Unit 4 – Transitions to Functions

Some information on this unit and the suggestions we are making:Functions are a unique set of mathematical relationships that students will study well into their college mathematics course work. Mathematically functions come with their own notation, which will be developed and utilized in this unit.

We have looked closely at module 3 and found the things we feel you should work on with your students. In the coming years this would have been integrated throughout the units thus far. We have outlined below the main outcomes that you will work towards in the unit. Your coach will also work closely with you.

The big questions in this unit: What’s the essence of a function? Why do functions serve as meaningful models? How do we explain the transformation of functions?

What do you want your students to understand (both enduring understandings and topical understandings), be able to do and think about at the end of this unit?

A nuanced understanding of a function How functions explain real world situations How to talk functions in relation to domain and range, increasing and decreasing,

What are the experiences you want your students to have that will lead them to attaining these understandings and skills?

Development of the meaning of a function (leading to understanding why the vertical line test is useful). Define domain and range in relation to individual functions Evaluating functions Comparing y = 2x versus f(x) = 2x.. Why do we use function notation? Discern when a function can be used and when it can’t Different notations for domain and range Graph functions where domain values are specific and restricted – The performance task is an example of this. Interpreting a function’s graph Transforming a function (e.g. absolute value)

1

Standards Addressed:F-IF.A.1 – Understand that a function from one 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 (𝑥) denotes the output of f corresponding to the input 𝑥. The graph of f is the graph of the equation 𝑦 = (𝑥).

F-IF.A.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.B.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.B.5 – Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

(Note to Teacher: In 8th grade students experienced a very narrow definition of what makes a function a function through the vertical line test. It is fine if they happen to bring it up, but in this unit we’re going deeper. It would not make sense to share it until it naturally comes out of the student experience. When you feel there is a good understanding of the essence of a function you could ask, “Why would a vertical make for a simple way to test whether or not a relationship is a function?”)

2

Lesson 1: Correspondence & Functions

(Note to Teacher: The purpose of the first lesson is to give students experiences that lead them to an intuitive understanding of how functions behave. At the end of this lesson (Activity 4) students will be asked to give language to describe what makes a function unique. This will be their beginning definition which will grow to be more comprehensive as the have more experiences in the subsequent lessons.)

3

Opening Activity: The Frustrating Vending Machine

(Note to Teacher: the opening activity is written so students begin to develop the meaning of a relationship that is not a function and one that can be a function. We will build on this activity throughout this lesson leading to the initial defining of a function.)

The Barclay Center is testing out the use of vending machines to sell Brooklyn Nets hats and jerserys. The vending machine has two buttons. One for hats and one for jerseys.

The Merchandise Office has received several complaints from frustrated fans about the machine. They’ve organized the complaints below:

Fans Wanted Received

Fan 1

Fan 2

Fan 3

Fan 4

4

Jersey

Hat

To try to make sense of this situation the merchandise manager made the following diagram:

Activity 1 Continued:

What do you think is wrong with the machine?

How would you fix the machine so these fans get what they want? Create a diagram similar to the merchandise manager’s to illustrate your new vending machine.

5

Hat

Jersey

MerchandiseButtons

Activity 2(Note to Teacher: The purpose of this activity is to further develop the concept of a function. Some of the examples below were taken from p. 103 in the teacher’s materials for module 3.)

Consider all the examples below and determine if they are like the frustrating vending machine or the new vending machines that were created.a. The assignment of the members of a football team to jersey numbers.

b. The relationship between time and a cooling cup of coffee.

c. The assignment of U.S. citizens to Social Security Numbers.

d. The assignment of zip codes to residences.

e. The relationship between the quantity of oranges purchased and cost of the oranges.

f. The assignment of residences to zip codes.

g. The assignment of a father to all his biological children.

h. The assignment of children to their biological father.

i. The relationship between Lebron James and the players he defends during a game.

6

Activity 3:Instructions: Sort the images in two groups. Make one group the representations that are like the frustrating vending machine and one group that are like the newly designed vending machines.After sorting these representations into groups explain why each representation belongs in the group.

