YOUNGSTOWN CITY SCHOOLS
MATH: ALGEBRA I
UNIT 2B: LINEAR AND EXPONENTIAL RELATIONSHIPS – PART I (4-5 weeks) 2013-2014
Synopsis: This unit will provide students with the knowledge to work with linear and exponential functions,
examining particular ones in real-life situations. The students will also be able to represent functions in
several different forms (algebraically, graphically, numerically, or verbally) and interpret the meanings of the
domain and range of the function.
STANDARDS
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.
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.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified
interval. Estimate the rate of change from a graph. 
F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using
technology for more complicated cases. 
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing
period, midline, and amplitude.
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.
MATH PRACTICES
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning
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LITERACY STANDARDS
L-2
L-6
Communicate using correct mathematical terminology
Represent and interpret data with an without technology
MOTIVATION
1. Teacher has students engage in an activity which will assist in understanding the
definition of a function.
a. Teacher explains the intuitive definition of a function – for each input there is
exactly one output.
b. Next introduce the following activity: The activity will begin by students getting into
groups by the color of their pants. Black slacks will be in one group, navy slacks in
another, etc. Each student will go into a specific group. So the x values of this
function are the individual students and the y values are the color of slacks. The x
values are not repeated however the y values are - - thus, making this a function.
c. This can be repeated again by using the color of shirts.
d. After that is reinforced, break the students up by their color of socks. Hopefully
some students will have socks on that are multicolored and will not know what
group to go to. At this point, explain to the students this is not a function because
some of the students (the x values) can go to various color groups (the y values)
so some of the x values are repeated with the same y value.
e. Something other than sock color can be used, such as dividing the students into
groups where the students with the same first letter of their last names go into
groups. When dividing into groups on the second round, make sure some of the
students will be able to go into several different groups.
f. Then use this later in the unit when defining a function, domain and range.
(F.IF.1, MP.1, MP.4, MP.7, L.2)
TEACHER NOTES
2. Preview expectations for end of Unit.
3. Have students set both personal and academic goals for this Unit.
TEACHING-LEARNING
TEACHER NOTES
Vocabulary
Function
Function notation
Non-function
Arithmetic sequence
Exponential function
Maximum
Domain
Set notation
F(x)
Geometric Sequence
Integers
Minimum
Range
Sequence
Recursive
Linear function
X and y intercepts
End behavior
1. Using the motivation activity, relate it to the definition of a function, a relationship where for each
x there is only one y. For example the color of pants would be a function, since for each student
there is only one color of pants. Also shirt color would also be a function. However the sock
color would not be a function, since some students have multi-colored socks and therefore
some of the x’s (students) would go to more than one color of socks. Hence for some x’s there
is more than one y (sock color). Discuss the definition of non-function, for each x there is more
than one y. As this activity progresses, record results and draw a mapping diagram for each.
(F.IF.1, MP.4, MP.7, MP.8, L.2)
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TEACHING-LEARNING
TEACHER NOTES
2. Extend this discussion to the following examples:
a. Amount of money earned when given number of hours worked in contrast to an employee
who works three different jobs for a company with three different rates per hour (they
could work 3 hours and get paid $30 (rate of $10 per hour), $45 (rate of $15 per hour) and
$60 (rate of $20 per hour). So 3 would be mapped to 30, 45, and 60. Here the x values
would be number of hours and the f(x) values would be amount of money earned.
Discuss which one is a function and which one is not and why.
b. A fee for a single person to enter an event in contrast to a single fee for a carload of
people to enter. Discuss which one is a function and which one is not and why.
c. Example of a non-function: weather
temp. amt. of snow
300 1″ snow
300 2″ snow
d. Example of non-function: # of dogs and number of puppies in a litter
# of dogs # of puppies
1
1
3 pups
6 pups
e. Create a function table for a person earning $5 per hour
input
output
No. of hours
Amt. earned
5 hrs.
10 hrs.
