Common Core Algebra I Unit 2 Starting Points
Unit 2: Linear and Exponential Relationships
Part 1: Representing Linear and Exponential Functions
Essential Questions:
o How can the given explicit or recursive formula be used to find the value of a specific
term?
o What are the benefits of explicit and recursive formulas? Which would be more useful in
a given situation?
o Is the given relationship an example of an arithmetic or geometric sequence? How can
this be justified?
o How do arithmetic and geometric sequences differ?
o How can a given set of data be modeled by a recursive or explicit formula?
o What type of sequence can be modeled by a linear equation? An exponential equation
model?
o In what situations are linear functions practical to use? In what situations are exponential
functions practical to use?
o When is it appropriate to use inequalities?
o What do the features of the graph reveal about the problem situation? Does this help in
the decision-making process? How do these features affect the symbolic and numeric
representations of the problem?
o What is the effect of adding or subtracting a constant to a function? How does this affect
the y-intercept?
o What is the effect of multiplying a function by a scalar? What happens if the scalar is a
negative number?
Curriculum Standards:
Understand the concept of a function and use function notation.
F.IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain
is a subset of the integers.
Build a function that models a relationship between two quantities.
F.BF.A.1 Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for calculation from a
context.
Construct and compare linear and exponential models and solve problems.
F.LE.A.1 Distinguish between situations that can be modeled with linear functions and with
exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals; and that
exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity change at a constant rate per unit interval
relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per
unit interval relative to another.
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
F.LE.A.2 Construct linear and exponential functions, including arithmetic and geometric
sequences, given a graph, a description of a relationship, or two input-output pairs (include
reading these from a table).
F.LE.A.3 Observe using graphs and tables that a quantity increasing exponentially eventually
exceeds a quantity increasingly linearly.
Interpret linear and exponential functions that arise in applications 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, and end behavior.
F.IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative
relationship it describes.
F.IF.B.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.
Analyze linear and exponential functions using different representations.
F.IF.C.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 exponential functions and show intercepts, maxima, and minima.
e. Graph exponential functions, showing intercepts and end behavior.
F.IF.C.9 Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions).
Build new functions from existing functions.
F.BF.B.3 Identify the effect of the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k)
for specific values of k (both positive and negative); find the values of k given the graphs.
Experiment with cases that illustrate an explanation of the effects on the graph using technology.
Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical
translation of a linear function to its y-intercept.
Interpret expressions for functions in terms of the situations they model.
F.LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context. Limit
exponential functions to those of the form f(x) = bx+ k.
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
Approximate Length of Unit (Representing Linear and Exponential Functions): 27 days
Standard(s)
F.IF.A.3
F.BF.A.1a
Days
3-4
Notes
Big Ideas:
Identify sequences as explicit or recursive.
Write and complete sequences given a set of integers,
graph, or table.
Write a formula for sequences given a pattern, graph, or
table of values.
Write a formula for sequences in a real-world setting.
Resources:
 Lesson: Recursive and Explicit Formulas
 Lesson: Recursive and Explicit Geometric Formulas
 Task: Discounting Tickets
 Task: Maria’s Quinceañera
 Web Resource: Function Machine applet
 Web Resource: Towers of Hanoi applet
 Resource:
Cooney, T. J., Beckmann, S., & Lloyd, G. M. (2010).
Developing Essential Understanding of Functions for
Teaching Mathematics in Grades 9-12. P. S. Wilson
(Ed.). Reston, VA: The National Council of Teachers
of Mathematics, Inc. "Definition of Function and
Matching Activity" p. 13-14.
PARCC Assessment Limits/Clarification:
 Major Content and will be assessed accordingly
Assessment Items:
 UTA Dana center Prototype Assessment Item: Golf
Balls in Water
 Illustrative Mathematics: Kimi and Jordan
 Illustrative Mathematics: Lake Algae
 Illustrative Mathematics: Susita’s Account
F.IF.A.3
F.LE.A.1.a
F.LE.A.2
3-4
Big Ideas:
Identify sequences as arithmetic or geometric.
Recognize arithmetic sequences are linear functions and
geometric sequences as exponential functions.
Identify common differences/ratios of linear and
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
exponential functions.
Resources:
 Lesson: Building Arithmetic and Geometric
Functions
 Lesson: Comparing Linear and Exponential Functions
 Task: Setting the Table
 Task: Towering Numbers
 Web Lesson: Curses and Re-curses! It’s happening
again
Assessment Items:
 Pre/Post Task Assessment: Generous Aunt College
Savings Plan
 Illustrative Mathematics: Equal Differences over
Equal Intervals I
 Illustrative Mathematics: Equal Differences over
Equal Intervals II
 Illustrative Mathematics: Equal Factors over Equal
Intervals
F.LE.A.1.a
F.LE.A.1.b
F.LE.A.1.c
F.LE.A.2
F.LE.A.3
F.IF.B.6
F.IF.C.7
F.BF.B.3
F.LE.B.5
15-17
Big Ideas:
Create linear and exponential functions given a table,
graph or verbal model.
Distinguish between real-world situations that can be
modeled with linear and exponential functions.
