Ratios, Rates, and Unit Rates
Lesson 3 of 20
Objective: Students will be able to understand and demonstrate the similarities and differences between ratios, rates, and unit rates.
The Curriculum Reinforcement task for today is very simple. It is simply for the purpose of keeping students' decimal calculation skills sharp and accurate.
In today’s opening ratio activity I will ask my students to simplify ratios. As I project the slide I will say, "See if you can simplify each ratio so that the denominator is 1 if you write the ratio as a fraction."
We practice this skill because today we will begin to talk about unit rates. The ratios that the students will simplify are as follows:
9/3 4.5/5 6/1.5
Once the students have simplified these three ratios, we will identify the fraction with a denominator of "1". I will use these examples to introduce the class to the definition of a unit rate. Then, I will ask, “How does the exercise that we just completed help us to understand how to determine a unit rate?”
Next, we will discuss some of the different types of ratio statements one encounters when solving problems. It is important for students to consider how the type of ratio can influence how a problem can be solved. For example, unit rates are helpful for comparisons. If a problem asks two compare two prices, calculating the unit rate indicates that we would necessarily start simplifying by identifying the greatest common factor. When finding a unit rate you will simplify the ratio by one of the two quantities being compared, so that one of those quantities is being compared to a quantity of one. This being the case, it is possible to end up with a fractions or decimal quantity being compared to 1 (MP7).
I will first discuss the meaning of a ratio based upon what we learned in the previous lesson.
A ratio is a comparison between two quantities by multiplication or division. I will ask, “What does this mean?” After receiving several answers, I will clarify what I want them to know by telling them that, “A ratio is usually characterized by having the same type of unit.”
- 4 apple pies to 3 cherry pies – They both have the same unit of pies… we are comparing pies to pies
- 2 apples to 5 oranges – They both have the same unit of fruit… we are comparing the quantity of one type of fruit to another type of fruit.
A rate is a comparison in which two quantities with different units are being compared. A rate is a special type of ratio
- 3 burgers in 2 minutes – In this case we are comparing the unit of burgers to the unit of minutes.
- Miles per hour, Miles per gallon, $ per hour are very common rates
A unit rate is special type of rate. In this case, the comparison is made with a denominator of 1, which is why it is called a unit rate. The word “UNIT” indicates that we are interested in a "Quantity per one of" comparison.
- 60 miles per (one) hour. This phrase, “60 miles per hour” means that something is moving at a rate of 60 miles for every 1 hour that it is moving
- 2 pencils per student -- note that the "one" is often omitted in stating a unit rate
Try It Out
To prepare students for independent exploration, we will complete an activity in which I guide students through practice with the concept of rates and unit rates. My students will complete the problems presented on the PowerPoint, converting given rates into unit rates.
I expect my students may have trouble determining the unit rate when the rate includes terms that do not have a common integer factor. When quantities being are not factors or multiples of each other, my students often lack an efficient strategy for determining the unit rate. To help them, I emphasize that, "a ratio is a comparison between two quantities. It is NOT A FRACTION." I help my students to understand that some of the constraints that apply to operations on fractions do not apply to ratios. For example, in fractions, the numerator and the denominator must be integers. This constraint does not apply to a unit like 1.29 per liter of milk. I often find it necessary to highlight this fact for my students: a ratio can be any type of quantity compared to any other type of quantity.
Next, my students will demonstrate their understanding of today’s lesson by completing an activity on their own. They will be given 15 minutes to complete this worksheet. I will ask them to work alone recording their work, at first. As I assess students progress, I will determine whether or not to give students an additional 10 minutes to collaborate with a partner to complete the problems or share answers and explanations.
The worksheet requires the students to find unit rates in different types of situations. Then, it asks students to compare ratios, rates, and unit rates in a brief essay that compares and contrasts the three types of numerical comparison.
I expect to give students time to compare their essays, making edits or additions as they listen to their partners' essays, or, their partners' feedback. When this works well, students discuss the mathematics involved in using ratios to solve problems. As students work, I am listening for this as well as conversations about difficulties they are having.
To close this lesson, I will select one student per problem from the Independent Exploration.
- The students selected for the first section will go to the board and demonstrate how they came to their solution.
- The students selected for the second section of the worksheet will present their problems under the document camera so that we can see what they did step by step without taking up too much time.
- The student selected for the last section of the worksheet will read their essay aloud. While they are reading their essay, I will make a bulleted list of the different characteristics of ratios, rates, and unit rates mention in their essay. The rest of the students in the class will need to critique the essay and my bulleted list (MP3).
As a Ticket Out The Door I ask my students to provide an example of a ratio, a rate, and a unit rate, explaining why each example fits the category.