Instructional Unit Framework
Part I: Unit Information
Unit Title: Ratio and Proportionality
Grade Level: 6
Time Frame: 3-4 weeks
Prerequisite Knowledge (where this unit sits in a scope and sequence):
Unit Overview
This ratio and proportionality unit focuses on developing three big ideas on ratios:
1. A ratio is a comparison of two quantities by division. Students recognize
the ratio as a relationship different from the quantities that compose it.
2. Proportions are multiplicative comparisons. Students recognize that
proportional relationships derive from multiplication or division rather than
addition or subtraction.
3. Proportional reasoning involves comparing and determining equivalence
as well as using proportions to mathematically model a variety of
situations.
The unit builds from introducing the different types of comparisons students can
make between quantities. Students will consider the differences between
absolute (or additive) comparisons and relative (or multiplicative) comparisons,
with a lens on which comparison is most appropriate in a given situation.
Students will then experience ratio situations involving both part-to-whole and
part-to-part comparisons with a lens on conceptual understanding (i.e., how to
define and set-up these comparisons). Finally, students will work to develop
proportional reasoning abilities by investigating comparisons, determining the
equivalence of ratio, and constructing and applying proportions in a variety of
meaningful contexts using a variety of representations and without
overdependence on algorithms or formulas. Key ideas in the instructional
pathway include:
1. Ratios can be represented by a fraction (e.g., ¼ is one to four)
2. Ratios can compare quantities that are not parts of the same whole (e.g.,
tables to chairs)
3. Ratios can compare two parts of the same whole (e.g., boys to girls in a
class)
4. Proportions are a statement of two or more equivalent ratios
5. Rates are ratios that compare two different quantities with different
measuring units (e.g., miles per gallon)
By the conclusion of this unit of study, students should be able to work fluently
with proportional situations involving a variety of different mathematical
comparisons and be able to recognize the distinction between proportional and
Instructional Unit Framework
non-proportional relationships.
Essential Question:
What is a mathematical comparison and how do I use different
mathematical comparisons to model various situations?
CCSS Mathematical Content Standards Addressed:
6.RP.1 Understand the concept of a ratio and use ratio language to described a
ratio relationship between two quantities
6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical
problems
6.RP.3c: Find a percent of a quantity as a rate per 100; solve problems involving
finding the whole, given the part and percent
CCSS Mathematical Practice Standards Addressed:
MP1: Make sense of problems and persevere in solving them
MP2: Reason abstractly and quantitatively
MP3: Construct viable arguments and critique the reasoning of others
MP4: Model with mathematics
MP5: Use appropriate tools strategically
MP6: Attend to precision
MP7: Look for and make use of structure
MP8: Look for and express regularity in repeated reasoning
*While all eight mathematical practices may not be used in the culminating task, they will all be used by
students at some point within the overall unit of instruction.
Concepts
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




Understanding quantities
Absolute comparisons versus
relative comparisons
Ratio
Proportion
Rate
Inflation
Skills/Performances






Critiquing opinion
Setting up and applying absolute
comparisons
Setting up and applying relative
comparisons
Setting up and applying proportions
Finding percent of a number
Describing the comparisons (including
Instructional Unit Framework
defending why a comparison fits a
given situation)
Part II: Evidence of Understanding
Culminating Performance Task: Budget Mystery
6.RP.1 Understand the concept of a ratio and use ratio language to described a
ratio relationship between two quantities
 The task presents three perspectives: a comparison using subtraction, a
proportion, and the impact of percent change. Students need to
understand ratio relationships in order to determine the comparisons
made by each constituent.
6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical
problems
 The problem context represents a typical budget situation that can apply in
many relevant contexts.
6.RP.3c: Find a percent of a quantity as a rate per 100; solve problems involving
finding the whole, given the part and percent
 Students use simple percent problems to calculate the percent of the
school’s budget that is spent on maintenance in 1993 and 1994 as well as
the effect the 4% inflation from 1993 to 1994 has on the maintenance
budget.
Practice Standards Addressed:
MP1: Make sense of problems and persevere in solving them
 Students must read and digest the problem to understand what is being
asked of them and must identify an entry point (or points) to reason
through each situation. The problem asks students to rectify three points
of view using mathematics to justify each. Student use the available data
to make their argument for each case.
MP2: Reason abstractly and quantitatively
 Students use reasoning to defend each of the three perspectives offered
in the task. They are asked to consider the quantities involved and how
they relate to each other (e.g., what effect does a 4% inflation have on the
maintenance budget?) and work beyond simple calculation
MP3: Construct viable arguments and critique the reasoning of others
 At its core, the task is asking the student to make an argument for each of
the three cases and to critique the reasoning that is set forth in the task. If
students discuss this problem with other students, they would have to
present their argument to their peers and critique their peers’ reasoning.
MP4: Model with mathematics
 This task uses mathematics to model a relevant situation and solve a
problem that could occur in everyday life.
MP6: Attend to precision
Instructional Unit Framework

