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Increasing Cognitive Demand in Algebra 2 March 14, 2014 VCTM Spring Conference 2014. Pat Gabriel Diane Leighty Nancy Taylor. Outline for Session. Introduction – 5 min Sorting Task in small groups – 20 min Share actual placements/additional discussion. 5 min - PowerPoint PPT Presentation
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Increasing Cognitive Demand in Algebra 2
March 14, 2014VCTM Spring Conference 2014
Pat GabrielDiane LeightyNancy Taylor
Outline for SessionO Introduction – 5 minO Sorting Task in small groups – 20 minO Share actual placements/additional discussion. 5
minO Features of the tasks discussion 5 minO Define Categories – 5 minO Provide 3 examples of rewrites of textbook
problems – 5 minO Pairs to work on rewrites of textbook problems – 15
minO Share rewrites with entire group –15 minO Wrap-up and additional website resources
Levels of Cognitive DemandO MemorizationO Procedures without connectionsO Procedures with connectionsO Doing mathematics
Sorting ActivityO In groups, sort the problems into the
categories to which they belong. O What are the characteristics of each
category?
Discussion – Features of the tasks
O Does a particular feature indicate that the task has a certain level of cognitive demand?
O Is there a difference between “level of cognitive demand” and “difficulty”?
O What effect does context (e.g. setting in which the task is used, students’ prior knowledge, etc) have on the level of cognitive demand required by a task?
Low LevelsO MemorizationO Procedures without connections
High LevelsO Procedures with connectionsO Doing mathematics
Definitions of Levels
O Handout (University of Pittsburgh 2001)
Why Higher Levels?
VA Standards of LearningNCTM Process Standards
Textbook
Rich TasksExamples:
Tickets to a concert cost $2.50 for students and $3.75 for adults. Suppose x represents the number of student tickets sold and y represents the number of adult tickets sold. Write an inequality that describes the condition that total receipts must exceed $1500.
Original Textbook Problem
REWRITE – increase levelTickets to a concert cost $2.50 for students and $3.75 for adults. If the total receipts must exceed $1500 in order to make a profit, and there are only 480 seats in the auditorium, how many of each kind of ticket need to be sold? Try to find several combinations that will work. Can you generalize your solution?
Textbook ProblemO Find the equation for the inverse
relation .
Rewrite of Textbook ProblemO Diane is doing an experiment where she must
collect temperatures and record them in degrees Celsius. She only has a thermometer that records degrees Fahrenheit. She goes to an encyclopedia and finds a formula to convert Celsius to Fahrenheit : . She
O converts this equation so that Celsius can be expressed in terms of Fahrenheit. What is her new equation? When she substitutes 32 for in her new equation, what does she get? Why?
Original Textbook Problem –
Recently the cost c, in dollars of shipping a FedEx Priority Overnight package weighing 1 lb or more a distance of 1001 to 1400 miles was given by c = 3.1w + 29.07, where w is the package’s weight, in pounds. Graph the equation and use the graph to estimate the cost of shipping a 6 ½ lb package.
Mailing Costs – Make “HOT”ter
Recently the cost of shipping a FedEx Priority Overnight package weighing 1 lb to Chicago was $32.17 while a package weighing 2 lbs shipped to Chicago cost $35.27, and a package weighing 5 lbs to Chicago cost $44.57. How much will it cost to ship a 6 ½ lb package to Chicago?
Original Textbook Problem 3 – A pizza establishment offers 6
kinds of meat toppings (pepperoni, sausage, bacon, ham, meatball and salami) and 5 kinds of vegetable toppings (onions, green peppers, mushrooms, tomatoes, and olives). In how many different ways can you choose one meat topping and one vegetable topping?.
Problem 3 RewriteO Currently, a pizza establishment offers 6 kinds of
meat/fish toppings (pepperoni, sausage, bacon, ham, meatball and anchovies) and 5 kinds of vegetable toppings (onions, green peppers, mushrooms, tomatoes, and olives). One of the competitors advertises “Over 10,000 varieties of pizza available!”. The owner wants to be able to exceed this claim and hires you to determine what the minimum number of additions he should make to his offerings in order to achieve his goal. Write a proposal, and be sure to include the calculations used to arrive at your solution.
Developing your own….O Each pair of teachers is given
problems to work on to try to make HOT!
O Share your problems and your rewrites with the entire group.
Rich Tasks Are: O Are accessible and extendable.O Allow individuals to make decisions.O Involve learners in testing, proving,
explaining, reflecting and interpreting.O Promote discussion and communication.O Encourage originality and invention.O Encourage “what if?” and “what if not?”
questions.O Are engaging and contain opportunity to
surprise.
Resources
O https://sites.google.com/site/mathematicscapstonecourseunits/
O Exeter Mathematics:
http://www.exeter.edu/academics/72_6539.aspx
O http://map.mathshell.org/materials/task
s.php
