Calculus AB APSI 2015 Day 3 Professional Development Workshop Handbook Curriculum Framework Calculus AB and BC Professional Development Integration, Problem Solving, and Multiple Representations Curriculum Module

Wednesday Morning (Part 1) Break Morning (Part 2) Lunch Introducing the Definite Integral Through the Area Model Investigating How to Find Area Using Riemann Sums and Trapezoids Developing an Understanding for a Definite Integrals Numerical Integration Break   Morning (Part 2) Fun Finding Volume Solids with Known Cross Sectional Area Fundamental Theorem of Calculus Activity Sheet for Understanding the 2nd Fundamental Theorem of Calculus Lunch   Afternoon (Part 1) Discussion of Homework Problems Free Response Problem 2014 AB5 Share an Activity Dan Meyer  Break  Afternoon (Part 2) Calculus Games Building Understanding for the Average Value of a Function Mean Value Theorem Curriculum Module: Motion (w/Smartboard) Big Idea 3 – The Integral and the Fundamental Theorem of Calculus

Wednesday Assignment - AB Multiple Choice Questions on the 2014 test: 1, 4, 8, 12, 14, 18, 26, 76, 77, 70, 80, 81, 83, 84, 85 Free Response: 2014: AB4, AB6 2015: AB4/BC4, AB5

Key Ideas to Cover on Integration A definite integral is the limit of a Riemann sum The definite integral is the net accumulation of a rate of change or

Concept of a Definite Integral All the important concepts related to definite integrals can be taught and understood without knowing antiderivatives.

Calculus AP should include opportunities for students to understand Area under a graph Riemann Sum – Definition of a Definite Integral Ways to Evaluate a Definite Integral Fundamental Theorem How integrals accumulate area How functions can be by integrals Techniques for finding indefinite integrals Applications of integrals

Deal with graphical and tabular sets of data to find area That can be represented by a bounded region. That can be approximated using several methods. Relationships between approximations. How more accurate approximation be found The units of measure for .

Introducing Integration through the Area Model

Introducing the Definite Integral Through the Area Model Figure 1 shows the velocity of an object, v(t), over a 3-minute interval. Determine the distance traveled over the interval . The area bounded by the graph of v(t) and the t-axis for represents the distance traveled by this object. The distance can be represented by the definite integral . Activity

The following chart gives the velocity of a particle, v(t), at 0 The following chart gives the velocity of a particle, v(t), at 0.5 second intervals. Estimate the distance traveled by the particle in the three seconds using three different methods. Each method is an approximation for . Activity

Investigating How to Find Area using Riemann Sums and Trapezoids Using the NUMINT program or LMRRAM and TRAPEZOID program on a TI84 Activity

Things You Should have Observed As the number of rectangles increases on monotonically increasing functions, the left hand sums increase, but remain less then the area.

Things You Should have Observed Which sums are always greater than the actual area? Which sums are always less than the actual area?

Things You Should have Observed The limit of the left hand sum equals the limit of the right hand sum and equals the area of the region. area of the region or

Students should be able to Set up and evaluate left, right and midpoint Riemann sums from analytical data, tabular data, or graphical data. Set up and evaluate a trapezoidal sum approximation from analytical data, tabular data, or graphical data. 0 3 9− 𝑥 2 𝑑𝑥

Determine Units of Measure The units of the definite integral are the units of the Riemann Sum The units of the function multiplied by the units of the independent variable.

Verbal Explanation Students need to be able to tell what a definite integral represents in the context of the problem and identify the units of measure. Very common AP question on Free Response Questions

Using Technology to Approximate the Definite Integral

Developing an Understanding for the Definite Integral Smartboard File

Fun Finding Volume

Solids with Known Cross Sectional Area Activity

Volume of a Solid with Known Cross Sectional Area Create a table and a sketch for 𝑓(𝑥)= 𝑥 scale for the grid is 0.5 cm on the x and y axes

Sketch the graph of f (x). The scale for this grid is 0.25 cm on both the x and y axes.

Select one of the figures Select one of the figures. Cut out the 9 shapes, keeping the tabs on the shape. Fold the trapezoidal tab. Glue the tab on the graph so that the edge of the shape is the f(x) segment. Face all the colored faces in the same direction.

Group Work Complete the Finding the Volume of the Solid Activity Sheet with your group members.

Volume of Solids of Revolution Section 9 in Notebook Smartboard File

Rotating about a Line Other than the x- or y-axis Pages 2 to 5

Rotating about a Line Above the Region Pages 5 to 7

Rotating about a Line to the Left of the Region Pages 10 and 11

Rotating about a Line to the Right of the Region Pages 8 and 9

Fundamental Theorem of Calculus Smartboard File

Activity Sheet for Understanding the 2nd Fundamental Theorem of Calculus Section 5 in Notebook Activity

Section 8 in Notebook

Areas, Derivatives, and the Fundamental Theorem of Calculus Worksheet 1: pages 8-11

Applying the Second Fundamental Theorem of Calculus: Finding Derivatives of Functions Defined by Integrals Worksheet 2: page 13

Applying the First Fundamental Theorem of Calculus: Definite Integrals as Total Change Worksheet 3: page 15-16

Examples of Multiple Choice Questions Worksheet 4: page 15-16

Applying the Fundamental Theorem of Calculus Exercises Worksheet 5: page 15-16

Tuesday Assignment - AB Multiple Choice Questions on the 2014 test: 9, 11, 15, 19, 21, 22, 23, 27, 28, 82, 88, 89, 90, 91, 92 Free Response: 2014: AB3, AB6 2015: AB2, AB3

Scoring Rubric 2014 AB3

2014 AB3

2014 AB6

Scoring Rubric-2014 AB6

2015 AB2

Scoring Rubric 2015 AB2

2015 AB3

Scoring Rubric 2015 AB3

Share An Activity from Your Classroom

Dan Meyer Taco Stand All Examples

Calculus Games

Building Understanding for the Average Value of a Function Activity

The Mean Value Theorem

The Mean Value Theorem - Smartboard What is guaranteed? What must be true for the guarantee? Can parts be true if the conditions are not met? How does it apply to real data?

Motion Smartboard File College Board has developed a Curriculum Module to assist you in teaching how to use Calculus to study motion. Dixie Ross Pflugerville High School Pflugerville, TX Motion Smartboard File

What You Need to Know about Motion Worksheet 1: page 5

Sample Practice Problems Numerical, Graphical, Analytical Worksheet 2: pages 7-9

Understanding the Relationship Among Velocity, Speed and Acceleration Worksheet 3: page 13-15

What You Need to Know about Motion Along the x-axis Worksheet 4: page 21

Sample Practice Problems Worksheet 5: page 23-26

Big Idea 3: Integrals and the Fundamental Theorem of Calculus Page 364-367

Wednesday Assignment - AB Multiple Choice Questions on the 2014 test: 1, 4, 8, 12, 14, 18, 26, 76, 77, 70, 80, 81, 83, 84, 85 Free Response: 2015: AB4, AB5 2014: AB4, AB6