5e 2 day lesson galvez kelley final

7/30/2019 5E 2 Day Lesson Galvez Kelley Final

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Step 1/2: Explorations inMathematics and Science Teaching

Title of Lesson: Functions and Basketball

UFTeach Students Names: Gabriela Galvez and Peter Kelley

Teaching Date and Time: 3/15 and 3/18; 12:50 and 1:45

Length of Lesson: two 50 minute lessonsGrade / Topic: 11th Grade Algebra II Functions and Inverses

Sources of the Lesson:Mathematics Formative Assessment: 75 Practical Strategies for Linking Assessment,Instruction and Learningby Page D. Keeley, Cheryl Rose Tobey; self-generated ideas.

Appropriateness for High School Students: This lesson is part of a PBI unit. This unit attempts to place whatstudents already know about functions and how to manipulate them into a real world situation. By doing so,students can see that what they learned about functions is not just a skill needed in the classroom. Showingthem that the skills they learn are used outside of the classroom will hopefully spark their interest in other areasof mathematics.

This lesson uses two formative assessments that address the needs of high school students. The Hot Seat

assessment utilizes the high school students tendency to socialize. During the assessment, selected studentswill be at the front of the class providing answers to questions asked by the teacher. Once the students at thefront respond, the rest of the class can agree or disagree with the answer, giving reasons to support theirdecision. Having this formative assessment helps keep students, specifically students who like to talk,conversations focused on the lesson at hand.

Concepts

Functions are a topic in mathematics that very often is approached from a very procedural standpoint. In otherwords, students learn the basic procedure of what they can do with functions and they leave school with asurface level understanding of functions. What should be done in a study of functions is for students tounderstand what the function they are dealing with actually means. They should be able to see how a change in

one variable affects change in the other one(s). Manipulating functions is a great skill to have; but manipulatingthem while understanding the meaning behind it can provide countless benefits.

This lesson will attempt to show students that functions can be used to show real life relationships between twomeasurable quantities. Moreover, the students will engage in a study of how they can write two functions usingthe same variables.

Florida State Standards (NGSSS):

MA.912.A.2.11 Solve problems involving functions and their inverses.

MA.912.A.2.6 Identify and graph common functions

Performance Objectives

1. Students will be able toidentify the independent and dependent variables in a function.2. Students will be able to evaluate functions given different values3. Students will be able to find the inverse of a function.4. Students will be able to graph the original function and the inverse of the function.5. Students will be able to tell if one function is the inverse of another algebraically.


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Materials List and Student Handouts

One basketball Copies of Florida Functions worksheet (one for each student) Copies of the Final Question Day 1 (one for each student) Copies of Graphing Functions (one for each student)

Advance Preparations

Teacher will need a power point for the engage

Safety

Students will be passing a basketball in the beginning of class. This could be a danger to some students

5E Lesson Templates

ENGAGEMENT Time:_____

What the Teacher Will Do Teacher Directions and Probing

Questions

Student Responses and Potential

Misconceptions

Begin class by playing a video ofthe Florida Gator Basketball teamas the class is walking in. Thisvideo will be shown to introducethe theme of the lesson and thedriving question, How well doesthe Florida Gator Basketball team

play together?

Teacher will give an introduction tothe class

Good afternoon everyone.My name is (teachers name, ifapplicable).How many of you like basketball?Or have played before?

I have. I have played before. I

have seen it before.

Okay great, so what were going todo today is use basketball as a wayto study functions. Were also

going to look at a new topic, theinverses of functions.Teacher will gather what studentsknow initially about functions andtheir meaning.

Teacher will write Functions onwhite board and make a list of all

So what can you all tell me aboutfunctions?What are they?How have you used them?Can they be used outside of theclassroom?

They are equations. Theyresomething where we plug in anumber. Weve graphed thembefore by plugging in x values andfinding y values. Functions haveend behavior.[Functions show a relationshipbetween two variables.]


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student responses to probingquestions.

Okay those are all great answers!What we want to do today is usewhat you have done with functionsand take it a little further. We willlook at what functions actuallymean and how we, in fact, can usethem outside of the classroom.

I think they can be used outside ofthe classroom. You can factorthem and find the zeroes.

Teacher will write Points, assists,turnovers, and fouls on the whiteboard.

