CK-12 Foundation
Solving Logarithmic Equations
Pupils follow a chain of reasoning in deconstructing a logarithmic equation step by step. Using their knowledge of logarithms, learners compare the solutions of logarithmic equations when the bases are changed.
CK-12 Foundation
Change of Base: River Logs
Using the answers to the challenge questions, class members work through simplifying a complex logarithmic expression that requires changing bases. Pupils drag values to fill in the steps to arrive at a numerical equivalent expression.
CK-12 Foundation
Existence: One-to-One Functions and Inverses
One-to-one means the answer is simple, right? Given four graphs, pupils use a vertical line to test each graph to find out if they are one-to-one. By using the resource, learners realize that not all one-to-one relations are functions....
CK-12 Foundation
Inverse Functions: Definition of Inverse Functions
Is the inverse of a function also a function? Pupils manipulate the graph of a function to view its inverse to answer this question. Using a horizontal and vertical line, class members determine whether the initial function is...
CK-12 Foundation
Inverse Functions
Provide a graphical view of inverses. Pupils manipulate points on a line and see the relationship of the graph with the graph of its inverse. Using the relationship between the graphs, scholars respond to questions concerning inverses...
CK-12 Foundation
Work and Force: Lifting a Bucket
How much work does it take to lift a bucket? An interactive presents a problem of lifting a bucket from the ground to the top of a building. Using their knowledge about work and integrals, pupils calculate the amount of work required to...
CK-12 Foundation
Volume by Disks: The Vase Case
Finding the volume is an integral characteristic of a vase. Using the idea that summing the areas of cross-sectional disks will calculate the volume of a rotational solid, pupils find the volume of a vase. Scholars determine the interval...
CK-12 Foundation
Area Between Curves: Income and Expenses
Use the area of polygons to calculate the area between curves. Pupils calculate areas under income and expense curves by filling the space with squares and right triangles. Using that information, they determine the profit related to the...
CK-12 Foundation
Trapezoidal and Midpoint Approximations: Area of a Skirt
When are trapezoids better than rectangles? Using trapezoids pupils approximate the area of fabric defined by a function. Just like with rectangles, learners realize the more trapezoids the more accurate the approximation. Scholars use...
CK-12 Foundation
Properties of Definite Integrals
Close your eyes and visualize a definite integral. An interactive provides a visualization of a function and definite integral on a closed interval. Pupils move one point of the interval in either the positive or negative direction....
Beyond Benign
Daphnia Bioassay LD50
De-icing materials may have a harmful effect on our environment; have your class perform an experiment to test the nature of these effects. Scholars monitor the survival rate of a sample of daphnia as the concentration of a de-icing...
CK-12 Foundation
Vector Projection: Hitting a Baseball
Hit a home run with a resource that is all fun and games. Learners apply an interactive to determine the horizontal component of the distance a baseball travels. They must consider vector projections for this task.
CK-12 Foundation
Vector Sum and Difference: The Country of Dreams
Find your way around using vectors. Scholars use an interactive to learn about vector addition. They answer a set of questions about modeling a route on a map using vectors.
CK-12 Foundation
Explicit Formulas: Tiles for Writing nth Term in a Sequence
Build an explicit formula using tiles. Pupils develop a tile representation of a term within a sequence given figures of previous terms. Using the diagrams, learners develop the explicit formula by recognizing the common difference and...
CK-12 Foundation
Logarithmic Differentiation: Graphing the Derivative of a Logarithm
Log the values of the derivative of a logarithm. The interactive plots the derivative of the natural logarithm. Learners first determine the derivative of natural logarithm and the general logarithm. Using the formulas for the...
CK-12 Foundation
Logarithm Properties: The Log Properties
Roll a log into an equivalent expression. Given four expanded logarithmic expressions, pupils write an equivalent condensed expression. They identify which properties allows for the simplification.
CK-12 Foundation
Definition of Inverse Functions
Investigate the definition of inverse functions graphically. Using the interactive, scholars create a graph of a function and view its resulting inverse. They then determine whether a group of functions have inverses that are also...
CK-12 Foundation
Differential Equations Representing Growth and Decay: Rice Legend
The legend of a wise man who asks a king for rice as a reward presents a context to study exponential solutions to differential equations. Pupils move quantities of rice to a chessboard and calculate the amount of rice for each day. To...
CK-12 Foundation
Length of a Plane Curve
Challenge your class to use straight lines when estimating the length of a curve. An engaging interactive allows individuals to place line segments one after another along the arc. Learners determine that the more lines used, the better...
CK-12 Foundation
Volume by Cross Section: Volume of the Cone
Discover another way to find the volume of a cone. Pupils explore how the area of a cross section changes as it moves through a cone. The interactive uses that knowledge to develop the integral to use to find the volume of the cone....
CK-12 Foundation
Method of Cylindrical Shells
Approximate the volume of a solid of revolution. Using a method similar to approximating the area under a curve, pupils investigate the volume of a solid of revolution. The learners use a given definite integral to find the volume of...
CK-12 Foundation
Area Sums: Estimation with Rectangles
The more rectangles, the better the estimate. Using the interactive, pupils explore estimating the area under a curve using left-hand sums. Learners respond to challenge questions on how to get better estimates using the same technique.
University of Minnesota
Homeostasis of Thermoregulation
Whether you're battling the flu or trying to warm up on a chilly day, your body's ability to react to temperature change is fascinating! Anatomy scholars discover the fantastic feedback loops that control body temperature in a rigorous...
CK-12 Foundation
Analyzing the Graphs of Functions: Analyzing a Rational Function
Shift the function and transform the key features of the graph. By translating the graph of the rational function, class members find out how the key features alter. Pupils determine the domain, range, asymptotes, and intervals of...