Communication is at the heart of this task. Students must not only
develop an argument to support (or refute) these cases, they must then
communicate their findings in a way that helps others understand their
viewpoint.
Interim/Formative Performance Tasks
Pre-Assessment: Types of Comparisons
Last week Lisa planted two flowers in her garden, a lily and a daffodil. The lily
was 12 inches tall when she planted it and it now measures 18 inches tall. The
daffodil was 9 inches tall when she planted it and is now 15 inches tall. Which
flower grew more? Use mathematics to support your conclusion.
There are two viable comparisons students can make. Using an absolute
comparison, students can note how much more each flower measures in height
(i.e., both flowers grew 6 inches). Using a relative comparison, students can
note the change in height as it relates to the overall height of the flower (i.e., the
1
2
lily grew by and the daffodil grew by , meaning the daffodil grew more
2
3
relative to its original height).
 Performance Task 1:
Interim
Part to Whole Comparisons
Maya and Janelle were discussing their class. Maya noted that there were more
boys then girls in the class. Janelle stated that there are 15 boys the class. Maya
noted there are 10 girls in the class. What is the ratio of boys to the class? What
is the ratio of girls to the class?
The ratio of boys to girls is 15 to 10. The ratio of girls to boys is 10 to 15. There
are 15 boys in a class of 25. There are 10 girls in a class of 25.
Interim Performance Task 2: Part to Part Comparisons
See Connected Mathematics Program Version 2, Grade 7, Comparing and
Scaling, Investigation 2.1: Mixing Juice
Students look at different part-to-part comparisons that will impact the taste of
juice based on the comparisons of concentrate to water and how to use these
relationships to scale mix to larger total quantities.
Interim Performance Task 3: The Van Problem
The sixth grade students at NF Prep are going on a trip to Zoar Valley. Because
the roads are winding and steep, they can only travel in mini-vans. The rental
company has a small van that can take 8 students or a large van that can hold 10
students. The ratio of small vans to large cans is 5 to 8. If there are exactly
enough vans to transport the 240 students, how many small and large vans are
there?
Instructional Unit Framework
In this problem, the relationship of students to buses needs to be considered
within the context of a total that can be satisfied maintaining all comparative
relationships.
Interim Performance task 4: Rates as Ratios
A father and son run on a track that is ¼ mile long. They each run at a steady
pace, but at different speeds. They start running at the same time from the same
point. The father completes one lap every 3 minutes. The son completes one
lap every 2 minutes. The father runs six laps. How far ahead is the son?
Justify your responses
A rate is a comparison of the measures of two different quantities that have two
different units of measure.
Interim Performance Task 5: Percent
Task 1: 7 million people visited Niagara Falls in 2010. This year there was a 15%
decrease in visitors. How many people visited this year?
Task 2: There are more visitors to Niagara Falls in the summer then in winter. If
the total number of visitors for 2010 was 7 million, and 4.5 million visited in the
summer, what percent of visitors visit in the summer?
Task 3: Visitors to the falls stay in the United States or in Canada. 70% of the
visitors stay in Canada. In 2010, 4.9 million visitors stayed in Canada. How many
total visitors were there?
In this task, students work with simple percent problems in relevant situations.
Part III: Instructional Pathway
Learning Map
Timeframe
Learning Objectives
Week 1: Ratio and
Equivalence
Types of comparisons,
Setting up comparisons,
Linking situations to
different types of
comparisons, Ratio
equivalence
Rates as ratios,
Proportions as a
statement of ratio
equivalence,
Ratio representations:
Week 2: Comparisons of
Ratios
Evidence of
Performance
Pre-Assessment
Interim Task 1
Interim Task 2
Interim Task 3
Interim Task 4
Instructional Unit Framework
Week 3: Scaling and
Complex Ratios
Week 4: Unit in Review
ratio tables, strip
diagrams, number lines,
etc.
Percents and other
complex comparisons
Culminating
performance
Interim Task 5
Culminating TaskBudget Mystery
Journal taskDevelopment of sample
situations that describe a
ratio and a proportion
Part IV: Scaffolding
Students need many opportunities to state comparisons and define these using
numbers and variables. Students need concrete models and diagrams of
comparative situations and to use the models to develop and evaluate situations,
thus developing a mature understanding of ratio and proportional reasoning.
Ratio and proportion are large concepts that frame relationships throughout
algebra. Describing and defining the relationships between quantities and
analyzing presentations of comparisons are all important representations.
Challenges and Barriers
Content / Concept
Barrier
Scaffold
Inability to define the ratio Seeing only absolute
Have students draw a
comparisons
picture of the comparison
and determine why an
absolute comparison isn’t
appropriate in a given
situation
In proportions, while the
Does not recognize that a Error analysis of
comparative units may be proportion must be
proportions.
different, the appropriate created strategically as a Defining proportion
corresponding
comparison of
situations in their own
components must be
corresponding
words
compared
components
Understanding the ratio
Students solve a ratio or
Students need to use the
as its own quantity
proportion problem but
ratio in a sentence
describing a comparison do not understand what
related to the context.
the ratio or proportion
represent.
Universal Design for Learning
Representation
Action and Expression Engagement
 Use pictures in
 Manipulatives
 Have students create
situations
their own ratio and
 Graphic organizers
proportion
 Have tactile
 Sentence starters and
manipulatives
 Reinforce classroom
language objectives
Instructional Unit Framework


