## Roads and Ramps: Slopes, Angles, and Ratios

In this unit, students work with concepts and skills involving angles, slopes, scale drawings, right triangles, proportions and the tangent ratio. Students begin by building a “slope-o-meter” to explore slope and angle relationships. This leads to work with the Pythagorean Theorem. Students then investigate slope as a ratio and use the idea of percent-grade of roads to study real-world slopes. They then work with tangent ratios and apply everything they have learned to design a road and build a 3-D model of it.

### NCTM standard: Geometry

• Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
• Use visualization, spatial reasoning, and geometric modeling to solve problems

### NCTM standard: Measurement

• Understand measurable attributes of objects and the units, systems, and processes of measurement
• Apply appropriate techniques, tools, and formulas to determine measurements

### NCTM standard: Algebra

• Understand patterns, relations, and functions
• Represent and analyze mathematical situations and structures using algebraic symbols

### Primary Mathematical Goals

• Make scale drawings involving angles of elevation
• Build three-dimensional scale models that are consistent with given dimensions
• Develop and apply the Pythagorean Theorem
• Use the tangent ratio to solve problems
• Understand slope as an angle, slope as a ratio, and the connection between these ideas
• Write and solving proportions