Lesson 5: 2-6 Ratios and Proportions

What must be true about the units in the numerators and denominators of ratios in a proportion?

Is there more than one way to set up a proportion?

How can you find a unit rate from a given rate?

ratio
proportion
means
extremes
rate
unit rate
scale
scale model

N.Q.1 Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

3A Functions
3B Functions
3C Functions

A ratio is a comparison of two numbers by division. A ratio is called a rate if the two numbers of a ratio represent measurements with different units, such as miles and hours. A proportion is an equation stating that two ratios are equal.

Bicycling: The gear on a bicycle is 8:5. This means that for every 8 turns of the pedals, the wheel turns 5 times. Suppose the bicycle wheel turns about 2435 times during a trip. How many times would you have to crank the pedals during the trip?

Point out that the definitions of extremes and means are not arbitrary. In the proportion a/b = c/d, a and d are the extremes, and b and c are the means. Remind students that the ratios can be written in the form x:y. If the proportions above are rewritten in this form, they are a:b = c:d. Looking at this proportion, a and d are the extremes because they are on the outside, and extreme is a synonym for outside. Similarly, b and c are the means because they are in the middle, and mean is often a synonym for middle.

If you have any doubt that your students have mastered the concept of proportions, then place students in small groups to work through the Check Your Understanding problems. Have a student from each group report on that group's progress and areas in which the group may need assistance. It is important for students to have a good understanding of writing and solving proportions before studying the next few lessons.

Extension: Present students with the following problem: Kala made a scale drawing of a 12 m × 15 m room. The scale drawing is 9.6 cm × 12 cm. What scale did she use? Explain. Justify your answer.

Exercises 51 - 60

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