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AF Math Algebra 1 Unit 2
UNIT OVERVIEW
Unit Title
Function Modeling
Unit Designers
Approximate Length
IA Period
16 days
1
Stage 1: Desired Results
ENDURING UNDERSTANDINGS AND BIG IDEAS
What do you want students to know in 10 years about this topic?
Enduring understanding/Big idea
What it looks like in this unit
Algebraic expressions and equations are used to model and makeIn this unit, students will use functions to model a variety of
sense of real-life problems and relationships. Many relationships
dependent relationships. They will begin to distinguish between
can be modeled linearly if there is a constant rate of change
relationships that are best modeled linearly versus non-linear (in
between the variables. Otherwise, a non-linear function such as a
this unit, just exponential) relationships.
quadratic or exponential can be used.
Algebraic relationships can be represented in multiple, equivalent
Students begin working with and seeing connections between
ways (verbally, analytically, numerically, and graphically). Each
multiple representations of functions.
representation is useful for different types of analysis.
In arithmetic, you calculate with numbers; in algebra, you reason
Students will build on their knowledge of properties of numbers
about numbers, however, the essential operations and properties
when evaluating functions for given inputs and when combining
of numbers are consistent.
functions.
An equation is a statement of equality between two expressions
It is important for students to begin, in this unit, to see the
whereas functions describe situations where one quantity
connections between equations and functions. Though this will be
determines another. However, expressions and equations can also
more the focus of Unit 4 when students work with two-variable
be viewed as defining functions if the input and output variables
equations.
are defined, and if there is just one output for each input.
Solving problems algebraically typically involves transforming one
equation to another equivalent equation, often in a standard form,
in order to find the solution.
ESSENTIAL QUESTIONS
AF Math Algebra 1 Unit 2
What question(s) will guide this unit and focus learning and thinking?

How is a function similar to and different from an equation or algebraic expression?
o In one-variable equations, the variable represents a fixed quantity that can be determined by solving the equation. In a
function such as f (x) = 2 x + 3 , f (x) represents a specific output value when the input value is x. In general, the function f as a
whole defines the set of output values far all possible input values. An expression could also be defined as a function of x
where the input value is the given value at which you are asked to evaluate the expression.
o Students should be able to explain why they can’t “solve” the equation f (x) = 2 x + 3 but why they could solve the equation if
they were given the input value and asked to find the output value or vice versa. This is all connected to and laying the
foundation for two-variable equations in Unit 4.
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What can the graph of a function tell us?
o Students should see the graph of a function as a different way of looking at the same relationship (rather than representing a
different relationship) and that the graph provides a quick way to see trends of the function or to look for specific behaviors.
For example, if you were looking for the x-intercept of a function, you could continue plugging in points until you found one
where the output was zero, you could solve for the input value when the output is zero, or you could check the graph of the
function and at least make a quick approximation. In order to make this point, students should use graphs to make
approximations of key points of functions that they are not ready to find algebraically. In this unit, the idea of graphing and why
it is useful is more important than having students create tables and plot points.
o Be sure to also include key graph features that are outside of the normal viewing window of the calculator.

How can you tell if a situation should be modeled linearly or exponentially?
o Students should be able to look for non-linear trends in the data given either as a table or graph to identify that a non-linear
model would be most appropriate. However, they should also be able to look for non-constant rates of change in a situation
given in context.
UNIT NARRATIVE
What is the purpose of this unit? How does it fit in the broader context of the course?
In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. In this unit,
students will learn function notation and develop the concepts of domain and range. They explore many examples of functions, including
sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and
understand the limitations of various representations.
AF Math Algebra 1 Unit 2
The goal of this unit is to develop students’ understanding of the broad themes of functions used to model real-life situations: the
relationship between two quantities, the domain and range over which those quantities are defined, evaluating multiple representations
of functions, and analyzing functions for given characteristics such as increasing rates of change, maximums and minimums, and
intercepts. The focus of this unit should be on conceptual understanding of these ideas rather than on computation and procedural
fluency.
STANDARDS FOR MATHEMATICAL PRACTICE
Which are the primary SMPs for this unit? How will they be incorporated and reinforced throughout this unit?
