Lessons
| Function - a predictable relationship between two quantities. Wages
earned for hours worked is usually a function, because you can predict how
much money you will have after working a certain number of hours. Without
functions, nothing in life would be predictable, and the world would be a
very different place!
Unit Graphics |
Lesson 1:
Relations and Functions
Duration: @ 1 class period
Standard information 2:
2.1
Recognize, extend and use geometric, numeric, linear, or visual patterns to
solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding:
A function is a special relationship.
Essential Question: How do functions show relationships?
Activities
-
Discuss with your classmates what it means for things to
be related. Use these questions to guide the discussion:
What does it mean that I am related to my daughter?
How am I related to my colleagues here at school?
Is my gas mileage related to the kind of car I drive?
Is my eye color related to my name?
Is your locker number related to your grade in school?
All of these associations could be called relations. Now
brainstorm other relations, then write a definition of a relation
in their own words. We are looking for something like “a relation is two
things that have some connection,” or “things that have something in
common.” Share your definitions with the class. Formalize vocabulary:
define “things” (or similar vocabulary used by students) as variables.
-
Discuss with your classmates how some relations are
predictable, but others are not. Use leading questions like:
Can you predict my age if you know my year of birth?
Can you predict my first name if you know my hair color?
Can you predict my occupation if you know my bank account balance?
Can you predict my make and model of car if you know my car’s fuel
efficiency?
Now brainstorm and come up with three relations that are predictable and
three relations that are not predictable. After students share their
ideas with a partner (think-pair-share), list selected predictable
relations on the board. These predictable relations are also
functions, and that a function is a special kind of
relation. It may help to include a Venn Diagram
to show that all functions are also relations, but that the opposite is
not true.
|
A function is constructed as a relation in which the
value of the second variable can be predicted based on the value of
the first variable. |
Lesson 2:
A Function is Predictable
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding:
A function
is predictable.
Essential Question: How do inputs affect outputs?
How are patterns and functions related?
Activities
-
Introduction -
This lesson will introduce the additional vocabulary of
input and output ,_____ is a function of _____, and
domain and range, and will begin to seed the idea of
one-to-one correspondence.
Discuss with your classmates examples of
relations hips in which the second variable is predictable
based on the first variable. that is called a function.
Consider how an electric juicer with fruit can demonstrate this concept.
If you put oranges into a juicer, you get out orange juice.
If you put apples into a juicer, you get out apple juice.
If you put grapefruit into a juicer, you get out grapefruit juice.
You can predict the kind of juice you will have based on the kind
of fruit you put into the juicer.
The domain of the function (the juicer) is fruit; the
range of the function is fruit juice.
If you put oranges into the juicer, you cannot get anything but orange
juice out.
The output (kind of fruit juice) depends on the input
(kind of fruit).
The kind of juice is a function of the kind of fruit.
-
Practice – Practice identifying functional
relationships between two variables.
Use the Student Worksheet:
A Function is a Predictable Relationship
-
Notice some pairs of variables may have a
functional relationship if the first variable is considered to be the
input and the second variable is considered to be the output, but the
opposite is not true. For example, Social Security Number, First Name
is a function if we consider Social Security Number to be the input,
because each Social Security Number is assigned to only one person.
However, if we take First Name as the input and Social Security Number
as the output, it is no longer a function because there are many people
with the same first name who all have different social security number.
-
Remember -It is also important to consider domains
when identifying functions. For example, at one high school, Locker
Number, Student may be a function if no students at that school share
lockers. However, at another school, students may share lockers, and
this relationship would no longer be a function. Similarly, a relation
like First Name, Student ID number could be a function inside a
particular classroom, for example, but not on a larger scale like the
entire freshman class.
Lesson 3:
Road Trip
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding:
A function tells a story.
Essential Question: How can the language of functions be used to
illustrate ideas? How can understanding patterns help us understand and
analyze data?
Activities
-
Introduction – As you enter the classroom, the
Car
Speed video (3:13) will be playing. Since a function tells a story,
how can we use functions to tell the story of speed over time?
in this lesson you will learn to
read and interpret time-distance graphs. Then you will see a
time-distance graph that is not a function, and will interpret it. This
graph is impossible to interpret accurately because it is not a
function; and a given time, there are two data points for distance.
