path of the birds should model a parabola. The equations for each flight path must be written in both VERTEX and STANDARD form. You will need to find a reasonable domain and range and be sure to label your x and y axis. You may work alone or in a group of 2.

Follow the checklist below to make sure your performance task meets the criteria.

Groups:● 1 person – 2 flight paths● 2 people – 3 flight paths

Checklist ______ Step 1: Determine the three points that each bird/flight path will need to go through. You will be selecting coordinate points of the pigs to help you with this step. For each bird you will determine the following (completed on sheets attached):● The maximum height● The axis of symmetry● Distance traveled● Reasonable domain and range

______ Step 2: Write equations for each bird/path that represents the parabola in both VERTEX and STANDARD form. All work must be done neatly and with steps clearly shown.

______ Step 3: Create a graph representing each flight path and clearly identify three points along the path. Each graph should be drawn on the same sheet of graph paper.

______ Step 4: An explanation about how the structures will fall when hit along the flight path.Explanation should be clear, accurate and include details.Flight Path Considerations:

Coach: Questions 1-2 allows students to gather three points in order to find the quadratic regression1. Where is your slingshot located? Answer should be given as a coordinate pair.2. What three coordinate points does your Angry Bird need to hit?

Coach: Question 3 gives student practice using the “completing the square” method.Decision maker/Learner: Some equations will result in decimals. Do you want to students to round to the nearest tenth? Whole number? What difference does it make when rounded? How does the parabola change? Will the pig still be hit?3. Create a quadratic equation that can represent this function.

a. Vertex Formb. Standard Form

Coach: Questions 4-5 gives students practice finding the vertex. There are different ways they could have obtain this (table, graph, formula)4. What is the maximum height that the bird travels? How do you know?5. What is the axis of symmetry of the flight path? How do you know?

Coach: Questions 6-7 allows the students to think of a reasonable domain and range in context of the problem.6. How much distance did your Angry Bird travel in total?7. Write a reasonable domain and range for your Angry Bird. Be sure to write it in correct format and with

inequalities.

Coach: Questions 8 allows students to understand the transformations of parabolas. Do they need to move their parabola (left, right or up, down)? Do they need to make their parabola wider or narrower in order to hit the pig? Note: all equations will have a negative “a” value.8. How will your angry bird hit the pig and knock over any obstacles? Answer in complete sentences with

detail and reasoning.

Decision Maker: If students have access to technology, will you allow them to use desmos.com/calculator? The online calculator allows students to input their three points and experiment with a, b, and c (a x2+bx+c as sliders or a, h, and k (a ( x−h )2+k) as sliders.

Quadratic UnitPerformance TaskStudent Version

Angry Birds'Take aim and shoot those naughty pigs!'

The pigs have stolen the birds’ eggs. That makes them angry, very angry. They take aim and launch themselves towards the pigs to get their revenge and reclaim their babies. Based on the classic angry birds game you will be guiding the birds to ensure that their aim is good. Each successive level gets more difficult as the information makes the calculation more challenging. Before attempting this game, you should know and understand the basic properties of the quadratic function, findings zeros, the vertex and it might be helpful to be able to solve simultaneous equations.

You have been given the job of designing your own Angry Bird level. Your Angry Bird level will need to show the birds, pigs, and any obstacles that are in the path. The flight path of the birds should model a parabola. The equations for each flight path must be written in both VERTEX and STANDARD form. You will need to find a reasonable domain and range and be sure to label your x and y axis. You may work alone or in a group of 2.

Follow the checklist below to make sure your performance task meets the criteria.

Groups:● 1 person – 2 flight paths● 2 people – 3 flight paths

Checklist ______ Step 1: Determine the three points that each bird/flight path will need to go through. You will be selecting coordinate points of the pigs to help you with this step. For each bird you will determine the following (completed on sheets attached):● The maximum height● The axis of symmetry● Distance traveled● Reasonable domain and range

______ Step 2: Write equations for each bird/path that represents the parabola in both VERTEX and STANDARD form. All work must be done neatly and with steps clearly shown.

______ Step 3: Create a graph representing each flight path and clearly identify three points along the path. Each graph should be drawn on the same sheet of graph paper.