In this unit, students will encounter challenging problems in which they must look for
MP1 Make sense of problems and
patterns, identify linear or exponential trends from context, and create and test (rather than
persevere in solving them
just identify) functions.
MP2 is a huge focus of this unit since students are constantly going to be moving back and
forth between verbal, algebraic, graphical, and numerical representations for the same
MP2 Reason abstractly and quantitatively
relationship. They should build on their work in the previous unit by continually considering
the meaning of the numbers and letters in terms of the situation.
Students will need to argue and defend their choices in this unit around whether or not a
MP3 Construct viable arguments and
relationship is a function, what makes sense as a reasonable domain for a given function,
critique the reasoning of others
and whether a linear or exponential model best fits a given relationship.
Each lesson of this unit students should see functions that model real-life situations,
MP4 Model with mathematics
including functions which can’t easily be defined algebraically (e.g. the dollar-bill change
machine or a soda vending machine).
Students should be using their graphing calculator when necessary to represent functions
MP5 Use appropriate tools strategically
given algebraically and look for key features of the graph.
The new notation and vocabulary gives students many opportunities to practice attending
MP6 Attend to precision
to precision.
Students will analyze patterns and also look for structures (algebraic, verbal, and graphical)
MP7 Look for and make use of structure
that indicate whether a function is best represented linearly or exponentially.
MP8 Look for and express regularity in
repeated reasoning
CONTENT STANDARDS
Major:
AF Math Algebra 1 Unit 2
Understand the concept of a function and use function notation.
F-IF.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 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.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example,
the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1 (n is greater than or equal to 1).
Interpret functions that arise in applications in terms of the 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.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.*
F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by
verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the
larger maximum.
Build a function that models a relationship between two quantities.
F-BF.1 Write a function that describes a relationship between two quantities.*
F-BF.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature
of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
Supporting:
Construct and compare linear, quadratic, and exponential models and solve problems.
F-LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.*
AF Math Algebra 1 Unit 2
F-LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly,
quadratically, or (more generally) as a polynomial function.*
Interpret expressions for functions in terms of the situations they model.
F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context.*
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ESSENTIAL SKILLS AND PROCEDURAL KNOWLEDGE
What do you want students to be able to do comfortably, accurately, and with flexibility?
Evaluate a function given algebraically or graphically using function notation
Identify the domain and range of a function given in any representation
Identify key features of a graph such as x- and y-intercepts, increasing, decreasing, positive, and negative intervals
Connect an algebraic representation to key features of the graph of the function
Describe differences in growth and rates of change between linear and exponential functions
Identify a recursive or explicit function that matches a given sequence
UNIT VOCABULARY
Vocabulary to Review (CC 8th grade math)
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Function
Function model
Initial value
Input
Linear
Output
Rate of change
New Vocabulary
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Continuous
Decreasing
Discrete
Domain
Domain restriction
Explicit formula
Exponential decay
Exponential growth
Function notation
Increasing
Interval
Negative
Positive
Range
Reasonable domain
AF Math Algebra 1 Unit 2
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Recursive formula
Relationship
Sequence
x-intercept
y-intercept
PREREQUISITE SKILLS
What skills will students need in order to be successful with this unit?
Students should be creating and interpreting exponential expressions and equations in this unit and therefore, they should be reviewing
basic exponent skills throughout the first several units of the course.
 Meaning of exponents: x 2 = x × x
 Basic properties of exponents (meaning of negative and zero powers, multiplying terms with same base, dividing terms with same
base, raising a power to a power)

Graphing points on a coordinate plane
AF Math Algebra 1 Unit 2
Stage 2: Acceptable Evidence
W RITTEN ASSESSMENTS
All topics:
1. Understand the concept of a function and use function notation
2. Interpret functions that arise in applications in terms of the context
3. Relate multiple representations of functions
4. Identify, analyze, and build a linear or exponential model
Unit 1 Exam
*There is currently no time allotted in the unit plan (14 teaching days) for a quiz on this unit given
that the unit is relatively short and that time will be precious, particularly in this transition year to CC
Algebra 1. (Students who have been through CC 8th grade will most likely be able to complete this
unit in a shorter amount of time than that allotted.) However, if teachers feel compelled to give a
quiz, there are some lessons that are spread over two days that could be compressed into a day and
a half, for example, or perhaps a take-home quiz or other time during the school day could be
utilized.