Since it is impossible to be in two places at once, the graph is not a
valid time-distance graph and is not a function. Use the
Student Worksheet:
Road Trip
-
Read Data Points - Read the first two pages of the worksheet and
interpret two different time-distance graphs. Share your interpretations
with your teacher to check for understanding. Share your answers with
other classmates or discuss in a think-pair-share format. The segments
of positive (and negative) slope on each graph do not mean that the
family is traveling uphill (or downhill) – this is a common
misconception. Also emphasize the similarities and differences between
the first graph (a total distance traveled graph) and the second graph
(distance from a fixed point). Finally, emphasize that since the graph
gives you a data point for each time listed, you can accurately
predict where the family was at that time.
-
Map the Data - On page 1 you are asked to read
data points off of the graph to complete a table. This is assumed to be
prerequisite knowledge, but students may need some guidance to complete
the table.
On page 2 you are asked to complete a mapping of the data given in the
graph. Many students may be unfamiliar with mappings, and will likely
need guidance to complete the mapping. The values are already given, so
students only need to draw in the arrows.
-
Experiment
-
The worksheet will give you a seemingly impossible time-distance graph
that is not a function. Try to complete a table with data from the
graph. You may be wondering which one to use since there are two
distances for several of the times listed. When you need help, drag your mouse over the
Help Box below.
|
Help Box
Nobody can be in two places at once, and that the graph you were
given is actually impossible. You cannot predict the
location at any given time if there are two different locations
given on the graph for that time. Since you cannot predict,
the graph does not represent a function. Recall what you
learned about one-to-one correspondence in Lesson 2. |
-
Practice - pages 3-5 will give you
a
chance to
practice
completing tables and mappings from graphs as well as determining
whether each relation is a function.
Student Worksheet:
Road Trip Practice
Differentiation
Extension: Watch the
Car
Speed video (3:13) and the
Bungee
Jumper video (1:43) to see how time distance graphs are used to display
and analyze speed.
Lesson 4:
Identifying
Functions
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding:
A function tells a story.
Essential Question: How can graphs be used to tell the story of a
function?
Activities
-
Introduction -
This lesson formalizes the Vertical Line Test for determining whether a
graph represents a function. The
Student Worksheet:
Vertical Line Test
will walk you through the use of the vertical line test, and then gives
you several opportunities to practice using the vertical line test.
Next, the
Student Worksheet:
Function or Not
gives you an opportunity to revisit tables and mappings and determine
whether each represents a function. You are encouraged to refer back to
the Road Trip – Practice Worksheet to examine tables and mappings which
are functions and those which are not.
-
Student Worksheet:
Vertical Line Test
- Student Worksheet:
Function or Not
Lesson 5:
Road Trip, Part 2
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding:
A function tells a story.
Essential Question: How can equations be used to demonstrate situations?
Activities
-
Introduction
- In this lesson you will be asked to create time-distance tables and
graphs from a description. Again you will be able to see if tables and
graphs represent functions and to explain your reasoning. This lesson
also includes practice with scaling axes. First, you'll be guided
through a series of questions to help you properly scale the axes of a
graph in order to plot a data set. Then, you will be given a series of
data sets, either as tables or mappings, and asked to graph them on a
blank grid. You will need to scale each grid appropriately.
-
Practice - Student Worksheet:
Road Trip – Part 2
-
Practice - Student Worksheet:
Scaling Practice
Lesson 6:
Road Trip, Part 3
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding: Functions include a specific and common vocabulary.
Essential Question:
What essential vocabulary must on know to be able to use functions to tell
stories?
Assessment:
Formative Assessment #1
Activities
-
Introduction
- In Road Tripping Part 3, you will be given a detailed written
description of a journey along with a graph, and asked to interpret
several pieces of the graph, including domain and range.