______ Step 4: An explanation about how the structures will fall when hit along the flight path.Explanation should be clear, accurate and include details.

Flight Path Considerations:

1. Where is your slingshot located? Answer should be given as a coordinate pair.

2. What three coordinate points does your Angry Bird need to hit?

3. Create a quadratic equation that can represent this function.a. Vertex Form

b. Standard Form

4. What is the maximum height that the bird travels? How do you know?

5. What is the axis of symmetry of the flight path? How do you know?

6. How much distance did your Angry Bird travel in total?

7. Write a reasonable domain and range for your Angry Bird. Be sure to write it in correct format and with inequalities.

8. How will your angry bird hit the pig and knock over any obstacles? Answer in complete sentences with detail and reasoning.

Lesson BookmarksQuadratic Unit

Lesson 23Teacher Guide

Lesson 23: Square Root Function

Opening ActivityNow you are going to use your understanding of quadratic functions to create other functions.In this activity you are going to think about two functions one that you know well y = x2 and one that you are going to get to know, y = √ x. Based on your knowledge how would you compare these two functions?

Use your graphing calculator to create a data table for the function y¿ x2 and y=√x for a variety of x-values. Use both negative and positive numbers and round decimal answers to the nearest hundredth.

Observe both columns from above. What do you notice about the values in the two y-columns?Why are all y-values for y=x2positive?Why do all negative x-values produce an error for y=√x?What is the domain of y=x2 and y=√x? What is the range of y=x2 and y=√x?

Compare the domain and range of y=x2and y=√x.What is the result if we take the square root of x2? Try making a third column in the chart to see if you can

come up with a rule for √ x2.

Coach: This purpose of this question is to help students understand the need for the ± when taking the square root of a variable expression.

Activity 2a. Now you are going to graph theses two functions. You know what y = x2 will look like. Draw a sketch of what you think y=√x will look like.b. Now place both of these graphs on your graphing calculator. Compare and contrast the two graphs. How are they similar and how are they different?

x y=x2 y=√x420

−2−4

Activity 3What affect does a have on f (x)=a∗√x?You are going to graph the following four functions on your calculator? Before you graph them make a prediction about the affect of a on the parent function f (x)=√x.

f (x)=√xf ( x )=2∗√ xf (x)=5∗√ x

f ( x )=0.5∗√xf ( x )=−√x

f ( x )=−0.5∗√x

a. Prediction:

b. What did happen?

What can you say about the affect of a on the parent function f (x)=√x?

Activity 4At an amusement park, there is a ride called The Centre. The ride is a cylindrical room that spins as the riders stand along the wall. As the ride reaches maximum speed, riders are pinned against the wall and are unable to move. The model that represents the speed necessary to hold the riders against the wall is given by the function s(r )=5.05√r, where s=¿ required speed of the ride (in meters per second) andr=¿ the radius (in meters) of the ride. In a competing ride called The Spinner, a car spins around a center post. The measurements in the table below show the relationship between the radius (r) of the spin, in meters, and the speed (s) of the car, in m/sec.

r(meters) s (meters per second)0 01 7.79422 11.0233 13.54 15.5885 17.428

Due to limited space at the carnival, the maximum spin radius of rides is 4 meters. Assume that the spin radius of both rides is exactly 4 meters. If riders prefer a faster spinning experience, which ride should they choose? Show how you arrived at your answer.

Quadratic UnitLesson 23

Student Version

Opening ActivityNow you are going to use your understanding of quadratic functions to create other functions.In this activity you are going to think about two functions one that you know well y = x2 and one that you are going to get to know, y = √ x. Based on your knowledge how would you compare these two functions?Use your graphing calculator to create a data table for the function y¿ x2 and y=√x for a variety of x-values. Use both negative and positive numbers and round decimal answers to the nearest hundredth.

Observe both columns from above. What do you notice about the values in the two y-columns?

Why are all y-values for y=x2positive?

Why do all negative x-values produce an error for y=√x?

What is the domain of y=x2 and y=√x?

What is the range of y=x2 and y=√x?

Compare the domain and range of y=x2and y=√x.

