Define the sharpest in the group. Given a section of a trail map, pupils determine a method to measure the sharpness of each turn in the path. Individuals then determine what modifications to their formulas to make to find the sharpness...

Group equations by their solutions. Given six different equations, pupils determine which have the same solutions. Scholars explain why some are the same and some are different.

Four 4s represent the counting numbers. Pupils attempt to write equivalent expressions to as many counting numbers as possible using only four 4s. Scholars then determine whether the same feat is possible using only three 3s.

Find an easier way to double it. Using the price of an item and the Consumer Price Index, learners determine how long it will be for the price to double. Scholars calculate the length of time it would take for the price to double using a...

What will that cost in the future? The scenario provides pupils with a growth as a Consumer Price Index. Learners create functions for a given item to determine future prices and graph them. Class members then compare their functions to...

Take a coordinated look at rectangles. The task asks pupils to plot the length and width of created triangles in the coordinate plane. Using their plots, scholars respond to questions about rectangles and their associated points on the...

Reflect upon the graphs of quadratic functions. Given a quadratic function to graph, pupils determine whether the graph after a horizontal and vertical reflection is still a function. The final two questions ask scholars to describe a...

Create polynomials with specific values. The task consists of writing three polynomial functions that evaluate to specific values for any given number. Scholars first find a polynomial that evaluates to one for a given value, then a...

Do not cross the line while graphing. Provided with several coordinate axes along with asymptotes, pupils determine two functions that will fit the given restrictions. Scholars then determine other geometrical relationships of asymptotes...

Sometimes the solution is all a matter of perspective. The short assessment task presents a problem to pupils that requires them to make sense of a diagram. Once learners see two similar triangles, the rest of the solution is solving a...

How many ways can you make 10? Class members tackle three problems to find all possible ways three numbers add to be 10. The first is with positive integers, secondly with non-negative integers, and finally with real numbers. Pupils also...

Graph growth in the US. Given population and area data for the United States for a period of 200 years, class members create graphs to interpret the growth over time. Graphs include population, area, population density, and population...

Let's give parameters a second try. Scholars take a second look at a system of linear equations that involve a parameter. Using their knowledge of solutions of systems of linear equations, learners describe the solution to the system as...

Chase the traveling solution. Pupils analyze the solutions to a system of linear equations as the parameter in one equation changes. Scholars then use graphs to illustrate their analyses.

Find the area as it slides. Pupils derive an equation to find the painted area of a section of a trapezoidal-shaped stage The section depends upon the sliding distance the edge of the painted section is from a vertex of the trapezoid....

Just how far can I see? The short assessment question uses the Pythagorean Theorem to find the distance to the horizon from a given altitude. Scholars use the relationship of a tangent segment and the radius of a circle to find the...

Ready to field questions about slope fields? An article on AP® Calculus teaching methods describes how to teach about slope fields to solve differential equations. It gives some sample problems to consider with the class and how to...

Local linearity isn't the first thing that comes to mind to start off an AP® Calculus course. A scholarly article discusses one possible beginning to the AP® Calculus course: investigating and introducing derivatives through activities...

All's fair in math and war. Scholars examine the Battle of Trafalgar using calculus. They set up and solve a system of differential equations to determine the number of ships remaining in each fleet over time.

Root for young mathematicians learning about functions. A set of two problems assesses understanding of polynomial functions and their roots. Scholars select values for a, b, and c, and then create two functions that meet given...

Sometimes close enough is appropriate. A curriculum document for AP® Calculus examines the importance of providing approximation questions throughout the course. It looks at approximating derivative values, approximating definite...

No need to go to extreme lengths to find resources on extrema. The central focus of an AP® curriculum module is on critical points and extrema, and how to cover these concepts throughout the course. A set of three worksheets helps assess...

Don't table the resource—use it now. An AP® Calculus curriculum module encourages the use of tabular data throughout the course. It provides some example topics, such as rate of change, net change, and average value of a function, where...

Motivation is key to learning. The author points out that it is important to engage intuition for aspects of statistics that pupils find counter-intuitive. Three strategies—using multiple representations, using intuitive analogies, and...