-
Practice - Student Worksheet:
Road Trip – Part 3
-
Notice - The
first graph given is a time vs. total distance traveled graph, and
students are asked to interpret several features of the graph. Be sure
you can accurately identify the key points on the graph, including the
beginning and end points as well as each point where the graph changes
direction. These key points represent specific details in the story, and
identifying these key points is essential to writing an accurate story
in the next part.
-
Apply What You
Have Learned - Next, interpret a time vs. distance from a fixed
point graph, and compare it to the first graph before writing a detailed
story to go along with the graph. Again, the key points where the graph
begins, ends, and changes direction should correlate to the main details
of the story.
-
Test Yourself
- Finally, students are asked to write a story to match a graph
without any context, so they must also decide on the context and
appropriately title and label the graph.
Formative Assessment #1
Lesson 7: Talking About Functions
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding:
Functions include a specific and common vocabulary.
Essential Questions:
What essential vocabulary must one know to be able to
use functions to tell stories?
Activities
-
Introduction
- This lesson will leads you to the discovery of linear and nonlinear
functions, discrete and continuous functions, and increasing,
decreasing, and constant functions.
-
Experiment -
You will
be asked to compare and contrast two graphs. Graph A is linear,
discrete, and increasing, and Graph B is nonlinear, continuous, and
decreasing. After that comparison is made, you will be given a series of
graphs which are all linear, and asked to find the similarities among
the graphs before writing their own definition of linear. The process is
repeated for graphs which are continuous, discrete, increasing,
decreasing, and constant.
Student Worksheet:
Talking About Functions
Lesson 8:
Discrete and Continuous Functions
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding:
Algebra is a language.
Essential Questions: What essential vocabulary must one know to be
able to use functions to tell stories?
Activities
-
Introduction
- Now that you have an idea of what a discrete function looks like,
discuss with your classmates why a graph might be discrete. In other
words, discussion should center around when it does not make sense to
draw a line between data points. Use the two situations given on the
Discrete and Continuous Functions Worksheet as a starting point for your
discussion. Then, complete the second page, on which they are asked to
determine whether each of a series of data pairs represents a discrete
function or a continuous function.
-
Practice - Student Worksheet:
Discrete and Continuous Functions
Lesson 9:
Domain and Range
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding:
Algebra is a language.
Essential Question:
What is the practical use of knowing the domain and range?
Activities
- Introduction
- This lesson emphasizes the differences in the domains and ranges of
discrete and continuous graphs. The first page gives concrete examples
of the domains and ranges of discrete and continuous graphs with an
explanation of the notation. Then, you will be given a set of exercises
in two parts. For the first part, you are asked to identify and list
domains and ranges for given graph. Next, you are asked to create graphs
with given domains and ranges. Remember to use appropriate scales on
your graphs.
-
Practice - Student Worksheets:
Domain and Range
Lesson 10:
Independent and Dependent Variables
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding: Algebra is a language. All of math requires
understanding patterns.
Essential Question: How do variables impact outcomes?
Activities
-
Introduction
- This lesson will expand on your prior knowledge of dependence to
introduce independent and dependent variables. Discuss
with your classmates the meanings of the terms independent and
dependent. Use guiding questions like:
What does it mean to depend on someone else?
What does it mean that I depend on my income?
How do you depend on your parents?
If you truly depend on something, you need it in order to exist. Discuss
other types of dependence:
What does the amount of money I spend at the grocery store depend on?
What does my car’s gas consumption depend on?
What does your grade in this class depend on?
-
Linking
Variables - This type of dependence links two or more variables. The
amount of money I spend at the grocery store depends on how much food I
buy. For example, “amount of money spent” and “amount of food purchased”
are both variables. So, we can say that the amount of money spent
depends on the amount of food purchased, so the variable “amount of
money spent” is the dependent variable. The variable it
depends upon, in this case “amount of food purchased,” is the
independent variable.
-
Brainstorm
- Think of three additional
relationships in which one variable depends upon the other. Have
students share their ideas with a partner, and then have each pair
select one to share with the entire group. Be sure to discuss with
classmates how sometimes, it is not clear that either variable is either
independent or dependent. For example, there is a clear relationship
between the year a person is born and his age, but it could make as much
sense to say “How old you are depends on when you were born” as it would
to say “When you were born depends on how old you are.”
