Learn on PengienVision, Mathematics, Grade 6Chapter 5: Understand and Use Ratio and Rate

Lesson 5: Understand Rates and Unit Rates

In this Grade 6 enVision Mathematics lesson from Chapter 5, students learn to define and work with rates and unit rates as special types of ratios that compare quantities with unlike units of measure. Students practice finding equivalent rates using ratio tables and multiplication, then calculate unit rates by writing equivalent rates with a denominator of 1 to solve real-world problems involving speed, pricing, and recipes. Aligned to Common Core standards 6.RP.A.2 and 6.RP.A.3, the lesson builds fluency with proportional reasoning skills essential for sixth-grade math.

Section 1

Write a rate as a fraction

Property

A rate compares two quantities of different units. A rate is usually written as a fraction. When writing a fraction as a rate, we put the first given amount with its units in the numerator and the second amount with its units in the denominator. When rates are simplified, the units remain.

Examples

  • If Bob drove his car 525 miles in 9 hours, the rate is written as 525 miles9 hours\frac{525 \text{ miles}}{9 \text{ hours}}, which simplifies to 175 miles3 hours\frac{175 \text{ miles}}{3 \text{ hours}}.
  • A price of 595 dollars for 40 hours of work is the rate 595 dollars40 hours\frac{595 \text{ dollars}}{40 \text{ hours}}.

Section 2

Understanding Unit Rate

Property

A rate is a ratio of two quantities.
The unit rate is the amount of one quantity that corresponds to 1 unit of the other quantity.
The designation of unit rate must be clear about the choice and order of the units.

For a ratio a:ba:b with b≠0b \neq 0, the unit rate is ab\frac{a}{b} units of the first quantity per 1 unit of the second quantity.

Examples

  • If you pay 9 dollars for 3 sandwiches, the unit rate is found by dividing: 9÷3=39 \div 3 = 3 dollars per sandwich.
  • A cyclist travels 30 miles in 2 hours. The unit rate for her speed is 30÷2=1530 \div 2 = 15 miles per hour.
  • A team scores 45 points in 3 quarters. Their unit rate is 45÷3=1545 \div 3 = 15 points per quarter.

Section 3

Two Unit Rates for Every Rate

Property

For any rate that compares two different quantities, you can calculate two different unit rates. If a rate compares quantity A to quantity B, the two unit rates are found by calculating:

quantity Aquantity Bandquantity Bquantity A \frac{\text{quantity A}}{\text{quantity B}} \quad \text{and} \quad \frac{\text{quantity B}}{\text{quantity A}}

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Write a rate as a fraction

Property

A rate compares two quantities of different units. A rate is usually written as a fraction. When writing a fraction as a rate, we put the first given amount with its units in the numerator and the second amount with its units in the denominator. When rates are simplified, the units remain.

Examples

  • If Bob drove his car 525 miles in 9 hours, the rate is written as 525 miles9 hours\frac{525 \text{ miles}}{9 \text{ hours}}, which simplifies to 175 miles3 hours\frac{175 \text{ miles}}{3 \text{ hours}}.
  • A price of 595 dollars for 40 hours of work is the rate 595 dollars40 hours\frac{595 \text{ dollars}}{40 \text{ hours}}.

Section 2

Understanding Unit Rate

Property

A rate is a ratio of two quantities.
The unit rate is the amount of one quantity that corresponds to 1 unit of the other quantity.
The designation of unit rate must be clear about the choice and order of the units.

For a ratio a:ba:b with b≠0b \neq 0, the unit rate is ab\frac{a}{b} units of the first quantity per 1 unit of the second quantity.

Examples

  • If you pay 9 dollars for 3 sandwiches, the unit rate is found by dividing: 9÷3=39 \div 3 = 3 dollars per sandwich.
  • A cyclist travels 30 miles in 2 hours. The unit rate for her speed is 30÷2=1530 \div 2 = 15 miles per hour.
  • A team scores 45 points in 3 quarters. Their unit rate is 45÷3=1545 \div 3 = 15 points per quarter.

Section 3

Two Unit Rates for Every Rate

Property

For any rate that compares two different quantities, you can calculate two different unit rates. If a rate compares quantity A to quantity B, the two unit rates are found by calculating:

quantity Aquantity Bandquantity Bquantity A \frac{\text{quantity A}}{\text{quantity B}} \quad \text{and} \quad \frac{\text{quantity B}}{\text{quantity A}}

Examples