UNIT 2: RATIOS AND PROPORTIONAL RELATIONSHIPS
Unit Description/ Topic Length: This 3-week unit focuses on the concept of ratios
and proportional relationships. Students will extend their understanding of ratios and
develop understanding of proportionality to solve real-world and mathematical
problems. They will engage in instructional tasks that provide them with opportunities
to recognize and represent proportional relationships between quantities in different
ways.
Essential Question:
How does ratio reasoning differ from proportional reasoning?
Big Ideas/Enduring Understandings
 A ratio is a multiplicative comparison
of two quantities, or it is a joining of
two quantities in a composed unit.
 Forming a ratio as a measure if a realworld attribute involves isolating that
attribute from other attributes and
understanding the effect of changing
each quantity on the attribute of
interest.
 A number of mathematical
connections link ratios and fractions:
o Ratios are often expressed in
fraction notation, although
ratios and fractions do not have
identical meaning.
o Ratios are often used to make
“part-part” comparisons, but
fractions are not.
o Ratios and fractions can be
thought of as overlapping sets.
o Ratios can often be
meaningfully reinterpreted as
fractions.
 Ratios can be meaningfully
reinterpreted as quotients.
 A proportion is a relationship of
equally between two ratios. In a
proportion, the ratio of two quantities
remains constant as the corresponding
values of the quantities change.
 Proportional reasoning is complex and
involves understanding that
o Equivalent ratios can be
created by iterating an/or
Guiding Questions:
1. What is a ratio?
2. What is a ratio as a measure of an attribute in a
real-world situation?
3. How are ratios related to fractions and division?
4. What is a proportion?
5. What are the key aspects of proportional
reasoning?
6. What is a rate and how is it related to
proportional reasoning?
7. What is the relationship between the crossmultiplication algorithm and proportional
reasoning?
8. When is it appropriate to reason proportionally?
partitioning a composed unit:
o If one quantity in a ratio is
multiplied or divided by a
particular factor, then the other
quantity must be multiplied or
divided by the same factor to
maintain the proportional
relationship; and
o The two types of ratioscomposed units and
multiplicative comparisons are
related.
 A rate is a set of infinitely many
equivalent ratios.
 Several ways of reasoning, all
grounded in sense making, can be
generalized into algorithms for solving
proportions problems.
 Superficial cues present in the context
of a problem do not provide sufficient
evidence of proportional relationships
between quantities.
NYS Common Core Learning Standards for Mathematics:
7.RP.1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and
other quantities measured in like or different units.
7.RP.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. For example, if total cost t is proportional to
the number n of items purchased at a constant price p, the relationship between the total cost and
the number of items can be expressed as t = pn.
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.
Content
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Unit Rates
Recognizing Proportional Relationships
Representing Proportional Relationships
Constant of proportionality (unit rate)
Vocabulary/ Key Terms
Ratio
Rate
Unit rate
Proportion
Proportional Relationship
Constant of proportionality
Complex fractions
Skills
 Compute unit rates associated with
ratios of fractions in like or different
units
 Determine if two quantities are in a
proportional relationship from a table, a
graph, an equation and a verbal
description
 Represent a proportional relationship
using a table, a graph, and an equation
 Identify the constant of proportionality
(unit rate) in tables, graphs, equations,
diagrams, and verbal descriptions of
proportional relationships
 Explain what a point (x, y) on the graph
of a proportional relationship means in
terms of the situation, including the
points (0, 0) and (1, r) where r is the unit
rate
ASSESSMENT EVIDENCE
Performance Task(s):
Please refer to the attached file named
Performance Assessment. You should find
the Grade 7 Performance-Based
Assessment on Proportional Reasoning in
the document.
Alignment to NYS Common Core
Standards for Mathematics:
Content Standards:
7.RP.1, 7.RP.2 (a, b, c, d)
Standards for Mathematical Practice:
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
Diagnostic and Pre/Post Assessments:
Students will respond to the essential question at the start of the unit and at the end of the
unit. (pre/post)
Formative Assessments:
1. Discussions
2. Daily reflections at the end of class
3. Class work
4. Practice quizzes
5. Student mini-showcase
Summative Assessments:
1.
2.
3.
4.
Quizzes (graded)
Interim assessments
Unit test
Portfolio assignments
TEACHING PLAN
Teaching and Learning Activities:
1. Administer the Pre-Test for Unit 2.
2. Pose this question: “How does ratio reasoning differ from proportional reasoning?”
Have students brainstorm based on previous knowledge.
3. Review ratios.
4. Discuss rate and unit rate. Then have students work in pairs to complete
instructional task/s.
5. Think-Pair-Share: “What is a proportion?”
6. Have students work on instructional tasks to help them learn the key aspects of
proportional reasoning.
7. Check for understanding: How is rate related to proportional reasoning?
8. Conduct the formative assessment lesson from MARS: Developing a Sense of Scale.
This lesson is intended to help assess whether students recognize relationships of
direct proportion and how well they solve problems that involve proportional
reasoning. (Please see attachments.)
9. Discuss how to represent a proportional relationship using a table, a graph, and an
equation.
10. Have students work on instructional tasks to help them identify the constant of
proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions
of proportional relationships.
11. Explain what a point (x, y) on the graph of a proportional relationship means in terms of
a given situation, including the points (0, 0) and (1, r) where r is the unit rate.
12. Have students complete the performance-based assessment task for the unit.
13. Use essential question as a post-assessment. (individual journal entry)
14. Administer the unit test.
Resources Needed:
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IMPACT Curriculum
GLENCOE Math
Chart Paper
http://www.projectpaced.com/index.html
NYC Common Core Library
CALENDAR
Time
Spent on
Standard
Standards
1 week
Compute unit rates
associated with ratios of
fractions, including ratios
of lengths, areas and other
quantities measured in like
or different units.


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.


3 days
Identify the constant of
proportionality (unit rate)
in tables, graphs,
equations, diagrams, and
verbal descriptions of
proportional relationships.

Unit Rate and the Constant of
Proportionality
Lesson 10.2
Proportions and Similarity
Pp. 505-510
3 days
Represent proportional
relationships by
equations. For example, if
total cost t is proportional
to the number n of items
purchased at a constant
price p, the relationship
between the total cost and
the number of items can be
expressed as t = pn.

Representing proportional
relationships by equations
Lesson 10.4
Rates
Pp. 540-550
1 week
Topics To Cover


Ratios of Fractions
Unit Rates
Equivalent ratios
Proportional Relationships
Graphing a proportional
relationship, with special
attention to the points (0,
0) and (1, r) where r is the
unit rate
Main Curriculum
Lesson 10.1
Ratios
Pp. 494-503
Lesson 10.4
Rates
Pp. 540-546
Lesson 8.1
Rates
Pp. 368-387
Lesson 10.1
Ratios
P. 495
Recognizing proportional
relationships in a table and
graphs on a coordinate plane
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5: unit 2: ratios and proportional relationships