Teacher will use a basketball to getthe students to explain the different

terms that will help them studyfunctions. Teacher will as studentsto demonstrate how each of thestatistics to the right is calculatedduring a basketball game.

So you all watched this video of theGator basketball team. Were

going to be talking about them, andusing functions to do so. First, weneed to define a few terms to helpus in this discussion.

So these four terms are all statisticsfor players in a basketball game;keeping track of these tracks howgood or bad a player is doing.Who can tell/show me what points

are?Assists?Turnovers?Fouls?

[Point is whenever you make a

basketAssist is whenever you pass to aplayer and he scores.Turnover is when you give the ballto the other teamFoul is whenever you hit a playeron the opposing team.]

Great! Now that we all have someworking knowledge of the terms inbasketball, lets talk about theGators.

What a lot of people are sayingabout the Gators this year is thatthey play well as a team.

Where as some teams have onegreat player and everyone else doesnothing, the Gators all depend on


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and feed off of one anothers play.So what we can do is see how wellthey actually play together usingfunctions.

Teacher will put up slide withpictures of Patric Young andScottie Wilbekin on it.

Here are a few of the key playerson the Gators that well be lookingat today, just to give us somenames.

EXPLORATION Time: _________

What the Teacher Will Do Teacher Directions and

Probing/Eliciting Questions

Student Responses and

Misconceptions

Pass out Florida Functionsworksheet and give brief directions

Okay everyone, Im passing out asheet that will help guide us

through the next part of ourdiscussion.This worksheet will get us startedfor today. It is dealing withevaluating functions and whatfunctions actually are. Todaysfocus is going to be primarily onthat, what functions are.

If we all look real quick at theworksheet, the directions are there

for each problem. It says toevaluate a couple of values, and thelast two I would like you to discusswith your groupmates

There are directions on the page foreveryone. You can work with yourgroup.

Please note the last question onworksheet is a discussion question.

I would like you all to talk aboutthe question with your groupbecause we are going to, as a class,talk about our answers.

While students are working, theteacher will need to move 3-5chairs/desks (depending on thenumber of students) to the front of

What are you finding about therelationship between theindependent and dependentvariables?


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the room.After teacher has moved chairs, heshould go around the room askingprobing questions

How did you evaluate number 4?Is it okay for us to plug in a numberfor the P(a)?

EXPLANATION Time: _________

What the Teacher Will Do Teacher Directions and

Probing/Eliciting Questions

Student Responses and

Misconceptions

Get students attention to move onto the next part of the lesson.Teacher will introduce Hot Seatactivity.

Okay everyone, were going to talkabout what we found in thisworksheet.As you can see, there are 3(possibly more) chairs here at thefront. Im going to call on a few ofyou and you are going to come to

the Hot Seat and were going to askyou some questions about the work.

Teacher will call on 3 (more ifnecessary) students.

Darrell, John, and Sam will be ourfirst contestants in the Hot Seat.

Note: this activity is designed to

spark discussion about the answersthe students arrived at and whetherthey are correct or incorrect.

Okay, I will ask a question to oneof you, and, when you answer it,the rest of the class will eitheragree or disagree. Once we all

agree on an answer, you can sitdown and someone else will takeyour place.

Teacher will ask first question.

Throughout this activity, the

teacher will be checking forunderstanding based on the studentresponses.

Darrell, what did your group saywas the function for Patric Youngspoints as a function of ScottieWilbekins assists?

Okay, are there any disputes to thatanswer?

Alright Darrell, that is correct. Youcan sit and Jennifer will take yourplace.

Sam, why is that the answer?Would you like to come to theWhite board and show us how youdid it?

[We said P(a) = 3a 2. ]

[No.]

[Because you can find the slope ofthe line between those two points.If you treat the assists like the xvalues and the points like y valuesyou can do y / x to find the


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Okay, are there any argumentsfrom the class?

Okay, David, can you explain tome how you got that? Would youlike to come to the board and showus?

Did anyone else do #1 like this?James?

Great explanation. That is exactlywhat were looking for here. Weneed to see the difference betweenwith points as a function of assistsand assists as a function of points.

Okay Sam, you can sit down. Dowe all see how the first answer isthe correct one for #1? So whatdoes the answer in #2 actually tell

us?

