```Mathematics
Ratio, Proportion, and Percent
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:
Questions to Focus Assessment and Instruction:
1. How can proportional relationships be represented using
tables, graphs, and algebraic equations?
2. How can proportions be used to find unknown quantities or
inaccessible measurements?
3. When quantities have different measurements how can they
be compared?
4. How do you determine the percentage change for a given
situation?
5. Why is ratio a good means of comparison?
Key Concepts:
constant of proportionality
Positive and Negative
Numbers
Ratio, Proportion, and Percent
proportion
direct variation
constant rate of change
percent
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.
Reason abstractly and quantitatively:
Solve problems using contexts involving
percents and proportions.
Model with Mathematics: Translate
among verbal, tabular, graphic, and
algebraic descriptions of directly
proportional relationships.
ratio
linear equation
Page 1 of 7
2/8/16
Mathematics
Ratio, Proportion, and Percent
Unit Abstract
Students studied rates and ratios in sixth grade. In this unit, students reason proportionally. They
use ratios as a basis of comparison between two sets of data. They observe related data in the
form of a table and look for patterns connecting these data values. Plotting the paired data points
to see a graphical representation, and writing an equation that shows the relationship of the data in
the table further strengthens this understanding. When the change observed in the table is
constant, students connect to a linear graph. This demonstrates a proportional relationship across
multiple representations and deepens the understanding of these characteristics. The unit rate
studied in grade six is now a focus of rate of change used in writing linear equations in grade
seven.
Other concepts in this unit include solving problems to find an unknown part of a proportion and
applying proportional reasoning to real-world contexts. Students think proportionally in such
situations as calculating sales taxes, interest, and commissions; scale drawings; and unit pricing.
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.
3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple
interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and
decrease, percent error.
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?”.
5. Decide when the most informative comparison is the difference between two quantities and
when it is a ratio between pairs of quantities. For example, should Biggest Loser contestants
Page 2 of 7
2/8/16
Mathematics
Ratio, Proportion, and Percent
be judged on the amount of pounds lost, or the percent of body weight lost?
Expressions and Equations (7.EE)_________________________________________________
Use properties of operations to generate equivalent expressions.
1. Apply properties of operations as strategies to add, subtract, factor, and expand linear
expressions with rational coefficients.
2. Understand that rewriting an expression in different forms in a problem context can shed light on
the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that
“increase by 5%” is the same as “multiply by 1.05.”
Solve real-life and mathematical problems using numerical and algebraic expressions and
equations.
3. Solve multi-step real-life and mathematical problems posed with positive and negative rational
numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply
properties of operations to calculate with numbers in any form; convert between forms as
appropriate; and assess the reasonableness of answers using mental computation and
estimation strategies. For example: If a woman making \$25 an hour gets a 10% raise, she will
make an additional 1/10 of her salary an hour, or \$2.50, for a new salary of \$27.50. If you want
to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will
need to place the bar about 9 inches from each edge; this estimate can be used as a check on
the exact computation.
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.
Geometry (7.G)_________________________________________________________________
Draw, construct, and describe geometrical figures and describe the relationship between
them.
1. Solve problems involving scale drawings of geometric figures, including computing actual
lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
Instructional Resources
NCTM Illuminations (http://illuminations.nctm.org)
Constant Dimension: 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.
http://illuminations.nctm.org/LessonDetail.aspx?id=L572
Understanding Rational Numbers and Proportion: In this lesson, students use real-world
models to develop an understanding of fractions, decimals, unit rates, proportions, and
problem solving. The three activities in this investigation center on situations involving
rational numbers and proportions that students encounter at a bakery.
Page 3 of 7
2/8/16
Mathematics
Ratio, Proportion, and Percent
http://illuminations.nctm.org/LessonDetail.aspx?id=L284
Bagel Algebra: A real-life example—taken from a bagel shop, of all places—is used to get
students to think about proportional reasoning. Students must decipher a series of
equations and interpret results to understand the point that the bagel shop’s owner is trying
to make.
http://illuminations.nctm.org/LessonDetail.aspx?id=L662
Measuring Up: This unit explores the concepts of proportional reasoning, ratio, and indirect
measurement. Students engage in a variety of activities that involve taking their own
measurements, exploring different ratios, and examining similar figures. Students convert
measurements in to customary and metric units. These activities immerse students in
problem solving, reasoning, and making connections to real-life situations.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L511
Scaling Away: Students will measure the dimensions of a common object, multiply each
dimension by a scale factor, and examine a model using the multiplied dimensions.
Students will then compare the surface area and volume of the original object and the
enlarged model.
http://illuminations.nctm.org/LessonDetail.aspx?id=L584
Walk the Plank: When one end of a wooden board is placed on a bathroom scale and the
other end is suspended on a textbook, students can "walk the plank" and record the weight
measurement as their distance from the scale changes. The results are unexpected— the
relationship between the weight and distance is linear, and all lines have the same
x‑ intercept. This investigation leads to a real world occurrence of negative slope, examples
of which are often hard to find.
http://illuminations.nctm.org/LessonDetail.aspx?id=L682
Texas Instruments
Step Up: In this activity, students will explore the concept of slope. They will measure the
vertical height (rise) and horizontal length (run) of a set of stairs then find the ratio to
describe the steepness. A graph will then be created to show the relationship .
647
What’s Your Benchmark: This activity introduces students to the importance of
benchmarks when estimating measurements. Students will also use the Convert menu to
change from one unit to another.
291
Two Friends’ Methods: Students will investigate the decrease in items as a percent
change while considering several methods for calculating percent change. They will also
develop a procedure that can be repeated to solve problems.
