1
Rigorous Curriculum Design
Unit Planning Organizer
Subject(s)
Grade/Course
Unit of Study
Unit Type(s)
Pacing
Middle Grades Mathematics
8th
Unit 5: Linear Functions
❑Topical
18 days
X Skills-based
❑ Thematic
Unit Abstract
In this unit, students will graph proportional relationships; interpret unit rate as the slope;
compare two different proportional relationships represented in different ways and use
similar triangles to explain why the slope is the same between any two points on a nonvertical line. Students will use the equation y = mx for a line through the origin; the
equation y = mx + b for a line intercepting the vertical axis at b; and interpret equations
in y = mx + b form as linear functions.
Common Core Essential State Standards
Domain: Expressions and Equations (8.EE), Functions (8.F)
Clusters: Understand the connection between proportional relationships, lines, and
linear equations.
Define, evaluate and compare functions.
Standards:
8.EE.5 GRAPH proportional relationships, INTEPRETING 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.
8.EE.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.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.
Revised 7/24/13
2
Standards for Mathematical Practice
1. Make sense of problems and persevere
in solving them.
2. Reason abstractly and
quantitatively.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
3. Construct viable arguments and critique 7. Look for and make use of structure.
the reasoning of others.
8. Look for and express regularity in
repeated reasoning.
ons and Equations
8.
Core Cluster
the connections between proportional relationships, lines, and linear equations.
“UNPACKED
STANDARDS”
ly proficient students communicate precisely by engaging
in discussion
about their reasoning using appropriate mathematical language. T
s should learn to use with increasing precision with this cluster are: unit rate, proportional relationships, slope, vertical, horizontal, sim
8.EE.5 Students build on their work with unit rates from 6th grade and proportional
ntercept
relationships
in 7th grade to compare graphs, tables and equations of proportional
Unpacking
n Core Standard
relationships.
identify mean
the unit
(or slope)
in graphs,
What Students
does this standard
thatrate
a student
will know
and betables
able toand
do?equations to
th
compare
two
proportional
relationships
represented
in
different
ways.
proportional
8.EE.5 Students build on their work with unit rates from 6 grade and proportional relationships in 7th grade to
interpreting the unit rate compare graphs, tables and equations of proportional relationships. Students identify the unit rate (or slope) in
Examplegraphs,
1:
f the graph. Compare
tables and equations to compare two proportional relationships represented in different ways.
Compare
the
scenarios
to determine which represents a greater speed. Explain your
proportional
Example
1:
choice including a written description of each scenario. Be sure to include the unit rates
represented in different
Compare the scenarios to determine which represents a greater speed. Explain your choice including a written
ample, compare ain your explanation.
description of each scenario. Be sure to include the unit rates in your explanation.
graph to a distanceto determine which of
Scenario
Scenario
2:
Scenario 1:
1:
Scenario
2:
bjects has greater speed.
y = 55x
y = 55x
x is time
hours
x is in
time
in hours
y is distance
in
milesin miles
y is distance
Solution: Scenario 1 has the greater speed since the unit rate is 60 miles per hour. The graph shows this rate si
60 is the distance traveled in one hour. Scenario 2 has a unit rate of 55 miles per hour shown as the coefficient
Solution:the
