## Search Results ### Law of Sines

9-12
In this lesson, students will use right triangle trigonometry to develop the law of sines. ### Law of Cosines

9-12
In this lesson, students use right triangle trigonometry and the Pythagorean theorem to develop the law of cosines. ### Optimal Strategies

9-12
In static nim, the set of possible move sizes remains the same during the play of the game. In various versions of dynamic nim, the rules are such that the maximum number of counters that can be removed on each turn changes as the game is played. This maximum can depend on the current size of the pile, the number of counters removed on the previous play, or the move number of the game. In this lesson, students will explore the second type, where each move determines the maximum move size of the next move. 9-12
In this lesson, students consider the costs of owning a car and ways to lessen those costs.  In particular, highway and city mileage are considered, and optimal mileage is calculated using fuel consumption versus speed data. ### Perplexing Parallelograms

9-12
A surprising result occurs when two line segments are drawn through a point on the diagonal of a parallelogram and parallel to the sides. From this construction, students are able to make various conjectures, and the basis of this lesson is considering strategies for proving (or disproving) one of those conjectures. ### Inscribed and Circumscribed Polygons

9-12
By calculating the areas of regular polygons inscribed and circumscribed about a unit circle, students create an algorithm that generates the never-ending digits of π, a common curiosity among high school students. ### Improving Archimedes' Method

9-12
Archimedes was the first mathematician to develop a converging series approximation to π. That highly influential discovery guided the development of calculus many hundreds of years later. However, his method only gives lower and upper boundaries that form intervals known to capture π, not a single numeric estimate of π. In this lesson, students ask, “Where is π located in those intervals?” They also discover an improvement to Archimedes' method that generates the infinite digits of π more efficiently and accurately. ### Capture–Recapture

6-8, 9-12
In this lesson, students experience an application of proportion that scientists actually use to solve real-life problems. Students learn how to estimate the size of a total population by taking samples and using proportions. The ratio of “tagged” items to the number of items in a sample is the same as the ratio of tagged items to the total population. ### Bean Counting and Ratios

6-8, 9-12

By using sampling from a large collection of beans, students get a sense of equivalent fractions, which leads to a better understanding of proportions. Equivalent fractions are used to develop an understanding of proportions.

This lesson can be adapted for lower-skilled students by using a more common fraction, such as 2/3. It can be adapted for upper grades or higher-skilled students by using ratios that are less instinctual, such as 12/42 (which reduces to 2/7).

Scaffold the level of difficulty in this lesson by going from a simple ratio such as 2/3 to more complicated ratios such as 2/7 or 5/9. ### Will the Best Candidate Win?

9-12
This lesson plan for grades 9‑12 is adapted from an article in the January 2000 edition of Mathematics Teacher. The following activities allow students to explore alternative voting methods. Students discover what advantages and disadvantages each method offers and also see that each fails, in some way, to satisfy some desirable properties.