A Coherent Progression
To develop an understanding of how ratios
and proportional relationships flow from 6th
grade through algebra and into calculus
Agenda
● Complete and discuss one 8th grade task involving ratio and
proportion
● Make connections between 6th, 7th and 8th grade tasks
● Discuss a high school functions task and the prerequisite
skills necessary from middle school
Goals
Participants will
● use the lens of rate of change to make connections between
middle grades courses and into high school courses
● use these connections to better assist students on their
paths toward a robust understanding of rate of change
A task to start
8.EE Sore Throats
Make connections
Two Tasks:
● 6.RP Equivalent Ratios and Unit Rates
● 7.RP Cider vs Juice
Make connections
A Coherence Map
Connect to high school
Ratios and Fractions:
Distinguishing the Difference
Objective:
Participantswillbeableto:
• accuratelycreateandidentifyaratio,
fractionandunitrate.
• explainthedifferencebetweenaratio,
fractionandunitrate.
What is the difference between
a ratio and a fraction?
• Let’s listen to how students and staff define
the difference in their own words.
• As you watch listen for any additional
misconceptions you hear in the student’s
responses.
Students and Staff's take on ratios vs. fractions
Video Response
What are additional misconceptions you
heard from students/staff?
Share with your elbow partner and using
#NCMath2
Evaluating Ratio Statements:
Juice Task
• In groups of three or four, determine if each
of the statements is true or false.
• Use blocks to help discover the validity of
the statements.
• On chart paper, explain how/why you
determined if each statement was true or
false.
Ratio
• Ratios arise in situations in which two or more
quantities are related
In our task, orange juice to pineapple juice.
• Quantities may be any measureable attribute, such
as distance, area, weight, income, population
• A measurable quantity can be described with a
number and unit:
Example: 3 cups
Ratio cont.
Quantities may have:
1. Same units: 3 cups orange juice and 2 cups
pineapple juice
2. Different units that measure the same attribute: 3
meters and 2 feet
3. Units that measure different attributes:
3 meters and 2 seconds
Ratio cont.
Ratios can be expressed in multiple ways
1.Words
• 3 to 2
• 3 for every 2
• 3 out of every 5
• 3 parts to 2 parts
• 3 cups of orange juice for every 2 cups
pineapple juice
2. Colon
3:2
Ratio cont.
Since ratios relate two quantities, they are
not a single number and cannot be placed
on a number line
Unit Rate
The unit rate tells:
”For each 1…”
“For every 1…”
In other words, it tells how many times as
large one quantity is compared to the other,
and is thus a number, and may be a fraction
Unit Rate
The ratio 3 cups orange juice to 2 cups pineapple
juice, has the unit rate:
3/2 cup orange juice for every 1 cup
pineapple juice
There is also an inverse rate, formed by
switching the order of numbers in the ratio
⅔ cup pineapple juice for every 1 cup
orange juice.
Fraction
A fraction is a single number that tells
how many parts we have and into how
many parts each whole has been
partitioned.
Since a fraction is a number, it can be
shown on a numberline.
Fraction
In our task, the 2 and 3 cups can be combined
since the units are the same unit of
measurement.
⅖ of the cups are pineapple juice.
⅗ of the cups are orange juice.
Fraction connected to Unit
Rate
A unit rate is also a fraction because it is a
single number derived from the ratio.
When you divide to find the unit rate, the
quotient is called the value of the ratio.
3 divided by 2 is 3/2
3/2 is called the value of the ratio
Creating A New Color
Directions:
1.Use two colors of playdough.
2.Determine and keep track of how many
tablespoons of each color you want to use.
3.Mix thoroughly, until you have a new color.
4. Write down your combination and develop a
name for your new color.
Creating a New Color
On a post-it note, identify the following:
-Part to part ratios
-Part to whole ratios
-Fractions
-Two unit rates:
“ For every 1 tablespoon _______, there is ________
tablespoons ___________.”
Equivalent Ratios and Fractions
Equivalent fractions name
the same number. When we
place equivalent fractions
on a number line, they
appear in exactly the same
place.
Equivalent ratios represent multiple
GROUPS that each have the same
composition. As we add groups, the
total number of cups grows, but for
every 1 group, ⅔ of the total is blue.
The ratios 2:3, 4:6, 6:9 all have the
same rate, ⅔, but represent different
amounts of the two measured
quantities.
Equivalent Ratios and Fractions
Equivalent fractions name the same
number. When we place equivalent
fractions on a number line, they appear in
exactly the same place.
Equivalent ratios represent
multiple GROUPS that each
have the same composition. As
we add groups, the total
number of cups grows, but for
every 1 group, ⅔ of the total is
blue.
The ratios 2:3, 4:6, 6:9 all have the same
rate, ⅔, but represent different amounts of
the two measured quantities.
Equivalent Ratios and Fractions
Equivalent fractions name the same
number. When we place equivalent
fractions on a number line, they
appear in exactly the same place.
Equivalent ratios represent multiple
GROUPS that each have the same
composition. As we add groups, the
total number of cups grows, but for
every 1 group, ⅔ of the total is blue.
The ratios 2:3, 4:6, 6:9 all
have the same rate, ⅔, but
represent different
amounts of the two
measured quantities.
