Mathematics
Meaning of Fractions, Decimals, Ratios, and Percents
Sixth Grade: Mathematics
Unit #2: Meaning of Fractions, Decimals, Ratios, and Percents
Overarching Question:
How can reasoning with ratios and rates help solve real world and mathematical problems?
Previous Unit:
Factors and Multiples
This Unit:
Meaning of Fractions, Decimals, Ratios,
and Percents
Next Unit:
Two-Dimensional
Geometry
Questions to Focus Assessment and Instruction:
Intellectual Processes (Standards for
Mathematical Practice):
1. How are ratios and percents alike? How are they different?
Model with mathematics: Solve
2. What process do you use to make a scale model of a real-life object?
problems involving comparisons of
3. How do you determine a unit rate given a table of values?
amounts and measurements.
4. What are the differences between converting measurements in the metric
Attend to precision: Convert units of
and standard measurement systems?
measure within the metric or standard
5. Given the quantity and price of two objects, how can you determine which
system with an appropriate degree of
one is the better buy?
precision.
6. How are fractions, decimals, and percents related?
Make sense of problems and persevere
in solving them: Use ratio and rate
reasoning to solve real-world and
mathematical problems
Key Concepts:
rate
ratio decimals
convert measurement units
equivalent fractions
rate language(per, for every) ratio language(a/b, a:b, a to b)
percent
unit rate
equivalent ratios
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Mathematics
Meaning of Fractions, Decimals, Ratios, and Percents
Unit Abstract
Relationships between fractions, decimals, and percents are developed in this unit. Students learn
how these forms are related to one another and make decisions about when to use each form.
Various models such as fraction strips, percent bars, and number lines are used to develop
conceptual understanding. There is an emphasis on understanding and applying these
relationships to solve real-world problems such as unit rate, measurement conversions, constant
rate, and comparison of quantities. Fractions are viewed as rates, ratios, or parts of a proportion to
provide underpinnings needed in seventh grade for work with proportional reasoning.
Common Core State Standards
Ratios and Proportional Relationships (6.RP)________________________________________
Understand ratio concepts and use ratio reasoning to solve problems.
1. Understand the concept of a ratio and use ratio language to describe a ratio relationship
between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo
was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received,
candidate C received nearly three votes.”
2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate
language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups
of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for
15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade
are limited to non-complex fractions.)
3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning
about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
a. Make tables of equivalent ratios relating quantities with whole number measurements, find
missing values in the tables, and plot the pairs of values on the coordinate plane. Use
tables to compare ratios.
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times
the quantity); solve problems involving finding the whole, given a part and the percent.
d. Use ratio reasoning to convert measurement units; manipulate and transform units
appropriately when multiplying or dividing quantities.
The Number System (6.NS)_______________________________________________________
Apply and extend previous understandings of numbers to the system of rational numbers.
6. Understand a rational number as a point on the number line. Extend number line diagrams and
coordinate axes familiar from previous grades to represent points on the line and in the plane
with negative number coordinates.
c. Find and position integers and other rational numbers on a horizontal or vertical number
line diagram; find and position pairs of integers and other rational numbers on a coordinate
plane.
*d. Develop an understanding of fractions, decimals, and percents and the relationships
between and among the concepts and their representations.
*e. Understand and use equivalent fractions.
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Mathematics
Meaning of Fractions, Decimals, Ratios, and Percents
*f. Compare and order fractions.
*g. Model situations involving fractions, decimals and percents and use physical models
(fractions strips, number line, partition, grid area, percent bar models) and drawings to
help reason about the situations.
*h. Develop and use benchmarks such as 0, ½, 1, and 1 ½ to help estimate the size of a
number or sum and relate different forms of representations for these benchmarks (e.g.
50% is the same as 0.5 and ½)
* Indicates items added to CCSS
Instructional Resources
NCTM Illuminations (http://illuminations.nctm.org)
Understanding Rational Numbers and Proportions: In this lesson, students use real
world models to develop an understanding of fractions, decimals, unit rates, proportions, and
problem solving.
http://illuminations.nctm.org/LessonDetail.aspx?id=L284
Bean Counting and Ratios: By using sampling from a large collection of beans, students
get a sense of equivalent fractions, which leads to a better understanding of proportions.
Equivalent fractions are used to develop an understanding of proportions.
http://illuminations.nctm.org/LessonDetail.aspx?id=L722
Feeding Frenzy: In this activity, students will multiply and divide a recipe to feed groups of
various sizes. Students will use unit rates or proportions and think critically about real world
applications of a baking problem.
http://illuminations.nctm.org/LessonDetail.aspx?id=L781
Capture-Recapture: In this lesson, students experience an application of proportion that
scientists actually use to solve real-life problems. Students learn how to estimate the size of
a total population by taking samples and using proportions. The ratio of “tagged” items to the
number of items in a sample is the same as the ratio of tagged items to the total population.
http://illuminations.nctm.org/LessonDetail.aspx?id=L721
Grid It and Percent It: In this lesson, students use a 10 × 10 grid as a model for solving
various types of percent problems. This model offers a means of representing the given
information as well as suggesting different approaches for finding a solution.
http://illuminations.nctm.org/LessonDetail.aspx?id=L249
Fractional Clothesline: In this lesson, a string will be stretched across the classroom and
various points will be marked for 0, 1, 2, 3, and 4. This classroom number line will be used to
show that all proper fractions are grouped between 0 and 1, and that improper fractions or
mixed numbers are all grouped above 1. Students clip index cards with various proper
fractions, improper fractions, and mixed numbers on the clothesline to visually see
groupings. Students then play an estimation game with groups using the same principle.
