Word Problems with Unit Rates

Introduction

In this lesson plan, your learner will solve word problems with unit rates by drawing models and interpreting the meaning of the unit rate. Before diving into rate and ratio word problems, it is important for your learner to grasp the concept of equivalent ratios, as these play a pivotal role in solving problems related to unit rates.

Key Concepts for Word Problems with Unit Rates

Here are a few concepts that are helpful to know for this lesson:

• Recognizing Ratio Word Problems: Ratio word problems describe relationships between two quantities. They often include phrases like "for every," "per," or "each." These problems require you to understand how one quantity changes in relation to another.
• Interpreting Ratios and Rates: Depending on the context of the word problem, it can be helpful to interpret the ratio or rate in different ways. For example, a juice recipe of 5 cups of lemonade and 2 cups of fruit punch can be interpreted in many ways such as:
  • 5 cups of lemonade / 2 cups of fruit punch
  • ⁵/₂ cups of lemonade per cup of fruit punch
  • 2 cups of fruit punch / 5 cups of lemonade
  • ²/₅ cups of fruit punch per cup of lemonade

Strategies for Solving Ratio Word Problems

There are many strategies that your learner can use to help them understand and solve ratio word problems. Here are a few for them to try:

• Double Number Lines: Double number lines help visualize the relationship between two quantities by aligning them on parallel lines, making it easier to see equivalent ratios.
• Tables: Tables organize information in rows and columns, allowing you to systematically compare ratios and find patterns.
• Graphs: Graphs visually represent ratios and rates. Graphs for ratios and rates follow a linear pattern which is useful for solving equations and word problems.
• Drawing Pictures: Drawing pictures or tape diagrams can simplify complex word problems by breaking them down into more manageable visual representations.

Teaching Plan

For this lesson plan, we will look at a few examples of ratio word problems and possible strategies for solving them. Each problem provides a unit rate that your learner should interpret and apply to finding the solution.

Examples and visuals to support the lesson:

Example 1: Ratios for Mixing Juice

A juice recipe calls for 5 cups of lemonade for every 2 cups of fruit punch. Interpret each of the following: 5 to 2, 2:5, 2/5, 5/2.

• The purpose of this example is to review the different ways that ratios can be written (using words like "to," using a colon, or as a fraction). It also emphasizes that the order of the numbers determines how it is interpreted since each number corresponds to a quantity in the word problem.
• Have your learner draw a model (such as a double number line or bar model) that shows the relationship between the quantities then interpret the meaning of each rate and ratio.
• The ratio 5 to 2 represents the 5 cups of lemonade for every 2 cups of fruit punch. While the ratio 2:5 can be interpreted as 2 cups of fruit punch for every 5 cups of lemonade. The relationships between the quantities is the same, just in a different order.
• Note that the fractions can be interpreted in multiple ways. For instance, 2/5 can be interpreted as a rate (2 cups of fruit punch/5 cups of lemonade) or as a unit rate (2/5 cups of fruit punch per 1 cup of lemonade). Being able to recognize the two interpretations of the fractions will help your learner find equivalent ratios and solve problems more easily.

Skill Check
I can interpret the meanings of ratios and rates that are written in different ways.

Example 2: Ratios for Mixing Paint

Brett uses 3/4 of a cup of blue paint for every cup of red paint to take a certain shade of purple paint. If he needs 21 cups of purple paint for an art project, how many cups of each, red and blue paint, does he need?

• Your learner may solve this problem using a variety of strategies. One way is to realize that the ratio of blue paint to red paint is 3 to 4. From here, they may draw a tape diagram made of 7 total boxes (3 blue boxes and 4 red boxes). From there they can reason that if the total amount needed is 21 cups, then each box of the tape diagram has a value of 3 cups of paint. Therefore Brett would use 9 cups of blue paint and 12 cups of red paint.
• Another strategy involves setting up a table with three columns (cups of blue paint, cups of red paint, and total cups of paint). Using the information from the word problem, we can write 3 for blue paint, 4 for red paint, and 7 for total paint. Since we want a total of 21 cups of paint, we can multiply each value by 3. The result is 9 cups of blue paint, 12 cups of blue paint, and a total of 21 cups of paint.

Skill Check
I can interpret and use unit rates to solve word problems.

Example 3: Rate of Snowfall

It snows ²/₃ of an inch per hour. If 6 inches of snow fell, how long was it snowing for?

• The ratio associated with this rate is 2 inches in 3 hours. Once your learner identifies this ratio, they may realize that multiplying both values by 3 shows that it would take 9 hours for 6 inches of snow to fall.
• A common mistake in this problem is for students to multiply 2/3 by 6 and say the answer is 4 hours. If your learner makes this error, have them draw a picture to model the scenario. They will realize that if it snows 2/3 of an inch for 4 hours, only 8/3 (or 2 2/3) inches would have accumulated.

Continue to provide your learner with word problems that have unit rates. Encourage them to draw models and carefully interpret the meaning of the unit rate before solving the problem. This step ensures that they understand the problem thoroughly and will be more confident when finding the solution.

Skill Check
I can use different strategies to solve word problems and explain my reasoning.

Summary

Through this lesson plan, your learner will develop a solid foundation in solving rate word problems using unit rates and drawing models to visualize relationships between quantities. Interpreting and understanding the meaning of unit rates equips them with valuable problem-solving skills applicable to many real-world scenarios. This skill will also prepare them for solving problems with proportions, percentages, and converting units of measurement.

Teaching Plan adapted from Utah Middle School Math Project under CC BY 4.0.