7

(1)

DEFG

ABC

1 32 53 74 35 56 7

(3)(2)

567

1234

(4) (5)

1 32 52 84 125 17

(7)

{(3,8) , (4,7) , (5,6) , (6,4)}

(8)

{(1,2) , (3,3) , (3,5) , (4,7)}

(9)

(6)

(10)

(11)

Activity 4: You are going to now give a definition for all the examples that are like the new vending machine. What makes them unique?(Note to Teacher: Help students put language to the beginning definition of a function.)

8

Lesson 2: Opening Activity:Looking at the sort we worked with yesterdayFind the non-functions then make a change so the non-function is now a function and be able to explain how your change has made it a function.

(Note to Teacher: This can be a lot of fun as there are many ways to turn the non-functions into functions. For example 5 they can cut the circle in half along any horizontal line. For example 9 you could eliminate the base line of the triangle, any two sides, or unfold the triangle, which makes it a line. The main goal of this activity is for you to see what students at this point understand a function to be.)

Actvity2:Now that all the representations are functions find the domain and range for each of the functions in the sort.

(Note to Teacher: If the language is new, help students understand a beginning meaning to domain and range. Here are a few things to be aware of: For example 6 (the parabola) the students can write it as Range: [the lowest point, ∞) or Range: lowest point < y < ∞) or Range: any real number > the lowest point (see pg. 25 in the Fall 2013 Sample Questions on EngageNY (http://www.engageny.org/sites/default/files/resource/attachments/sample_items_fall_2013.pdf))

Activity 3:(Note to Teacher: The big step in this activity is moving to an understanding of the complexity of recognizing if an equation is a function. This activity is intended to get this conversation started. It is not the intent of this activity to formalize an exact answer to how we determine if a relationship is a function based on the equation. Thus, this question should be revisited multiple times throughout the rest of the year as they have experiences with different types of functions.)

So far we have considered what a function looks like when represented in different ways. Now we want to specifically identify the clues for determining from a table, graph, and equation when a relationship is a function.

How would you determine whether a relationship is a function from a table?

How would you determine whether a relationship is a function from a graph?(Note to Teacher: This could lead to a discussion of the vertical line test and how a procedure arises from conceptual understanding.)How would you determine whether a relationship is a function from an equation?

9

(Note to Teacher: What question(s) should a student ask himself or herself if they are going to determine whether a given equation is a function?)

To help you better answer the question about an equation, consider the following:

Which do you think are functions? Why?y = | x |y = 3X

y = sin(x)y = x2

y = 3√ x−2y = 9/xy > x

(Note to Teacher: Some of these you’ll want to see if the student can determine if it’s a function without graphing it, but others they will need a graph and will need your help in some way. One recommendation we would make for you helping them is to use a website like Desmos (https://www.desmos.com/) to simplify this process. This site has many useful features. For instance you could create and share the Desmos files with the students if they have laptops or iPads they can use during the class. Otherwise you’ll need to be prepared to equip your students to do this on a graphing calculator.)

After looking at each equation comment on this question: What might help you to recognize if a given equation is a function?

Activity 4:Note to Teacher: It is now time to introduce function notation. The end of the previous activity should serve as a catalyst of why we need function notation. As a result of Activity 4, we want to students to leave with these three understandings: (a) by using f(x) rather than y the reader is told immediately that what you will now be thinking about is a function (b) y = 3x + 4 is the same as f(x) = 3x + 4 so if you graph them or look at their table they will be identical (c) using f(x) allows for a more elegant way to evaluate the relationship at a particular value. That is to say, f(3) naturally moves one to substitute 3 for x, resulting in the value of the function at 3. This could be illustrated in the following way:

If we want to know the value of the relationship at 3with y = 3x + 4 we’d have to express the followingx=3y=3x+4y=3(3)+4y=13

but with f(x) = 3x + 4 we can writef(x) = 3x + 4f(3) = 3(3) + 4f(3) = 13

Why might we want a different notation for function? To help you in your thinking look at: y = 3x + 4 and f(x) = 3x + 4

What do they each mean? (Note to Teacher: In most classes you will need to help students to read the functions and help them make sense of its meaning.)

Create a table and graph for each one. What do you observe?

10

Activity 5:Now let’s see if you can evaluate functions.