Encouraging students to look at fractions in various ways will help foster their conceptual
fraction sense.
http://illuminations.nctm.org/LessonDetail.aspx?id=L784
Now and Then: In this activity, students work in groups as contestants on a fictitious game
show, Now and Then. They use their knowledge of percent computations and percent
change to answer each game show question. As a culminating activity, the students create
their own game show and host it in front of the rest of the class.
http://illuminations.nctm.org/LessonDetail.aspx?id=L837
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Mathematics
Meaning of Fractions, Decimals, Ratios, and Percents
Real Estate Tycoon: This article describes a year-long project. Students design, "build,"
and "sell" a house; after which they simulate investment of the profits in the stock market.
Along the way, students make scale drawings, compute with fractions and decimals in
various contexts, and even solve simple equations. This lesson plan was adapted from an
article by David B. Smith, which appeared in the September 2000 edition of Mathematics
Teaching in the Middle School.
http://illuminations.nctm.org/LessonDetail.aspx?id=L279
Equivalent Fractions: Using this applet, students create equivalent fractions by dividing
and shading squares or circles, and match each fraction to its location on the
number line.
http://illuminations.nctm.org/activitydetail.aspx?id=80
Fraction Models: Use this computer applet to explore different representations for fractions
including improper fractions, mixed numbers, decimals, and percentages. Additionally,
there are length, area, region, and set models. Adjust numerators and denominators to see
how they alter the representations and models. Use the table to keep track of interesting
fractions.
http://illuminations.nctm.org/activitydetail.aspx?ID=11
Texas Instruments
Different Looks, Same Values (TI-73): In this activity, students explore the meaning and
purpose of equivalent fractions. They also practice writing fractions that meet a given criteria.
http://education.ti.com/calculators/timiddlegrades/US/Activities/Detail?s=Math&id=11731
What’s the Unit of Choice (TI-73): In this activity, students will select appropriate units for
given situations. Then they will use the Convert menu to convert measures from metric to
customary and from a small unit to a larger unit of the same system.
http://education.ti.com/calculators/timiddlegrades/US/Activities/Detail?s=Math&id=12290
Percents – Small and Large (TI-73): In this activity, students will explore percents greater
than 100% and less than 1%. Students will use a 100 grid to represent percents and convert
between fractions, decimals, and percents.
http://education.ti.com/calculators/timiddlegrades/US/Activities/Detail?s=Math&id=11268
TEXTEAMS Rethinking Middle School Mathematics: Proportionality
http://www.mansfieldisd.org/curriculum/mathematics/pdf/6th/texteams%20Proportionality.pdf
Here are several examples of the activities available:
Trains: Investigate properties of proportional relationships using tables, graphs, and
equations.
http://math.springbranchisd.com/middle/resources/proportionality/trains.pdf
Lost and Gained: Manipulatives, tables, and graphs are used to demonstrate the
proportional relationship between equivalent measures.
http://www.crowley.k12.tx.us/curriculum_and_instruction/Math/7TH/data_files/TT_Activit
ies/lostandgained.pdf
Thinking Rationally about Fractions, Decimals, and Percent: Instructional Activities for
Grades 4 through 8: This resource, developed by the Virginia Department of Education, has a
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Mathematics
Meaning of Fractions, Decimals, Ratios, and Percents
large number of activities and lessons to provide instruction to middles grades students in
fractions, decimals and percent.
http://www.doe.virginia.gov/instruction/mathematics/elementary/think_rationally_fraction
s_dec_prcnt.pdf
Other Resources
Percent Bars: This site provides a model for students to make a foldable manipulative
comparing fractions and percents.
http://lib.store.yahoo.net/lib/yhst-29588664682132/percent-bar-lesson.pdf
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
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/
Proportion Land: This activity provides a variety of unit rate activities in a carnival setting.
Students use proportions during investigations at the carnival. A class could be divided into 8
groups with each group completing an investigation and reporting the results to the rest of
the class. Students complete tables and graph the results.
http://www.aptv.org/aptplus/flashlibrary/main-menu.htm
Kool-Aid Experiment: Middle School students have had wide-ranging experiences
with fractions and operations on fractions, but ratios and rates may be new concepts
for them. This activity introduces students to the idea of rates via a familiar and
tangible product: Kool-Aid. Students will add varying amounts of Kool-Aid powder to a
set amount of water, thereby producing a sensory experience of rates. The Kool-Aid
example also provides a base experience upon which other examples can build.
http://alex.state.al.us/lesson_view.php?id=9860
Pennies, Pennies, Everywhere!: During this lesson, students will use ratios and
proportions to find perimeter, area, and volume. Students will estimate the number of
pennies it takes to fill their classroom, then perform a series of measurements and
calculations involving perimeter, area, and volume. They will then use these measurements
and calculators to help them determine the room's penny capacity.
