Exploration 1: Polygon Perambulations

The word "perambulate" means to "walk around." In this activity, you figuratively (and maybe literally!) perambulate polygons, looking at their angles from many different perspectives. As you develop and compare a variety of strategies for drawing regular polygons, you naturally learn about central, interior, and exterior angles—not just their definitions, but relationships between them. Eventually, you discover and justify important algebraic formulas enabling you to calculate angles in polygons without having to measure them.

Exploration 2: Impossible Polygons

Can you make a concave quadrilateral? How about a pentagon with four right angles? In this investigation, you are challenged to create shapes satisfying certain geometric conditions. As you try to discover which drawings are possible and which are not, you explore deep ideas about mathematical definitions.

Exploration 3: Starstruck!

Draw intriguing star-shaped designs and explore their angles. There are countless discoveries to make and patterns to explore—and lots of room for creativity and surprise! If you are ready to take the activity to the next level, you will connect your geometric discoveries to advanced algebraic concepts.

Exploration 4: Geoboard Squares

Explore connections between area and algebra as you search for squares on a geoboard or dot paper. At first, it may seem easy, but watch out—there is more to this challenge than meets the eye!

Exploration 5: Creating Area Formulas

In this exploration, you are not satisfied with simply believing what you are told. Memorizing and calculating are useful skills, but creating formulas and understanding where they come from require the kinds of thinking skills that real mathematicians use. The formulas that you discover may not always be exactly like the ones in your textbook, but if you can prove that they work, they will belong to you!

Exploration 6: A New Slant on Measurement

By extending some of the ideas from Exploration 4, you create methods for calculating lengths of diagonal segments on a grid. By the time you complete this work, you will have discovered a famous mathematical formula, and you will be prepared to take on countless new mathematical challenges in the rest of this exploration and beyond!

Exploration 7: Ladders and Saws

Top mathematicians know that one of the best ways to solve a difficult problem is to find a way to make it easier. Some geometrical problems are challenging mainly because of their complexity—the large number of objects and relationships that you must pay attention to. In this exploration, you practice isolating simple patterns within complex drawings and applying them to make surprising and useful discoveries. You may even learn why the interior angles in a triangle must always add to 180°.

Exploration 8: Designing Nets

Create your own design for a special type of pyramid. Build it and use your model to find the pyramid's surface area and volume. Next, use a set of design specifications to engineer a cone. After you construct it, test your model to ensure that it has the correct volume. If you are ready to keep exploring, use your construction process to discover a formula for the surface area of cones.

Exploration 9: Measuring Oceans

From the comfort of your chair, estimate the amount of water in the Earth's oceans. You may be surprised at how close you can come! Then reach back into the mind of an ancient Greek mathematician to learn where the formula for the volume of a sphere came from and how we can be sure that it is correct.