Common Core Mathematics 8 (CCM-8) Starting Points
Unit 3: Analyzing Functions and Equations
Essential Questions:
o What is a function? Describe what it means for a situation to have a functional
relationship.
o What is the relationship between the input and output of a function?
o In what ways can different types of functions be used and altered to model various
situations that occur in life?
o What units, scales and labels must be applied to accurately represent a linear function in
the context of a problem situation?
o Can students represent a function using real world contexts, algebraic equations, tables,
and with words?
o What are the advantages of representing the relationship between quantities
symbolically? Numerically? Graphically?
o Are students able to compare the properties of multiple functions, given a linear function,
and determine which function has the greater rate of change?
o Can students construct a function to model a linear relationship between two quantities,
and determine the rate of change and initial values of the functions?
o Can students relate and compare graphic, symbolic, and numerical representations of
proportional relationships?
o In what way(s) do proportional relationships relate to functions and functional
relationships?
o Are students able to calculate the slope of a line graphically, apply direct variation,
differentiate between zero slope and undefined slope, and understand that similar right
triangles can be used to establish that slope is a constant for a non-vertical line?
o Do students have the knowledge to solve multi-step equations using simple cases by
inspection, one solution, infinitely many solutions, or no solution?
o Are students able to solve systems of linear equations numerically, graphically, or
algebraically using substitution or elimination?
o Do students have the ability to discuss efficient solution methods when solving a system
of equations?
o Can students use a system of equations to solve real-world problems and interpret the
solution in the context of the problem?
Curriculum Standards:
8.F.A Define, evaluate, and compare functions.
1. Understand that a function is a rule that assigns to each input exactly one output. The graph
of a function is the set of ordered pairs consisting of an input and the corresponding output.
2. Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions).
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.
8.EE.B Understand the connections between proportional relationships, lines, and linear
equations.
5. Graph proportional relationships, interpreting the unit rate as the slope of the graph.
Compare two different proportional relationships represented in different ways.
6. Use similar triangles to explain why the slope m is the same between any two distinct points
on a non-vertical line in the coordinate plane; derive the equation y  mx for a line through
the origin and the equation y  mx  b for a line intercepting the vertical axis at b.
8.F.A Define, evaluate, and compare functions.
3. Interpret the equation y  mx  b as defining a linear function, whose graph is a straight line;
give examples of functions that are not linear.
8.F.B Use functions to model relationships between quantities.
4. Construct a function to model a linear relationship between two quantities. Determine the
rate of change and initial value of the function from a description of a relationship or from
two (x, y) values, including reading these from a table or from a graph. Interpret the rate of
change and initial value of a linear function in terms of the situation it models, and in terms
of its graph or a table of values.
5. Describe qualitatively the functional relationship between two quantities by analyzing a
graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a
graph that exhibits the qualitative features of a function that has been described verbally.
8.EE.C Analyze and solve linear equations and pairs of simultaneous linear equations.
7. Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many
solutions, or no solutions. Show which of these possibilities is the case by
successively transforming the given equation into simpler forms, until an equivalent
equation of the form x = a, a = a, or a = b results (where a and b are different
numbers).
b. Solve linear equations with rational number coefficients, including equations whose
solutions require expanding expressions using the distributive property and collecting
like terms.
8. Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two variables
correspond to points of intersection of their graphs, because points of intersection
satisfy both equations simultaneously.
b. Solve systems of two linear equations in two variables algebraically, and estimate
solutions by graphing the equations. Solve simple cases by inspection.
c. Solve real-world and mathematical problems leading to two linear equations in two
variables.
Approximate Length of Unit: 60-70 days
Standard(s)
8.F.A.1
8.F.A.2
Days
10
Notes
Big ideas:
Define a function.
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.
Explore multiple representations (tables of values, graphs,
equations, scenarios) of functions.
Find inputs and outputs of functions.
Resources:
 Lesson: Introduction to Functions
 Task: Earning Incentives
 Learner.org: Tiles Lesson
 Task: Growing Dots
 Task: Towering Numbers
 Lesson: Show Me the Money
 Dan Meyer Lesson: Stacking Cups
 Task: Music Dilemma
 Lesson: Comparing Offers
 Lesson: My Summer Options
 Lesson: Perimeter Patterns
Assessment Items:
 Illustrative Mathematics: Foxes and Rabbits
 Illustrative Mathematics: Function Rules
 Illustrative Mathematics: US Garbage, Version 1
 Illustrative Mathematics: Battery Charging
8.EE.B.5
8.EE.B.6
10-12
Big ideas:
Connect understanding of unit rate (constant of
proportionality) and proportional relationships to concept of
slope (rate of change).
