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.

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.

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.

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.

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.

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.

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.

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.

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