A rate is a special ratio that compares two quantities measured in different units. It is a comparison of two different quantities. For example, miles per hour, price per pound, or words per minute are all examples of rates.

To calculate a rate, you need to divide one quantity by the other. The formula for calculating rate is:

**Rate = Quantity 1 / Quantity 2**

When expressing a rate, it is important to include the units. For example, if the rate is 60 miles per hour, both "miles" and "hours" should be included to indicate the units being compared.

A unit rate is a rate with a denominator of 1. It is often used to compare the prices of items. For example, if a 2-pound bag of rice costs $4, the unit rate is $2 per pound.

Rates can be used in proportions to solve problems. For example, if 3 gallons of juice cost $6, you can set up the proportion 3 gallons / $6 = x gallons / $12 to find the cost of 6 gallons.

Rates are used in many real-life situations, such as speed (miles per hour), unit pricing in stores, and performance measures like words per minute in typing or reading speed.

- Calculate the rate: 360 miles in 6 hours.
- Determine the unit rate: 15 apples for $6.
- Use a proportion to solve: If 5 shirts cost $75, how much would 8 shirts cost?

Study GuideSimilarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand and use ratios and proportions to represent quantitative relationships.

Compute fluently and make reasonable estimates.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Geometry (NCTM)

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.

Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Apply transformations and use symmetry to analyze mathematical situations.

Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.

Measurement (NCTM)

Apply appropriate techniques, tools, and formulas to determine measurements.

Solve problems involving scale factors, using ratio and proportion.

Grade 8 Curriculum Focal Points (NCTM)

Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle

Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.