Ratio and Unit Rates

RATIO AND UNIT RATES

The objective for this lesson on ratios and unit rates is, the student will understand and apply the concepts of ratios and unit rate and use ratio and unit rate language to describe the relationship between two quantities.

The skills students should have in order to help them in this lesson include, interpreting data from a table and equivalent fractions.

We will have three essential questionsthat will be guiding our lesson. Number one, how can I describe a relationship between two quantities? Number 2, how can I write a ratio using numbers and words? Or number and symbols? And number three, how can I describe and determine a unit rate using a ratio?

Begin by completing the warm-up answering questions using data displayed in a table to prepare for a ratios and unit rates in this lesson.

SOLVE INTRODUCTION

The SOLVE problem for this lesson is, Eric’s mother is making punch for his school party. She needs to serve seventy two students and four teachers. If the recipe calls for four cups of water for every two cups of juice, how many cups of water will be needed for one cup of juice?

To start we will Study the Problem. First we need to identify where the question is located within the problem and underline the question. How many cups of water will be needed for one cup of juice? Now that we have identified the question we want to put this question in our own words in the form of a statement. This problem is asking me to find the number of cups of water needed for each cup of juice.

During this lesson we will learn how to determine and use ratios and unit rates. We will use this knowledge at the end of the lesson to complete this SOLVE problem.

RATIO – CONCRETE

For this activity you will need counters and toothpicks. Place one counter on the workspace. Place two toothpicks beside the counter. Now compare the number of counters to toothpicks. There is one counter and two toothpicks on the workspace. If the counter represents the first quantity, identify the first quantity. The first quantity is one. If the toothpicks represent the second quantity, identify the second quantity. The second quantity is two. Explain the relationship between the counter and the toothpicks. There is one counter to two toothpicks. Now explain the relationship between the toothpicks and the counter. There are two toothpicks to one counter. The number of objects to be compared will be referred to as a quantity. A quantity is how much there is of something. It has a number value.

Now place three counters on the workspace. And place five toothpicks beside the counters. If the counters represent the first quantity, identify the first quantity. The first quantity is three. There are three counters. If the toothpicks represent the second quantity, identify the second quantity. The second quantity is five. There are five toothpicks. So explain the relationship between the counters and the toothpicks. There are three counters to five toothpicks. Now explain the relationship between the toothpicks and the counters. There are five toothpicks to three counters.

RATIOS – PICTORIAL

We are now going to represent our relationships with pictures. Begin by using the counters and the toothpicks to set up the relationship. We will place one counter on the workspace. And we will place two toothpicks beside the counter. Now let’s draw a picture. Remove the counter and draw a picture of it. Next remove the toothpicks and draw a picture of them. Identify the number of counters in the set. There is one counter. And identify the number of toothpicks in the set. There are two toothpicks. So determine the relationship of the counter and the toothpicks. And fill in the blanks for Question number one. The relationship of the counters to the toothpicks is one counter to two toothpicks. Let’s record this information in our graphic organizer. The first quantity will represent the counters. And the second quantity will represent the number of toothpicks. We have one counter, so our first quantity is one. And we have two toothpicks, so our second quantity is two. For the relationship between the quantities we know that there is one counter to two toothpicks.

Identify the number of toothpicks in the set. There are two toothpicks. Now identify the number of counters in the set. There is one counter. This time determine the relationship of the toothpicks to the counters. And fill in the blanks for Question two. The relationship of the toothpicks to the counters is two toothpicks to one counter. Let’s take a look at the graphic organizer. This time the first quantity is represented by the toothpicks. And the second quantity is represented by the counter. Since there are two toothpicks, the first quantity is two. And since there is one counter, the second quantity is one. So the relationship between the quantities is that there are two toothpicks to one counter.

So what observations do you have regarding the first two relationships we completed? The values are the same with different order. The first time we found the relationship with the counters to the toothpicks. And the second time we found the relationship of the toothpicks to the counters. So the order changed. We are comparing the same two items. Both times we compared counters and toothpicks to each other. The quantities do not change. In each relationship there was one counter and two toothpicks. Do you have any other observations?