PERFORMANCE ASSESSMENTS
In this unit, several tasks are aligned in the learning plan. Most of these are meant to be instructional but some tasks or parts of some of
the tasks could be turned in to more formal, individual performance assessments.
OTHER EVIDENCE
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Exit tickets
CFUs
Informal interviews
Homework
AF Math Algebra 1 Unit 2
Stage 3: Learning Plan
LEARNING PLAN
Lesson #
Standards
2.01
F-IF.1, F-IF.3
MP1, MP2, MP3, MP4, MP7,
MP8
Aim/Exit Ticket
SWBAT define a functional relationship
Examine the tile pattern below.
Topic 1: Understand the
concept of a function and use
function notation
a. Sketch Figures 4 and 5.
b. How does the pattern grow? Explain your thinking.
c. How many tiles will there be in Figure 0 (the figure before Figure
1)? Explain how you know.
d. Make a table showing the number of tiles for Figure 0 through
Figure 9.
Key Points, Resources and Notes
An extra day is included at the beginning of this
unit to provide time for students to work on an
investigation and to begin exploring the properties
of functions without getting into all of the
vocabulary yet. One possible task is the Square
Pool Problem (there are several versions of this
problem but this one includes excellent extensions
that could be brought back to have students
discuss at the end of the unit) Students should be
asked to create multiple representations of the
functional relationship and to discuss questions
such as: if you know the side length of a pool will
you always know the number of tiles? Is it possible
for a pool with a given side length to have more
than one answer for the number of tiles around it?
When you graph the points of the function should
you connect the dots? Can you guess the rule of
this function? If I gave you a square of side length
x, could you say how many tiles would surround
it? If a pool had 44 tiles around it, could you tell
me the side length?
Students can also draw this function as a function
machine and should be able to define the input
and output, if not the function rule. You might
informally introduce function notation over these
first few days but it will be formally used in Lesson
2.04.
AF Math Algebra 1 Unit 2
2.02
F-IF.1, F-IF.2, F-IF.3
MP2, MP6
Topic 1: Understand the
concept of a function and use
function notation
SWBAT use a functional relationship to identify input and output values
1. Zack charges $2.50 per hour for baby-sitting one child. He charges
$.75 per hour for each additional child. Which table should represent
Zack’s hourly charges for baby-sitting?
Starting here and then throughout the unit,
students should use function notation, see
functions in multiple representations, work with
sequences etc.
A.
The goal of this day is to get students more
comfortable with some of the language and
notation around functions. Guess my Rule (see
misconceptions section) would be a good starting
activity for this lesson. Students could also define
and discuss and create multiple representations of
a dollar bill change machine function.
C.
Core Connections Algebra by College Preparatory
Mathematics has some great introductory
activities using function machines. (See page 12,
activity 1.2. You may want to adjust the functions,
particularly if students haven’t been through CC
8th grade math.)
2. One wedding-cake design has a base layer that is 5 inches tall, and
each additional layer is 3 inches tall. What is the height of a 4-layer
cake made in this design?
Throughout this unit, students might revisit unit
conversions by building functions to convert units
and then discussing domain, range, and graphical
representations.
AF Math Algebra 1 Unit 2
2.03
F-IF.1
MP3, MP6
Topic 1: Understand the
concept of a function and use
function notation
SWBAT determine if a given relationship is a function
1. Decide whether each description represents a function. Explain your
thinking.
a. The input is a day of the year. The output is the average
temperature in Barcelona on that day.
b. The input is the speed of a car. The output is the time it takes for a
car moving constantly at that speed to travel 100 miles.
c. The input is a positive number. The output is a number whose
absolute value is the input.
d. The input is a year. The output is the population of the United
States during that year.
2. Consider the relationship graphed below.
a. Explain how you know that the relationship is a function of x.
b. What point could you add to the relationship that would make
it no longer a function? Explain your reasoning.
The key question for students today is: is the
output predictable? See the discussion of possible
approaches to this lesson in the misconceptions
section on the vertical line test.
