Grade 8 - Math
2016-17 Unit 3
Title
Suggested Time Frame
1st
Proportional Linear Relationships, Slope & Functions
*CISD Safety Net Standards: 8.4BC, 8.5GI
Big Ideas/Enduring Understandings
•
•
2nd
&
Six Weeks
Suggested Duration: 25 days
Guiding Questions
Proportional and non-proportional relationships can be
• How are different forms of proportional and non-proportional
relationships used to determine slope and y-intercept?
modeled in different forms to represent slope and y-intercept.
• How are proportional and non-proportional relationships used to
Proportional and non-proportional relationships are used to
develop foundational concepts of functions?
develop foundational concepts of functions.
Vertical Alignment Expectations
*TEKS one level below*
*TEKS one level above*
TEA Vertical Alignment 5th Grade – Algebra I
Sample Assessment Question
8.4B
2016 STAAR #34
STAAR 2016 #5
2016 STAAR #42
Grade 8 Math- Unit 3
Updated April 6, 2016
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Grade 8 - Math
2016-17 Unit 3
8.4C
2016 STAAR #12
2016 STAAR #39
2016 STAAR #47
Grade 8 Math- Unit 3
Updated April 6, 2016
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Grade 8 - Math
2016-17 Unit 3
8.5G
2016 STAAR #25
2016 STAAR #28
2016 STAAR #56
8.5I
2016 STAAR #36
2016 STAAR #14
2016 STAAR #54
Grade 8 Math- Unit 3
Updated April 6, 2016
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Grade 8 - Math
2016-17 Unit 3
The resources included here provide teaching examples and/or meaningful learning experiences to address the District Curriculum. In order to address the TEKS to the proper
depth and complexity, teachers are encouraged to use resources to the degree that they are congruent with the TEKS and research-based best practices. Teaching using only the
suggested resources does not guarantee student mastery of all standards. Teachers must use professional judgment to select among these and/or other resources to teach the
district curriculum. Some resources are protected by copyright. A username and password is required to view the copyrighted material. District Specificity/Examples TEKS
clarifying examples are a product of the Austin Area Math Supervisors TEKS Clarifying Documents.
Ongoing TEKS
Math Processing Skills—To be embedded and used all year long throughout all concept student expectations.
8.01 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The
student is expected to:
(A) apply mathematics to problems arising in everyday life, society, and the workplace;
(B) use a problem-solving model that incorporates analyzing given information, formulating a plan or
strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the
reasonableness of the solution;
(C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and
techniques, including mental math, estimation, and number sense as appropriate, to solve problems;
•
Focus is on application
•
Students should assess which tool to apply
rather than trying only one or all
•
Students should evaluate the effectiveness
of representations to ensure they are
communicating mathematical ideas clearly
Students are expected to use appropriate
mathematical vocabulary and phrasing
when communicating ideas
Students are expected to form conjectures
based on patterns or sets of examples and
non-examples
Precise mathematical language is expected.
(D) communicate mathematical ideas, reasoning, and their implications using multiple representations,
including symbols, diagrams, graphs, and language as appropriate;
(E) create and use representations to organize, record, and communicate mathematical ideas;
•
(F) analyze mathematical relationships to connect and communicate mathematical ideas; and
•
(G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in
written or oral communication
•
Grade 8 Math- Unit 3
Updated April 6, 2016
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Grade 8 - Math
2016-17 Unit 3
Knowledge and Skills
with Student
Expectations
District Specificity/ Examples
Vocabulary
8.04 Proportionality. The
student applies
mathematical process
standards to explain
proportional and nonproportional relationships
involving slope. The student
is expected to :
8.04(A) use similar right
triangles to develop an
understanding that slope, m,
given as the rate comparing
the change in y-values to the
change in x-values, (y2y1)/(x2-x1), is the same for
any two points (x1, y1) and
(x2, y2) on the same line.
8.04A
Make a connection that 𝒎𝒎 = 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔; 𝒎𝒎 =
points on a line.
𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓
; 𝒎𝒎
𝒓𝒓𝒓𝒓𝒓𝒓
=
𝒚𝒚𝟐𝟐 −𝒚𝒚𝟏𝟏
𝒙𝒙𝟐𝟐 −𝒙𝒙𝟏𝟏
for any two
Graph two or more right triangles on a coordinate plane where the
hypotenuse of the right triangles falls on the same line.
𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓
Write the ratio of
for each triangle and compare the ratios to show
𝒓𝒓𝒓𝒓𝒓𝒓
that the slope of the right triangles represent a proportional relationship.