Scenario
equation.1 has the greater speed since the unit rate is 60 miles per hour. The
graph shows this rate since 60 is the distance traveled in one hour. Scenario 2 has a
unit rate of
55 miles
per hour
shown as relationship,
the coefficient
in the
equation.
Given
an equation
of a proportional
students
draw
a graph of the relationship. Students recognize
the unit rate is the coefficient of x and that this value is also the slope of the line.
equation
of a proportional
relationship,
students
draw
a graph ofbetween
the them. Using a graph, stud
milar triangles toGiven an8.EE.6
Triangles
are similar when
there is a constant
rate of
proportionality
he slope m is the same
construct triangles between two points on a line and compare the sides to understand that the slope (ratio of rise
Revised
7/24/13
wo distinct points
on a
run) is the same between any two points on a line.
e Mathematics Unpacked Content
Page 13 !
3
relationship. Students recognize that the unit rate is the coefficient of x and that this
value is also the slope of the line.
8.EE.6 Triangles are similar when there is a constant rate of proportionality between
them. Using a graph, students construct triangles between two points on a line and
compare the sides to understand that the slope (ratio of rise to run) is the same
between any two points on a line.
Example 1:
The triangle between A and B has a vertical height of 2 and a horizontal length of 3.
The triangle between B and C has a vertical height of 4 and a horizontal length of 6.
The simplified ratio of the vertical height to the horizontal length of both triangles is 2
2
to 3, which also represents a slope of
for the line, indicating that the triangles
3
are similar. Given an equation in slope-intercept form, students graph the line
represented.
l height of 2 and a horizontal length of 3.
l height of 4 and a horizontal length of 6.
onthe
both triangles
is 2
thehorizontal
coordinatelength of
Example
1:
equation
= mx for thatThe
triangle between A and B has a vertical height of 2 and a horizontal length of 3.
the line, yindicating
the triangles
origin and the
The triangle between B and C has a vertical height of 4 and a horizontal length of 6.
+ b for a line
The simplified ratio of the vertical height to the horizontal length of both triangles is 2
ertical axis at b.
2
3, which also represents a slope of for the line, indicating that the triangles
students graph the line to
represented.
3
are similar.
x for lines going through the origin, recognizing that m represents the
Given an equation in slope-intercept form, students graph the line represented.
Students write equations in the form y = mx for lines going through the origin,
recognizing
represents
slope
of for
thelines
line.going through the origin, recognizing that m represents the
Studentsthat
writemequations
in thethe
form
y = mx
slope of the line.
Example 2:
Example 2:
Write Write
an equation
to to
represent
graph
an equation
represent the
the graph
to to the right.
the right.
Solution: y = -
3
x
2
x + b for lines not passing through the origin, recognizing that m
3
ntercept.
Solution: y =
x
2
Students write equations in the form y = mx + b for lines not passing through the origin, recognizing that m
represents(6,the
2) slope and b represents the y-intercept.
(3, 0)
Revised 7/24/13 2
Solution: y =
3
(6, 2)
x–2
(3, 0)
for the school year. Write the rule for the total cost (c) of renting a calculator as a function of the number of
for the school
3 year. Write the rule for the total cost (c) of renting a calculator as a function of the number of
months (m).
Solution:
y=- x
months (m).
c = 10 + 5m 2
c = 10 + 5m
4
Solution: Function 1 is an example of a function whose graph has a negative slope. Both functions have a
