# Proportional Reasoning: Difficult to Learn Challenging to Teach

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```					      Dr. Betty Long
Appalachian State University
Sheila Brookshire
Buncombe County Schools
July 9-13, 2012
Norms
• Be on time
• Silence cell phones
• Limit sidebar conversations to
activities
• Be respectful
• Be actively engaged
• Have Fun!
Goals
• Participants will:
• Become more familiar with the
Common Core State Standards for grade
6-8 and how they progress horizontally
and vertically
• Engage in tasks that can be used to
develop students’ use of the Standards for
Mathematical Practice
• Connect concepts among Common Core
State Standards
Icebreaker
How do you “picture” the
Standards for Mathematical
Practice?
Jelly Bean Video
What is Proportional Reasoning?
• Ability to scale up and down in appropriate situations and to
situations involving simple direct and inverse proportions
(Lamon)
• Understanding:
1. Equivalent ratios can be created by iterating and/or
partitioning a composed unit
2. If one quantity in a ratio is multiplied or divided by a
particular factor, then the other quantity must be multiplied or
divided by the same factor to maintain the proportional
relationship
3. The 2 types of ratios – composed units and multiplicative
comparisons – are related (NCTM)
Critical Components of
Reasoning
•   Rational Number Interpretations
•   Relative Thinking
•   Measurement
•   Quantities and Covariation
•   Reasoning Up and Down
•   Unitizing
•   Sharing and Comparing
Rational Number Interpretations
Using a pizza as the unit, draw the following
fractional parts:
• 1 (3/8 pizza)
• 3 (1/8 pizzas)
• 1 ½ (1/4 pizzas)
Relative and Absolute
Thinking
On a certain evening at a local restaurant, the
five people sitting at Table A and the 7 people
sitting at Table B ordered the drinks listed
below. Later, the waitress was heard referring
to one of the groups as “the root beer
drinkers.” To which table was she referring?
Table A: 2 Root Beers and 3 Colas
Table B: 3 Root Beers and 4 Colas
Measurement
This figure has an area of 1 square inch. Draw a
figure that has an area of 2/3 inches squared..
Quantities and Covariation
For each situation, list the changing
Quantities:
•You are filling your car’s gas tank
•You travel by car from Milwaukee to Chicago
•You make several credit card purchases
•You are scuba diving
•You are draining the water in your bathtub
Reasoning Up and Down
• Draw a picture for ½

• If the unit is

represents ?

•If the unit is

represents ?
Unitizing
We are going to solve a problem that
uses a case of soda (24 cans). What
sort of mental picture do you get?

Unitizing refers to the process of constructing
chunks in terms of which to think about a given
commodity.
Sharing and Comparing
In Mr. Trent’s science class, students sit 3 to a table.
One day for mid-morning break, Mr. Trent put 2
candy bars on each table and told the students to
share them. “But before you do,” he said, “please
push all 4 tables together.” If you are a candy lover,
would you like to get your share of the candy before
or after the tables are pushed together?
Bonita
Bonita was given a problem about a leaky
faucet through which 6 ounces of water
dripped in 8 minutes. She needed to
figure out how much water dripped in 4
minutes.

```
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 views: 0 posted: 10/4/2012 language: English pages: 17