2012 CLIME Summer Math Institute Curriculum
1. Day 1: Introduction to Ratios
 Situating Proportional Reasoning
 Defining Terms
 Relative vs. Absolute Change
 Classifying /Expressing Ratios
 Part-part-whole
 Associated sets
 Rates/well-known measures
 Growth (stretching and shrinking situations)
 Percents
 Ratios vs. Rates
 Representing ratios (percents too)
 Tape diagrams
 Double number line diagrams
 Tables
 Pie chart diagrams
 Graphs
 Equations
2. Day 2: Comparing Ratios
 Equivalent ratios vs. ratios that are not equivalent
 Equivalent ratios vs. equivalent fractions.
 Distinguishing between proportional and non proportional situations
 How do I know it when I see it?
 Defining proportions
 Solving proportion problems
3. Proportional Reasoning and Percents/Fractions
 Revisiting percents as ratios
 Solving percent, ratio and fraction word problems using proportional reasoning
4. Pedagogical Content Knowledge for Proportional Reasoning
 Types of Proportional reasoning questions:
There are four broad types of tasks that can be used to assess proportional reasoning:







Missing Value Problems
Numerical Comparison Problems
Qualitative Prediction Problems
Qualitative Comparison Problem.
Assessing proportional reasoning (see blue folder for bookmarked article)
Levels of Thinking (see 30 page paper and bookmarked page on blue folder)
Analyzing students’ work
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Proportional Reasoning
Title
Goals
Standard
Addressed
by the Unit
Introduction to Ratio Sense
By the end of this session, the teachers will be able to:
 Situate proportional reasoning in the curriculum.
 Distinguish relative from absolute change.
 Distinguish ratios from rates.
 Utilize multiple representations to represent ratios
California Standards:
Common Core Standards: Understand ratio concepts and use ratio
reasoning to solve problems.
6.RP.1 Understand the concept of ratio and use ratio language to
describe a ratio relationship between two quantities.
6.RP.2 Understand the concept of unit rate a/b associated with a:b with
b ¹ 0 , and use rate language in the context of ratio relationships.
6.RP.3Use ratios and rate reasoning to solve real-word and mathematical
problems, e.g., be reasoning about tables of equivalent ratios, tape
diagrams, double number line diagrams, or equations.
a. Make tables of equivalent ratios relating quantities with wholenumber measurements, find the missing values in the tables,
and plot the pairs of values on the coordinate plane. Use tables
to compare ratios. Use tables to compare ratios.
b. Solve unit rate problems including those involving unit pricing
and constant speed.
c. Find a percent of a quantity as a rate per 100; 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.
Handouts (Smart board), Color Tiles, Grid paper, Ruler
Materials
for Teacher
Materials
Color pencils, Color Tiles, Grid paper, Ruler
for Students
Introduction to Ratio Sense
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What is Proportional Reasoning?
 Research into student understanding of rational numbers indicates that students who learn in a
rich environment that requires problem solving, conversations about mathematics, and higher
cognitive demands learn well (Lamon, 2001).
 Students who study rational numbers in an environment where learning is by rote or limited
meanings are less able to transfer or extend their learning.
 Fractions fall in a general category of reasoning called “Proportional Reasoning” which has the
components shown on the diagram below.
Figure 11: Components of Proportional Reasoning
 Researchers claim that it is impossible to teach proportional reasoning directly because “… in
short, the whole is greater than the sum of parts” (Lemon, 1999).
 The development of proportional reasoning is not an all-or-nothing affair; rather, competence in
this kind of reasoning grows over a long period of time and in several dimensions (Lamon, 1999).
 The operating theory for teaching proportional reasoning is that “by providing children
experiences with some of the critical components proportional reasoning before proceeding to
the more abstract, formal presentations, we increase their chances of developing proportional
reasoning” (Lamon, 1999)
1
Source: Lamon, S.J. (1999). Teaching Fractions and Ratios for Understanding.