Aligned Tasks: The Customers (IM)
AF Math Algebra 1 Unit 2
2.04
F-IF.2
MP2, MP3, MP4, MP6
SWBAT use and interpret function notation algebraically and in context
Topic 2: Interpret functions that
arise in applications in terms of
the context
2. Wilson is buying a new television for $3750. To pay for the television,
he makes an initial down payment and then pays off a certain amount
each month. Let f (t) be the amount of money that he still owes after t
months. Explain the meaning of each of the following statements in
everyday language.
a. f (0) = 3000
1. What is
b.
c.
d.
2.05
F-IF.1, F-IF.2, F-IF.4
MP2, MP6
Topic 1: Understand the
concept of a function and use
function notation
f (-2) for the function f (x) = -3x +8 ?
b.
c.
Instead of just identifying or writing ideas using
function notation, now students are given function
notation and a context and asked to interpret the
meaning. If they didn’t already, students could go
back and interpret statements in function notation
about the Square Pool Problem and the dollar bill
change machine.
f (2) > f (4)
f (5) =1750
f (a) = 0
SWBAT use and interpret function notation graphically
1. Use the graph (for example, by marking specific points) to illustrate the
statements in parts (a)-(c). if possible, label the coordinates of any
points you draw.
a.
Aligned Tasks: The Random Walk (IM), Random
Walk II (IM), Yam in the Oven (IM)
f (-3) = f (3) = f (9) = 0
f (2) = g(2)
g(x) > f (x) for x > 2
The goal of this lesson is for students to
understand function notation in terms of the graph
and be able to find both output and possible input
values graphically. Students are going to learn a
lot of new vocabulary around graphs in this unit
and this day would be a good place to introduce
some new terms such as x or y intercept,
increasing interval, positive interval, etc. Make
sure that some of the graphs students see have a
context so that they can continue to make
meaning out of the questions they are being
asked.
AF Math Algebra 1 Unit 2
2. On the coordinate plane below, sketch the graph of a function that
meets all of the following criteria.
 The output values of the function f are positive for all input
values.
 The function f has a maximum where f (2) = 5 .
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2.06
F-IF.1, F-IF.3
MP2, MP6
Topic 1: Understand the
concept of a function and use
function notation
f (0) = f (4) =1
The function is increasing for
x <0.
SWBAT identify the domain and range of a function
1. The tables below show three sets of input-output values.
Table A
Input
Output
2
9
-3
½
-1
100
4
8
12
3
4
7
Table B
Input
Output
4
13
7
-2
4
11
9
3
16
-4
9
0
a. Does each table define a function? Explain.
Table C
Input
Output
-1
0
1
4
-3
-1
1
2
3
4
5
6
Continue to bring in many different types of
functions and many different representations,
including sequences, functions that don’t have a
clearly defined rule, and functions that are given
solely verbally without an algebraic or graphical
representation.
If students have been consistently discussing
input and output values then defining the domain
as the set of all possible input values and the
range as the resulting set of output values should
be a smooth transition. Have students practice
several methods of writing the domain and range.
AF Math Algebra 1 Unit 2
b. For each table that defines a function, state the domain and range
of the function.
2. On the coordinate grid below sketch a function that has:
Domain: −2 ≤ 𝑥 ≤ 5
Range: 𝑦 ≤ 4
2.07
F-IF.1, F-IF.5
MP2, MP3, MP4, MP6
Topic 2: Interpret functions that
arise in applications in terms of
the context
SWBAT identify the reasonable domain and associated range for a
function in context
1. A balloon shaped like a sphere contained a volume of 8 liters of air.
Over a period of 6 hours, all of the air leaked out of the balloon. In this
situation, V, the volume in liters of the balloon, is a function of t, the
number of hours. Which of these describes the range of the function?
A
0£t
B
0£t £6
Ensure to discuss that sequences are distinct from
other continuous functions in that they have
limited inputs.
AF Math Algebra 1 Unit 2
C
V £8
D
0 £V £8
2. An equilateral triangle is shown below.
a. If l represents the length of one side of an equilateral triangle, write
the equation for the perimeter, P of the triangle.
𝑃=
b. If the perimeter of the triangle is 36 meters or less, give the
domain and range for the perimeter function.
c.
Topic 2: Interpret functions that
arise in applications in terms of
the context
SWBAT describe the graph of a function in context
1. A swimmer is at one end of a pool. She starts swimming at a constant
rate, touches the other side of the pool, and then swims back at a
constant rate. Sophie draws the graph below to represent the distance
of the swimmer over time.
