Understanding ratios and their applications is fundamental in solving various real-world problems involving proportions and comparisons. In this lesson plan, your learner will engage in word problems that require calculating ratios and unit rates. These activities will set the stage for deeper explorations into unit rates and proportional relationships.
Here are a few concepts that are important to know for the lesson:
Rate: A rate is a comparison of two quantities with different units. It expresses how one quantity changes in relation to another. Rates are often used to describe how one quantity changes with respect to another, such as speed (miles per hour), price (dollars per pound), or efficiency (tasks completed per hour).
Unit Rate: A unit rate is a specific type of rate that compares a quantity to one unit of another quantity. It represents the amount of one quantity corresponding to one unit of the other quantity. For example, if the price of jelly beans is $2 per pound, the unit rate is $2 per pound, indicating the cost of one pound of jelly beans.
Two Ways to Write Unit Rates: Unit rates can be written in two forms: as a fraction a/b or b/a . The first form a/b indicates the quantity of the first unit for one unit of the second, while the second form b/a represents the quantity of the second unit for one unit of the first. For instance, if the ratio of miles to hours is 60/1 , the unit rate can be written as 60/1 miles per hour or 1/60 hours per mile.
Unit Price vs. Unit Rate: In scenarios involving price, it is customary to find the unit price, which is the cost per one unit of a specific item (e.g., dollars per item). However, in this introductory lesson, we focus on unit rates without distinguishing between unit price and unit rate. The emphasis is on understanding how to calculate and interpret ratios with different units.
Teaching Plan
The following activities will help your learner become confident in calculating ratios and unit rates.
Examples and visuals to support the lesson:
1. Reasoning Strategies for Ratios
Many strategies can be used to solve problems with ratios. This activity will give you an opportunity to observe how your learner intuitively approaches ratios.
Begin with a word problem that requires your learner to reason about ratios to find the solution. Observe the strategies that your learner uses. Encourage them to explain their reasoning strategies.
For example: Chris and Danika are shopping for jelly beans to fill eggs. Danika needs to buy 8 pounds of jelly beans. How much will 8 pounds cost?
Here are examples of strategies that your learner may use: Since the jelly beans cost $2 per pound then 8 pounds would cost $16. This strategy uses the unit rate which we will explore further in the next activity.
Another strategy is to use multiplication and/or division to create equivalent ratios, then find the ratio that corresponds to 8 pounds. Your learner might also notice that in this scenario, the cost is always twice the number of pounds.
Follow up with another question: Chris has $8. How many pounds of jelly beans can he buy for $8? Your learner will likely apply strategies similar to the first question and find that Chris can buy 4 pounds of jelly beans for $8.
Skill Check
I can use reasoning strategies to solve ratio word problems.
2. Defining Unit Rate
Once your learner has explored strategies for solving problems with rates, introduce them to the concept of unit rates.
Explain to your learner that sometimes, when solving word problems involving ratios, it is helpful to find the unit rate. The unit rate is the amount of one quantity that corresponds to 1 unit of the other quantity.
In the jelly bean example, you can find the unit rate for cost per pound (dollars/pound). This tells us the cost of 1 pound. You can also find the unit rate for pounds per dollar (pounds/dollar ). This tells us the number of pounds we can buy for $1.
Have your learner find the unit rate in dollars/pound and pound/dollars for the jelly bean scenario. Ask your learner: What is the unit rate in dollars/pound? In other words, how much will you pay for 1 pound of jelly beans? (Solution: $2 per pound.) Encourage them to draw models or write their steps to show their strategy.
Next ask your learner: What is the unit rate in pounds/dollar? In other words, how many pounds of jelly beans can you get for 1 dollar? (Solution: 1/2 pound per dollar.)
Skill Check
I can write unit rates to represent word problems in two ways.
3. Calculating Unit Rates
This activity will give your learner an opportunity to practice interpreting and calculating unit rates to solve word problems.
Provide your learner with additional word problems for them to practice finding unit rates. Have them find both unit rates. Note that at some point in this lesson, your learner may realize that the unit rates associated with a given ratio are reciprocals of each other.
Remind your learner of the importance of writing the units with the values in the ratios. For example, writing 2/1 does not help us understand what the rate represents. It can be interpreted, for example, as $2 for 1 pound of jelly beans or 2 pounds of jelly beans for $1.
Your learner may discover that the unit rate can be found by dividing the numerator of a ratio by its denominator. It might be tempting to show your learner this method since it is the most straightforward way to calculate unit rates. However, your learner should use their own strategies during this lesson to get a firm grasp on the underlying concepts.
Skill Check
I can use what I know about unit rates to solve ratio word problems.
The concept of unit rate emerges as a powerful tool for solving ratio word problems efficiently. By calculating ratios and unit rates, your learner will gain insight into the relationship between quantities and their costs or values. Encourage your learner to explore multiple strategies and articulate their reasoning to enhance their understanding of ratios. Through continued practice and exploration, your learner will confidently apply ratio strategies to solve a wide range of problems, building a strong foundation for more advanced math skills.
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Skill Check