Grades 4 – 5 Fractions
Grades 6 – 7 Ratios and
Proportional Relationships
Grade 8 Functions
Progressions for the Common
Core State Standards in
Mathematics (draft)
Ratio Proportional Progression 6 – 7
 ©The Common Core Standards Writing
Team 26 December 2011

Examples of Topic Progression
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Fractions  Ratios  Proportions  Rate of Change
Geometry - Similar figures, trigonometric ratios
Science - Rate of change, average rate of change in
Calculus, slope, speed, acceleration, density, quickness of
technology
Everyday - cooking, tips, tax, miles per gallon, discounts
Statistics – demographics, economics, birth rate, body
mass index, rain fall, medicine dosing
Developing Essential
Understanding of Ratios,
Proportions & Proportional
Reasoning
Grades 6 – 8
National Council of Teachers of Mathematics
Important Transitions - Ratios
• Reasoning from one quantity to two
• Moving from additive comparisons to
multiplicative comparisons
• Progressing from ratios as composed units to
having a multiplicative relationship
• Moving from iterating with composed units to
creating infinitely many equivalent ratios through
multiplication (rate)
Rates

Previous definition of rate: a comparison
of two quantities of different units

Alternate definition of rate: a set of
infinitely equivalent ratios or a ratio in
which one of the quantities is time
Reasoning From
One Quantity to Two
Orange concentrate example:
Sixth graders were shown a large and a
small glass of orange juice filled by the
same carton and asked if they thought
both glasses would taste equally orangey
or if one was more orangey than the
other.
Reasoning with Two Quantities
Ramp example (p. 25):
In order to use a ratio you have to isolate
the attribute being measured.
Ex. A ramp is composed of base length and
height, but the attribute being measured is
steepness of the incline.
Reasoning with Two Quantities
Ramp example (p. 25):
Effects of changing one quantity.
Ex. What happens to the steepness:
 base length is increased…decreased
 height is increased…decreased
Moving from Additive to
Multiplicative Comparisons
When students focus on only one quantity
they wrongly interpret ratios using additive
comparisons.
e.g. When asked to compare lengths they
assume the question refers to “how much
more” or “how much less” rather than “how
many times bigger” or “What part is one
compared to the other”
Moving from Additive to
Multiplicative Comparisons

Use tables and repeated reasoning
1
2
3
4
5
3
6
9
12
15
Moving from Additive to
Multiplicative Comparisons

Use of double number lines and repeated
reasoning
m
0
5
10
15
sec
0
2
4
6
Ratios as Composed Units
(iterating and partitioning)
Composed units refers to the joining of two
quantities to create a new unit.
Consider mixing juices. 3 apple and 2 grape
Here the composed unit refers to one batch.
So now you can say let us iterate (multiply) or
partition (divide)
Ratios as a Fixed
Number of Parts
Consider mixing juices. 3 apple and 2 grape
Out of 5 parts, 3 are apple, 2 are grape.
This leads us to one is 3/2 of the other and one is 2/3
of the other
Which leads us to y = cx where c is the constant of
proportionality (rate of change/slope)
Geometric Version

Consider two similar triangles.
6
3
4
8
Ratios Having a Multiplicative
Relationship
(infinitely many equivalent ratios)
Consider the length of Worm A is 6 in and the
length of Worm B is 4 in
A:B = 6:4 = 1.5:1
A is 1.5 times as big as B
B:A = 4:6 = 2:3 = 1:1.5
A is 1.5 times as big as B
Ratios and Fractions –
do not have identical meaning

Ratios are often used to make “part-part”
comparisons while fractions are “partwhole”

Ratios can involve more than two terms
while fractions do not
Ex. Ratio of types of milk in a store
Ratios and Fractions –
fractions reinterpreted
Fractions can be reinterpreted as a point
on a number line or an operator such as a
scale factor
Ex. Ratio of 2:5 as two-fifths of something
 Fractions as quotients can be
reinterpreted as sharing
Ex. Ratio of 2:5 as sharing two ounces
among five minutes

Recognize and Describe Ratios

“for each” “for every” “per”

“Two pounds for a dollar” needs more
explaining. Is it every two pounds costs
me a dollar or are we talking about a
discount for every two pounds?
Recognize and Describe
Proportions
If a factory produces 5 cans of dog food
for every 3 cans of cat food, then when
the company produces 600 cans of dog
food, how many cans of cat food will it
produce?
 If a factory produces 5 cans of dog food
for every 3 cans of cat food, then how
many cans of cat food will the company
produce when it produces 600 cans of
dog food?

Recognize and Describe Multistep
Problems
After a 20% discount, the price of a
SuperSick skateboard is $140. What was
the price before the discount?
 A SuperSick skateboard costs $140 now,
but its price will go up by 20%. What will
the new price be after the increase?
 The solutions are different because the
20% refers to different wholes.