Distance
2.08
F-IF.4
MP1, MP2, MP4
Is the perimeter function for an equilateral triangle a discrete or
continuous function? How do you know?
Aligned Tasks: Telling a Story with Graphs (IM)
This lesson should connect back to the graph
stories that students looked at in Unit 1. We are
now just creating a way to communicate and
analyze those graphs rather than just draw them.
Make sure to continue talking about and having
students discuss the meaning of
increasing/decreasing intervals, positive/negative
intervals and intercepts, and rate of change on a
AF Math Algebra 1 Unit 2
given interval.
Time
Explain why this graph is not an accurate representation of the
swimmer’s distance over time. Sketch a graph that would be an
accurate representation.
2. The graph below shows the velocity of a car in meters per second as a
function of time, measured in seconds.
2.09
F-IF.4, F-IF.9
MP2, MP5, MP6
Topic 3: Relate multiple
representations of functions
a. For approximately how long was the car traveling faster than
15 meters per second? Explain your reasoning.
b. Over what interval of time is v(t) decreasing? Explain what
this means in terms of the situation.
c. Interpret the meaning of the point v(24) = 0 .
SWBAT analyze and compare two functions presented in different ways
Both Ronny and Tara are growing bean plants for their gardening club.
Ronny wrote the following function to model his plant’s growth 𝑓(𝑡) =
2.5𝑡 + 4. Tara wrote the following table to model her plant’s growth:
0
1
2
3
4
t
5
7
9
11
13
f(t)
Ronny and Tara noticed that their representations of their plant growth
were different. What are two similarities and two differences they might
These two days of lessons are intended to
reinforce and connect together all that students
have been learning in this unit. They should be
comparing attributes of functions (see list from
2.08) given in many different representations,
some with context and some without.
Students should be making comparisons by using
graphs and tables, by approximation, and by
AF Math Algebra 1 Unit 2
find as they take a close look at their functions? Please be specific and
use appropriate function vocabulary.
2.10
F-IF.4, F-IF.9
MP2, MP5, MP6
Topic 3: Relate multiple
representations of functions
2.11
F-BF.1b, F-LE.1, F-LE.3, F-LE.5
MP1, MP2, MP4, MP7
Topic 4: Identify, analyze, and
build a linear or exponential
model
calculating (for example, by solving to find the
intercepts or plugging values to find the rate of
change on a given interval). Students won’t be
asked to find the y-intercept of a linear function
given as a table of points where the intercept is
not already provided in the table until the next unit.
Although, if they are ready and time allows, that
would be a great task for them here.
SWBAT analyze and compare two functions presented in different ways
Damian found the following graph of a function modeling how many
seconds it took a group of high school sprinters to run 100 meters after
being part of an intensive Olympic training program for many months.
Damian’s friend Jorge says the function should be 𝑓(𝑚) = −0.5𝑚 + 20 if
the runners could run 100 meters in 20 seconds at the beginning of the
program.
a. Compare the graph of the function Damian found to his friend
George’s written function. Name two similarities and two
differences in the functions.
b. Which model do you think is more realistic? Why?
SWBAT identify the best model (linear or exponential) for a given situation
1. For each part below, state whether the quantity is changing in a linear
or exponential fashion.
a. A savings account, which earns no interest, receives a deposit
of $723 per month.
b. The value of a machine depreciates by 17% per year.
c. Every week, 9/10 of a radioactive substance remains from the
Students should have seen both linear and
exponential functions throughout the unit and in
these two days will be adding vocabulary to
communicating and analyzing the functions as
well as working on identifying which function
would make the best model for a given situation.
AF Math Algebra 1 Unit 2
beginning of the week.
d. A liter of water evaporates from a swimming pool every day.
e. Every 124 minutes, ½ of a drug dosage remains in the body.
Students should be able to identify that the rate of
change of linear functions is constant while
exponential functions have a non-constant and
increasing or decreasing rate of change. They
should also be able to distinguish that exponential
functions grow much faster and will eventually,
always, exceed linear functions.
Students should look at multiple representations of
functions, especially functions given verbally, in
order to distinguish between linear and
exponential models.