-
Practice -
Use this time with your teacher to practice identifying independent and
dependent variables. Student Worksheets:
Independent and Dependent Variables
Lesson 11:
Describing Functions
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding: Algebra is a language.
Essential Questions: What are key features of graphs?
Activities
-
Introduction - This
lesson will give you an opportunity to
practice your skills with identifying key features of graphs and
describing those features. It provides an opportunity for a check for
understanding before exploring independent and dependent variables.
-
Practice - Student Worksheets:
Describing Functions
Lesson 12:
x and y Intercepts
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding: Algebra is a language.
Essential Questions:
What information is revealed by knowing multiple intercepts? How is it
useful and practical in real world situations?
Assessment:
Formative Assessment #2
Activities
-
Introduction
- This lesson introduces
intercepts by having you read and interpret real-world graphs with
multiple intercepts. You will then have an opportunity to practice
reading intercepts off of graphs. Be sure to remember that a graph which
has more than one y-intercept cannot be a function!
-
Practice - Student Worksheet:
Reading Intercepts
-
Test Yourself -
Formative Assessment #2
Lesson 13:
What’s in a Name?
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding: Algebra is a language.
Essential Questions:
What information is revealed by knowing multiple intercepts? How is it
useful and practical in real world situations?
Activities
-
Introduction - Lesson 13 introduces function notation to students.
We use function notation as “names” to distinguish different functions
from one another. “f” is the name of
f(x) and “g” is
the name of the separate function g(x).
-
Class Discussion - Discuss
why things are named. Use leading questions like:
Why do we have names?
What would the world be like if we didn’t have names?
What would the world be like if we all had the same name?
What do our names say about us?
-
Drawing Conclusions - Did you draw the
conclusion that names are
there to distinguish amongst ourselves, avoid confusion, and can
sometimes give clues about a person? For example, a person named
Elizabeth is most likely female. However, this is not always the case,
because some names are androgynous. Explain that in order to
differentiate between functions and avoid confusion, we give functions
names. These names sometimes tell us something about the function and
sometimes do not, just as a person’s name can sometimes tell us if that
person is male or female but sometimes cannot.
-
Revisit - _____ is a function of _____ language from previous
lessons.
Discuss the following:
We have seen that, in general, we can always say that the output
is a function of the input
the dependent variable
is a function of
the independent variable
y
is a function of x .
With function notation, we write this in shorthand.
You can think of the “f” as the name of the function. We could
also write
output = f (input)
Dependent variable = f (independent variable)
We don’t always have to use “f” to name a function. We commonly
use “g” and “h” as well, but it can be any letter. This
way, we know that f (x)
g(x) and
h(x) are all
separate, different functions.
-
Remember - Sometimes,
we also use letters from the independent and dependent variables to help
us name our functions. These names are more descriptive and tell us
sometime about the function. We might name a function
d (t)
to show that distance is a function of time, for example.
Practice - Complete
Student Worksheet:
Norah’s Growth.
This worksheet gives you an opportunity to practice using function
vocabulary and gives an example of a situation-specific name for a
function.
Lesson 14:
Function Notation on a Graph
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding: Algebra is a language.
Essential Questions:
How is algebra useful and practical in real world situations?
Assessment:
Formative Assessment #2
Activities
-
Introduction -
Now that you have an idea
of what function notation looks like, you have an opportunity to
practice reading function notation on a graph. First, we'll walk through
some examples with a familiar graph. Then you will be asked to interpret
several elements of another graph.
-
Practice - Student Worksheets:
Reading Function Notation
Lesson 15:
Order of Operations
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding:
All systems have an established order.
Essential Questions:
How does order in mathematics insure reliability?
Materials Needed:
loaf of bread (or part of a loaf) in its plastic bag, a small amount of
butter, a small jar of jelly, a butter knife, and a napkin.
Activities
-
Introduction - this lesson
gives you some rationale for the importance of following the correct
order of operations, and reviews that correct order before giving the
students an opportunity to practice using the order of operations.
