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Grade 8 Mathematics, Quarter 1, Unit 1.3
Proportional Reasoning
Overview
Number of instruction days:
10–12
Content to Be Learned
Develop arguments to establish facts about the
angle sum and exterior angle of triangles.
Develop arguments about the angles created
when parallel lines are cut by a transversal.
Understand similar triangles using angle-angle
criterion.
Investigate graphs of proportional
relationships.
Develop an understanding of unit rate as the
slope of the graph.
Investigate two proportional relationships
using different representations.
Investigate similar triangles to explain slope.
Develop the equations y = mx and y = mx + b.
Understand examples of linear equations in one
variable that have one solution, infinite
solutions, or no solution.
Mathematical Practices to Be Integrated
2 Reason abstractly and quantitatively.
Analyze the results of parallel lines cut by a
transversal.
Determine if two lines are parallel.
Determine if triangles are similar using angleangle criterion.
Compare two different proportional
relationships represented in different ways.
Use similar triangles to explain why the slope
is the same between any two distinct points in a
coordinate plane.
4 Model with Mathematics.
Graph proportional relationships and interpret
rate as the slope of the graph.
8 Look for and express regularity in repeated
reasoning.
Find relationships among the angles of a
triangle.
Discover characteristics of parallel lines cut by
a transversal.
Use similar triangles to derive the equation
y = mx + b.
Essential Questions
What does the graph of a proportional
relationship look like?
How can knowing the unit rate in a relationship
help you create a graph, table, and equation?
Without measuring angles, how can you tell if
two triangles are similar?
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What can you tell about the angles formed by
two parallel lines intersected by a transversal
not at 90 ?
How many pairs of angles in a triangle have to
be congruent to determine that the triangles are
similar?
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Grade 8 Mathematics, Quarter 1, Unit 1.3
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Proportional Reasoning (10–12 days)
Standards
Common Core State Standards for Mathematical Content
Expressions and Equations
8.EE
Understand the connections between proportional relationships, lines, and linear equations.
8.EE.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.
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.
Geometry
8.G
Understand congruence and similarity using physical models, transparencies, or geometry
software.
8.G.5
Use informal arguments to establish facts about the angle sum and exterior angle of triangles,
about the angles created when parallel lines are cut by a transversal, and the angle-angle
criterion for similarity of triangles. For example, arrange three copies of the same triangle so
that the sum of the three angles appears to form a line, and give an argument in terms of
transversals why this is so.
Common Core State Standards for Mathematical Practice
2
Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations.
They bring two complementary abilities to bear on problems involving quantitative relationships: the
ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own, without necessarily attending to their referents—
and the ability to contextualize, to pause as needed during the manipulation process in order to probe into
the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent
representation of the problem at hand; considering the units involved; attending to the meaning of
quantities, not just how to compute them; and knowing and flexibly using different properties of
operations and objects.
4
Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in
everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition
equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a
school event or analyze a problem in the community. By high school, a student might use geometry to
solve a design problem or use a function to describe how one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are comfortable making assumptions
and approximations to simplify a complicated situation, realizing that these may need revision later. They
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Proportional Reasoning (10–12 days)
Grade 8 Mathematics, Quarter 1, Unit 1.3
Version 4
are able to identify important quantities in a practical situation and map their relationships using such
tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their mathematical results in the context of
the situation and reflect on whether the results make sense, possibly improving the model if it has not
served its purpose.
8
Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods
and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating
the same calculations over and over again, and conclude they have a repeating decimal. By paying attention
to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope
3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way
terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them
to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically
proficient students maintain oversight of the process, while attending to the details. They continually
evaluate the reasonableness of their intermediate results.
Clarifying the Standards
Prior Learning
In Grade 4, students drew right, acute, and obtuse angles as well as parallel lines. In Grade 5, students
identified relationships between corresponding terms. Students also interpreted coordinate values of
points in the context of the situation. In Grade 6, students used rate language in the context of a ratio
relationship. They understood the concept of a unit rate a/b associated with a ratio a:b. Sixth-grade
students also solved unit rate problems involving unit pricing and constant speed. In Grade 7, students
solved real-world and mathematical problems using numerical and algebraic expressions and equations.
Students also solved word problems leading to equations of the form px + q = r and p(x + q) = r and
solved these equations fluently. Seventh-grade students also focused on constructing triangles from three
measures of angles or sides, and they solved problems involving supplementary, complementary, vertical,
and adjacent angles.
Current Learning
Students establish facts about the angle sum and the exterior angles of triangles as well as the angles
created when the triangles are cut by a transversal. Students understand similar triangles to explain slope
and by using angle-angle criterion. They investigate graphs of proportional relationships to informally
understand unit rate. Understanding the connections among proportional relationships, lines, and linear
relationships is a major cluster; therefore, this is a major focus for assessment.
Future Learning
In Algebra I and Algebra II, students will create equations that describe numbers or relationships (linear,
quadratic, and exponential). In Geometry, students will prove theorems involving similarity and
congruence. Slope of a linear function is used in all subsequent mathematics courses. The concept of a
line tangent to a point on a curve is the derivative of a function in Calculus.
