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BPRIME Proportional Reasoning

Date post: 05-Dec-2014
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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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Title: Nov 15-11:27 PM (1 of 9) Photo Source: Eternal Light How Students Learn Proportional Reasoning
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Page 1: BPRIME Proportional Reasoning

Title: Nov 15-11:27 PM (1 of 9)

Photo Source: Eternal Light

How Students Learn

Proportional Reasoning

Page 2: BPRIME Proportional Reasoning

Title: Nov 15-9:27 PM (2 of 9)

Students’ Errors and Misconceptions Based on Previous LearningStudents 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.

Source: How Students Learn

(Principle 1) TEACHERS MUST ENGAGE STUDENTS’ PRECONCEPTIONS

x =

Photos Source

Page 3: BPRIME Proportional Reasoning

Title: Nov 15-10:14 PM (3 of 9)

The Knowledge Network: New Concepts of Numbers and New ApplicationsWhat 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?

(Principle 2)

Source: How Students Learn

UNDERSTANDING REQUIRES FACTUAL KNOWLEDGE AND CONCEPTUAL FRAMEWORKS

Photo source

Page 4: BPRIME Proportional Reasoning

Title: Nov 15-10:14 PM (4 of 9)

Metacognition and Rational NumberA metacognitive approach to instruction helps students monitor their understanding and take control of their own learning. The complexity of

(Principle 3)

Source: How Students Learn

A METACOGNITIVE APPROACH ENABLES STUDENT SELF-MONITORING

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

Page 5: BPRIME Proportional Reasoning

Title: Nov 15-9:33 PM (5 of 9)

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.

An AP Calculus Problem Requiring Proportional Reasoning

AP Calculus AB 2007 Exam Question 2

Page 6: BPRIME Proportional Reasoning

Title: Nov 15-9:33 PM (6 of 9)

(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.

Page 7: BPRIME Proportional Reasoning

Title: Nov 15-9:33 PM (7 of 9)

(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.

Page 8: BPRIME Proportional Reasoning

Title: Nov 15-9:33 PM (8 of 9)

(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.

Page 9: BPRIME Proportional Reasoning

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

Solutions and Scoring Guide


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