Compare the ratios of
𝒚𝒚𝟐𝟐 −𝒚𝒚𝟏𝟏
𝒙𝒙𝟐𝟐 −𝒙𝒙𝟏𝟏
using the end points in the form of an ordered
pair of the hypotenuse of the right triangle.
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.
Grade 8 Math- Unit 3
Updated April 6, 2016
• Data
• Graph
• Line
• Points (x1, y1)
and (x2, y2)
• Proportional
relationship
• Rate (comparing
change in yvalues to change
in x-values)
• Rate of change
• Similar right
triangles
• Slope, m
• Table
• Unit rate
• X-intercept
• Y-intercept
• Y=mx + b
Suggested Resources
Resources listed and categorized to
indicate suggested uses. Any
additional resources must be aligned
with the TEKS.
McGraw-Hill
Chapter 3
Lessons 1-6
(Lessons 7 & 8 are not
addressed in TEKS)
McGraw-Hill
Chapter 4
Lessons 1-6
(Lesson 2 is not addressed
in TEKS)
Web Resources:
Region XI: Livebinder
NCTM: Illuminations
Slope Dude Video
Rise Up, Run Out Video
Slope Video by Colin
Dodds
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Grade 8 - Math
2016-17 Unit 3
Students understand that slope, m, compares the change in any y-values to
the change in x-values.
(y2 - y1) / (x2 - x1) when (x1, y1) and (x2, y2) are points on the same line.
Common Errors:
Students will put change in x-values over the change in y-values.
Students may use (y1 -y2) / (x2 - x1) where they reverse the order of the
points.
Example:
*CISD Safety Net*
8.04(B) graph proportional
relationships, interpreting
the unit rate as the slope of
the line that models the
relationship.
Graph Shop (y=mx + b)
Video
Graph! Video (y=mx+b)
Rap
Y=mx+b (ymca) song
video
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
𝟐𝟐
triangles is 2 to 3, which also represents a slope of for the line, indicating
𝟑𝟑
that the triangles are similar.
8.04B
Unit rate is the slope of a line.
Student understands that the graph of a proportional relationship will
always have a y-intercept (intersect with)at the origin (0, 0).
Student understands that the unit rate in a given situation will become the
slope of their graph.
Grade 8 Math- Unit 3
Updated April 6, 2016
Student will graph one point at (0, 0) and another at (1, y), where y is their
unit rate and extend the line through future points (2, 2y), (3, 3y).
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Grade 8 - Math
2016-17 Unit 3
Common Misconceptions
Unit rate is amount per one unit.
Slope isn’t always a whole number.
Example:
The table shows the distance Ms. Long had traveled as she went to the
beach. Use the data to make a graph. Find the slope of the line and
explain what it shows.
𝟑𝟑
The slope is which means that for every 4 minutes Ms. Long drives, she
𝟒𝟒
travels 3 miles. She is driving 45 mph.
Grade 8 Math- Unit 3
Updated April 6, 2016
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Grade 8 - Math
2016-17 Unit 3
Rosa earns $5 per hour for babysitting. What is the rate of change? What
is the slope of the line?
Think of a relational situation in your life that changes at a steady rate.
Create a table showing five ordered pairs. Write the rate as the slope of the
line, and graph the linear relation. Explain how the slope of the line
compares to the rate of change for the data.
Extra example problems and answers
Problems
For each of the following problems, draw the graph of the proportional
relationship
between the two quantities and describe how the unit rate is represented
on the graph.
1. An Elm tree grows 8 inches each year.
2. Davis adds $3.00 to his savings account each week.
3. Bananas are $2.40 per pound.
4. Lunches in the cafeteria are $2.25 each.
*CISD Safety Net*
8.04(C) use data from a
table or graph to determine
the rate of change or slope
and y-intercept in
mathematical and realworld problems.
Grade 8 Math- Unit 3
Updated April 6, 2016
Answers
1. The graph of y = 8x , which is a line passing through (0, 0) with a slope of
8; the slope 8 is the rate of change of the tree each year.
2. The graph of y = 3x , which is a line passing through (0, 0) with a slope of
3; the slope 3 is the rate of change of Davis’ account each week.
3. The graph of y = 2.4x , which is a line passing through (0, 0) with a slope
of 2.4; the slope 2.4 is the unit rate of each pound of bananas.
4. The graph of y = 2.25x , which is a line passing through (0, 0) with a slope
of 2.25; the slope 2.25 is the unit rate of change of each lunch.
8.04C
Determine the slope (constant rate of change) from a table by finding the
common difference.
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Grade 8 - Math
2016-17 Unit 3
Determine the slope from a graph.
