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### Constant Dimensions

6-8

Students will measure the length and width of a rectangle using both standard and non-standard units of measure. In addition to providing measurement practice, this lesson allows students to discover that the ratio of length to width of a rectangle is constant, in spite of the units. For many middle school students, this discovery is surprising.### The Ratio of Circumference to Diameter

6-8

Students measure the circumference and diameter of circular objects. They calculate the ratio of circumference to diameter for each object in an attempt to identify the value of pi and the circumference formula.### Discovering the Area Formula for Circles

6-8

Using a circle that has been divided into congruent sectors, students will discover the area formula by using their knowledge of parallelograms. Students will then calculate the area of various flat circular objects that they have brought to school. Finally, students will investigate various strategies for estimating the area of circles.### Discovering the Area Formula for Triangles

6-8

In this lesson, students develop the area formula for a triangle. Students find the area of rectangles and squares, and compare them to the areas of triangles derived from the original shape.### Finding the Area of Parallelograms

6-8

Students will use their knowledge of rectangles to discover the area formula for parallelograms.### Finding the Area of Trapezoids

6-8

Students discover the area formula for trapezoids, as well as explore alternative methods for calculating the area of a trapezoid. ### Finding the Area of Irregular Figures

6-8

Students will estimate the areas of highly irregular shapes and will use a process of decomposition to calculate the areas of irregular polygons.### Scaling Away

6-8

Students will measure the dimensions of a common object, multiply each dimension by a scale factor, and examine a model using the multiplied dimensions. Students will then compare the surface area and volume of the original object and the enlarged model. ### Sticks and Stones

3-5, 6-8

Students will play *Sticks and Stones*, a game based on the Apache game "Throw Sticks," which was played at multi-nation celebrations. Students will collect data, investigate the likelihood of various moves, and use basic ideas of expected value to determine the average number of turns needed to win a game.

### Learn the Game

6-8

In the following lessons of this unit, students will have an opportunity to fully explore the patterns that result from the Paper Pool game. In this lesson, however, students will only spend time learning the rules, playing the game, and collecting data.