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Description of Components of Proportional Reasoning (Adapted from Lamon, 1999)
Components of Proportional Reasoning
Relative Thinking
Description
Relative thinking can be described as thinking
multiplicatively. It is important for students to
understand both absolute change and relative
change. Making the transition from absolute
change to relative change is an important step in
transitioning from additive to multiplicative
reasoning.
Sam
Sally
Partitioning
The partitioning of an object is the process of
dividing the object into a number of disjoint parts
that collectively make the whole.
!
Unitizing
Example: Sam the snake is 4 feet long. When he is
fully grown, he will be 8 feet long. Sally the snake
is 5 feet long. When she is fully grown, she will be
9 feet long. Which snake is closer to being fully
grown? Explain how you know.
When dealing with fractions, to determine
fractional parts, one partitions the object into
parts of equal size.
Unitizing is the cognitive process used to assign a
unit to a given quantity.
Example: When asked to think about a case of
soda, do you picture 24 cans, two 12- packs, or
four 6-packs [Lamon, 2006]?
It is desirable for students to build flexibility in
determining the size of the chunk of a quantity that
they use for a unit, as different situations may call
for different sized chunks.
Asking students to explain their choices can help
encourage this flexibility
Introduction to Ratio Sense
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Ratio Sense
We will spend a lot of our time on this!
Rational Numbers
Ratio Sense involves the ability to think
flexibly in problem situations involving
ratios.
Rational Numbers are numbers of the
a
form , where b ¹ 0 and a and b are
b
integers. That is, numbers that can be
written as fractions.
More importantly, reasoning with rational
numbers requires the ability to reason
flexibly with fractions, ratios, rates, and
percents, and the operations on them.
Quantities and Change
Quantitative reasoning involves the ability
to interpret and operate with change. This
may require operating with constant or
varying rates of change.
Ratio
Ratios arise from situations in which two
(or more) quantities are related.
A ratio is a quotient of two numbers or
quantities. Ratios can compare similar
units of measure or unlike units of
measure (e.g., 100 miles per 2 hours).
Ratios that compare unlike units are called
rates.
Guys : Girls
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Proportion
A proportion is the equality of two ratios.
Blue Cups Total Cups
2
3
4
6
6
9
Directly Proportional (Direct Variation)
We will explore these later in the week!
Indirectly Proportional (Indirect Variation)
We will explore these later in the week!
Introduction to Ratio Sense
Two quantities, y and x, are said to be
directly proportional if their ratio is some
y
nonzero constant k. That is, = k .
x
Alternatively, this means that y = kx . The
constant k is often referred to as the
constant of proportionality or the quantity
that is invariant, while y and x are said to
be covariant.
Two quantities, y and x, are said to be
indirectly proportional if their product is
some nonzero constant k. That is, xy = k .
k
Alternatively, this means that y = . The
x
constant k is often referred to as the
constant of proportionality or the quantity
that is invariant, while y and x are said to
be covariant.
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Revisiting Relative Change (Absolute Vs. Relative Change)
[Also referred to as Additive vs. Multiplicative]
1. For a science experiment, Quinn planted some sunflower seeds in two pots and began to make some
observations. She controlled the amount of light that each plant received while making certain to hold
other important variables constant (e.g., the amount of water that each plant receives, the amount of
nutrients that each plant receives, the soil conditions) in an effort to determine how the amount of
sunlight each plant receives affects the height of each plant after various weeks. Here are some of
Quinn‘s observations.
Week 3
Week 4
Plant A
Plant B
5 inches
8 inches
7 inches
10 inches
Which plant grew more between weeks 3 and 4?
2. Keeping the Problem Situation the Same, Varying the Types of Questions (Adapted from
Lamon, 1999)
Look at the number of cookies that Marcus has and the number of cookies that Nadia has.
Marcus’s cookies
Absolute Thinking
Nadia’s cookies
Relative Thinking
Who has more cookies, Marcus or Nadia?
How many times would you have to stack up Marcus‘
cookies to get a pile as high a Nadia‘s cookies?
How many more cookies does Nadia have
than Marcus?
What fraction of a dozen cookies does Nadia have?
How many fewer cookies does Marcus
have than Nadia?