1. Let f ( x )=6x−3 , and let g ( x )=0.5 (4 )x. Find the value of each function for the given input.

f (0 ) g (0 )

f (−10 ) g (−1 )

f (2 ) g (2 )

f ( 53 ) g( 12 ) (optional)

f (0.01 ) g (2 )+g(1)

f (6 )−f (2 )

11

Lesson 3: Understanding the Symbols and Notation of FunctionsOpening Activity:

(Note to Teacher: The Opening Activity is an opportunity for you to hear how your students make sense of a new notation for defining a function. Can they identify what is the domain and range of the function? Also, during the course of the discussion be sure to ask them why they think this is a function. Once you think students have made sufficient sense of this have them go to Activity 2. This activity is taken from p. 101 teacher lesson guide from Module 3.)

Part 1: Please read the description below and put in your own words what is being explained to you.

Given f as described below:f :{wholenumbers }→{wholenumbers }Assign each whole number to its largest place value digit.

For example, f (4 )=4, f (14)=4, and f (194)=9.

Part 2:

a. What is the domain and range of f ?

b. What is f (257)?

c. What is f (0)?

d. What isf (999)?

e. Find a value of x that makes the equation f (x)=7 a true statement.

Activity 3:(Note to Teacher: You want to see how students interpret information when it is given to them in a new way. Do they talk about the domain and range? Does anyone rewrite the function as f(x) = x2? Does anyone ask about real numbers and what would happen if it changed to integers or whole numbers?)

Part !:Please read the description below and put in your own words what is being given to you.

The function is defined as follows:

12

Let f : X→Y be the function such that x↦ x2, where X is the set of all real numbers.

Part 2:Now that you’ve made sense of how this function works consider the following:

f. What are the values of f(0), f(3), f(.02), and f(√3)?g. If f(x) = 9, what is the value of x?h. Create a question for your neighbor to answer.

Activity 4:Part 1:Please read the description below and put in your own words what is being given to you.

Let X={1 ,2,3 ,4 } and Y={2,4,6,8,16 }. f and g are defined below.

f : X→Y g :X→Y

f={ (1,2 ) , (2,4 ) , (3,6 ) , (4,8 )} g={(1 ,2 ) , (2 ,4 ) , (3 ,8 ) , (4,16 ) }

Part 2:

a. Compare f and g with as much detail as possible.

(Note to Teacher: Some things you hope students might see (1) both f and g are functions (2) the domain and range is limited (finite) (3) the first function is f(x) = 2x with a finite domain g(x) = 2x.)

b. Graph each function on a coordinate plane. Should the points be connected for each function? Why?(Note to Teacher: This is important for students to think about when the points are and are not connected.)

Activity 5:You’ve looked at defining functions in several different ways. Make up your own function using whatever notation you’d like and pose three questions for your classmates to answer.(Note to Teacher: Have the students put their function definitions on a poster. Use a gallery walk as away for students to see one another’s functions and have them answer the questions on the poster.)

13

Lesson 4: Performance TaskThe Situation: Scientists on a space observatory were discussing two recently discovered planets, each

inhabited by different types of beings. The planets’ populations grow in different ways than ours.

The scientists need assistance in understanding the population growths on each planet and would like you use

the information about each planet and help them answer some questions.

Planet 1: Xerses

Xerses: The total population started as 2,520,000. After the 1st month the total population was 2,600,000. After the 2nd month the total population was 2,680,000.

Planet 2: Orodes

The total population of Orodes is modeled by the following function:

f(x) = 3( x+1)−12

, where x represents number of months since the planet was discovered.

The scientist know that Xerses will have more inhabitants in the beginning months, but they want to know when the number of inhabitants on Orodes will surpass the number of inhabitants on Xerses. Justify your answer with two representations.

(Note to Teacher: Here are some things you might bring into a discussion at some point: How would you represent Xerses as a function? What is the domain and range for each function? How does the context affect the domain and thus the range? Since graphing these functions is difficult, because of the size of the numbers a question to consider would be, how would you best scale the axes?)

14

Lesson 5: Max’s Mints1) Max purchased a box of green tea mints. The nutrition label on the box stated that a serving of three mints contains a total of 10 Calories. A full box of mints has 180 Calories.

How many mints are in the box?

Is the relationship between the mints and calories a function? Why? Support your thinking with two representations.