Important Understandings Proportions
• A proportion is a relationship of equality
between two ratios
- Equivalent ratios by iterating or
partitioning composed units
- If one quantity of a ratio is multiplied or
divided, the same must happen to the
other quantity to maintain the
proportional relationship
-The two types of ratios (composed
units and multiplicative comparison) are
related
A Proportion is a Relationship of
Equality Between two Ratios
• Understand what the equal sign in a proportion
means
Ex. A clown walks 10 cm in 4 sec. A frog walks 20
cm in 8 sec. In the proportion
10/4 = 20/8 the equal sign means that the two
speeds are the same.
So the constant rate of change is the same or in
other words both share the same unit rate
Equivalent Ratios by Iterating or
Partitioning Composed Units
• Understanding begins with iterating with the
composed unit by doubling, tripling, etc.
• Next comes partitioning the composed unit by
simplifying or finding the unit rate
Maintaining the Proportional
Relationship
•Iterating as multiplication or repeated groups of
both parts of the ratio
• Partitioning as division or the repeated sharing
of both parts of the ratio
Composed Units and Multiplicative
Comparison are Related
• Use composed units to find both unit rates
• Interpret the unit rates as the multiplicative
comparisons
• Use the multiplicative comparisons to represent
the relationship in an equation
Example: Orange Juice
3 concentrates 4 waters
1.
2.
3.
Not as a ratio: Purely counting 3,
counting 4 and putting them in fractional
form
As a ratio: Forming a composed unit of
3:4, iterating 6:8, and understanding that
it tastes equally orangey
As a rate: The student can use 1:4/3 to
find any quantity of orange juice with
equivalent strength
Ratios, Rates, Proportions and
Graphing
Grade 6 – Use a table of equivalent ratios to
plot the pairs of values on the coordinate
plane
 Grade 7 – Use a graph to decide if two
quantities are proportional (linear through
the origin). Represent proportional
relationships in the equation y = mx. Explain
that the upoint (1, r) represents the unit rate
and (0, 0) represents the starting point.
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Functions and Graphing

Grade 8 – Interpret the equation
y = mx + b as a linear function where m
is the rate of change and b is the starting
point. Understand that when b is
nonzero the function is not proportional.
Use slope to determine greater or lesser
rates of change.
Example of a Function
Developing the Various Parts of a
Function through Activities
Functions 8.F 1

Understand that a function is a rule that
assigns to each input exactly one output.
The graph of a function is the set of
ordered pairs consisting of an input and
the corresponding output.
Algebraic Ups and Downs

In ratios we discover the relationship
between the two or more given quantities

In functions we discuss the relationship
between x and y
y is a function of x
(y depends on what happens with x)

Inquiry Lab –
Relations and Functions

Understanding the specific relationship
that defines a function

Compare different typical explanations
Graphing Linear Equations

Practice input/output

Discover the equation y = mx + b

Discuss how changing m or changing b
effects the graphic representation and the
verbal representation
Zap It 1

Keep practicing, but in a different way
Extend the Matching Game

In 7th grade we used a game where each
student had a different representation of
a proportion (verbal, graphical, table, etc.).

Extension: Have proportional, nonproportional and non-linear examples
Applicable Problem

Working your way back to the unit price
Number of
Pencils
Rule
Total Cost of the
Pencils
3
?
$0.75
4
?
$1.00
5
?
$1.25
x
y
Write the rule as an equation using x and y
Functions 8.F 2

Compare properties of two functions
each represented in a different way
(algebraically, graphically, numerically in
tables or by verbal descriptions). For
example, given a linear function
represented by a table of values and a
linear function represented by an
algebraic expression, determine which
function has the greater rate of change.
Pledge Plans

Real world context to help understand
what changing the y-intercept really
means

Connections to Statistics
Applicable Problem

Determining the greater rate of change
y = –2x
y = 4x
y=x
y = –3x
How does this relate to absolute value?
Problem Solving Connections

Bringing all the ideas of the unit together

Real world context
Functions 8.F 3

Interpret the equation y = mx + b as
defining a linear function, whose graph is a
straight line; give examples of functions
that are not linear. For example, the
function A = s2 giving the area of a square
as a function of its side length is not linear
because its graph contains the points (1,
1), (2, 4) and (3, 9) which are not on a
straight line.
Rising Towers

Comparing geometric quantities to
discover proportional versus nonproportional connections
Applicable Problem

Discern linear from non-linear situations
Jose decided to hike up a mountain last Saturday.
It took him the same amount of time to hike up
the mountain as it did to hike back down.
Label as linear or nonlinear:
 A graph comparing time x and elevation y
 A graph comparing time x and distance y
Functions 8.F 4

Construct a function to model a linear
relationship between two quantities.
Determine the rate of change and initial
value of the function from a description of a
relationship or from two (x, y) values,
including reading these from a table or from
a graph. Interpret the rate of change and
initial value of a linear function in terms of
the situation it models, and in terms of its
graph or a table of values.
Zap it 2 and 3

Focuses on working backwards to find an
initial starting point for non-proportional
functions
Applicable Problem


The prices for entry into a science center are
recorded below. Assuming the relationship is linear
find the rate of change and initial value.
Number of People (x)
Total Cost (y)
2
$65
3
$80
4
$95
5
$110
Explain the meaning of the rate of change and initial
value in the context of the question.
Functions 8.F 5

Describe qualitatively the functional
relationship between two quantities by
analyzing a graph (e.g. where the function
is increasing or decreasing, linear or
nonlinear). Sketch a graph that exhibits
the qualitative features of a function that
has been described verbally.
Books Upon Books

Statistics connection

Identify independent and dependent
variables in a real world context

Introduction to solving systems of linear
equations
Applicable Problem


Explain the function in context
The graph shows the number of gallons of water in a
bathtub after filling it for a certain number of minutes.
20
Number of Gallons
18
16
14
12
10
8
6
4
2
0
0
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1
2
3
4
5
6
Number of Minutes
7
8
9
10
Write an equation to represent the situation.
Why is the slope increasing?
What would the situation be if the slope were
decreasing?
The Activities Resources
AIMS
 NCTM Navigating Series
 On Core Mathematics
 Glencoe
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Additional Resources
Zeroing in on Number and Operations
Grades 7-8
Anne Collins & Linda Dacey
Additional Resources

Illuminations:
◦ Function Matching
◦ Circle Tool

Math Playground:
◦ Thinking Blocks
◦ Function Machine
◦ Equivalent Fractions

Balanced Assessments:
◦ Bicycle Rides (functions)
◦ Pen Pals (measurement conversions)