Students should also discuss parameters of
functions given algebraically in terms of the
context that they represent.
2.12
F-BF.1b, F-LE.1, F-LE.3, F-LE.5
MP1, MP2, MP4, MP7
SWBAT identify the best model (linear or exponential for a given situation
1. The snowflake pattern shown below is called the Koch Snowflake.
Level 0
Level 1
Topic 4: Identify, analyze, and
build a linear or exponential
model
The snowflake pattern is generated by drawing an equilateral triangle,
removing the middle third of each side, and replacing each of these middle
thirds with two sides of an equilateral triangle.
a. Draw the level 2 snowflake.
b. Suppose the original triangle has a side length of 1 and a
perimeter of 3. Find the perimeters of the level 1 and level 2
snowflakes.
c. Is the function for the perimeter, P(l) where l is the level number,
AF Math Algebra 1 Unit 2
a linear or exponential function? Explain your reasoning.
2. Ly is spending the day riding her bicycle in the countryside. She
knows that she will travel 8 miles per hour on unpaved roads and
will travel 12 miles per hour on paved roads. She plans to travel
a total of 48 miles.
a. If Ly travels x hours on unpaved roads, then ____x
represents the distance she travels on unpaved roads.
b. If Ly travels y hours on paved roads, then ____y
represents the distance she travels on paved roads.
c. Combine your answers to (a) and (b) to write an equation
that models this situation, using the form 𝑎𝑥 + 𝑏𝑦 = 𝑐,
where c is total miles traveled.
2.13
F-BF.1b, F-LE.1, F-LE.3, F-LE.5
MP1, MP2, MP4, MP7
Topic 4: Identify, analyze, and
build a linear or exponential
model
SWBAT build an equation to represent a given functional situation
1. In the picture below, you see one paper cup and five paper cups that
are stacked together. The table shows the heights of stacks with
different numbers of cups.
Aligned Tasks: A Sum of Functions (IM)
Students will have more practice creating
equations on their own in the next unit. The focus
of this unit should still be more on getting students
to understand what the different parts of a function
represent in the problem situation and how
combining those different parts builds a model of
the whole problem.
Connect this lesson to creating equations from the
AF Math Algebra 1 Unit 2
Which function rule gives the height of a stack of cups, h, in terms of
the number of cups in a stack, c?
A
B
C
D
c = 2 +8h
c = 8+ 2h
h = 2 +8c
h = 8+ 2c
2. City Bank pays a simple interest rate of 3% per year, meaning that
each year the balance increases by 3% of the initial deposit. National
Bank pays a compound interest rate of 2.6% per year, compounded
monthly, meaning that each month the balance increases by one
twelfth of 2.6% of the previous month’s balance.
a. Which of these functions represents City Bank’s interest plan?
Please circle your answer.
i. 𝐶(𝑦) = 10,000 (
1.03 𝑦
12
)
.03 𝑦
ii. 𝐶(𝑦) = 10,000 (1 + )
12
iii. 𝐶(𝑦) = 10,000(1.03) 𝑦
b. Which of these functions represents National Bank’s interest
plan? Please circle your answer.
i. 𝑁(𝑚) = 10,000 (
1.026 𝑚
12
)
.026 𝑚
ii. 𝑁(𝑚) = 10,000 (1 +
)
12
iii. 𝑁(𝑚) = 10,000(1.026)𝑚
c.
Which bank will provide the largest balance if you plan to
invest &10,000 for 10 years? For 15 years?
last unit. If students are still working on
identification from lessons 2.11 and 2.12, this
lesson can become an extension of those two
days by scaffolding the tasks (either by providing
equations that students pick from or by looking for
elements of linear or exponential functions as full
credit without expecting complete accuracy).
AF Math Algebra 1 Unit 2
2.14
F-IF.3, F-BF.1b, F-LE
MP2, MP4, MP6
SWBAT identify the explicit or recursive formula for a given sequence
1. Which recursive formula is represented by the graph where n is an
integer and g(0) =1?
Topic 4: Identify, analyze, and
build a linear or exponential
model
A
g(n) = g(n -1)+1
B
g(n) = g(n -1)+ 2
C
g(n) = 2g(n -1)
D
g(n) = g(n -1)2
Given problems in context, introduce idea of
explicit and recursive functions. include both linear
and exponential sequences. Make sure to bring
back the discussion of domain and have students
practice writing domain restrictions.