Solution: Function 1 is an example of a function whose graph has a negative slope. Both functions have a
positive starting amount; however, in function 1, the amount decreases 3.50 each week, while in function 2, the
Students
in the form
y = mx
+ bamount
for lines
not passing
through
the origin,
positive write
startingequations
amount; however,
in function
1, the
decreases
3.50 each
week, while
in function 2, the
amount increases 5.00 each month.
amount
increases
eachform
month.
recognizing
that m5.00
represents
and
b not
represents
the y-intercept.
Students
write
equations
in the
y the
= mxslope
+ b for
lines
passing through
the origin, recognizing that m
represents
the slope could
and b be
represents
theiny-intercept.
NOTE: Functions
expressed
standard form. However, the intent is not to change from standard form
NOTE: Functions could be expressed in standard form. However, the intent is not to change from standard form
to slope-intercept form but to use the standard form to generate ordered pairs. Substituting a zero (0) for x and y
to slope-intercept form but to use the standard form to generate ordered pairs. Substituting a zero (0) for x and y
will generate two ordered pairs. From these ordered pairs, the slope could be determined.
will generate 2two ordered pairs. From these ordered pairs, the slope could be determined.
Example 3:
(6, 2)
Solution:
Exampley3:= x - 2
2x + 3y = y6= 2 x –3 2
Solution:
2x + 3y =36
Let x = 0:
2(0) + 3y = 6
L et y = 0: 2x + 3(0)
=6
0)
Let x = 0:
2(0) + 3y = 6
L et y = 0: 2x(3,
+ 3(0)
=6
3y = 6
2x=6
3y = 6
2 x=6
3y = 6
2x = 6
3y = 6
2x = 6
3
3
2
2
3
3
2
2
y=2
x=3
y=2
x=3
8.F.3
Students
that linear functionsOhave
constant
Ordered
pair: (0,understand
2)
rderedapair:
(3, 0) rate of change between
Ordered
(0,
2)
Ocomparisons
rdered pair: with
(3, 0)another function.
Using
2)points.
and pair:
(3, 0)
students
could
find the slope
and make
any (0,
two
Students
use equations,
graphs
and
tables to categorize
functions as
Using (0, 2) and (3, 0) students could find the slope and make comparisons with another function.
y = mx + b 8.F.3
Students
understand
that
linear
functions
have
a
constant
rate
of
change
between
any
two points. Students
linear or non-linear.
ation y = mx + b 8.F.3 Students understand that linear functions have a constant rate of change between any two points. Students
n, whose
use equations, graphs and tables to categorize functions as linear or non-linear.
nction,
whoseContent
use equations, graphs and tables to categorize functions as linear or non-linear.
Unpacked
Page 14 !
ecsexamples
Example
;ear.
giveFor
examplesExample 1: 1:
ot2 giving
linear. the
For Determine
Example
1: functions listed below are linear or non-linear. Explain your reasoning.
if the
if2ifthe
functions listed below are linear or non-linear. Explain your reasoning.
Aon=ofs2 its
giving the Determine
Determine
1. y = -2x
+the
32 functions listed below are linear or non-linear. Explain your reasoning.
2
1. y = 1.-2xy =+-2x
3 +3
unction
cause
its of its
2. y = 0.25 + 0.5(x
– 2)
2.
y
=
0.25
+
r
because
its
2.
y
=
0.25
+
0.5(x
– 2)
2 0.5(x–2)
1,1), (2,4)
3. A = ∏
r
2
2
nts
(1,1), (2,4) 3. A =3. r A = ∏r
a straight
5.
ot on a straight 4. 4. 4.
55..
X
Y
X
Y
1
12
1
12
2
7
2
7
3
4
4 3 3 4
5 4 4 3
6 5 7 4
6
7
s Unpacked Content
matics Unpacked Solution:
Content
1.
2.
3.
4.
5.
Page 21 !
Page 21 !
Non-linear
Linear
Non-linear
Non-linear; there is not a constant rate of change
Non-linear; the graph curves indicating the rate of change is not constant.
Revised 7/24/13
5
“Unpacked” Concepts
(students need to know)
8.EE.5
 Proportional relationships
“Unwrapped” Skills
(students need to be able to do)


