Mathematics
Ratio, Proportion, and Percent
Seventh Grade: Mathematics
Model Lesson for Unit #2: Ratio, Proportion, and Percent
Overarching Question:
How do you use proportional reasoning to make sense of mathematical and real-world problems?
Previous Unit:
Similar Figures
This Unit:
Next Unit:
Ratio, Proportion, and Percent
Questions to Focus Assessment and Instruction:
1. How can proportional relationships be represented using
tables, graphs, and algebraic equations?
2. When quantities have different measurements how can they
be compared?
3. Why is ratio a good means of comparison?
Key Concepts:
constant of proportionality
proportion
direct variation
constant rate of change
Positive and Negative
Numbers
Intellectual Processes (Standards for
Mathematical Practice):
Look for and make use of structure:
Use tables to find the constant of
proportionality for real-world problem
situations.
Model with Mathematics: Translate
among verbal, tabular, graphic, and
algebraic descriptions of directly
proportional relationships.
ratio
linear equation
Lesson Abstract
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Mathematics
Ratio, Proportion, and Percent
Students will measure the length and width of a rectangle using both standard and non-standard
units of measure. In addition to providing measurement practice, this lesson allows students to
discover that the ratio of length to width of a rectangle is constant, in spite of the units. For many
middle school students, this discovery is surprising.
Common Core State Standards
Ratios and Proportional Relationships (7.RP)________________________________________
Analyze proportional relationships and use them to solve real-world and mathematical
problems.
1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and
other quantities measured in like or different units. For example, if a person walks 1/2 mile in
each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently
2 miles per hour.
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.
4. Use ratios, fractions, differences, and percents to form comparison statements in a given
situation, such as: “Which model of car has the best fuel economy?”, “What percent of girls play
basketball?”, “What is the ratio of boys to girls?”.
Expressions and Equations (7.EE)_________________________________________________
Solve real-life and mathematical problems using numerical and algebraic expressions and
equations.
4. Use variables to represent quantities in a real-world or mathematical problem, and construct
simple equations and inequalities to solve problems by reasoning about the quantities.
Instructional Resources: Rulers (both cm and in), alternate units of measure (pennies, beads,
paper clips, dice, pattern blocks (all the same shape), M & M’s, bingo chips, sticks of gum,
unsharpened pencils, crackers)
Sequence of Lesson Activities
Lesson Title: Constant Dimensions (http://illuminations.nctm.org/LessonDetail.aspx?id=L572)
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Mathematics
Ratio, Proportion, and Percent
Selecting and Setting up a Mathematical Task:

By the end of this lesson
what do you want your
students to understand,
know, and be able to do?
•
Measure with both rulers (cm and in) and alternate units.
•
Use a linear graph to model, analyze, and make predictions.
•
Draw conclusions about the relationship of two dimensions based on
collected data.
•
Identify proportional relationships.
•
Plot two variables as points on a scatterplot and identify the meaning of
these points.
•
Determine a rule relating two variables that are proportional.

In what ways does the task
build on student’s previous
knowledge?
•
Students have studied ratios in grade 6. They will use their knowledge of
ratios to determine equivalent relationships when various objects are used
to measure the length and width of a rectangle.

What questions will you ask
to help students access
their prior knowledge?
•
How can you compare two numbers mathematically? What information will
you learn in each case? Which of these ideas will give you the most
information about the connection between pairs of data?
•
Give students rulers that have both English and metric units. Choose an
object that all students have access to, such as their math book. Ask
students to measure the shortest side of this object. Collect and display
student answers. Discuss discrepancies in the values, by asking “Why
didn’t everyone get the same number?” What could be some reasons for
differences in the measurements?
•
Provide each student with a stack of pennies and have them measure the
same object using them. Select students to demonstrate how they did their
measuring. Did anyone get a different result or use the pennies differently
to get their measurement? Encourage other students to share their
method. How is measuring with pennies the same as measuring with a
ruler? What is different about measuring with the pennies? Did you have
any problems measuring? If so, what did you do to resolve the problem?
Launch:

How will you introduce
students to the activity so
as to provide access to all
students while maintaining
the cognitive demands of
the task?
Supporting Student’s Exploration of the Task:


What questions will be
asked to focus students’
thinking on the key
mathematics ideas?
What questions will be
asked to assess student’s
understanding of key
mathematics ideas?
•
How does the size of the measuring object relate to the location of the
(length, width) pair on the scatterplot?
•
Do the points on your graph appear to form any pattern?
•
What remains constant even as the units of measurement change?
•
What algebraic rule could be used to represent the ordered pairs? What
equation shows how the length and width are related for this rectangle? Can
this rule be written in the form length : width = _____ : _____?
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Mathematics
Ratio, Proportion, and Percent

How will you extend the
task to provide additional
challenge?
•
•
Give students a scatterplot or table of values of (length,width) and ask them
to determine an algebraic rule that represents the relationship.
Provide other sizes of rectangles and ask students if the ratio is the same
for every rectangle. Students should see that rectangles with the same ratio
have the same shape, leading to the concept of similarity.
Sharing and Discussing the Task:

What specific questions will
be asked so that all
students will:
o
o

•
Although more pennies were used than M&M’s when measuring the
width, did the size of the width actually change? Although more pennies
were used than M&M’s when measuring the width, did the size of the width
actually change?
Make sense of the
mathematical ideas
that you wanted them
to learn?
•
Expand on, debate,
and question the
solutions being
shared?
If someone used gumballs to measure the length and width, and their
ordered pair were placed at (22, 10), would we suspect that they made a
good measurement? What if the ordered pair had the coordinates (16,
10.5)? What is your reasoning?
•
If the length of the rectangle is 13 wooches, what is the width of the
rectangle in wooches using (1) your line of best fit and (2) your algebraic
rule?
•
What unit of measure could be used that would have produced a
point very close to the origin? Why?
o
Make connections
between the different
strategies that are
presented?
o
Look for patterns?
o
Begin to form
generalizations?
What will be seen or heard
that indicates all students
understand the
mathematical ideas you
intended them to learn?
Students should recognize that the relationship between the sides of an object
remains constant regardless of the unit of measure. This constant of
proportionality is a key component in this unit.
Formative Assessment:
Provide students with a number of graphs similar to the one they created, but using different ratios. Have students
write a summary of conclusions that can be drawn from each graph.
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