**Name** |
**Description** |

Toilet Roll: | The purpose of this task is to engage students in geometric modeling, and in particular to deduce algebraic relationships between variables stemming from geometric constraints. |

Coins in a circular pattern: | Using a chart of diameters of different denominations of coins, students are asked to figure out how many coins fit around a central coin. |

The Lighthouse Problem: | This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat. |

Solar Eclipse: | This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth. |

Seven Circles III: | This provides an opportunity to model a concrete situation with mathematics. Once a representative picture of the situation described in the problem is drawn (the teacher may provide guidance here as necessary), the solution of the task requires an understanding of the definition of the sine function. |

Running around a track II: | The goal of this task is to model a familiar object, an Olympic track, using geometric shapes. Calculations of perimeters of these shapes explain the staggered start of runners in a 400 meter race. |

Running around a track I: | In this problem, geometry is applied to a 400 meter track to find the perimeter of the track. |

How thick is a soda can? (Variation II): | This problem solving task asks students to explain which measurements are needed to estimate the thickness of a soda can. |

How thick is a soda can? (Variation I): | This problem solving task challenges students to find the surface area of a soda can, calculate how many cubic centimeters of aluminum it contains, and estimate how thick it is. |

How many leaves on a tree? (Version 2): | This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree. |

How many leaves on a tree?: | This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree. |

How many cells are in the human body?: | This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body. |

Hexagonal pattern of beehives: | The goal of this task is to use geometry to study the structure of beehives. |

Global Positioning System II: | Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems. |

Use Cavalieri’s Principle to Compare Aquarium Volumes: | This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere. |

Tennis Balls in a Can: | This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a "double-naped cone" with vertex at the center of the sphere and bases equal to the bases of the cylinder |

Regular Tessellations of the Plane: | This task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. These tessellations are studied here using algebra, which enters the picture via the formula for the measure of the interior angles of a regular polygon (which should therefore be introduced or reviewed before beginning the task). The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible. |