## 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