Proportional Reasoning: Difficult to Learn Challenging to Teach

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Proportional Reasoning: Difficult to Learn Challenging to Teach Powered By Docstoc
					      Dr. Betty Long
Appalachian State University
    Sheila Brookshire
 Buncombe County Schools
      July 9-13, 2012
• Be on time
• Silence cell phones
• Limit sidebar conversations to
• Be respectful
• Be actively engaged
• Have Fun!
• 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
How do you “picture” the
Standards for Mathematical
Jelly Bean Video
What is Proportional Reasoning?
• Ability to scale up and down in appropriate situations and to
  supply justifications for assertions made about relationships in
  situations involving simple direct and inverse proportions
• 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
  3. The 2 types of ratios – composed units and multiplicative
  comparisons – are related (NCTM)
Critical Components of
•   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
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
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
•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 ?
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
      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 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

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