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### Princess Dido and the Ox Skin: Investigating Maximizing Area Using a Literature-Based Model

6-8, 9-12

This lesson is based upon a story from Virgil's *Aeneid.*Students work in groups to investigate maximizing area with a fixed length of rope. They investigate which figure results in the greatest area by real-life experimentation as well algebraically. Students gain an understanding of quadratic functions, the isoperimetric principle, and parabolas.

### Increasing Portions, Expanding Waistlines: Exploring the Relationship Between Calories Consumed and Weight

9-12

This
activity uses a 2009 study based on the book *The Joy of Cooking*that found portion sizes in the iconic cookbook had expanded 60% since 1936 and 33% since 1996. Students use linear functions to create models of weight as a function of calories and time, then use the models to make projections about what impact increased portion sizes may have on weight.

### Ice Cream Puddle

9-12

Many
problems in mathematics classes are exactly defined and require very specific
solutions. This lesson is intended to be an open-ended problem with multiple solutions
and multiple entry points. Students use volume formulas and have some decisions
to make about how to use them. After making an initial estimate, students will
use modeling with more concrete parameters to help them refine their choices.### Gerrymandering: Is It or Isn't It?

9-12

Congressional districts are redrawn after a census year
(conducted by the federal government every 10 years) to reflect shifting
populations. Political scientists say there is a major advantage for whichever
political party has control of a state legislature when districts are redrawn.
However, during the process, the minority party tends to make accusations that
districts are being gerrymandered, or drawn in unnatural ways to manipulate
voter populations that would favor the majority. In this lesson, students
explore how mathematics can be used to make decisions about whether a given
congressional district is or is not being gerrymandered. They also use
objective measurements to explore the ambiguity that is inherent in the
process.### Mars Orbit

9-12

Astronomers now know that the planets in our solar system travel around the Sun in roughly circular orbits. However, people used to believe that the Sun and other planets orbited the Earth. If that were the case—if the Earth were the center of our solar system—what would the orbit of the other planets look like relative to Earth? That is the question that students will explore in this lesson. Students will generate parametric equations to describe the position of planets relative to the Sun; then, they will combine the equations to describe the position of Mars relative to Earth.### Maya Constructions

6-8, 9-12

In this lesson, students learn to use a compass and a straight edge to
construct rectangles of leg ratios 1:1; 1:√2; 1:√3; 1:2; and 1:√5. The
lesson culminates with the class constructing a full size façade of a
house using the proportions of the Ancient Maya.### Measuring Uncertainty

9-12

By measuring long jump results, students will discover how to determine the appropriate number of digits they should report when taking a linear measurement. Students will realize that measured numbers are never exact and researcher skill and tools can be analyzed to determine the precision of a measurement.### Analyzing the Data

9-12

In the third lesson, students identify a function that models the data. An interactive graphing tool can be used to determine a function that fits the data points.

### Do You Notice Sum-Thing?

6-8, 9-12

This lesson invites students to investigate the patterns when a "plus sign" (a cross-shaped arrangement of five squares) is placed on the board in various locations. Students will conjecture about the pattern of the five displayed numbers, the sum of the five numbers, and any other patterns that they notice. Students may also explore similar patterns when other shapes, such as a 2 × 1 rectangle, are placed on the hundreds board.### Don't Freeze the Engine

9-12

In this lesson, students will learn to read an antifreeze chart to determine the necessary mix of antifreeze and water for protection against various temperatures. Students will then numerically analyze a particular cooling system to determine how much fluid to drain in order to get the desired percent concentration of antifreeze. Finally, students will set up a system of equations to solve the same system.