What is the result if we take the square root of x2? Try making a third column in the chart to see if you can come up with a rule for √ x2.

x y=x2 y=√x420

−2−4

Activity 2:a. Now you are going to graph these two functions. You know what y = x2 will look like. Draw a sketch of what you think y=√x will look like.

b. Now place both of these graphs on your graphing calculator. Compare and contrast the two graphs. How are they similar and how are they different?

Activity 3:What affect does a have on f (x)=a∗√x?You are going to graph the following four functions on your calculator? Before you graph them make a prediction about the affect of a on the parent function f (x)=√x.

f (x)=√xf ( x )=2∗√ xf (x)=5∗√ x

f ( x )=0.5∗√xf ( x )=−√x

f ( x )=−0.5∗√x

a. Prediction:

b. What did happen?

c. What can you say about the effect of a on the parent function f (x)=√x?

Activity 4:At an amusement park, there is a ride called The Centre. The ride is a cylindrical room that spins as the riders stand along the wall. As the ride reaches maximum speed, riders are pinned against the wall and are unable to move. The model that represents the speed necessary to hold the riders against the wall is given by the function s(r )=5.05√r, where s=¿ required speed of the ride (in meters per second) andr=¿ the radius (in meters) of the ride. In a competing ride called The Spinner, a car spins around a center post. The measurements in the table below show the relationship between the radius (r) of the spin, in meters, and the speed (s) of the car, in m/sec.

r(meters) s (meters per second)0 01 7.79422 11.0233 13.54 15.5885 17.428

Due to limited space at the carnival, the maximum spin radius of rides is 4 meters. Assume that the spin radius of both rides is exactly 4 meters. If riders prefer a faster spinning experience, which ride should they choose? Show how you arrived at your answer.

Lesson BookmarksQuadratic Unit

Lesson 24Teacher Guide

Lesson 24: Comparing f(x)=x2 and y=x3

In this lesson you are going to think about two functions one that you know well y= x2 and one that that you are going to get to know y= x3. Based on your knowledge and your own intuition how would you compare these two functions?

Use your graphing calculator to create a data table for the functions y=x2 and y=x3 for a variety of y-values.

Observe both columns from above. What do you notice about the values in the two y-columns?Why are all y-values for y=x2 positive but there are values for y=x3 that are both positive and negative?What is the domain of and y=x3? What is the range of y=x2 and y=x3?

Activity 2a. Now you are going to graph these two functions. You know what y=x2 will look like. Draw a sketch of what you think y=x3 will look like?b. Now place both of these graphs on your graphing calculator. Compare and contrast the two graphs. How are they similar and how are they different?

Activity 3What affect does a have on y = ax3 ?You are going to graph the following four functions on your calculator? Before you graph them make a prediction about the affect of a on the parent function y=x3.

f ( x )=x3

f ( x )=2 x3

f ( x )=0.5∗x3

f ( x )=−0.5∗x3

a. Prediction:

b. What did happen?

c. What can you say about the affect of a on the parent function f ( x )=x3?

x y=x2 y=x3

420-4-2

Activity 4What effect will c have on the function f ( x )=x3+c?

What effect did c have on the function f ( x )=x2? Do you think the c will have the same effect on the functionf ( x )=x3? Create an experiment to test this out and be ready to present this to your classmates.

What conclusion can you make about the effect of c on f ( x )=x3?

Quadratic UnitLesson 24

Student Version

In this lesson you are going to think about two functions one that you know well y= x2 and one that that you are going to get to know y= x3. Based on your knowledge and your own intuition how would you compare these two functions?Use your graphing calculator to create a data table for the functions y=x2 and y=x3 for a variety of y-values.

Observe both columns from above. What do you notice about the values in the two y-columns?

Why are all y-values for y=x2 positive but there are values for y=x3 that are both positive and negative?

What is the domain of and y=x3?

What is the range of y=x2 and y=x3?

x y=x2 y=x3

420-4-2

Activity 2:a. Now you are going to graph these two functions. You know what y=x2 will look like. Draw a sketch of what you think y=x3 will look like?

b. Now place both of these graphs on your graphing calculator. Compare and contrast the two graphs. How are they similar and how are they different?