$25
$50
f. Create a function table for a person making $120 per day regardless of number of hours
worked:
Hours amt. earned
1
2
3
$120
$120
$120
g. Mapping diagrams: graph these and discuss the vertical line test
Function:
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3
0
-1
5
6
2
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TEACHING-LEARNING
Non-function
3
TEACHER NOTES
4
7
6
2
(F.IF.1, MP.1, MP.2, MP.4, MP.5, MP.7, L-2, L-6 )
3. Skills practice 4-6 (attached on pages 7-8). (F.IF.1, F.IF.2, MP.1, MP.2, MP.4, MP.7 MP.8, L2, L-6)
4. Teacher addresses domain and range with function notation using previous examples.
a) Set notation x {… -1, 0, 1, 2 …}
b) For f(x) = -2x, if you are given the domain: {x | x < 1}, the range would be: {f(x) | f(x)
> -2}
c) Discuss multiple ways to denote functions:
Rule
Mapping diagram
Ordered pairs
Table
d) Function notation: f(x) = 2x + 1, make a table with x and f(x), can also use g(x),
h(n), etc., connect back to the section 2-e.
e) Discuss evaluating functions with f(x) = 2x + 1. For instance, find f(3), f(-2), and f(5),
set up in ordered pairs.
f) Either use the additional examples found on websites or use your own examples
(F.IF.1, F.IF.2, MP.MP.1, MP.2, MP.4, MP.5, MP.7, MP.8)
http://algebra-i--20-p2abyr8.woodbridgems.schools.pwcs.edu/modules/locker/files/get_group_file.phtml?gid=2818808&fid=8864487
http://nlvm.usu.edu/en/nav/frames_asid_191_g_4_t_2.html?from=category_g_4_t_2.html
5. At this time, the student should be able to identify a function, give the definition of a function,
name the domain and range, use function notation and evaluate a function for a given value.
(F.IF.1, I.IF.2, MP.2, MP.4, MP.5, MP.7, MP.8, L-2, L-6)
RECOMMENDED: STOP FOR A TEACHER GENERATED ASSESSMENT
NOTE TO TEACHER: This is an introduction to these topics and will be discussed again in unit 2C .
6. Teacher introduces the students to a linear function, which they should be familiar. Look at a
special linear function, arithmetic sequence. Perform this activity with students to introduce
them to arithmetic sequences. Students will count by 2’s and list numbers on the board. They
will then count by 5’s listing the numbers on the board. Then they will count be 7’s listing the
numbers on the board. Teacher will discuss the difference between the terms in each. Since
the difference between the terms is the same and the number of terms are denoted by 1, 2, 3,
etc., this is an arithmetic sequence. To generate the terms, the formula an=a1 + (n – 1) * d is
used where an is the nth term, a1 is the first term, n is the number of term and d is the difference
between the terms. Reinforce this with several examples. To denote in function notation, look at
f(n) = a1 + (n – 1) * d where a1 and d are given. The domain contains the number of terms and
the range contains the values of the sequence. (F.IF.2, F.IF.3, MP.2, MP.4, MP.7, MP.8, L.2)
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TEACHING-LEARNING
TEACHER NOTES
7. Teacher explains exponential functions: f(x) = 2x, where the independent variable is
an exponent. Look at a special exponential function, geometric sequence.
No. of term
Value of
term
1
2
3
4
5
6
1
3
9
27
81
243
This geometric sequence has equation f(n) = 3n. Teacher examines the sequence 1, , ,
, … Another example is to discuss the teams in the NCAA. It starts with 64 teams and half
are eliminated leaving 32, then half are eliminated again leaving 16, etc. This is a geometric
regression. Refer to p. 438-439 in textbook (2012 edition) and also attached pages 9-10. The
formula for a geometric sequence is tn = a1 * r n-1. (F.IF.2, F.IF.3, MP.1, MP.2, MP.4, MP.7,
MP.8, L.2, L.6)
8. Students should realize that sequences are functions at this time. Discuss the Fibonacci
sequence: 1, 1, 2, 3, 5, 8, 13, … which is generated by adding the two previous terms.
