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Geometry

Power of Points

9-12
In many curricula, the Power of Points theorem is often taught as three separate theorems: the Chord-Chord Power theorem, the Secant-Secant Power theorem, and the Tangent-Secant Power theorem. Using a dynamic geometry applet, students will discover that these three theorems are related applications of the Power of Point theorem. They also use their discoveries to solve numerical problems.
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Geometry

Law of Sines

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

Law of Cosines

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

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.
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Data Analysis and Probability

Paying for Your Wheels

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.
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Geometry

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.
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Geometry

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.
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Geometry

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.
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Number and Operations

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.
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Number and Operations

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.