What starts as a basic question of division and remainders quickly turns abstract in this question of related ratios and radii. The class works to surround a central coin with coins of the same and different values, then develops a formula to relate the radii of the central coin and size / number of the surrounding coins. Extensive teaching notes provide different methods and supports for solving, should redirection of learner efforts be necessary.

Concepts

circles, radius, trigonometric equations, math modeling

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Additional Tags

math

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Instructional Ideas

• Rather than providing the given table, have students either measure coins to create their own table, or perform the arranging with actual coins 
• Look up coin measurements from other countries' currency to develop other patterns before finding the abstract ratio for r/s
• Use combinations of coins to see if a pattern can be developed if all of the surrounding circles are not the same size

Classroom Considerations

• If not used in sequence with Seven Circles I, II, or III the class might need some help setting up the triangles and ratios needed for the calculations in the final part
• May require sensitivity in requiring the class to provide coins as manipulatives, given economic considerations
• Use as a refresher lesson on triangle trigonometry before introducing unit circle trigonometry to help strengthen links between the two

Pros

• Creative use of common physical objects to develop a theoretical geometric problem
• Answer key and teacher notes provided outline more than one correct solution method, appealing to diverse learner strengths

Cons

• Class worksheet not separated from teacher notes and answer key

Common Core

HSG-SRT.C.8 HSG-MG.A.1 HSG-MG.A.2 HSG-MG.A.3