Determine the y-intercept from a table and graph given that y-intercept is
the ordered pair (0, y)
Connect the meanings of slope and y-intercept in real world problem
solving situations.
Student understands that the graph of a proportional relationship will
always have a y-intercept (intersect with) at the origin (0, 0).
Student determines the rate of change (slope) from a proportional graph by
finding the value of the y-coordinate when the x-coordinate is one.
Student determines the rate of change (slope) from a proportional table by
finding the value of the output when the input is one.
Student understands that slope, rate of change, and unit rate mean the
same thing.
Student understands that in a non-proportional linear relationship, the yintercept will not be at (0,0).
Common Misconceptions.
Students think that constant rate of change automatically means it is
proportional.
Examples:
In this table, the rate of change is 2 since 8/4, 6/3 etc.
All have the same value in this graph, the rate of change would be 60 since
240/4 and 60/1 have the same value.
Grade 8 Math- Unit 3
Updated April 6, 2016
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Grade 8 - Math
2016-17 Unit 3
8.05 Proportionality. The
student applies
mathematical process
standards to use
Grade 8 Math- Unit 3
Updated April 6, 2016
8.05A
Susan is able to bike 18 miles in 3 hours. Create a table, graph, and
equation to represent and distance if her rate stays constant.
• Constant rate of
change
• Direct variation
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Grade 8 - Math
2016-17 Unit 3
proportional and nonproportional relationships
to develop foundational
concepts of functions. The
student is expected to:
8.05(A) represent linear
proportional situations with
tables, graphs, and
equations in the form of y =
kx.
8.05(B) represent linear nonproportional situations with
tables, graphs, and
equations in the form of y =
mx + b, where b ≠ 0.
8.05(E) solve problems
involving direct variation.
8.05B
Slope intercept form is written in the form of 𝒚𝒚 = 𝒎𝒎𝒎𝒎 + 𝒃𝒃; where m=slope
and b=y-intercept.
• Equation (y = kx)
• Equation (y = mx +
b, b ≠ 0)
• Function
• Linear nonproportional
situation
• Linear
proportional
situation
• Linear relationship
• Mapping
• Non-proportional
• Ordered pair
• proportional
Joshua buys film through the mail. The standard shipping cost is always
$5.50, regardless of how many rolls of film he buys. The cost of each roll of
film equals $7.50.
8.05E
Two variables show direct variation provided 𝒚𝒚 = 𝒌𝒌𝒌𝒌 and 𝒌𝒌 is not equal to
zero. 𝒌𝒌 is the constant of proportionality or constant of variation.
The equation is read as 𝒚𝒚 varies directly as 𝒙𝒙. Direct variation problems
represent proportional relationships.
Grade 8 Math- Unit 3
Updated April 6, 2016
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Grade 8 - Math
2016-17 Unit 3
8.05(F) distinguish between
proportional and nonproportional situations using
tables, graphs, and
equations in the form y = kx
or y = m + b, where b ≠ 0.
8.05F
Graphs of proportional relationships are linear and go through the origin.
Tables of proportional relationships have a constant equivalent ratio.
Equations of proportional relationships can be written in the form of 𝒚𝒚 =
𝒚𝒚
𝒌𝒌𝒌𝒌 where 𝒌𝒌 = .
𝒙𝒙
Graphs of non-proportional linear relationships have a y-intercept that
does not go through the origin.
*CISD Safety Net*
8.05(G) identify functions
using sets of ordered pairs,
tables, mappings, and
graphs.
8.05(H) identify examples of
proportional and nonproportional functions that
arise from mathematical and
real-world problems.
*CISD Safety Net*
8.05(I) write an equation in
the form y = mx + b to
model a linear relationship
between two quantities
using verbal, numerical,
tabular, and graphical
representations.
Grade 8 Math- Unit 3
Updated April 6, 2016
Tables of non-proportional relationships have a constant difference;
𝒚𝒚
however, the ratio of are not equivalent.
𝒙𝒙
Equations of non-proportional relationships can be written in the form of
𝒚𝒚 = 𝒎𝒎𝒎𝒎 + 𝒃𝒃 where 𝒃𝒃 ≠ 𝟎𝟎.
8.05G
Identify whether a relation is a function by using ordered pairs, tables,
mapping diagrams, and graphs.
8.05H
Compare and contrast proportional and non-proportional relationships
using table, graphs, equations, and real world problem solving situations.
8.05I
Determine the slope (𝒎𝒎) and y-intercept (𝒃𝒃) from a verbal, numerical, table
and graphical representation in order to write the equation in 𝒚𝒚 = 𝒎𝒎𝒎𝒎 + 𝒃𝒃.
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