6sa%3d1008%26t%3d1130%26d%3d1%26size%3d15%26page%3d4
Other Resources
Calculation Nation: The games of Calculation Nation® are organized around content from
the upper elementary and middle grades math curriculum. By becoming a citizen of
Calculation Nation®, your child or student will play online math strategy games that allow
Page 4 of 7
2/8/16
Mathematics
Ratio, Proportion, and Percent
them to learn about fractions, factors, multiples, symmetry and more, as well as practice
important skills like basic multiplication and calculating area — all while having fun.
http://calculationnation.nctm.org/
Figure This! Math Challenges for Families has a variety of problems for students to solve.
There is a Math Index, which sorts the problems by math content.
http://www.figurethis.org/index.html
Middle School Portal: The problems here deal with ratio, in the concrete as well as the
abstract. Seventh grade students will make actual scale models with paper or clay and find
percentages in real-world situations. But they will also work hands-on with online images
that make visual the abstractions of ratio and percentage.
http://msteacher.org/epubs/math/math15/ratio.aspx
Math Forum: Clearinghouse of ratio and proportion activities for 7th grade.
http://mathforum.org/mathtools/cell/m7,9.10,ALL,ALL/
Math.com: An online lesson with explanation, examples, and practice problem. Topics
include ratio, proportion, distance -rate-time, and similar figures. A unit assessment is also
included.
http://www.math.com/school/subject1/lessons/S1U2L1GL.html
Ratios for All Occasions: Features resources on the concept of the ratio as encountered
in middle school: as rates in real-world problems, percents in relation to fractions, scale
factors in building models, and comparisons of lengths in geometry. Most of these digital
resources are activities that can serve as supplementary or motivational material.
http://msteacher.org/epubs/math/ratios/math.aspx
Scale Drawings Resources: This resource provides twenty-four different activities for use
with middle school students to allow them to experience scale factors (dilations) in realworld contexts.
http://msteacher.org/return_list_math.aspx?id=366
Proportional Relationships and Unit Rates: The activities that follow interpret
proportional relationships using a unit-rate approach.
http://www.cehd.umn.edu/rationalnumberproject/89_4.html
TEXTEAMS Rethinking Middle School Mathematics: Proportionality
Trains: Investigate properties of proportional relationships using
tables, graphs, and equations.
http://math.springbranchisd.com/middle/resources/proportionality/trains.pdf
Restless Rectangles: Investigate the constants of proportionality within similar shapes as
“shape ratios” and the scale factors between pairs of similar shapes as “size ratios” for a set
of similar rectangles and use these ratios to solve problems.
http://math.springbranchisd.com/middle/resources/proportionality/restless_rectangles.pdf
Jet Ski Rental: Distinguish between a proportional and non-proportional situation using the
Page 5 of 7
2/8/16
Mathematics
Ratio, Proportion, and Percent
characteristics of a proportional relationship.
eeks/Proportionality/jetskirental.pdf
Lost and Gained: Use ratios and unit rates as conversion factors between measurement
units and rates.
http://www.crowley.k12.tx.us/curriculum_and_instruction/Math/7TH/data_files/TT_Acti
vities/lostandgained.pdf
Professional Resources
NCTM (www.nctm.org)
Yearbook: NCTM's 2002 Yearbook emphasizes that although fractions, ratios, and
proportions are pivotal concepts in the middle school, their development and
understandings begin in the elementary school. The companion booklet presents activities
that illustrate some of the ideas in the yearbook and that go beyond the content of the
yearbook itself. Teachers' notes and handouts are designed to bring the yearbook to life in
the classroom.
http://www.nctm.org/catalog/product.aspx?ID=12244
Essential Understanding Series: Developing Essential Understanding of Ratios,
Proportions, and Proportional Reasoning for Teaching Mathematics: Grades 6-8. This book
goes beyond a simple introduction to ratios, proportions, and proportional reasoning. It will
help broaden and deepen your mathematical understanding of one of the most challenging
topics for students.
http://www.nctm.org/catalog/product.aspx?ID=13482
Articles from National Council of Teachers of Mathematics (www.nctm.org)
Articles are available as free downloads to NCTM members, or for a fee to non-members.
Seeley, C., and Schielack, J., (2007). A Look at the Development of Ratios, Rates, and
Proportionality. Mathematics Teaching in the Middle School, 13(3), 140-142. Retrieved
March 7, 2011 from
http://www.nctm.org/eresources/article_summary.asp?from=B&uri=MTMS2007-10-140a
Attia, L., (2003). Using School Lunches to Study Proportion. Mathematics Teaching in the
Middle School, 9(1), 17-21. Retrieved March 7, 2011 from
http://www.nctm.org/eresources/article_summary.asp?from=B&uri=MTMS2003-09-17a
Lanius, C. and Williams, S., (2003). Proportionality: A Unifying Theme for the Middle
Grades. Mathematics Teaching in the Middle School, 8(8), 392-396. Retrieved March 7,
2011 from
http://www.nctm.org/eresources/article_summary.asp?URI=MTMS2003-04-392a&from=B
Chapin, S., and Anderson, N., (2003). Crossing the Bridge to Formal Proportional
Reasoning. Mathematics Teaching in the Middle School, 8(8), 420-425. Retrieved March 7,
2011 from
http://www.nctm.org/eresources/article_summary.asp?from=B&uri=MTMS2003-04-420a
Page 6 of 7
2/8/16
Mathematics
Ratio, Proportion, and Percent
Buhl, D., Oursland, M., and Finco, K., (2003). The Legend of Paul Bunyan: An Exploration in
Measurement. Mathematics Teaching in the Middle School, 8(8), 441-448. Retrieved March
7, 2011 from
http://www.nctm.org/eresources/article_summary.asp?from=B&uri=MTMS2003-04-441a
Page 7 of 7
2/8/16
```