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April 2013, Volume 106, Issue 8

FEATURES

Galileo, Gauss, and the Green Monster
Dan Kalman and Daniel J. Teague
Using ideas of Galileo and Gauss but avoiding calculus, students create a model that predicts whether a fly ball will clear the famous left-field wall at Fenway Park.

Geometric Reasoning about a Circle Problem- FREE PREVIEW!
Gloriana González and Anna F. DeJarnette
An open-ended problem about a circle illustrates how problem-based instruction can enable students to develop reasoning and sense-making skills.
Second Look:
Velocity

Building Sinusoids
Mara G. Landers
A measurement-based activity can help students struggling to understand trigonometric functions.

Examples of worksheets and student solutions for Building Sinusoids

Students as Mathematics Consultants
Jennifer L. Jensen
Five problems—relating to gas mileage, the national debt, store sales, shipping costs, and fish population—require students to use functions to connect mathematics to the real world.

Is That Square Really a Circle?
Christopher E. Smith
Considering circles in taxicab geometry helps students with Euclidean concepts.

Elementary Algebra Connections to Precalculus
Roberto López-Boada and Sandra Argüelles Daire
Students use elementary algebra concepts to solve trigonometric and logarithmic equations and systems.

Second Look - Velocity

Tangent Lines without Calculus
A problem that can help high school students develop the concept of instantaneous velocity and connect it with the slope of a tangent line to the graph of position versus time. It also gives a method for determining the tangent line to the graph of a polynomial function at any point without using calculus. It encourages problem solving and multiple solutions.

Creating and Exploring Simple Models
Students manipulate data algebraically and statistically to create models applied to a falling ball. They also borrow tools from arithmetic progressions to examine the relationship between the velocity and the distance the ball falls. A supplemental option is to use a Computer Based Laboratory (CBL) for this activity. Students use graphing calculators to manipulate the sequences (in the statistics editor). Students can plot their data and superimpose their model on the calculator.

Bouncing Balls and Graphing Derivatives
Using motion detectors, calculus, and physics, students create height-, velocity-, and acceleration-versus-time graphs of a bouncing ball.

Illuminations Lesson: Varying Motion
This lesson helps students clarify the relationship between the shape of a graph and the movement of an object. Students explore their own movement and plot it onto a displacement-vs.-time graph. From this original graph, students create a velocity-vs.-time graph, and from there create an acceleration-vs.-time graph. The movement and how to interpret each type of graph are emphasized through the lesson, which serves as an excellent introduction to building blocks of calculus.

Illuminations Applet: Vector Investigation: Boat to the Island

Move the boat around the water by changing the magnitude and direction of the boat's speed (blue vector) or the magnitude and direction of the water current (red vector). Try to land the boat on the island — but be careful not to hit the walls!


Measurement Standard for Grades 9-12

Instructional programs from prekindergarten through grade 12 should enable all students to—

Understand measurable attributes of objects and the units, systems, and processes of measurement

Expections: In grades 9–12 all students should—

  • make decisions about units and scales that are appropriate for problem situations involving measurement.
     

Apply appropriate techniques , tools, and formulas to determine measurements

Expections: In grades 9–12 all students should—

  • analyze precision, accuracy, and approximate error in measurement situations;
  • understand and use formulas for the area, surface area, and volume of geometric figures, including cones, spheres, and cylinders;
  • apply informal concepts of successive approximation, upper and lower bounds, and limit in measurement situations;
  • use unit analysis to check measurement computations.