1. Mathematical Practices
Students should engage in
Reasoning with definitions and theorems
Connecting concepts
Implementing algebraic/computational processes
Connecting multiple representations
Building notational fluency
Communicating
Mathematical Practices for AP Calculus

2. Mathematical Pratices
Students should engage in
Reasoning with definitions and theorems
Connecting concepts
Implementing algebraic/computational processes
Connecting multiple representations
Building notational fluency
Communicating
Mathematical Practices for AP Calculus

3. Where do your students Reason with definitions and theorems?
Mathematical Practices for AP Calculus

4. How does the area change?
A 6th grade class where the teacher’s goal was that students should understand the relationship between the area of a parallelogram and the length of its sides.
How does the area change?
From 3:54 to 5:20
Park City Mathematics Institute
Reflecting on Practice

5. Where do your students
use definitions and theorems to build arguments, justify conclusions, prove results;
confirm that hypotheses have been satisfied in order to apply the conclusion of a theorem;
apply definitions and theorems to solve a problem;
develop conjectures based on exploration with technology;
produce examples and counterexamples to clarify understanding of definitions, investigate whether converses of theorems are true, or test conjectures
Mathematical Practices for AP Calculus

6. Definitions & Theorems
Why is the sum of an even number and an odd number odd?
Find the length of the missing side.
(3x4y2)3
Sin(ø) =y x 5 3

7. 4. Assume that y = log2 (8x) for each positive real number x
4. Assume that y = log2 (8x) for each positive real number x. Which of the following is true? A) If x doubles, then y increases by 3. B) If x doubles, then y increases by 2. C) If x doubles, then y increases by 1. D) If x doubles, then y doubles. E) If x doubles, then y triples.
Algebra and Precalculus Concept Readiness Alternate Test (APCR alternate) – August 2013

8. Where do students build notational fluency?
know and use a variety of notations;
connect notation to definitions
connect notation to different representations (graphical, numerical, analytical, and verbal); and
assign meaning to notation, accurately interpreting the notation in a given problem and across different contexts.
Mathematical Practices for AP Calculus

9. Notational fluency ¼ 1÷ 4 2(1/4) 2 ¼ A=lw A =bh A=ba A = Bh
¼ 1÷ 4 2(1/4) 2 ¼
A=lw A =bh A=ba A = Bh
Rate of change
f(b)-f(a) Δy y2-y1 m f ’(x) dy
b-a Δx x2-x dx
Functions
f(x); f; f(x)=y; f(g(x))=h(x)
Standard deviation: √Σ(xi-x)2

10. Notational fluency If h(x)=f(g(x)) – 6, h(2) = f-1(6) = g-1(f(1)) =
x f(x) g(x)
1 6 2
2 9 3
3 10 4
4 -1 6
If h(x)=f(g(x)) – 6,
h(2) =
f-1(6) =
g-1(f(1)) =

11. How can you engage your students in
Reasoning with definitions and
theorems?
Connecting concepts?
Implementing algebraic/computational
processes?
Connecting multiple representations?
Building notational fluency?
Communicating?
Mathematical Practices for AP Calculus

12. References
Algebra and Precalculus Concept Readiness (APCR). (2013). Using Research to Shape Instruction and Placement in Algebra and Precalculus MAA Director of Placement Testing, Bernard Madison. Mathematics Association of America.
College Board (2005). Course and Exam Description for AP Calculus AB and AP Calculus BC Including the Curriculum Framework. (Effective ).New York NY
Putting it All Together (2012). Video clip from T-Cubed Common Core State Standards Professional Development Workshop. Brennan, B., Olson J. & the Janus Group. Curriculum Research & Development Group. University of Hawaii at Manoa, Honolulu HI.