There are 6 cookies in a package of cookies. What part of
a package of cookies does Marcus have? Nadia have?
Introduction to Ratio Sense
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Classifying/Expressing Ratios
1. Part-Part-Whole:
a.
The gym class has 27 girls and 25 boys.
i. What is the ratio of boys to girls in the gym?
ii.
b.
What is the ratio of girls to the whole gym class?
Mrs. Jones put her students into groups of five. Each group had three girls. If she has 25 students,
how many girls and how many boys does she have in her class?
2. Associated sets: Ellen, Jim, and Steve bought 3 helium-filled balloons and paid $2.00 for all three
balloons. They decided to go back to the store and buy enough balloons for everyone in the class. How
much did they pay for 24 balloons?
3. Rates/Well-known measures: Dr. Day drove 156 miles and used 6 gallons of gasoline. At this rate, can
he drive 561 miles on a full tank of 21 gallons of gasoline?
4. Growth (stretching and shrinking situations): A 6"´8" photograph was enlarged so that the width
changed to 8"´12".
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Ratios vs. Rates
Some authors assign a special name to ratios compare two (or more) quantities measured in
different units. For example these authors will say:
a. The ratio of the length and the width of a rectangle measured is 3:5. Since both the length and
the width can be expressed using the same units (e.g., meters or feet), then we will call this a
ratio.
b. A bus covers a distance of 390 miles in 6 hours. Write distance to time a ratio. Since the
distance is measured in miles and the time is measured in hours, then we will call this ratio a
rate.
Source: Lamon (1999, p. 165)
Definition:
In particular:
— A rate is a ratio that compares two quantities measured in different units.
— A unit rate is the rate for one unit of a given quantity. Unit rates have a denominator of 1.
Examples:
a. Suppose rectangles ABCD and RECT are similar. Further, the length and width of ABCD is 8 cm
and 12cm respectively. If the width of RECT is 2cm, what is the length of RECT?
Introduction to Ratio Sense
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b. If one sale said that grapes were 4 pounds for $2.99 and another sale was 5 pounds for $4.40. Which
is the better buy?
c. George and Juan compared the fuel economy of their cars and found these rates:
George’s car went 580 miles on 20 gallons of gas.
Juan’s car went 450 miles on 15 gallons of gas.
Assuming that both are highway miles, whose car has a better gas mileage?
Introduction to Ratio Sense
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Representing Ratios
Common Core Standard:
6.RP.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,
tables, graphs, or equations
Tape Diagram (CCSS): Tape diagram. A drawing that looks like a segment of tape, used to
illustrate number relationships. Also known as a strip diagram, bar model, fraction strip, or length
model.
Example: Slimy Gloopy mixture is made by mixing glue and liquid laundry starch in a ratio of 3 to
2. How much glue and how much starch is needed to make 85 cups of Slimy Gloopy mixture?
Double Number Line 101
Double number lines: Double number line diagrams are best used when the quantities have
different units (otherwise the two diagrams will use different length units to represent the
same amount). Double number line diagrams can help make visible that there are many, even
infinitely many, pairs
the same
including those
with rational number entries.
• in
There
areratio,
24 students
in a classroom
and 6 large round tables. How many
children should be seated at each table
Example: There are 24 students in a classroom and 6 large round tables. How many children
if there must be the same number of
should be seated at each table if there must be the same number of children at each
children at each table?
table?
Tables: We can also use tables to represent ratios.
Example: Amy walks 3 meters every 2 seconds. Let d be the number of meters Amy has walked after t
seconds. Complete the table below and explain how you “see” the ratio (rate) in each (d, t)
ordered pair.
d meters
t seconds
Introduction to Ratio Sense
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Use this graphic organizer in your journal throughout the week, continually adding to it as we go through the
unit on proportional reasoning
Proportional reasoning is …
Things that I’m unclear about regarding proportional
relationships/What I want to learn …
Students’ misconceptions with
proportional reasoning include …
Various ways to solve proportional problems include …
Proportional relationships are …
Things I learned include …
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Introduction to Ratio Sense
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