Mathematics
Exploring Linear and Inverse Relationships
Eighth Grade: Mathematics
Model Lesson for Unit #1: Exploring Linear and Inverse
Relationships
Overarching Question:
What does it mean when we see constant and predictable changes in a table of data or a
graph?
Previous Unit:
This Unit:
Next Unit:
Exploring Linear and Inverse
Relationships
Questions to Focus Assessment and Instruction:
1. What do the slope and y-intercept of a line represent in a realworld situation?
2. How can data in a table or scatterplot be used to predict a
future outcome?
Key Concepts:
y-intercept
slope
scatterplot
direct variation
line of best fit
This document is the property of MAISA.
Right Triangle
Relationships
Intellectual Processes(Standards for
Mathematical Practice)
Use appropriate tools strategically:
Use graphing tools to model linear
change in tables, graphs, and
equations.
Model with mathematics: Identify
proportional relationships and make
connections to linear functions.
constant of proportionality
solution
proportional relationship
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Mathematics
Exploring Linear and Inverse Relationships
Lesson Abstract
Students participate in two data collection activities which introduce direct and inverse variation. In
the pre-assessment, students categorize numerical, graphical and verbal representations as linear
or nonlinear and determine which linear representations are also direct variation relationships. In
the post-assessment activity, students convey their understanding of linear, nonlinear, direct and
inverse relationships through verbal and written communication. This three-part lesson provides
opportunities for students to work individually, with partners and in small groups. During activities,
monitor student progress in order to make informal assessments of student understanding and to
provide intervention, when necessary. This lesson plan focuses on the direct variation portion of
the lesson.
Common Core State Standards
Expressions and Equations (8.EE)_________________________________________________
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. For example, compare a
distance-time graph to a distance-time equation to determine which of two moving objects has
greater speed.
Functions (8.F)________________________________ ________________________________
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.
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 the points (1,1),
(2,4) and (3,9), which are not on a straight line.
Use functions to model relationships between quantities.
3. 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.
4. 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.
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Mathematics
Exploring Linear and Inverse Relationships
Statistics and Probability (8.SP)___________________________________________________
Investigate patterns of association in bivariate data.
2. Know that straight lines are widely used to model relationships between two quantitative
variables. For scatter plots that suggest a linear association, informally fit a straight line, and
informally assess the model fit by judging the closeness of the data points to the line.
3. Use the equation of a linear model to solve problems in the context of bivariate measurement
data, interpreting the slope and intercept. For example, in a linear model for a biology
experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each
day is associated with an additional 1.5 cm in mature plant height.
Instructional Resources: timers or stopwatches
Sequence of Lesson Activities
Lesson Title: X Marks the Spot
(https://dnet01.ode.state.oh.us/ims.itemdetails/lessondetail.aspx?id=0907f84c8053269d)
This link takes you to a three part lesson. Teachers can begin reading on page 1, but the X Marks
the Spot Activity actually begins on page 4.
Selecting and Setting up a Mathematical Task:



By the end of this lesson
what do you want your
students to understand,
know, and be able to do?
In what ways does the task
build on student’s previous
knowledge?
What questions will you ask
to help students access
their prior knowledge?
•
Students will identify linear and non-linear relationships in tables, graphs,
equations, and verbal descriptions and defend their choices.
•
Students will graph a scatterplot from collected data and look for patterns in
the table and graph that connect the two variables used.
•
Students can explain the meaning of direct variation as well as comparing
similarities and differences of direct variation and linear relationships.
•
Students will look for the constant of proportionality and connect it to the
table of values, graph, and equation.
•
Students will draw on their previous study of linear relationships to
categorize graphs, tables, equations, and verbal descriptions to determine
whether the relationship is linear or non-linear.
•
Students should come to the task with an understanding of proportional
relationships and the constant of proportionality.
•
Make a table of values that demonstrates a linear relationship. What key
characteristic(s) did you use to show a linear relationship?
•
What does a graph of a linear relationship look like? Why does the graph
look this way? What numerical relationship between variables makes linear
graphs appear different from one another? The same as one another?
•
What specific symbols are needed for an equation to make a linear graph?
•
In describing the connection between two variables, what about their
relationship makes it linear in nature?
This document is the property of MAISA.
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Mathematics
Exploring Linear and Inverse Relationships
•
What does it mean for two variables to be proportional?
•
Students will be given a collection of linear and non-linear situations in a
variety of representations (table, graph, equation, verbal description).
Individually have them sort these into two categories, linear and non-linear.
Next have them compare with another student. Have class members share
their results including an explanation of how they determined which
category to place them in.
•
In small groups ask students to think about and list situations that model
linear and non-linear relationships and respond to the following question:
What about the situation you provided made the situation linear or nonlinear?
•
Have students look at their linear collection and see if there is a way to sort
them further. Prompt them to consider y-intercepts. Ask what they think a
direct relationship would mean. Have students subdivide the linear group
into direct and non-direct categories.
•
Students will be able to distinguish between linear and non-linear
relationships and describe characteristics such as “straight-line graph”,
constant rate of change, always changes by the same amount, equation
could be a single number(constant) or contains a variable such as x .
•
In distinguishing between direct and linear relationships, they should
recognize that all direct variation problems are linear graphs that must pass
through the origin.
Launch:


