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How Students Learn
       Proportional Reasoning




                                  Photo Source: Eternal Light



Titl...
(Principle 1) TEACHERS MUST ENGAGE STUDENTS’ PRECONCEPTIONS
   Students’ Errors and Misconceptions Based on Previous Learn...
UNDERSTANDING REQUIRES FACTUAL
   (Principle 2)           KNOWLEDGE AND CONCEPTUAL FRAMEWORKS
   The Knowledge Network: Ne...
A METACOGNITIVE APPROACH
   (Principle 3)          ENABLES STUDENT SELF-MONITORING
   Metacognition and Rational Number
  ...
An AP Calculus Problem Requiring Proportional Reasoning
  The amount of water in a storage tank, in gallons, is
  modeled ...
(i) The rate at which water enters the tank is


   gallons per hour for 0 ≤ t ≤ 7.

   (ii) The rate at which water leave...
(i) The rate at which water enters the tank is


   gallons per hour for 0 ≤ t ≤ 7.

   (ii) The rate at which water leave...
(i) The rate at which water enters the tank is


   gallons per hour for 0 ≤ t ≤ 7.

   (ii) The rate at which water leave...
Solutions and Scoring Guide




Title: Nov 15-11:57 PM (9 of 9)
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BPRIME Proportional Reasoning

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Slides for my 10 min introduction to our afternoon discussing Proportional Reasoning across the grades from 7 to 12. This talk features an AP Calculus problem and explores the proportional reasoning skills students need to understand and successfully solve it.

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BPRIME Proportional Reasoning

  1. 1. How Students Learn Proportional Reasoning Photo Source: Eternal Light Title: Nov 15-11:27 PM (1 of 9)
  2. 2. (Principle 1) TEACHERS MUST ENGAGE STUDENTS’ PRECONCEPTIONS Students’ Errors and Misconceptions Based on Previous Learning Students come to the classroom with conceptions of numbers grounded in their whole-number learning that lead them astray in the world of rational numbers; e.g. multiplying always makes numbers bigger. x = Source: How Students Learn Photos Source Title: Nov 15-9:27 PM (2 of 9)
  3. 3. UNDERSTANDING REQUIRES FACTUAL (Principle 2) KNOWLEDGE AND CONCEPTUAL FRAMEWORKS The Knowledge Network: New Concepts of Numbers and New Applications What are the core ideas that define the domain of rational numbers? What are the new understandings that students will have to construct? How does a beginning student come to understand rational numbers? Source: How Students Learn Photo source Title: Nov 15-10:14 PM (3 of 9)
  4. 4. A METACOGNITIVE APPROACH (Principle 3) ENABLES STUDENT SELF-MONITORING Metacognition and Rational Number A metacognitive approach to instruction helps students monitor their understanding and take control of their own learning. The complexity of rational number—the different meanings and representations, the challenges of comparing quantities across the very different representations, the unstated unit—all mean that students must be actively engaged in sense making to solve problems competently. We know, however, that most middle school children do not create appropriate meanings for fractions, decimals, and percents; rather, they rely on memorized rules for symbol manipulation. Photo source Source: How Students Learn Title: Nov 15-10:14 PM (4 of 9)
  5. 5. An AP Calculus Problem Requiring Proportional Reasoning The amount of water in a storage tank, in gallons, is modeled by a continuous function on the time interval 0 ≤ t ≤ 7, where t is measured in hours. In this model, rates are given as follows: (i) The rate at which water enters the tank is gallons per hour for 0 ≤ t ≤ 7. (ii) The rate at which water leaves the tank is The graphs of ƒ and g, which intersect at t = 1.617 and t = 5.076, are shown in the figure. At time t = 0, the amount of water in the tank is 5000 gallons. (a) How many gallons of water enter the tank during the time interval 0 ≤ t ≤ 7? Round your answer to the nearest gallon. (b) For 0 ≤ t ≤ 7, find the time intervals during which the amount of water in the tank is decreasing. Give a reason for each answer. (c) For 0 ≤ t ≤ 7, at what time t is the amount of water in the tank greatest? To the nearest gallon, compute the amount of water at this time. Justify your answer. AP Calculus AB 2007 Exam Question 2 Title: Nov 15-9:33 PM (5 of 9)
  6. 6. (i) The rate at which water enters the tank is gallons per hour for 0 ≤ t ≤ 7. (ii) The rate at which water leaves the tank is The graphs of ƒ and g, which intersect at t = 1.617 and t = 5.076, are shown in the figure. At time t = 0, the amount of water in the tank is 5000 gallons. (a) How many gallons of water enter the tank during the time interval 0 ≤ t ≤ 7? Round your answer to the nearest gallon. Title: Nov 15-9:33 PM (6 of 9)
  7. 7. (i) The rate at which water enters the tank is gallons per hour for 0 ≤ t ≤ 7. (ii) The rate at which water leaves the tank is The graphs of ƒ and g, which intersect at t = 1.617 and t = 5.076, are shown in the figure. At time t = 0, the amount of water in the tank is 5000 gallons. (b) For 0 ≤ t ≤ 7, find the time intervals during which the amount of water in the tank is decreasing. Give a reason for each answer. Title: Nov 15-9:33 PM (7 of 9)
  8. 8. (i) The rate at which water enters the tank is gallons per hour for 0 ≤ t ≤ 7. (ii) The rate at which water leaves the tank is The graphs of ƒ and g, which intersect at t = 1.617 and t = 5.076, are shown in the figure. At time t = 0, the amount of water in the tank is 5000 gallons. (c) For 0 ≤ t ≤ 7, at what time t is the amount of water in the tank greatest? To the nearest gallon, compute the amount of water at this time. Justify your answer. Title: Nov 15-9:33 PM (8 of 9)
  9. 9. Solutions and Scoring Guide Title: Nov 15-11:57 PM (9 of 9)

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