Transform linear and exponential functions.
o Highlight vertical translation.
Use function notation to build a new function from an
existing function.
Resources:
 Lesson: Constant Rate Exploration
 Lesson: Exponential Decay Exploration
 Lesson: Chain Letter
 Lesson: Getting Paid for School
 Lesson: Rate of Change
 Lesson: Graphing linear and exponential functions
 Lesson: Comparing multiple representations of
functions
 Lesson: Rolling Marbles
 Web Lesson: Overrun by skeeters- exponential growth
 Web Lesson: Skeeter populations and exponential
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.





growth
Task: To Babysit or Not to Babysit?
Task: Lacrosse Tournament
Web Resource: Online Grapher applet
Web Resource: Linear Functions on a graphing
calculator
Web Resource: Linear Functions
Assessment Limit/Clarification:
 Tasks are limited to constructing linear and
exponential functions in simple context (not multistep)
 Tasks have a real-world context. Tasks are limited to
linear functions, quadratic functions, square root
functions, cube root functions, piece-wise defined
functions (including step functions and absolute value
functions), and exponential functions with domains in
the integers. (Focus on linear and exponential now.
This standard will be revisited in later units.)
PARCC Prototype Assessment Item/UTA Dana Center Items:
 High School Functions
 Golf Balls in Water
Assessment Items:
 Illustrative Mathematics: What Functions do Two
Points Determine?
 Illustrative Mathematics: Linear or Exponential?
 Illustrative Mathematics: In the Billions and Linear
Modeling
 Illustrative Mathematics: Identifying Functions
 Illustrative Mathematics: Linear Functions
 Illustrative Mathematics: US Population 1982-1988
 Illustrative Mathematics: Extending the Definitions of
Exponents, Variation 2
 Illustrative Mathematics: Comparing Exponentials
 Illustrative Mathematics: Exponential Functions
 Illustrative Mathematics: US Population 1790-1860
 Illustrative Mathematics: Carbon 14 Dating, Variation
2
 Illustrative Mathematics: Exponential Growth Versus
Linear Growth I
 Illustrative Mathematics: Exponential Growth Versus
Linear Growth II
 Illustrative Mathematics: Illegal Fish
 Illustrative Mathematics: Taxi!
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.




Illustrative Mathematics: Rumors
Illustrative Mathematics: Sundia’s Aerial Tram
Illustrative Mathematics: Basketball Rebounds
Illustrative Mathematics: Two Points Determine an
Exponential Function I
 Illustrative Mathematics: Two Points Determine an
Exponential Function II
 Illustrative Mathematics: Do Two Points Always
Determine a Linear Function?
 Illustrative Mathematics: Temperature in Degrees
Fahrenheit and Celsius
F.IF.B.4
F.IF.B.5
F.IF.C.7.a
F.IF.C.7.e
2-3
Big Ideas:
Identify key features of a graph (intercepts, maxima,
minima, end behavior).
Understand how these key features relate to real-life
situations.
Create an equation, graph, table or verbal description
given key properties.
Resources:
 Lesson: Domain
*Note: Students can review these skills by identifying
intercepts and end behavior in linear and exponential
functions.
Assessment Items:
 Illustrative Mathematics: Influenza Epidemic
 Illustrative Mathematics: Warming and Cooling
 Illustrative Mathematics: How is the Weather?
 Illustrative Mathematics: Telling a Story with Graphs
 Illustrative Mathematics: Oakland Coliseum
 Illustrative Mathematics: Average Cost
 Illustrative Mathematics: The High School Gym
 Illustrative Mathematics: Mathemafish Population
 Illustrative Mathematics: Temperature Change
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
Common Core Algebra I Unit 2 Starting Points
Unit 2: Linear and Exponential Relationships
Part 2: Modeling Data with Linear and Exponential Functions
Essential Questions
o How can we model two-variable data and use models to make predictions?
o When are linear models appropriate for two-variable data? When are exponential models
appropriate?
o How can the correlation coefficient and a plot of the residuals be used to justify the
strength of a linear model?
o What is the difference between correlation and causation?
Curriculum Standards:
Summarize, represent, and interpret data on quantitative variables.
S.ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the
variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context
of the data. Use given functions or choose a function suggested by the context.
Emphasize linear and exponential models.
b. Informally assess the fit of a linear function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
Interpret linear models.
S.ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model
in the context of the data.
S.ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.
S.ID.C.9 Distinguish between correlation and causation.
Approximate Length of Unit (Modeling Data with Linear and Exponential Functions): 18
days
Standard(s)
S.ID.B.6
S.ID.C.9
Days
2
Notes
Big Ideas:
Distinguish the difference between correlation, or association,
and causation.
Create a scatter plot using two sets of quantitative data.
Analyze the correlation of the scatter plot and predict which type
of model would best fit the data.
Resources:
 Lesson: Correlation or Causation
Assessment Items:
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
 Illustrative Mathematics: Golf and Divorce
 Illustrative Mathematics: High Blood Pressure
S.ID.B.6.a
S.ID.B.6.b
S.ID.B.6.c
S.ID.C.7
S.ID.C.8
16
Big Ideas:
Find the correlation coefficient for a set of data
Find the residuals for a set of data.