Activity 3:What affect does a have on y = ax3 ?You are going to graph the following four functions on your calculator? Before you graph them make a prediction about the affect of a on the parent function y=x3.

f ( x )=x3

f ( x )=2 x3

f ( x )=0.5∗x3

f ( x )=−0.5∗x3

a. Prediction:

b. What did happen?

c. What can you say about the effect of a on the parent function f ( x )=x3?

Activity 4:What effect will c have on the function f ( x )=x3+c?

What effect did c have on the function f ( x )=x2? Do you think the c will have the same effect on the functionf ( x )=x3? Create an experiment to test this out and be ready to present this to your classmates.

What conclusion can you make about the effect of c on f ( x )=x3?

Lesson BookmarksQuadratic Unit

Lesson 25Teacher Guide

Lesson 25: Comparing y = x3 and y= 3√x

Opening Activity:In your own words describe, as best you can the meaning o each of these functions y=x3 and y= 3√x. What is happening? How is x affecting y?

Activity 2:Now we are going to create a table for these two functions. What observations can you make from the tables

Observe both columns from above. What do you notice about the values in the two y-columns?What is the domain for each function?What is the range for each function?How would you compare the growth reflected in both functions?

Activity 3:a. Draw a sketch of each function based on your tables.

b. Then place each of the functions on your graphing calculator and compare your sketch with the precise graph? What observations can you make about the graphs?

Activity 4:What effect will a and c have on the parent function y= 3√x? What do you hypothesize the effect will be?Create an experiment to test out your hypothesis.

What results did you get?

x y=x3 y= 3√x. -4-3-2-101234

Quadratic UnitLesson 25

Student Version

Opening Activity:In your own words describe, as best you can the meaning o each of these functions y=x3 and y= 3√x. What is happening? How is x affecting y?

Activity 2:Now we are going to create a table for these two functions. What observations can you make from the tables

Observe both columns from above. What do you notice about the values in the two y-columns?

What is the domain for each function?

What is the range for each function?

How would you compare the growth reflected in both functions?

x y=x3 y= 3√x. -4-3-2-101234

Activity 3:a. Draw a sketch of each function based on your tables.

b. Then place each of the functions on your graphing calculator and compare your sketch with the precise graph? What observations can you make about the graphs?

Activity 4:What effect will a and c have on the parent function y= 3√x?

What do you hypothesize the effect will be?

Create an experiment to test out your hypothesis.

What results did you get?

Unit 3: Quadratics Table of Contents

Lesson Bookmarks

Appendix A: Use these lessons if you do not want to use algebra tiles Lesson 7: Doing and Undoing – Factoring Quadratics

Teacher GuideLesson Objectives: How do we factor a trinomial as a beginning to understanding how to solve for the x-intercepts?Why can we call the multiplying two binomials and factoring a trinomial doing and undoing?

Opening Activity:

In the last lesson left off with this dilemma: how do we solve y = x2 – 6x – 1?

We are going to come back to this one, but first we’ll examine an equation that can be solved by looking at graph or table.

This is Quentin. He likes solving quadratic equations because he figured out how to do it on his own.

Examine his work. Try to make sense of it.

A. What do you observe? What questions do you have?

____________________________________________________________________________

____________________________________________________________________________

Activity 2:

Let’s examine each part of Quentin’s work.

B. What did Quentin do here? Why is okay for him to do this? Learner: Do students connect this to what they did with x-intercepts and zeros in previous lessons?

___________________________________________________________________________

___________________________________________________________________________

C. Quentin replaced x2 + 4x + 3 with (x + 1) (x + 3). How can he do that?Note to teacher: we want students to understand that they are equivalent expressions.

___________________________________________________________________________

___________________________________________________________________________

D. What observations can you make about x2 + 4x + 3 and (x + 1) (x + 3)?

Note to teacher: What do your students say? You might list any observations on a parking lot to be revisited later.

To understand fully why x2 + 4x + 3 and (x + 1) (x + 3) are equivalent, we are going to go through a few activities. You are going to begin with the following.

Activity 3: Evaluate the following expressions.Coach: Have a conversation about the process. Can the distributive property be used to multiply any pair of polynomials? Explain.