Excellent! Jordan, why dont youtake Sams spot on the Hot Seat.

Okay, John. Explain what you didfor numbers 2,3, and 4.

And why did you do that?

slope. Then from there we canmake a y = mx + b formula andsolve for b using the x and y fromassists and points.]

We said it was P(a) = a/3+ 2/3.

Yes. We said that y/x was 1/3because 7-4 / 19-10. And fromthere we solved for b and found itwas 2/3.

[Yes, you asked for this to bepoints as a function of assists. Thatmeans points depend on assists, sopoints is the dependent variable andassists is the independent variable.So assists are like our x values andpoints are like our y values.]

[That tells us assists as a functionof points]

[Well, for 2 and 3 I just plugged in5 and 10 and got 13 and 28.]

[Since p(a) is a function describing


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Are there any disagreements in the

class? Does anyone have anotherway of finding 2-4?

points as a function of assists I canplug values for the assists in andget a prediction of points.]

[NO]

Great, okay John you can sit down.And Laruen can you take his placeon the Hot Seat please?

Okay Jennifer, what did your grouphave to say about numbers 5 and 6?

Okay, thats a good answer. Doesanyone in the class have a differentopinion?Jennifer, why did your group saythat?

Are there any other explanations?

Does everyone see what she issaying?

Okay, Hot Seat contestants, youcan all sit down. Were going to

discuss the last problem all as agroup.

The last question asks us to write afunction showing assists as afunction ofpoints. That is reversefrom the initial question.

First what are my independent anddependent variables going to be in

[For the first one we said it wasindependent and the second onewas dependent]

[Well, since you asked for points asa function of assists, the number ofpoints depend on the number ofassists. So the points are dependentand the assists are independent. ]

(possible other explanations, but allshould follow along those samelines)

[yes]

[in this one assists are dependent


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this last one? Maura?

Great, so what does that make my xand my y values? Bryan?

Excellent. Can anyone walk methrough how we can find thefunction? Yes, Caitlin?

Great work, and b will end upbeing 2/3. So our function thatdescribes assists as a function ofpoints is a = p/3 + 2/3.And that is what David got for hisanswer.

and points are dependent]

[Points are going to be like the x

values and assists will be the y]

[So we can still do y/x with thechange in assists divided by thechange in points. That will give us3 divided by 9 which is 1/3. So theslope of this functions graph willbe 1/3. Using y = x/3 + b we cansolve for b.]

Teacher will write the twofunctions up on the board P(a) = 3a 2 and a(P) = P/3 + 2/3.

Now were going to look atsomething a little different. Lookat these two functions. Do youthink it is possible to comparethem?

Why not?

[no]

Because one of them is a = and theother is p =[Because one of them is points as afunction of assists and the other isassists as a function of points.]

Teacher switches variables in the

second function to have P(a) = a/3+ 2/3 and P(a) = 3a 2

Teacher writes p-1(a)

Exactly, so the new thing we aredoing today is called the inverses offunctions.Looking at these two functions,what if for the second one we got,

we simply switched a and p?

Now we have two functions that wecan compare. Why can wecompare them?

Yes! The function that we changedthe variable in is called the inverseof the first function. We can writethat as p-1(a).

Because they are both p =[Because they are both points as afunction of assists]


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Before we move on, we need todiscuss what was just done. Justbecause we switched the variablesP(a) and a to find the inverse

doesnt mean that P(a) and a areequivalent variables.

With that in mind, Gabriel, are theinverse and the original functionequal to one another?

Right, it is hard to compare twofunctions that do not have the sameindependent and dependentvariables. So what flipping the

variables allows us to do is tocompare them. But the functionand its inverse are not the same.

Yes[No, we are just writing twoequations that are related to oneanother and in the same terms.]

Teacher will illustrate the stepswhile narrating.

So to summarize this problem wejust did: we found a function thatwas p as a function of a. Tofind its inverse, we write a as afunction of p and then just switchthe variables.

(teacher will wait so students cansee how the process works)

What questions do you have?

Yes. After you have done the restof the process.

All you do at the end is switch thevariables?

Teacher will illustrate on the whiteboard while he speaks.

Here, lets look at another example.This example will show us a moreprocedural way to find the inverseof a function.