AF Math Algebra 1 Unit 2
2. The value of a certain antique appreciates (or increases) by 20% for
every decade older that it gets. The value of the antique over the last
several decades is shown in the table below.
Year
1950
1960
1970
1980
1990
Value
$80,000
$96,000
$115,200
$138,240
$165,888
Which explicit formula could be used to represent the value of the antique
after n decades, assuming that 1950 is given by n = 0 .
A
f (n) = 80, 000(0.2)n
B
f (n) = 80, 000(1.2)n
C
f (n) = 80, 000 +1.2n
D
f (n) = 80, 000(0.2)(n)+80, 000
AF Math Algebra 1 Unit 2
Misconception
It is only a function if it is
written as f (x) .
State if the graph is a
function means: use the
vertical line test.
COMMON MISCONCEPTIONS
What misconceptions will prevent scholars from reaching mastery?
Clarification
Students should certainly see functions with a variety of letters used to represent the input and output
values. They should also understand that f (x) = 2 x + 3 and f (t) = 2t + 3 represent the same relationship. In
addition, students should be exposed to and asked to analyze functions that can’t be clearly written in an
algebraic formula. For example, when you press the brake pedal on a car (the input), the car slows down
(the output). Students could discuss whether or not they thought this might be a linear relationship. (Or
some other non-algebraic functions might be better since braking a car may not be relatable yet for
students!)
The Functions Progression emphasizes that while students still need to be able to define and identify a
function as distinct from a non-functional relationship, this should not devolve into an exercise in searching
lists of points for a repeated x value or in simply memorizing and universally applying the vertical line test.
The scenario at the beginning of this article below gives an example of how the vertical line test can
confuse students by over-simplification.
To avoid this misconception, focus on the definition of a function as a predictable rule where each input
value defines just one output value. Then have students apply this definition across a variety of settings and
situations. For example, Guess My Rule (where teachers ask for input values and give the output value
after a secret rule is applied; students are asked to figure out the rule) is a great activity to play with
students as they are beginning to learn about functions and, once students get the hang of the game, can
be followed by a Guess My Rule where the teacher doesn’t play fair and is giving different outputs for the
same input which can then lead to a great discussion about why this is unfair.
Or, students can think about a dollar-bill change machine that sometimes gives you 5 quarters when you
put in a dollar and sometimes gives you 3 quarters when you put in a dollar. I have also found it helpful to
draw every function as a function machine because the visual of the machine giving two outputs for the
same input helps students identify the problem. Students can also look at graphs of a function of points and
talk about why a certain relationship can’t be a function if graphically they can see that one input value
leads to two different output values. In other words, they can learn the vertical line test in terms of the
definition instead of learning it as “the vertical line test.” They could also be asked, if I defined the y-values
as the input values and the x-values as the output values, would this still be a function?
The key here is to have students always return to the definition in a variety of situations where they have to
think flexibly so that they don’t get stuck in one way of identifying whether or not a given relationship is a
AF Math Algebra 1 Unit 2
Functions that represent
sequences are drawn
continuously.
A functional
correspondence could be
arbitrary and limited.
function.
Context is helpful here as well as a variety of functions in each lesson. For example, can I put a dollar and a
half into the dollar bill change machine? So should I draw points in between one and two dollars?
Not everything should be in context! While modeling is a focus of the course, students should also be
exposed to functions that are just a defined set of points or that are defined graphically as just a few lines
(connected or not). Students need to also understand that the mathematics applies even when a functional
relationship is arbitrary and out-of-context.
Monday
9/23/13
2.01
Tuesday
9/24/13
2.02
ANTICIPATED LESSON CALENDAR
Wednesday
9/25/13
2.03
Thursday
9/26/13
2.04
Friday
9/27/13
2.05
9/30/13
2.06
10/1/13
2.07
10/2/13
2.08
10/3/13
2.09
10/4/13
2.10
10/7/13
2.11
10/8/13
PD Day
10/9/13
2.12
10/10/13
2.13
10/11/13
2.14
10/14/13
Holiday—No School
10/15/13
Flex/review Day
10/16/13
Unit 2 Exam