Examine defining a function recursively - - that is each succeeding term is formulated from one
or more previous terms. The Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1)
= f(n) + f(n-1). The arithmetic sequence and the geometric sequence can also be denoted
recursively. The arithmetic sequence: 1, 4, 7, 10… can be denoted recursively as f(1) = 1,
f(n+1) = f(n) + 3. The geometric sequence: 1, 3, 9, 27, … can be denoted recursively as f(1) =
1 f(n+1) = 3*f(n). (F.IF.3, MP.2, MP.4, MP.7, MP.8, L.2, L.6)
RECOMMENDED: STOP FOR A TEACHER GENERATED ASSESSMENT
9. Practice graphing linear and exponential functions showing key features such as x and y
intercepts, increasing or decreasing, positive or negative, and end behavior. For graphs of
lines, discuss average rate of change of a function which is slope (presented symbolically or as
a table.) Interpret this in relation to story problems. For example: An item purchased for $6000
depreciates linearly at the rate of 5% per year. The equation representing this is f(x) = 6000 –
300x. Have students explain what the rate of change is and what it means in this problem.
Students also need to be able to relate the graph to the solution to a problem. An equation for a
population growth in an area might be P(x) = 1200*4x, where 1200 is the initial population, x is
the number of years and P(x) is the population. The students are to graph this and interpret the
problem to be able to answer questions using the graph about the population, such as “What
would the population be in a certain year?” or “In how many years would the population be
100,000?” Practice graphing and then using the graphs to answer questions about the problem.
Also discuss appropriate domains and ranges for a given problem. (F.IF.4, F.IF.5, F.IF.6,
F.IF.7, MP.1, MP.2, MP.4, MP.5, MP.7, L.2, L.6)
10. As a culmination of this unit, discuss comparing properties of similar functions each
represented in a different way (algebraically, graphically, numerically in tables, or by verbal
descriptions). For example, take two lines, one whose equation is given and another whose
graph is given. Ask students to compare slopes and discuss rates of change, y intercepts and x
intercepts. (F.IF.4, F.IF.6, F.IF.7a, F.IF.7b, F.IF.9, MP.1, MP.2, MP.4, MP.7, L.2)
DISTRICT ASSESSMENT
TRADITIONAL ASSESSMENT
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TEACHER NOTES
Algebra 1 Unit 2B Linear and Exponential Relationships – Part 1 2013-2014 5
1. Paper-pencil test with M-C questions
TEACHER CLASSROOM ASSESSMENT
TEACHER NOTES
1. Quizzes
2. In-class participation and practice problems for each concept
AUTHENTIC ASSESSMENT
TEACHER NOTES
1. Students are to either Create or find on the internet two real-life problems: One
that depicts a linear function and one that depicts an exponential function. Graph
each explaining the graphs in terms of domain, range, x and y intercepts,
increasing or decreasing, positive or negative, and end behavior and compare and
contrast the two functions. Rubric attached below (F.IF.4, F.IF.5, F.IF.7a,
F.IF.7b, F.IF.9, MP.1, MP.2, MP.4, MP.5, MP.7, L.2)
PROJECT CRITERIA
Found or created a real-Life
Problem that depicts a linear
function
0
Did not find or create
real-life linear
function problem
Found or created a real-life
problem that depicts an
exponential function
Did not find or create
real-life exponential
function
Graph linear function
Was not able to
graph the linear
function
Was not able to
graph the
exponential function
Could not
compare/contrast
the two functions
Graph Exponential function
Compare/contrast the two
functions
Explain Graph in terms of:
Domain
Range
x and y intercepts
Increasing/decreasing
Positive or negative
End behavior
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0
Was not able to
explain domain
Was not able to
explain range
Was not able to
explain x and y
intercepts
Was not able to
explain increasing /
decreasing
Was not able to
explain positive or
negative
Was not able to
explain end behavior
5
Found or created a
real-Life Problem that
depicts a linear
function
Found or created a
real-life problem that
depicts an
exponential function
Correctly graphed the
linear equation
Correctly graphed the
exponential equation
Made accurate
comparison / contrast
statements about the
two functions
1
Made some valid
points about domain
Made some valid
points about range
Made some valid
points about x and y
intercepts
Made some valid
points about
increasing/decreasing
Made some valid
points about positive
or negative
Made some valid
points about end
behavior
2
Explained domain
Explained range
Explained x and y
intercepts
Explained increasing
/ decreasing
Explained positive or
negative
Explained end
behavior
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