Say I have the function s(t) = 4t 8And I am asked to find the inverseof it. First, I need to realize that iss as a function of t. that meansmy first step is to write t as afunction of s.

So first I will write this as s = 4t 8. James what should I do next?

[You should add 8 to the otherside.]


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Why?

Excellent, Then I will add 8 to the

other side.

Then I can divide both sides by 4.

So I am left with s/4 + 2 = t, whichis t as a function of s. Finally, Iswitch the variables. So I am leftwith t/4 +2 = s

So the inverse of our s(t), is s-1(t) =t/4 +2.

Are there any questions?

[Because we want to write t =something. So we need to moveeverything to the other side.]

Teacher will work through someexamples with the class aboutinverses to be sure they a

Alright now on your papers, tell methe inverse of this next function:F(x) = 9x +12

(teacher will allow for work time)

Okay I need three volunteers tocome to the board and show us how

they got the inverse of this f(x).How about Matthew, Sara, andKasey.(Teacher will allow students timeto write their work on the board).

Excellent work! x/9 x/3 = f-1(x)is correct. What questions do youhave? Did anyone get anythingdifferent?

(students writing at board)

Teacher will do an inverse problemusing a squared term. Lets do one more together. Thisone will be a little more difficult.Heres the function r(t) = t2 4.Find the inverse. And I will ask forthree more volunteers in a fewminutes.

(Teacher will allow for work time)


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Okay stop working. And I needthree volunteers. How aboutDaniel, Jarvis, and Julie? Come upto the board and show us how youfound the inverse.

(Teacher will allow for work time)

Very good. The inverse for this r(t)would be r-1(t) = t+4. The onlything that is missing from this oneis the + or - . I know we havetalked about that in previouslessons, so lets not forget to usethat.

SECOND ENGAGEMENT Time:_____The teacher will use this activity tocheck for students conceptualunderstanding

Okay everyone, were going to anactivity similar to the game ThreeCorners.

The way this will work is I will ask aquestion and give 3 or 4 answerchoices. Those answer choices willcorrespond to one of the corners ofthe room. Your job is to go to theanswer corner that you think is

correct.Are there any questions?Okay, first question: do a functionand its inverse express the samevariable as a function of the other?The answer choices are yes, no, orunsure.

(Allow students to choose theiranswer)

Okay, lets hear from some of thepeople in each corner. John, what didyou say?

Alright, lets hear from someone in

I said no because when we weredoing the work together, we endeup having points as a function ofassists and assists as a function ofpoints. And those arent the samrepresentation.

I said yes because at the end of th


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the yes corner. Alan?

Very good Alan. John, do you see

what Alan is saying? We switch thevariables at the end which makes thefunctions expressed the same.

process we switched the variableand that was the inverse. So theinverse and the original wereexpressed the same way.

Teacher will write the function r(t)= 3t 22 on the board

Okay heres the second question: Inthis function, which variable is theindependent variable? The cornersare R t and unsure.

(teacher will allow time for thestudents to pick their answer)

Okay lets hear from someone in theR corner. Jessica?

Okay and someone from the t corner?Danielle?

I said R because it is standing all itself in the equation.

I said t because r is being shown a function of t. So r depends on tand that means t is independent.

ELABORATION Time: _________

What the Teacher Will Do Probing/Eliciting Questions Student Responses and

Misconceptions

Teacher will start off by going backto the summary of inversefunctions.

Teacher will hand out the GraphingFunctions worksheets. [one perstudent]

Okay so last time we hadsummarized what an inversefunction is. Today we will begraphing the functions and theirinverses.

We will start off by going back tothe first function we found. Cananyone tell me what the firstfunction was? Can anyone tell mewhat the inverse function was?

Correct! So now you will begraphing these functions on acoordinate plane.

You will have 10 minutes to workwith your groups. What questionsdo you have before we begin?

[p(a)= 3a-2 and p-1(a)= a/3 + 2/3]

No questions.


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*Teacher will circulate theclassroom as students work

together in their groups on theassignment.After the ten minutes are up teachercalls attention to the class.Teacher goes over the GraphingFunctions worksheets with theclass.

Okay so now we will be going overthe worksheets you have justworked on.