Ratio, Fraction, Unit Rate
In your small groups, use the given notecards to
brainstorm your current thoughts about each of the
given terms:
Fraction
Ratio
Unit Rate
Reference using words, mathematics, etc. from the
Juice Task and Playdough Color task when possible.
-Tape your notecards to the bottom of your posters.
Definitions
From the Appendix of the Progression document:
•
A ratio is a pair of non-negative numbers, A : B, which are not both 0.
•
When there are A units of one quantity for every B units of another
quantity, a rate associated with the ratio A : B is A/B units of the first
quantity per 1 unit of the second quantity (Note that the two quantities
can have different units).
•
The associated unit rate is A/B. Example: 6 meters per 2 seconds
would have a unit rate of 6 meters/2 seconds or 3 meters for every 1
second.
•
The value of A:B is the quotient A/B, which may be expressed as a
decimal, percent, fraction, or mixed number.
•
The value of the ratio A:B tells how A and B compare multiplicatively –
it tells how many times as big A is as B.
Fractions
A fraction is a single quantity – a number that can be
shown on a number line.
Fractions definition from the Grade 3 standards:
CCSS.MATH.CONTENT.3.NF.A.1
Understand a fraction 1/b as the quantity formed by 1 part
when a whole is partitioned into b equal parts; understand
a fraction a/b as the quantity formed by a parts of size 1/b.
CCSS.MATH.CONTENT.3.NF.A.2
Understand a fraction as a number on the number line;
represent fractions on a number line diagram.
Ratios & Proportional Relationships Playlist
The progression of representations in the RP domain
Goals
1. Develop an understanding of the progression of visual
representations in the CCSS Ratio and Proportional
Relationships Domain through illustrative tasks.
2. Develop an understanding of the depth of the
expectations in sixth and seventh grade ratio and
proportional relationships domain through an analysis of
the language of the CCSS.
Let’s do some math!
A: Tomato Tomahto
B: Feeding a Crowd
C: Equivalent Ratios
D: Centimeters &
Millimeters
E: Art Class
F: Same and
Different
1. At your tables, split into
2 groups
2. Each group, deal out a
task to each person
Let’s do some math!
A: Tomato Tomahto
B: Feeding a Crowd
C: Equivalent Ratios
D: Centimeters &
Millimeters
E: Art Class
F: Same and
Different
1. Do the task…
•
•
•
What do you notice?
What’s it getting at?
How might students
approach this?
2. Share your
observations with your
group mates
Personal Reflection
• What’s the role and
development of visual
representations in the
ratios and
proportional
relationships domain?
• Write down notes...
“What I think” column
Create a Playlist
A: Tomato Tomahto
B: Feeding a Crowd
C: Equivalent Ratios
D: Centimeters & Millimeters
E: Art Class
F: Same and Different
Playlist Considerations
• How would you order these tasks, and why?
–Consider the numbers
–Consider the representations
–Consider the language
• Which tasks are appropriate for grade 6? Which are
appropriate for grade 7? Why?
Be prepared to share your group’s rationale.
Let’s dive in!
Reading Marks
!- surprising
*- important
?- I wonder
Ratios and Proportional
Relationships
Ratios and Proportional
Relationships
6.RP.A: Understand ratio concepts and use ratio
reasoning to solve problems.
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.
7.RP.A: Analyze proportional relationships and use
them to solve real-world and mathematical problems.
2. Recognize and represent proportional relationships
between quantities.
Our Order
1. Tomato Tomahto
2. Art Class
3. Same and Different
4. Equivalent Ratios
5. Feeding a Crowd
6. Centimeters and Millimeters
Our Reasoning
• Tomato Tomahto
–Very simple, almost literal diagrams.
–Introduces students to the idea of
representing a context with a ratio.
• Art Class
–Double tape diagram helps transition from literal
or set diagrams to double number lines.
–Purposefully expands use of ratio and rate
language.
Our Reasoning
Same and Different
• Double number lines are more
abstract than previous diagrams
but still support reasoning using
the geometric structure and the
intuitive understanding of
equivalent ratios.
• Focuses attention on the three
relevant quantities in the context
(quantity A, quantity B, unit rate
A/B).
Equivalent Ratios
• Introduces a more
abstract definition of
equivalent ratio (e.g., a : b
is equivalent to any sa : sb
for positive s).
• Students now have
access to all diagram types
to support understanding of
this more abstract definition.
Our Reasoning
Feeding a Crowd
• Introduces ratio tables—the
most abstract representation.
Requires an abstract
understanding of equivalent
ratios in order to use with
understanding.
• Draws attention to the two
possible unit rates in a context.
Centimeters and Millimeters
• Takes students to the next
level of abstraction:
proportional relationships.
• This is the only task that is
clearly grade 7. All other tasks
are appropriate in grade 6, but
could be adapted for grade 7 if
they are used to help students
understand proportional
relationships.
Fine-grained Progressions
Grade 6
• Ratio
• Equivalent ratios
characterized by scale
factor definition
• Equivalent ratios
characterized by unit
rates
Grade 7
• Proportional relationships
Personal Reflection 2
• What’s the role and
development of visual
representations in the
ratios and proportions
domain?
• Write down notes...
“What I now know”
column
Building Capacity
Next Steps
• As facilitators of this
module, how would you
use this module?
• Write down notes...
“Where I go from here”
column