http://www.thinkport.org/Tools/ContentViewer/ContentPreview.aspx?ContentID=c6d2d31f7ef2-46e2-9d52-5371666019ab
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
Geogebra Applets: A variety of applets to explore decimals, fractions, and percents
http://206.110.20.132/~dhabecker/geogebrahtml/#FDP
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Mathematics
Meaning of Fractions, Decimals, Ratios, and Percents
Seeing Math: This online activity asks students to resize objects to fit into a dollhouse.
http://seeingmath.concord.org/Interactive_docs/PR_Activity.pdf
Math Forum: Clearinghouse of ratio and proportion activities for 6th grade.
http://mathforum.org/mathtools/cell/m6,8.9,ALL,ALL/
Middle School Portal: Here you will find games, problems, and lesson ideas that can help
6th grade students extend their understanding of equivalent fractions and their mastery of
whole-number multiplication and division to the concepts of ratio and rate.
http://msteacher.org/epubs/math/math14/ratios.aspx
Students explore ratios, proportions, and percents apply these in solving real-life problems.
http://www.saskschools.ca/curr_content/mathcatch/ratio/ratio_c/body/image_student.html
Visual Fractions: A tutorial that models fractions with number lines or circles.
http://www.visualfractions.com/
Proportional Reasoning Math Skills- Scale City: This site is for grades 6-8 and provides a
variety of fun activities using proportional reasoning. Each activity provides an entertaining
video of real-life uses for proportions related to the name of the activity. There is also an
assessment covering all the ideas in the site, as well as assessments in the teacher
resource section for each activity. Activities include: Dinosaur World (one-dimensional
scaling), World of Mural Painting (two dimensional scaling), Miniature Land (three
dimensional scaling), World Chicken Festival (scaling recipes), Louisville Slugger Museum
(Similar Triangles), Sky-Vue Drive-In (Introducing Inverse Proportions), Belle of Louisville
(proportions and Music), Kentucky Horse park (Speed, Distance, and Time)
http://www.ket.org/scalecity/index.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
Use a jeopardy style game format for from one to four players to convert between decimals,
fractions, and percents.
http://www.math-play.com/Fractions-Decimals-Percents-Jeopardy/fractions-decimalspercents-jeopardy.html
Fresh Baked Fractions: Out of a group of 4 fractions, select the one that is not equivalent.
http://www.funbrain.com/fract/
Assessments
This website provides practice problems and randomly generated questions for the students
to test their understanding of a wide variety of sixth grade topics (including rates, ratios, and
proportions.)
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Mathematics
Meaning of Fractions, Decimals, Ratios, and Percents
http://www.math6.org/test_prep.htm
Helpwithmath.com: Writing ratios.
http://www.helpingwithmath.com/printables/worksheets/ratio_proportion/6rp2ratio_rates01.htm
http://www.helpingwithmath.com/printables/worksheets/ratio_proportion/wor0601ratio01.htm
Assessment questions working with ratio and rate.
http://www.helpingwithmath.com/printables/worksheets/ratio_proportion/6rp2ratio_rates02.htm
Algebra Lab: From Algebra Lab, an eight question assessment on the topic of rate of
change per unit.
http://www.algebralab.org/practice/practice.aspx?file=Algebra1_2-8.xml
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.
Moss, J. and Caswell, B. (2004). Building Percent Dolls: Connecting Linear Measurement to
Ratio and Proportion. Mathematics Teaching in the Middle School, 10(2), 68-74. Retrieved
March 7, 2011 from
http://www.nctm.org/eresources/article_summary.asp?from=B&uri=MTMS2004-09-68a
Dwyer, N., Causey-Lee, B., and Irby, N. (2003). Conceptualizing Ratios with Look-Alike
Polygons. Mathematics Teaching in the Middle School, 8(8), 426-431. Retrieved March 7,
2011 from
http://www.nctm.org/eresources/article_summary.asp?from=B&uri=MTMS2003-04-426a
Drum, R. and Petty, W. (2001). Miniature Toys Introduce Ratio and Proportion with a RealWorld Connection. Mathematics Teaching in the Middle School, 7(1), 50-54. Retrieved
March 7, 2011 from
http://www.nctm.org/eresources/article_summary.asp?from=B&uri=MTMS2001-09-50a
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Mathematics
Meaning of Fractions, Decimals, Ratios, and Percents
Sharp, J., and Adams, B. (2003). Using a Pattern Table to Solve Contextualized Proportion
Problems. Mathematics Teaching in the Middle School, 8(8), 432-439. Retrieved March 7,
2011 from
http://www.nctm.org/eresources/article_summary.asp?from=B&uri=MTMS2003-04-432a
Weinberg, S., Hammrich, P., and Bruce, M. (2003). The Giants Projects. Mathematics
Teaching in the Middle School, 8(8), 406-413. Retrieved March 7, 2011 from
http://www.nctm.org/eresources/article_summary.asp?from=B&uri=MTMS2003-04-406a
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