Graph and compare different proportional relationships when
given scenario, equation, and/or table of values.
Use similar triangles to explain why the slope m is the same
between any two distinct points on a non-vertical line in the
coordinate plane.
Identify direct variation situations from other linear
situations.
Resources:
 Task: Drops in a Bucket
 Lesson: NFL Football and Direct Variation
 Lesson: Similar Triangles and Slope
Assessment Items:
 Illustrative Mathematics: DVD Profits, Variation 1
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: Find the Change
 Illustrative Mathematics: Proportional Relationships,
Lines, and Linear Equations
 Illustrative Mathematics: Coffee by the Pound
 Illustrative Mathematics: Comparing Speeds in
Graphs and Equations
 Illustrative Mathematics: Peaches and Plums
 Illustrative Mathematics: Sore Throats, Variation 2
 Illustrative Mathematics: Who has the Best Job?
8.EE.B.6
8.F.A.3
5-7
Big ideas:
Interpret the slope-intercept equation of a line and describe
what the equation reveals about the graph.
Identify whether a representation of a function produces a
linear function or a nonlinear function.
Resources:
 Lesson: Function Families
 Task: Sorting Functions
Assessment Items:
 Illustrative Mathematics: Equations of Lines
 Illustrative Mathematics: Introduction to Linear
Functions
8.F.A.4
8.F.A.5
8-10
Big ideas:
Construct a linear function to model a relationship.
Find the rate of change (slope) and initial value from various
contexts and representations.
Describe slope and initial value in the context of the
situation.
Describe key features of a linear or nonlinear graph,
including when graph is increasing, decreasing, has a
maximum or minimum, etc.
Resources:
 Task: Filling the Pool
 Lesson: Cell Phone Plan
 Learner.org Task: Crossing the River
 Lesson: Functions and Volumes of Vases
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: Modeling with a Linear
Function
 Illustrative Mathematics: Heart Rate Monitoring
 Illustrative Mathematics: Baseball Cards
 Illustrative Mathematics: Chicken and Steak,
Variation 1
 Illustrative Mathematics: Chicken and Steak,
Variation 2
 Illustrative Mathematics: Delivering the Mail,
Assessment Variation
 Illustrative Mathematics: Distance around the
Channel
 Illustrative Mathematics: Video Streaming
 Illustrative Mathematics: Downhill
 Illustrative Mathematics: Bike Race
 Illustrative Mathematics: Riding by the Library
 Illustrative Mathematics: Tides
 Illustrative Mathematics: Velocity vs. Distance
8.EE.C.7
10-12
Big ideas:
Write and solve equations from real-world scenarios.
Solve linear equations in one variable, including equations
with one solution, infinitely many solutions, or no solutions.
Solve linear equations with rational number coefficients,
including equations whose solutions require expanding
expressions using the distributive property and collecting
like terms.
Resources:
 Lesson: Multi-Step Equations
 Lesson: Solving Multi-Step Equations
Assessment Items:
 Illustrative Mathematics: Coupon versus Discount
 Illustrative Mathematics: Sammy’s Chipmunk and
Squirrel Observations
 Illustrative Mathematics: Solving Equations
 Illustrative Mathematics: The Sign of Solutions
8.EE.C.8
15-20
Big ideas:
Analyze and solve pairs of simultaneous linear equations.
Solve real-world and mathematical problems leading to two
linear equations in two variables.
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.
Solve systems by graphing and identify the solution as the
point of intersection. Describe the solution in the context of
the problem.
Understand why algebraic strategies for solving systems
work. Solve systems of two linear equations in two variables
algebraically (using substitution or linear combination), and
estimate solutions by graphing the equations. Solve simple
cases by inspection.
Resources:
 Lesson: Pizza Lesson
 Task: Baseball Shop
 Lesson: Solving Systems with Substitution
 Lesson: Introduction to Linear Combination
 Learner.org Lesson: Right Hand Left Hand
 Task: Let’s Race
Assessment Items:
 Illustrative Mathematics: Two Lines
 Illustrative Mathematics: The Intersection of Two
Lines
 Illustrative Mathematics: Quinoa Pasta 1
 Illustrative Mathematics: Summer Swimming
 Illustrative Mathematics: Fixing the Furnace
 Illustrative Mathematics: How Many Solutions?
 Illustrative Mathematics: Kimi and Jordan
Culminating Tasks: Illustrative Mathematics: High School Graduation
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.