available to model
comparative
situations
Use color to highlight
comparison words in
situations
Have students model
comparisons

Culture of a “complete
answer”


Language
Language Objectives:
Sentence Frames:
Academic Language:
Comparison
Ratio
Percent
Proportion
Relationship
Quantity
Rates
Part V: Resources
Texts:
 Connected Math: Comparing and Scaling
 Impact Mathematics Course 2
norms
Ask students to draw
diagrams
Do not always ask for
answers
Instructional Unit Framework
Culminating Task
The Budget Mystery
In 1993, the maintenance budget for a large middle school was $300,000
out of a total budget of $2,500,000. In 1994, the figure was $306,000 out
of a total budget of $2,550,000. Inflation between 1993 and 1994 was 4%.
Different people have different reactions to the 1994 budget:



Parents complain that the money spent on maintenance increased.
The maintenance manager for the school complains that the money
spent for maintenance has decreased.
The principal maintains that, in fact, there has been no change in
the spending patterns at the school.
Could everybody be right?
How could each party use mathematics to justify his/her claim?
Instructional Unit Framework
Budget Mystery Rubric
Criteria of Standard
Evidence of
Meeting Standard
6.RP.1 Understand the
concept of a ratio and use
ratio language to described
a ratio relationship
between two quantities
Accurately compares
the quantities
Uses language of
ratios: compare, “to”,
per.
6.RP.3 Use ratio and rate
reasoning to solve realworld and mathematical
problems
Sets up the correct
ratio comparisons for
each claim.
6.RP.3c: Find a percent of
a quantity as a rate per
100; solve problems
involving finding the whole,
given the part and percent
Is able to use
percent in a ratio
problem to
understand the
impact of inflation on
the value of the
quantities.
Supports each claim
with mathematics
and forms a
conclusion.
MP1 Make sense of
problems and persevere in
solving them
MP2 Reason abstractly
and quantitatively
Sets up the
comparisons
between all
quantities accurately.
MP3 Construct viable
arguments and critique the
reasoning of others
Provides a
justification to
support the overall
question of could
everyone be right,
and evaluates each
claim to support their
conclusion.
Uses mathematical
formulas, diagrams
and connections to
support their claim.
MP4 Model with
Mathematics
Evidence of
Approaching
Standard
Identifies the
quantities to
compare but does
not use the
language of ratio
to support the
comparisons.
Evidence of Below
Standard
Uses the same
comparison for all
quantities.
No evidence of
academic language.
The ratio
comparisons are
incomplete or not
accurate for all
claims.
Correctly
computes the
percent but is
unclear of what
role inflation has
in a ratio.
Ratio comparisons are
incomplete and
inaccurate. May use
subtraction instead of
ratio.
Does not compute
percent or recognize
percent as a ratio.
Supports some
claims with
mathematics but
does not support
all claims.
Student does not
come to a
conclusion.
Comparisons
between
quantities are set
up but not all are
accurate, or not
all comparisons
are made.
Critiques the
reasoning behind