Creating your own function2) Now that you have looked at different examples of functions in the real world create your function situation (or problem). Answer the following in your write-up

Why is this a function situation? How would you represent this situation algebraically, and graphically? What is the domain and range of this function?

(Note to teacher: This activity is as important as any as you can learn about the depth and nuance of your students’ understanding of functions. If they are able to create a situation and talk about it that is sign of some real understanding.)

15

Lesson 6-8(Note to Teacher: Some of your students will come in with a different memories of absolute value but others will not remember that they worked with it in 8th grade. In this opening activity you want to find out what students know and simultaneously develop a beginning definition of absolute value.)

Opening Activity:(Note to Teacher: Place the word ‘absolute value’ and/or ‘|-3|’ on the board. Ask the students to comment on what these two things mean to them if anything. The purpose in doing this is to find out what understandings, misconceptions, or shallow definition they possess. We recommend using what students say as the beginning definition for absolute value. We want students to leave this discussion or experience you create with the sense that absolute value represents the distance from the origin and not an operation that somehow drops the sign or where they’ve mistaken the absolute value to be the opposite.)

Activity 2:(Note to Teacher: Students need to understand that if no domain is specified that it is assumed that the domain is all real numbers and the range is a subset of all real numbers. This point is made explicitly on pg. 106 in Module 3.)

1. Create the table and then the graph of f(x) = |x|

2. Observe the table and the graph, so that you can describe what is happening in this particular function.

3< How does the absolute value function compare to a linear function?

(Note to Teacher: Here you have an opportunity to discuss increasing and decreasing functions or portion of functions. You want students to recognize that linear function is either increasing or decreasing, while and absolute value function is both increasing and decreasing for parts of the domain. So you may want to ask the students, “Over which part of the domain increasing? Decreasing? How would we represent this?”)

4. What do you think we would need to do to f(x) = |x| to move the graph to another location on the coordinate plane? For instance, what would if we wanted to move it up or down or left or right.

(Note to Teacher: This is just to get students thinking about transformations. Last year students spent a month on geometric transformations. The students looked at translations, reflections, dilations, and rotations. There will be many opportunities in the next two lessons to connect their thinking about geometric transformations to the transformation of functions.)

16

Activity 3:(Note to Teacher: This activity should not be done by hand. Students need to use a piece of technology to create the graphs (calculator or Desmos). Thus, this activity could be done either individually, in small groups, or as a whole class. If it’s done as a whole class be sure to gather new conjectures before changing the graph using a new multiplier.)

1) You are going to be building off the function you just graphed by multiplying it by different values. How do you think it will transform the graph to multiply |x| by a number >1? Or a number between 0 and 1? Or a number less than 0?

You will eventually have all these on one graph, but add them one at a time. This will give you the chance to talk about the effect of the multiplier on the function.f(x) = |x|g(x) = 2*|x|h(x) = 4*|x|k(x) = 0.1*|x|m(x) = -|x|n(x) = -2*|x|

a. What observations do you have of the effect of the multiplier?(Note to Teacher: Facilitate a class discussion about the effects of the multiplier.)

b. What is the domain and range of m(x) = -|x|?

c. Describe the domain over which the function is increasing and decreasing.

d. How would you describe the affect on the function if you multiply |x| by a negative number?(Note to Teacher: Questions like this, ‘Is 2*f(x) the same as g(x)?’ would be good questions to have on hand to challenge your students thinking.)

Activity 4:Do you think that n(x) = -2*|x| is the same as p(x) = |-2x|? Why?(Note to Teacher: Facilitate a class discussion about the effects of the multiplier inside the absolute value symbol.)

In this activity you are going to sort out whether or not n(x) = p(x).

You will eventually have all these on one graph, but add them one at a time. This will give you the chance to talk about the effect of the multiplier on the function.f(x) = |x|g(x) = |2x|h(x) = |3x|k(x) = |0.1*x|

17

m(x) = |-3x|

a. What observations do you have of the effect of the multiplier?(Note to Teacher: Facilitate a class discussion about the effects of the multiplier.)

b. What is the domain and range of m(x) = |-3x|?

c. Describe the domain over which the function is increasing and decreasing.

d. After this investigation, how would you now compare n(x) and p(x)?