Note: If students struggle with this last one, it’s okay. We will be introducing a new model for helping students to think about the distributive property and multiplication.

E. 2(x+ 3)

F. 3(x2 + 2x + 1)

G. x(x + 3)

H. x(x2 + 2x + 1)

I. (x + 3) (x + 2)

Activity 4: Coach: The big idea of this part of the lesson is that the distributive property and factoring “undo” each other. You want students to see that in Quentin’s work. Make sure students are able to articulate that (x + 1)(x + 3) is an equivalent expression to x2 + 4x + 3. Make it clear to your students that the next few lessons will be spent learning different methods for factoring so that they may answer the question – can the x-intercepts of a quadratic equation be determined algebraically?

Let’s re-examine Quentin’s work, which he called factoring.

Look at Step 3.

Step 1: y = x2 + 4x + 3Step 2: 0 = x2 + 4x + 3Step 3: 0 = (x+1)(x+3)

J. How might an understanding of the distributive property help us to make sense of factoring?

Activity 5:K. Evaluate the following:

(x + 3) (x + 4) =

(x – 3) (x – 2) =

(x + 4) (x – 3) =

(x – 3) (x – 4) =

Look at your four results.

L. You do see a connection between the original factors and the trinomial answer?

M. If you were given the trinomial and where asked to find the factors, how might you do it?

Learner: How are your students developing their metacognition? Are they able to articulate their thinking/reasoning?

N. Try these two on your own:

x2 + 9x + 20 x2 + 2x - 15

Journal Write: Why do we say the process of multiplying two binomials (which gives us a trinomial) and factoring a trinomial (which gives us two binomials) is described as “doing” and “undoing”?

Unit 3: Quadratics Table of Contents

Lesson 8: Doing and Undoing – More Ideas in Factoring

Teacher Guide

Opening Activity: Quentin and his classmate Quincy were given these two problems. They are struggling with how to factor them.

x2 – 7x + 12 x2 + 7x + 12

A. Help them by giving them suggestions to show them what to do.Learner: What do your students do here?

Activity 2: B. Create a trinomial that you can factor but you think might be challenging to the rest of the class.

Learner: Which of your students do and undo? If students are struggling with creating a trinomial then they aren’t understanding the do and undo process.

Activity 3:

Qunicy tried solving the multiplying (x-3)(x + 3) and got the answer x2 – 3x + 3x – 9, which he knew was x2 – 9.

His answer was not a trinomial, but a binomial.

C. Was this a special case, or can you find another example?

D. Can you make a conjecture about when a binomial multiplied by a binomial produces a trinomial?

Note to teacher: This should lead to a discussion about the difference of two squares.

E. Based on what you just thought about, what are the factors of x2 – 64?

F. Why would we describe this process as the “difference between two squares”?Activity 4:

Note to teacher: We are introducing the idea of GCF.

Quentin saw this problem and factored it the following way:

G. Can you explain mathematically what he did?

H. What permitted Quentin to remove the 2 and put it outside the parenthesis?

I. How could you check to see that the original expression is equivalent to the new factored expression? Show your work.

2x2 + 12x + 162(x2 + 6x + 8)

2(x + 4) (x + 2)

Activity 4:

Now that you’ve learned how to factor, evaluate the following:

J. 3x2 – 15x + 18

K. 2x2 – 32

L. -2x2 – 14x – 24

Closing Activity/Journal Writing:

M. You have now experienced multiple ideas related to factoring, including

a) doing and undoingb) difference of two squaresc) using GCF

discuss, in your own words, what it means to factor and when you might use these different ideas.

APPENDIX B Lesson Bookmarks

Calculator Toolkit Topic: Entering a List of Data

Habit: Each time I turn on my calculator I have to clear its memory so that the problems other students were working on won’t affect my work.

Press to open the Statistics Menu

Highlight EDIT.

Press to select Edit…

In the L1 column you will enter the x-values and in the L2 column you will enter the y-values.

After typing in a number press to move to the next value in the row of the table.

Use the right arrow key to move to the next column to enter data.

Special Notes:If you want to delete a whole column of data, highlight the column title (L1, for example) and press .

1

STAT

ENTER

DELETE