Providence Public Schools
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Grade 8 Mathematics, Quarter 1, Unit 1.3
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Proportional Reasoning (10–12 days)
Additional Findings
While students are building their understanding of proportional reasoning, misconceptions may occur
because of the lack of quantitative scheme of measurements. (A Research Companion to Principles and
Standards for School Mathematics, p. 101)
Students may have difficulty understanding the representations of linear functions. Therefore, they
“should have frequent experiences in modeling situations with equations with the form y = kx, such as
relating the side lengths and the perimeters of similar shapes.” (Principles and Standards for School
Mathematics, p. 227)
Assessment
When constructing an end-of-unit assessment, be aware that the assessment should measure your
students’ understanding of the big ideas indicated within the standards. The CCSS for Mathematical
Content and the CCSS for Mathematical Practice should be considered when designing assessments.
Standards-based mathematics assessment items should vary in difficulty, content, and type. The
assessment should comprise a mix of items, which could include multiple choice items, short and
extended response items, and performance-based tasks. When creating your assessment, you should be
mindful when an item could be differentiated to address the needs of students in your class.
The mathematical concepts below are not a prioritized list of assessment items, and your assessment is
not limited to these concepts. However, care should be given to assess the skills the students have
developed within this unit. The assessment should provide you with credible evidence as to your students’
attainment of the mathematics within the unit.
Identify angle measures of all angles formed when triangles are cut by a transversal and use viable
arguments to support their thinking.
Determine the slope of a graph using their understanding of unit rate.
Compare two proportional relationships.
Explain slope using similar triangles.
Given problem situations, write the equations y = mx and y = mx + b and explain the relationship
between the terms in the equation and the context of the problem.
Demonstrate understanding of linear equations in one variable that have one solution, infinite
solutions, or no solution.
Compare proportional relationships, lines, and linear relationships and identify the connections.
Demonstrate their ability to answer all essential questions of the unit.
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Proportional Reasoning (10–12 days)
Grade 8 Mathematics, Quarter 1, Unit 1.3
Version 4
Instruction
Learning Objectives
Students will be able to:
Use informal arguments to establish facts about the angles created when parallel lines are cut by a
transversal.
Use informal arguments to establish facts about the angles created when parallel lines are cut by a
transversal.
Use informal arguments to establish facts about the angle-angle criterion for similarity of triangles
and understand that a two-dimensional figure is similar to another if the second can be obtained by a
sequence of transformations.
Graph proportional relationships, interpret the unit rate as the slope of the graph, compare different
representations of proportional relationships, and derive the equations y = mx and y = mx + b.
Simplify a linear equation in one variable to determine whether it has no solution, one solution, or
infinitely many solutions.
Understand that a function is a rule that assigns a unique output to each input, and that a graph of a
function is a set of ordered pairs consisting of each input and corresponding output.
Apply understanding of proportional relationships and functions in new contexts.
Resources
Connected Mathematics 2, Common Core Investigations, Pearson/Prentice Hall, 2011 and
www.phschools.com under the “worksheet center” tab.
CMP2 Additional Lessons Topic 1: Functions
Investigation 2: Functions
Student Materials (pages 5-12)
Teacher’s Guide
Additional Practice and Skills (Teacher’s Guide)
Investigation 4: Geometry Topics
Student Materials (pages 23 – 34)
Teacher’s Guide
Additional Practice and Skills (Teacher’s Guide)
Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the
Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations.
Materials
Student notebooks, student pages, straightedge, lined paper, grid paper angle ruler or protractor, index
card or business envelope (optional), spaghetti (optional)
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Grade 8 Mathematics, Quarter 1, Unit 1.3
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Proportional Reasoning (10–12 days)
Instructional Considerations
Key Vocabulary
alternate angles
interior angles
corresponding angles
proportional
exterior angles
transversal
function
vertical angles
Planning for Effective Instructional Design and Delivery
Reinforced vocabulary taught in previous grades or units: supplementary angles, complimentary angles,
vertical angles, and adjacent angles.
In seventh grade, students recognized and represented proportional relationships and unit rates with
tables, graphs, and equations. Additionally, they used variables to represent quantities in real-world
problems, and constructed and solved linear equations and inequalities in order to solve problems.
In this unit of study, students’ understandings of proportions, expressions, and equations are reinforced
and expanded. The concept of function is also formally introduced here. However, the Common Core
State Standards do not require function notation in Grade 8.
In later units of study, students will learn about a variety of linear and nonlinear functions and will
represent increasingly more sophisticated expressions and equations. Additionally, they will solve
systems of equations in two variables.
This unit of study also focuses on students investigating a variety of geometric topics, including angle
relationships, angle sums, and triangles. As students investigate the relationships between the angles
formed when parallel lines are cut by a transversal, they should use a nonlinguistic representations
strategy to represent knowledge. Students should work with pictographic representations in order to
determine the angles formed and the relationships among the angles. They can also model with pencils
and/or spaghetti.
It is important to develop vocabulary so that students can use mathematical terms to identify angles and
relationships. It helps if definitions are written in student language using their own descriptions. This will
help them understand the meaning at their level of verbal sophistication. These definitions can be refined
as they gain new insight and explore new examples.
Be sure to ask questions about similarity among figures. Students should notice that triangles are similar
after a sequence of transformations. Even though the questions in these investigations do not ask
specifically about transformations between figures, this is an excellent place for students to make
connections.
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Grade 8 Mathematics, Quarter 1, Unit 1.3
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Notes
Providence Public Schools
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Proportional Reasoning (10–12 days)
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