18

Activity 5:(Note to Teacher: This activity should not be done by hand. Students need to use a piece of technology to create the graphs (calculator or Desmos). Thus, this activity could be done either individually, in small groups, or as a whole class. If it’s done as a whole class be sure to gather new conjectures before changing the graph using a new multiplier.)

We’re going to look at another way to change the absolute value function. This time we are going to add a constant to the variable inside the absolute symbols bars ( |x+b| ). How do you think it will transform the graph to add a number to the x within the absolute value symbol? Or subtract a number from the x within the absolute value symbol?

You will eventually have all these on one graph, but add them one at a time. This will give you the chance to talk about the effect of the number added within the absolute value argument.f(x) = |x|g(x) = |x + 2|h(x) = |x – 2|k(x) = |3 + x|

a. What observations do you have of the effect of the adding a number inside the absolute value symbol? (Note to Teacher: Facilitate a class discussion about the effects of adding a number to the x within the absolute value symbol.)

b. What is the domain and range of h(x) = |x – 2|?

c. Describe the domain over which the function is increasing and decreasing.

d. What do you think would happen if you graphed m(x) = |2 – x|? Would it be the same as h(x) = |x – 2|? Why?

Activity 6: Adding a Constant

Do you think that n(x) = |x|+2 is the same as m(x) = |x + 2|? Why?

In this activity you are going to sort out whether or not n(x) = m(x).

19

You will eventually have all these on one graph, but add them one at a time. This will give you the chance to talk about the effect of the multiplier on the function.f(x) = |x|n(x) = |x| + 2h(x) = |x| – 2k(x) = |x| + 3

a. What observations do you have of the effect of the adding a number inside the absolute value symbol?(Note to Teacher: Facilitate a class discussion about the effects of adding a number to the x within the absolute value symbol.)

b. What is the domain and range of k(x) = |x| + 3?

c. Describe the domain over which the function is increasing and decreasing.

d. After this investigation, how would you now compare n(x) and m(x)?

20

Activity 7: Bringing it all together(Note to Teacher: In the table below we have set and grouped the ranges for a, b, c, and d. This eliminates some of the conversation that may be important to you so please feel free to edit the table as you see fit.)Now we want to bring together all the different ideas for transforming an absolute value function based on the investigations.

Describe for each of the following functions what will happen to the absolute value function:

f(x) = |x|

What happens to the graph if we transform f(x) = |x| to f(x) = a*|x|?

a > 1 or 0 < a < 1

-1 < a < 0 or a < -1

What happens to the graph if we transform f(x) = |x| to f(x) = |b*x|?

b > 1 or b < -1

0 < b < 1 or -1 < b < 0

What happens to the graph if we transform f(x) = |x| to f(x) = |x + d|?

d > 0

d < 0

What happens to the graph if we transform f(x) = |x| to f(x) = |x|+ c?

c > 0

c < 0

21

Activity 8:(Note to teacher: activity 8 and 9 will be challenging yet another opportunity to see what sense the students made out of the different investigations. How do you want to group students for this activity? How might you alter it for some of the struggling students?)

Based on what you have done with transformations

What do you predict will happen with each of these and make a sketch of what you think it should look like?

f(x) = 2|x+3|

g(x) = 2|x| + 3

h(x) = |2x| + 3

m(x) = |2x + 3|

(Note to Teacher: Once the student finish their sketches put them up by function for everyone to see and have a brief discussion about the similarities and differences between the sketches. Then bring up each of the graphs digitally using a projector or have the graphs each prepared on a poster so there can be a discussion about these graphs in comparison to the student sketches.)

22

Activity 9:(Note to Teacher: For this activity you have options on how you might setup this activity. (1) You can just give the students the graphs and ask them to come up with the function and develop an approach to figuring out how to determine the function. (2) If you think that might be too difficult for your students you can give all the graphs and the functions. Then ask them to match the graph with the function and how you know they belong together…..or some version of an activity like this.)

f(x) = -|x| :: Download graph from -

23

f(x) = |x| - 2 :: Download graph from –

f(x) = |x+3| :: Download graph from -

24

f(x) = 2|x| :: Download graph from -

f(x) = -3|x|+2 :: Download graph from –

25

f(x) = .5|x+2| - 3 :: Download graph from –

26