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### Northwestern Crows

9-12

Sea gulls and crows feed on various types of mollusks by lifting them into the air and dropping them onto a rock to break open their shells. Biologists have observed that northwestern crows consistently drop a type of mollusk called a whelk from a mean height of about 5 meters. The crows appear to be selective; they pick up only large-sized whelks. They are also persistent. For instance, one crow was observed to drop a single whelk 20 times. Scientists have suggested that this behavior is an example of decision-making in optimal foraging.### Analyze the Data

9-12

Sea gulls and crows feed on various types of mollusks by lifting them
into the air and dropping them onto a rock to break open their shells.
Biologists have observed that northwestern crows consistently drop a
type of mollusk called a whelk from a mean height of about 5 meters.
The crows appear to be selective; they pick up only large-sized whelks.
They are also persistent. For instance, one crow was observed to drop a
single whelk 20 times. Scientists have suggested that this behavior is
an example of decision-making in optimal foraging.### Looking Back and Moving Forward

9-12

As you review student work in this unit, it is important to remember
the mathematical objectives/expectations of this Unit Plan that are
stated in *Principles and Standards for School Mathematics*.

### Components of a Vector

9-12

In this lesson, students manipulate a velocity vector to control the movement of a car in a game setting. Students learn that vectors are composed of two components: magnitude and direction.### Sums of Vectors and Their Properties

9-12

This example illustrates how using a dynamic geometrical representation
can help students develop an understanding of vectors and their
properties, as described in the Number and Operations Standard.
Students manipulate a two vectors to control the movement of a plane in
a game-like setting. Students extend their knowledge from the first
lesson to further investigate the system of vectors.### Pi Line

9-12

Students measure the diameter and circumference of various circular
objects, plot the measurements on a graph, and relate the slope of the
line to π, the ratio of circumference to diameter.### Pedal Power

9-12

In this lesson, students investigate slope as a rate of change. Students compare, contrast, and make conjectures based on distance-time graphs for three bicyclists climbing to the top of a mountain. ### Caesar Cipher

9-12

In this lesson, students will investigate the Caesar substitution cipher. Text will be encoded and decoded using inverse operations. ### Exploring Measurement, Sequences, and Curves with Stringed Instruments

9-12

Students will measure lengths on stringed musical instruments, and discuss how the placement of frets on a fretted instrument is determined by a geometric sequence.### Fretting

9-12

In the second lesson of this unit, students will use their discoveries from the first lesson to place frets on a fretless instrument. They will then compare geometric sequences with exponential functions.