How will you introduce
students to the activity so
as to provide access to all
students while maintaining
the cognitive demands of
the task?
What will be heard that
indicates that the students
understood what the task is
asking them to do?
Supporting Student’s Exploration of the Task:


What questions will be
asked to focus students’
thinking on the key
mathematics ideas?
What questions will be
asked to assess student’s
understanding of key
mathematics ideas?

What questions will be
asked to encourage all
students to share their
thinking with others or to
assess their understanding
of their peer’s ideas?

How will you extend the
task to provide additional
•
What about the relationship between the two variables, time and number of
“X’s,” causes the graph to appear as it does?
•
What about the collection of data may cause the graph to not be perfectly
linear?
•
What is the difference in a table of values between linear and non-linear
relationships? In conducting this experiment, what are some factors that
you could change so that the table results would model a non-linear
relationship?
•
If you were to draw a line of best fit to model your data, are there any
specific points you would want to include in your line? Why?
•
Have students generate a list of ideas about how to make the outcome of
the experiment non-linear. Ask students to explain why they think each
scenario listed would cause the desired effect. What in the description of
the experiment leads to a non-linear relationship?
•
Have students give an explanation about how the data collection process
could be modified for specific cases, e.g. a negative linear relationship, an
This document is the property of MAISA.
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Mathematics
Exploring Linear and Inverse Relationships
challenge?
exponential growth pattern, a random pattern, etc. Is there something in the
rate of change that was affected? What about the manner in which the data
was collected changed the pattern?
Sharing and Discussing the Task:

What specific questions will
be asked so that all
students will:
o
o
o

•
Why did this activity produce data with a linear-like relationship between
variables?
Make sense of the
mathematical ideas
that you wanted them
to learn?
•
Does everyone’s graph look the same? What differences might you expect
to see because of different people writing the X’s?
•
What factors may have contributed to the data not being perfectly linear?
Expand on, debate,
and question the
solutions being
shared?
•
How does a direct relationship vary from a linear relationship? What about
this experiment leads to a direct relationship?
•
What proportional relationship exists in the table of data you collected?
How can this be seen in your scatterplot?
•
Once you draw a line of best fit on your scatterplot, how could the
proportional relationship of the two variables be used to write an equation
for the line of best fit?
Make connections
between the different
strategies that are
presented?
o
Look for patterns?
o
Begin to form
generalizations?
What will be seen or heard
that indicates all students
understand the
mathematical ideas you
intended them to learn?
Formative Assessment:
Teachers could use questions from the “Got Mail?” assessment in the lesson. These questions should focus on the
characteristics of direct variation. Another focus should be on the linear and non-linear patterns of change and how
these are expressed verbally, in tables of values, graphs, and equations.
Sample questions from “Got Mail?”
1. Explain why the “X” Marks the Spot” activity represents a direct variation. Explain how the activity could be
changed so that it no longer results in a direct variation relationship.
2. Describe a situation, other than “X” Marks the Spot” that represents a direct variation relationship.
4. Complete the table so that it models a direct variation relationship and write an equation that produces the table’s
values.
x 1 2 3
y 4
12
5.
6.
Draw a graph that represents a direct variation.
Draw a second graph that is linear, but does not represent a direct variation.
This document is the property of MAISA.
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