For the first question, Jessica willyou please come up to the boardand show us what your groupgraphed. (student will come up tothe white board and graph their

function)

For the first graph what didJessicas group label their axes?Does anyone else have anythingdifferent?

Now for the second graph, James

will you come up to the board andshow us how your group graphedthe function. (Student will come upto the white board and graph theirfunction)

Now for this graph what didJamess group label their axes?Did anyone else get somethingdifferent?

What do you notice about the twographs and their functions?Yes they are positive, what else doyou notice about the graphs?So we see that both graphs havedifferent slopes and also havedifferent axes. So what can you tellme about the graphs?

Can someone explain how the

[The x-axis are the assists and they-axis are the points.]No.

[The x-axis are the points and they-axis are the assists]

No.

They are both positive.[They have different slopes. Theiraxes are also different][The graphs are inverses of eachother]

[In the first graph, the x-axis are the


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they are both same, meaning thatthey are inverses of each other.

Now I will give you anotherexample lets say that p(a)=2a+4 is

our original function.Can someone give me the inversefunction?

Will someone tell me, what are thesteps I need to take to make surethat these two functions areinverses?

Good job! Ok so now that we knowhow to verify whether or not thefunctions are inverses of eachother, we will now go back to thetwo graphs we started with. Wefind that the original function hasan inverse which is the secondgraph.

We proved by working backwardswith the inverse function that it didturn out to be the inverse of theoriginal equation.

Now looking at the graph what elsedo you notice between these twofunctions?

Yes they both do have different

slopes. We notice that both havedifferent slopes but they are similarin a way. The second graph is a flipof the first graph over the y=x line.With this meaning that the inverseis a linear function of the originalfunction flipped over the y=x line.

Once we find the inverse graph, wenow have to test whether or not the

[p-1(a)= a/2 2 is the inverse of thefunction]

[First you have to add the two tothe other side. Then you multiplyby 2 on both sides, canceling the on the right side. Distribute the 2 to(p-1(a)+2). We get p(a)=2a+4 whichis the same as the original functionmeaning that both of the functions

are inverses.]

They both have different slopes.


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inverse is a function.

Does anyone have any idea of howwe could test to see if the inversefunction is a function?

Close, we could use the verticalline test for the original function,but for the inverse function we usethe horizontal line test.

To use the horizontal line test wedraw a horizontal line across theinverse function and if only onepoint touches the line then theinverse is a function.

Looking back at the inverse graphis the graph a function and why?

Good job! So now we know how toverify an inverse and also how tograph its inverse and see whetherit is or isnt a function.

The vertical line test.

[Yes, because only one pointtouches the line which makes itpass the horizontal line test.]

EVALUATION Time: _________

What the Teacher Will Do Assessment Student Responses

Disburse Evaluation worksheet.

[1 per student]

*Remind students that they are towork individually on thisassignment.*Collect Evaluations.

Now that we are at the end of the

lesson, I would like to see how wellI taught you today. [Reassure thestudents that this is not a test just anassessment.]


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Florida FunctionsName:_____________________________Date:____________________________Directions: Answer all of the following questions. Show your work.

1) In the table below we are given Patric Youngs points and Scottie Wilbekins assists

for two games. Based on this information, find a function that shows Pats points asa LINEAR FUNCTION of Scotties assists .


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2) Using the function you just found predict how many points Patric will score if

Scottie has 5 assists?

3) Predict how many points Patric will score if Scottie has 10 assists?

4) Predict how many assists Scottie will have if Patric scores 16 points?

Circle the correct term.5) In this example, Scotties assists (a) are the independent/dependent variable.

6) In this example, Patrics points (P) are the independent/dependent variable.

7) DISCUSSION: Write a function that shows Scottie Wilbekins assists as a function of

Patric Youngs points?

Points

Assists

19 7

10 4


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Graphing FunctionsName:_____________________________Date:____________________________Directions: Graph the functions indicated below.

1) P(a)=3a-2

2) A(p)= p/3 + 2/3

1) What do you notice from these two graphs?

2) Is the second graph also a function? Explain.


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Evaluation Day 2Name:_____________________________Date:____________________________

1) Find the inverse of the following function:

f(x)= 5x-6

2) Graph the function and its inverse.


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3) Explain why the graph of the inverse is or isnt a function.

5e 2 day lesson galvez kelley final

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