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.

1. How Students Learn
Proportional Reasoning
Photo Source: Eternal Light
Title: Nov 15-11:27 PM (1 of 9)

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. 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. 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. 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. (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. (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. (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. Solutions and Scoring Guide
Title: Nov 15-11:57 PM (9 of 9)