-
Class Discussion -
Why does the order matter?
We can use function notation to find values of a function. For every
input, we can find one unique output for a function by doing a series of
mathematical operations. It is important that the operations go in order
and that the operations follow certain rules so that the result is
always predictable.
-
Demonstration -
What will happen if you don’t do things
in the correct order? Use the following materials if you have them, and
follow the list of steps. Identify the process as you follow directions.
Take out two pieces of bread.
Use a knife to spread butter.
Use a napkin to clean the knife.
Use the knife to spread jelly.
Put the two pieces together.
Once you have identified this list as
the basic steps to making a butter and jelly sandwich, scramble the
steps to put them in this order:
Use a napkin to clean the knife.
Use the knife to spread jelly.
Take out two pieces of bread.
Use the knife to spread butter.
Put the two pieces together.
Perform the steps in this scrambled
order, which will make a bit of a mess but won’t make a butter and jelly
sandwich!
Clean the (already clean) knife with the napkin.
Use the knife to spread jelly (on the table, on a plate, on the outside
of the bag of bread, etc.)
Take two pieces of bread out of
the bag.
Use the knife to spread butter (again, somewhere other than the bread)
Put the two pieces of bread back together.
Observe the
resulting mess, and summarize two
things: steps need to be done in the correct order, and there need to be
specific rules about what each step entails.
-
Order of Operations – GEMDAS (or
PEMDAS) Mathematicians have agreed to follow a certain order when they
do mathematical operations so that the result is always the same. This
order also tells us where to start when performing mathematical
operations. The acronym GEMDAS (or PEMDAS) tells us the order in which
operations are performed.
|
G |
Simplify all operations within Grouping
symbols (parentheses and fraction bars). If you have multiple
groupings, you start the order of operations (GEMDAS)
over within each group. |
|
E |
Simplify all Exponents. Remember
that exponents mean the number is multiplied times itself that
many times. |
M
D |
Multiply
and Divide from left to right. |
A
S |
Add
and Subtract from left to
right |
See Examples.
-
Practice - Student Worksheet:
Order of Operations.
This sheet
gives you a chance to practice and apply the order of operations.
Lesson 16:
Evaluating Functions
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding:
Algebra is a language.
Essential Question: How can functions be evaluated?
Assessment:
Formative Assessment #3
Formative Assessment #4
Practice Test
Activities
-
Introduction – This lesson builds on the order
of operations exercises in Lesson 16, and leads students through
evaluating functions. Emphasize that a function determines what happens
to the input. Remember the juicer from Lesson 2. Discuss the
actions the juicer must take to get juice from an orange: the juicer
must peel the orange, grind up the orange, and extract the pulp from the
orange. So, if the juicer is our function and the orange
is our input, we could write:
juicer (orange) = peel (orange) + grind (orange) + de-pulp (orange) The
function, juicer, tells what happens to the input, orange.
A function in general can tell us what happens to the input. Let’s look
at a complicated one:
f (x) = 2X2 + 5X - 13
Our input here is x. The function tells us what happens to x,
following the order of operations:
square x
multiply the result times two
multiply x times 5
Add the result from step 2 to the result from step 3
subtract 13 from the result in step 4.
-
Example -
If our input was a number, say 4, we
would follow the same set of steps:
| 1. square 4 |
42
= 16 |
| 2. multiply the result from step
1 times two |
16 x 2 = 32 |
| 3. multiply 4 times 5 |
4x5 = 20 |
| 4. add the result from step 2 to the result
from step 3 |
32 + 20 = 52 |
| 5. subtract 13 from step 4 |
52 - 13 = 39 |
So, we find out that f (4) = 39. This is what it means to “Evaluate a
Function” for a given input. Sometimes you will hear this called
“plugging in,” because you are replacing the generic x in your
function with a specific input, or plugging that value in as x.
-
Evaluating Functions Practice -
Student Worksheet:
Evaluating Functions
- Test Yourself
-
Formative Assessment #3
Formative Assessment #4
-
Practice Test
|
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