Pre-Algebra (CCM-7 & CCM-8) Unit 3 Starting Points
Unit 3: Analyzing Functions and Linear Equations (8.F/7.RP/8.EE)
Essential Question:
● What is a function? Describe what it means for a situation to have a functional
relationship.
● What is the relationship between the input and output of a function?
● In what ways can different types of functions be used and altered to model various
situations that occur in life?
● What units, scales and labels must be applied to accurately represent a linear function in
the context of a problem situation?
● How can a function be represented using real world contexts, algebraic equations, tables,
and with words?
● What are the advantages of representing the relationship between quantities
symbolically? Numerically? Graphically?
● How do the properties of multiple linear functions compare? How can it be determined
which function has the greater rate of change?
● Can a function be used to model a linear relationship between two quantities? How can
the rate of change and initial values of the functions be determined?
● How can proportional relationships be used to represent authentic situations in life and
solve actual problems?
● In what way(s) do proportional relationships relate to functions and functional
relationships?
● How can the slope of a line be calculated graphically, apply direct variation? What is the
difference between zero slope and undefined slope? Can similar right triangles can be
used to establish that slope is a constant for a non-vertical line?
● How can systems of linear equations be solved numerically, graphically, or algebraically
using substitution or elimination?
● What are the most efficient solution methods when solving a system of equations?
● How can systems 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.
8.F.A.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.
8.F.A.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.
7.RP.A Analyze proportional relationships and use them to solve real-world and
mathematical problems. (7.RP)
7.RP.A.2 Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for
equivalent ratios in a table or graphing on a coordinate plane and observing whether the
graph is a straight line through the origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams,
and verbal descriptions of proportional relationships.
c. Represent proportional relationships by equations.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of
the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
8.EE.B Understand the connections between proportional relationships, lines, and linear
equations.
8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph.
Compare two different proportional relationships represented in different ways. Note: Build on
student understanding of constant of proportionality (7.RP.2) to build understanding of slope.
8.EE.B Understand the connections between proportional relationships, lines, and linear
equations.
8.EE.B.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.
8.F.A.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. For example, the function A=s2, giving the
area of a square as a function of its side length is not linear because its graph contains points (1,
1), (2, 4), and (3, 9), which are not on a straight line.
8.F.B Use functions to model relationships between quantities.
8.F.B.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. Note: Introduce concepts of independent and dependent variables.
8.F.B.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.
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.C Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.C.1 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 days: 50-55
Standard(s)
Days
Notes
8.F.A.1
8.F.A.2
4-5
Big ideas:
Define a function.
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
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.
7.RP.2a-d
7-10
Big ideas:
Determine whether or not two quantities are proportional.
Identify the constant of proportionality (unit rate) of
proportional relationships given tables, graphs, equations,
diagrams and verbal descriptions.
Represent proportional relationships by equations.
Graph proportional relationships. (Informally understand unit
rate as slope.)
Resources:
 Task: Photo Sizes
 Task: Ball Bounce
 Lesson: Hair and Nails…How fast do they grow?
 Lesson: Graphing Proportional Relationships
 Learner.Org: Direct Variation Resources
http://www.learner.org/workshops/algebra/workshop7
/index.html
 NCTM Illuminations: Feeding Frenzy
 TI Lesson: Spring Scales
 Lesson: Shrinky Dinks
 Lesson: Pancake Breakfast
PARCC Assessment Limit/Clarification:
This standard is part of the major content cluster assessed on
PARCC. Students will build on prior understanding of
proportional relationships from Grade 6. Student in Grade 7
grow in their ability to recognize, represent, and analyze
proportional relationships in various ways, including by using
tables, graphs, and equations.
Assessment Items:
 PARCC Prototype Item: Speed
 Illustrative Mathematics: Art Class
 Illustrative Mathematics: Art Class, Variation 2
 Illustrative Mathematics: Sore Throat, Variation 1
 Illustrative Mathematics: Robot Races
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.5
8.EE.B.6
8-10
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
 Illustrative Mathematics: Find the Change
 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
2-3
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:
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: Introduction to Linear
Functions
8.F.A.4
10-12
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.
Resources:
 Task: Filling the Pool
 Lesson: Cell Phone Plan
 Learner.org Task: Crossing the River
 Activity: Determine equations and represent with
graphs
 Lesson: Functions and Volumes of Vases
Assessment Items:
 Illustrative Mathematics: Modeling with a Linear
Function
 Illustrative Mathematics: Proportional Relationships,
Lines, and Linear Equations
 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
8.F.A.5
2-3
Big Ideas:
Describe key features of a linear or nonlinear graph, including
when graph is increasing, decreasing, has a maximum or
minimum, etc.
Resources:
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.
 Lesson: Functions and the Volume of Vases
 Van de Walle, J.A., & Lovin, L.H. (2006). Teaching
Student-Centered Mathematics. Boston: Pearson.
Activity 10.7 Sketch a Graph (p. 303) and
Expanded Lesson - Create a Journal Story (p. 306307)
Assessment Items:
 Illustrative Mathematics: Heart Rate Monitoring
 Illustrative Mathematics: Bike Race
 Illustrative Mathematics: Riding by the Library
 Illustrative Mathematics: Tides
 Illustrative Mathematics: Velocity vs. Distance
8.EE.C.8
10-14
Big ideas:
Analyze and solve pairs of simultaneous linear equations.
Solve real-world and mathematical problems leading to two
linear equations in two variables.
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
 Activity: Basic Look at Systems
http://www.figurethis.org/challenges/c30/challenge.ht
m
 Lesson: Solving Systems with Substitution
 Lesson: Introduction to Linear Combination
 Learner.org Lesson: Right Hand Left Hand
 Task: Let’s Race
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.
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
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.