8.EE.6
 Points in a linear function



8.F.3
 Linear and nonlinear functions

COGNITION
DOK
I can identify unit rate (slope) in
a graph, table or equation.
I can compare proportional
relationships represented in two
different ways.
I can demonstrate that points
that lie on the same line have the
same slope
I can develop the equation of a
line that passes through the
origin. ( y=mx )
I can develop the equation of a
line not passing through the
origin. ( y=mx + b)
I can determine if a table, graph
or function represents a linear or
non-linear function and explain
my reasoning
Essential Questions
8.EE.5
 How can I determine the unit rate, rate
of change, slope or constant of
proportionality from a data set, table,
graph or function?
2
2
2
2
2
2
Corresponding Big Ideas

Students will work with data sets,
tables, graphs, and functions to
determine the unit rate, rate of change,
slope, or constant of proportionality.

Students will compare proportional
relationships represented in two
different ways.
8.EE.6
 How can I demonstrate that points on
the same line have the same slope?

Students will demonstrate that points
on the same line have the same slope.


Students will write the equation of line

How can I compare proportional
relationships represented in two
different ways?
How can I write the equation of line
Revised 7/24/13
6
that passes through the origin on the
coordinate plane?

How can I write the equation of line
that does not go through the origin on
the coordinate plane?
8.F.3
 How do I know when a data set, data
table, function or graph is linear or
nonlinear?
that passes through the origin on the
coordinate plane.

Students will write the equation of line
that does not go through the origin on
the coordinate plane.

Students will be able to recognize that a
constant rate of change represents a
linear function, the equation can be
written as y = mx + b and the graph is a
straight line.
Vocabulary
unit rate, proportional relationships, slope, rate of change, vertical, horizontal, similar
triangles, y-intercept, linear, non-linear
Language Objectives
Key Vocabulary
8.EE.5 –
8.EE.6
8.F.3
8.EE.5
Students will be able to define, give an example of and
use the key vocabulary when working with linear
functions: unit rate, proportional relationships, slope, rate
of change, vertical, horizontal, similar triangles, yintercept, linear, non-linear
Language Function
SWBAT recognize that unit rate, rate of change, slope,
and constant of proportionality are all equivalent.
Students will also be able to calculate each measure for a
set of data.
Language Skill
8.EE.6
Revised 7/24/13
SWBAT demonstrate through graphic models that similar
triangles used to represent vertical change compared to
horizontal change between two points will produce points
7
on the same line.
8.F.3
SWBAT construct oral or written arguments to show that a
data set is linear if there is a constant of proportionality
between points.
Language Structures
8.EE.6
SWBAT demonstrate through graphic models, to their
partner or whole class that similar triangles used to
represent vertical change compared to horizontal change
between two points will produce points on the same line.
Language Tasks
8.EE.5
SWBAT explain to a partner or whole class that unit rate,
rate of change, slope, and constant of proportionality are
the same in terms of their meaning, by examining data
sets, data table, functions and graphs and demonstrating
that the calculations produce the same number and
meaning in the context of the given situation.
8.F.3
SWBAT construct oral or written arguments to show that a
data set is linear if there is a constant of proportionality
between points.
8.EE.6
8.F.3
Language Learning Strategies
SWBAT write an equation of a line that passes through
the origin on the coordinate plane. Students will justify
their equation using the correct vocabulary.
SWBAT determine if a data set, data table, function or
graph is linear or nonlinear by looking for proportionality to
determine a constant rate of change and justifying their
answers using correct vocabulary.
Information and Technology Standards
8.TT.1.1 Use appropriate technology tools and other resources to access information
(search engines, electronic databases, digital magazine articles).
8.TT.1.2 Use appropriate technology tools and other resources to organize information
(e.g. graphic organizers, databases, spreadsheets, and desktop publishing).
Revised 7/24/13
8
8.RP.1.1 Implement a project-based activity collaboratively.
8.RP.1.2 Implement a project-based activity independently.
Instructional Resources and Materials
Physical
Technology-Based
Connected Math 2 Series
 Common Core Investigation 2
 Thinking With Mathematical
Models, Inv. 2, ACE
 Say It With Symbols, Inv.4
WSFCS Math Wiki
Partners in Math Materials
 Sticks and No Stones I & II
Georgia Unit
Lessons for Learning (DPI)
 Perplexing Puzzle
 Non-Linear Functions
NCDPI.Wikispaces Eighth Grade
Illuminations.NCTM Walk the Plank
Education.ti.com/calculators/Activities=5875
Shodor.org/LinearFunctMachine/
Mathforum.org/cgraph/cslope/
Mathematics Assessment Project (MARS) ENLVM.usu.edu/eqns_lines
2.edc.org/mathpartners
MSteacher.org/linear
Book
 A Visual Approach to Functions by
Frances Van Dyke
NSDL.org/commcore/G.8
Math.fullerton.edu/Linear_Equations
Themathpage.com/alg/
UEN.org/Lessonplan/Grade=8
KATM.Flip Book8l
Graniteschools.org/Pre-AlgebraLessons
Revised 7/24/13