Analyze the correlation coefficient and residuals to determine the
appropriate model of best fit.
Find a linear and/or exponential model of best fit.
Use correlation coefficient and residuals to justify if the model of
best fit is appropriate.
Interpret slope and y-intercept for line of best fit in the context of
the problem.
Make predictions using linear/exponential regression models.
Resources:
 Lesson: Residuals 1
 Lesson: Residuals 2
 Lesson: Interpreting slope and intercept of a line
 Lesson: Correlation Coefficient
 Teacher Resource: Calculator Steps for Correlation
Coefficient
 Task: Statistical Facebook Analysis
Assessment Limits/Clarifications:
Tasks have a real-world context. Exponential functions are
limited to domains of integers.
Assessment Items:
 Illustrative Mathematics: Olympic Men’s 100-Meter
Dash
 Illustrative Mathematics: Used Subaru Forrester I
 Illustrative Mathematics: Used Subaru Forrester II
 Illustrative Mathematics: Texting and Grades II
 Illustrative Mathematics: Coffee and Crime
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
Common Core Algebra I Unit 2 Starting Points
Unit 2: Linear and Exponential Relationships
Part 3: Systems of Equations and Inequalities
Essential Questions:
o What types of real-world problems would systems of equations be used to solve?
o What is the meaning of a solution to a system of linear equations in the context of a
problem? How is this solution represented algebraically and graphically?
o How are solutions to systems of linear equations different from solutions to systems of
linear inequalities?
o What is the benefit of having multiple methods to use to solve systems? When is it
appropriate to use each method?
o How can multiplication be used to eliminate a variable in a system of equations?
o Why does changing the equations in a system by a multiple not change the solution?
o Explain the meaning of having no solution or infinite solutions for a system of linear
equations in the context of a real-world problem/scenario. What does these solutions
mean about the graphs of the system?
Curriculum Standards:
A.REI.C.5 Prove that, given a system of equations in two variables, replacing one equation by
the sum of that equation and a multiple of the other produces a system with the same solutions.
A.REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs),
focusing on pairs of linear equations in two variables.
A.REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y =
f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions
approximately, e.g., using technology to graph the functions, make tables of values, or find
successive approximations. Include cases where f(x) and/or g(x)are linear and exponential
functions.
A.REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding
the boundary in the case of strict inequality), and graph the solution set to a system of linear
inequalities in two variables as the intersection of the corresponding half-planes.
Total Approximate Length of Unit (Systems of Equations and Inequalities): 15 days
Pre-Assess systems of equations for student understanding
Standard(s)
Days
Notes
Big Ideas:
A.REI.C.5
3-5
Pre-assess for student understanding of systems of equations from
A.REI.C.6
Grade 8.
A.REI.D.11
Solve a system of equations algebraically using elimination or
substitution.
Students should be able to justify their method for solving and the
solutions produced
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
Determine the most efficient/practical method for solving a
system.
Determine a system of linear equations from the context of a
problem and interpret the solution.
Resources:
 Task: Main Street Festival
 Lesson: Dupont Circle
Assessment Limits/Clarifications:
Tasks have a real-world context. Tasks have hallmarks of
modeling as a mathematical practice (less defined tasks, more of
the modeling cycle, etc.).
Tasks that assess conceptual understanding of the indicated
concept may involve any of the function types mentioned in the
standard except exponential and logarithmic functions. Finding the
solutions approximately is limited to cases where f(x) and g(x) are
polynomial functions. Focus on linear examples for now. This
standard will be revisited in later units.
A.REI.C.5
A.REI.C.6
A.REID.11
3-5
Big Ideas:
Extend understanding of solving a system of equations
algebraically.
Solve a 3x3 system of equations algebraically.
Assessment Limits/Clarifications:
Tasks have a real-world context. Tasks have hallmarks of
modeling as a mathematical practice (less defined tasks, more of
the modeling cycle, etc.).
Tasks that assess conceptual understanding of the indicated
concept may involve any of the function types mentioned in the
standard except exponential and logarithmic functions. Finding the
solutions approximately is limited to cases where f(x) and g(x) are
polynomial functions. Focus on linear examples for now. This
standard will be revisited in later units.
Assessment Items:
 Illustrative Mathematics: Cash Box
 Illustrative Mathematics: Accurately Weighing Pennies II
 Illustrative Mathematics: Quinoa Pasta 2
 Illustrative Mathematics: Quinoa Pasta 3
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
 Illustrative Mathematics: Parts of Whole Numbers
 Illustrative Mathematics: Find a System
 Illustrative Mathematics: Population and Food Supply
A.REI.D.12
5-6
Big Ideas:
Graph the solutions to a linear inequality on a coordinate plane.
Solve a system of inequalities graphically.
Apply appropriate scales/windows when using graphical
representation.
Determine a system of linear inequalities from the context of a
problem and interpret the solution.
Resources:
 Lesson: Graphing Linear Inequalities
Assessment Items:
 Illustrative Mathematics: Fishing Adventures 3
 Illustrative Mathematics: Solution Sets
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.