# Advanced Common Core Math Explorations: Ratios, Proportions, and Similarity

The Advanced Common Core Math Explorations series creates and nurtures mathematical adventurers! I am especially excited about this new book in the ACCME series. We are using a new format inspired by Ian Byrd of byrdseed.tv. The problems have been separated onto individual pages with simplified, bulleted directions. Each problem is introduced with a "teaser" that encourages students to come up with their own questions and observations before receiving the directions. The discussion guides have been rewritten as "I notice" statements and "I wonder" questions, and there are vignettes showing examples of classroom discussion. The look and feel of the pages is open and inviting, and the writing is less formal with some of the "messy" details removed. (Some of these details will be available on this page for those who would still like them!) The new format enhances the open-ended, inquiry-based nature of the problems. Incidentally, Ian has created wonderful videos for a few of the activities in the ACCME series, with more planned. Please check them out at his website!

Exploration 1: The Incredible Shrinking Universe

Create scale models of our Solar System and calculate the travel time to another star. Explore the mathematics behind the "Cosmic Calendar" from the TV series Cosmos.

Exploration 2: Ramps, Paints, and Hot-Air Balloons

Learn about proportional relationships by exploring paint mixtures and studying science experiments on the expansion of heated air. Begin to learn about slope by helping a company design multi-purpose ramps.

Exploration 3: Gear Up!

Analyze the gears on a 20-speed bike and figure out how to use gear ratios to calculate a bike's speed. Help a cycler design a cycling calculator for her new website.

Exploration 4: Perplexing Percentages

Learn the ins and outs of percentages by solving a set of increasingly challenging real-world problems. Draw diagrams that connect percentage concepts to proportional relationships, and use algebraic reasoning to make predictions and prove them.

Exploration 5: Scaling a Tower

Use scale drawings to help Derek and Beth estimate the height of a transmission tower near their home. In the process, begin investigating size and angle relationships in similar triangles. If you continue to the final stage of the activity, you will explore the tangent function from trigonometry and develop strategies for solving a system of algebraic equations.

Exploration 6: Keep it in Proportion

Discover and explain what causes cross product relationships in proportions. Explore examples of "tricky" relationships that appear to be proportional but are not.

Exploration 7: Grab Bag

Explore a variety of real-world scenarios that deepen and extend your understanding of proportional relationships: counting wildlife populations, equalizing population densities in two schools, and calculating average rates of travel.

Exploration 8: Expanding and Contracting

Explore angle, length, and area relationships between similar figures by creating and analyzing complex drawings and solving challenging problems. In the final part of the activity, you apply your new knowledge of ratios and similarity to explore the famous Fibonacci sequence and the golden ratio.

Exploration 9: Pythagorean Connections

Connect your knowledge of similarity to the Pythagorean theorem in order to study properties of square roots and to design multiple strategies for solving a challenging geometry problem in minimizing lengths of segments.

Exploration 10: Twist and Shrink

Unearth surprising and beautiful connections between centroids (centers) of polygons and similarity. As your discoveries accumulate, you will expand your knowledge to include polygons with more and more sides. Advanced students may be challenged to prove predictions algebraically.