Calculus 12 - Curriculum Document
Calculus 12 - Outcomes At-a-Glance
Calculus 12 - Pacing Guide - This pacing guide replaces the previous yearly plan. It has been updated to reflect removed outcomes and provide flexibility for responsive instruction.
Calculus 12 - Desmos Activity Collection - A collection of online student Desmos activities organized by unit.
D1 apply and understand how Riemann’s sum can be used to determine the area under a polynomial curve
D2 demonstrate an understanding of the meaning of area under the curve
D3 express the area under the curve as a definite integral
D4 compute the area under the curve using numerical integration procedures
C7 solve initial value problems of the form dy/dx = f(x), y_0 = f(x_0), where f(x) is a function that students can recognize as a derivative.
B18 apply rules for definite integrals
B19 apply the Fundamental Theorem of Calculus
C8 understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus
C9 construct antiderivatives using the Fundamental Theorem of Calculus
C10 find antiderivatives of polynomials, e^(kx), and selected trigonometric functions of kx
B20 compute indefinite and definite integrals by the method of substitution
B21 (optional) apply integration by parts to evaluate indefinite and definite integrals
B22 solve problems in which a rate is integrated to find the net change over time
D5 apply integration to calculate areas of regions in a plane
Additional Resources and Activities for Unit 4
Estimating with Finite Sums
Estimating Areas - "Which shape has the bigger area? If you were a calculus student, which tool would you choose to explore this?" She used this activity to talk about "What is Calculus?" and ideas of accumulation and integration. Would quarters or dimes be a better/more accurate tool? What about really small circles like beads? As the item gets smaller (towards infinity), you can get a more accurate area. This idea was based on Which One Has a Bigger Area?
Riemondrian - Create a piece of calculus art inspired by the work of Piet Mondrian. Draw a curve and draw rectangles for the upper and lower Riemann sums for random subintervals. Additional random rectangles are placed, and random colors in the style of Mondrian are added to complete the effect. Riemondrian jewelery was shown at the 2019 bridges conference fashion show.
Definite Integrals and The Fundamental Theorem of Calculus
Antiderivative Jeopardy! - One way to introduce antiderivatives is to use the example of the TV game show Jeopardy. In this show, contestants have to come up with the question for a given answer. This is exactly what antiderivatives are... the answer to a differentiation question. So if the answer on the screen is x^2, the question would be "What is the derivative of 1/3 x^3?".
Definite Integral Auction - Students bid for components of a definite integral. Each group tries to make the largest value they can with the components they have purchased. More detail can be found here.
Fundamental Theorem of Calculus Escape the Apartment activity - The handout links students to this Google slides is in the format of a "choose your own adventure" book. Students take the elevator from floor to floor and from room to room solving question in order to find the correct path to the exit (similar to an escape room). More information about this activity can be found at Girl Math.
Rowing Machine Areas - Ask students to record data from a rowing machine (this could also be done with other exercise equipment such as a treadmill or stationary bicycle). Record the speed every 10 seconds for 2 minutes. Then graph speed-time and calculate the distance using Riemann sums. Students can check to see if they calculated the same answer as the digital readout gave for distance.
A Parabola in a Hexagon warm-up question - Parabola inside a regular hexagon. What fraction is colored red? from Henk Reuling.
Math Market Integration Review - Student buy questions from the market and sell solutions back for a profit. A fun activity to review of a variety of techniques of integration including substitution and integration by parts.
Jack's Beanstalk Desmos Activity - An introduction to velocity time graphs, area under a curve and definite integral. Jack's beanstalk is growing while he sleeps. Can you figure out how tall it will be given a velocity time graph of its growth rate?
Burning Daylight Desmos Activity - In this activity, students use sinusoids to model daylight data for two US cities (Fairbanks, AK and Miami, FL). They predict which city has more total daylight during a given year, and then use their model to calculate an answer to that question. (They may be in for a surprise!) Calculus students have an opportunity here to practice defining and calculating definite integrals, while students in earlier courses will need to take advantage of the fact that the average value of a sinusoid is equal to its midline.
Integration by Substitution
Introduction to Integration by Substitution puzzles - Students create an integration equation using pieces. Each puzzle should use all the pieces. This can be a launching activity to start talking about u-substitution.
Integration by Substitution matching activity - Match each integral with an appropriate substitution. There may be more than one! From Mr Southern Maths.
U-Substitution Scavenger Hunt - Students work through a series of u-substitution integration problems. The answer for each question leads to the next problem. When correctly done, students will end up back at the initial question. More information about this activity can be found at Girl Math.
Unit 4 Cumulative Review
Integral Struggle Game - Students form two teams. One team is trying to create definite integrals with a positive area and one team is trying to create integrals with a negative area. Teams alternate placing numbers into the definite integrals to adjust the values. The team that "captures" 2 of the 3 graphs for their side wins. Instructions handout, Instructions slide show, Functions, Graphs, Number tiles