some claims with
mathematics but
not all claims.
Uses the same
strategy to compare all
claims regardless of
whether the strategy
supports the
mathematics.
Uses procedures
to support claims.
Uses procedures
inaccurately and does
not model with
mathematics
Uses comparisons that
are not representative
of the quantities.
Does not support claim
or critique all claims.
Offers no conclusion
Instructional Unit Framework
MP6 Attend to precision
Uses academic
language and
mathematics to
clearly articulate
their claim. There
may be minor
arithmetic errors but
the overall claim is
justified.
Offers a
statement to
defend a point of
view but does not
connect all claims
to mathematics or
support all points
of view.
Offers a statement that
does not align with the
mathematics presented
or the problem. Does
not support all claims.
Instructional Unit Framework
INSERT SAMPLE STUDENT WORK
Instructional Unit Framework
Student Work Annotation
Budget Mystery
Content Standards Addressed:
6.RP.1: Understand the concept of a ratio and use ratio language to describe a
ratio relationship between two quantities.
 Students use ratios to compare the various quantities in the problem
situation.
6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical
problems
 Students can use ratio reasoning to sole a relevant applied problem
6.RP.3c: Find a percent of a quantity as a rate per 100; solve problems involving
finding the whole, given the part and percent
 Students use simple percent problems to calculate the percent of the
school’s budget that is spent on maintenance in 1993 and 1994 as well as
the effect the 4% inflation from 1993 to 1994 has on the maintenance
budget.
Practice Standards Addressed:
MP1: Make sense of problems and persevere in solving them
 Students must read and digest the problem to understand what is being
asked of them and must identify an entry point (or points) to reason
through each situation. The problem asks students to rectify three points
of view using mathematics to justify each. Student use the available data
to make their argument for each case.
MP2: Reason abstractly and quantitatively
 Students use reasoning to defend each of the three perspectives offered
in the task. They are asked to consider the quantities involved and how
they relate to each other (e.g., what effect does a 4% inflation have on the
maintenance budget?) and work beyond simple calculation
MP3: Construct viable arguments and critique the reasoning of others
 At its core, the task is asking the student to make an argument for each of
the three cases and to critique the reasoning that is set forth in the task. If
students discuss this problem with other students, they would have to
present their argument to their peers and critique their peers’ reasoning.
MP4: Model with mathematics
 This task uses mathematics to model a relevant situation and solve a
problem that could occur in everyday life.
MP6: Attend to precision
 Communication is at the heart of this task. Students must not only
develop an argument to support (or refute) these cases, they must then
communicate their findings in a way that helps others understand their
viewpoint.