Now take a look at the third box in the graphic organizer for Question one. Do you have any ideas how we could shorten the way we describe and write relationships? Take a moment to discuss. Let’s take a look at the statement for Question one. There is one counter to two toothpicks. How could we write the relationship using only the values of the two quantities? We could say one to two. We could also write the relationship using a colon! One colon two. The third way we can write the relationship is by using a fraction bar, one over two. So there are three ways to represent this relationship between the quantities. We could say one to two; one colon two; or one over two. What did you notice about the order of the quantities? The number of counters was first, and the number of toothpicks was second. Explain why the relationship was written this way. They relationship we were describing was counters to toothpicks. So the counters need to come first and the toothpicks need to come second.

Now let’s take a look at the statement for Question two. There are two toothpicks to one counter. How could we write the relationship using only the values of the two quantities? We could write two to one. What was the second way we could write the relationship? We can write the relationship using a colon, two colon one. And what was the third way that we could write the relationship? We could use a fraction bar, two over one. So when we’re representing the relationship between the quantities, we can write this relationship as two to one; two colon one; or two over one. What did you notice about the order of the quantities? The number of toothpicks was first, and the number of counters was second. Explain why the relationship was written this way. The relationship we were describing was toothpicks to counters. Since toothpicks comesfirst in this relationship we represent the quantity of toothpicks first. And since the counters came second in the relationship we represent the quantity of the counters second.

The three ways we wrote a relationship between the two numbers are: Using the word “to.” Separating with a colon. And writing the values as a fraction. The word we can use to describe that relationship is ratio. Record in the graphic organizer for Questions one and two. The word used to describe this relationship is ratio. What is a ratio? A ratio is a comparison of one quantity to another quantity. A ratio tells how one number is related to another number. How is a ratio written? A ratio uses the word “to” when comparing the quantities, places a colon between the quantities, or the two quantities are written as a fraction.

RATIOS – ABSTRACT

Now let’s take a look at the table. The table includes the names of six students and the number of books read by each of the students. What is different about how the information for the relationship between quantities is presented? There is no picture. The information this time is given in a chart. What is the relationship between the number of books Elisa read and the number of books Tomeka read? What does the first quantity represent? Elisa’s name came first in the relationship, so the first quantity represents the number of books Elisa read. Let’s label the first quantity Elisa. Why is this the first quantity? It is listed first in the question. How many books did Elisa read? Elisa read three books. What does the second quantity represent? The second quantity represents the books that Tomeka read. Let’s label the second quantity Tomeka. Why is it the second quantity? It is listed second in the question. Let’s look back at the chart. How many books did Tomeka read? Tomeka read one book. Let’s record this information in the graphic organizer. Elisa read three books, so the first quantity is three. And Tomeka read one book, so the second quantity is one. The relationship between the quantities is that for every three books Elisa read, Tomeka read one book. How is this relationship written as a ratio? Rememberthat we can write it as a ratio three different ways, using the word to, using a colon, or writing the ratio as a fraction. The ratios for this relationship are three to one, three colon one, or three over one.

Let’s take a look at another relationship using the table. What is the relationship between the number of books Andy read and the number of books Elisa read? We can see from the table that Andy read two books. And we know from the previous example that Elisa read three books. So what is the relationship between the quantities? For every two books Andy read, Elisa read three books. How can this relationship be written as a ratio? Remember, that there are three ways to write the ratio, two to three, two colon three, or two over three.

Let’s talk about another relationship. What is the ratio of the number of books Mario read and the total number of books read? How is Question three different from Question two? Question two shows the ratio between two of the parts, and Question three shows the ratio between a part and the whole group. So let’s get the information that we need from the table. How many books did Mario read? Mario read two books. And what was the total number of books read? To figure that out we need to add up the number of books read by each of the six students. The total number books, is twelve. So how can we write the ratio for this relationship? The number of books that Mario read to the total number of books read. We can write the relationship as two to twelve, two colon twelve, or two over twelve. Why are the numbers written in that order? That is the order of the quantities in the relationship. It is important that we keep the order the same when we write the ratio.

UNIT RATES – CONCRETE AND PICTORIAL

We are now going to talk about unit rates. Again, you will need your toothpicks and counters to start this activity. Place nine toothpicks on the workspace. Next place three counters beside the toothpicks. How many toothpicks are in the set? There are nine toothpicks. And how many counters are in the set? There are three counters. Let’s identify the ratio of toothpicks to counters. We can represent this ratio as nine to three, nine colon three, or nine over three. How can we make groups with only one counter in each group? Well, let’s start with another question. How many groups can we make with one counter in each group? We have three counters, so if we place one counter in each group, we can see that there is a total of three groups. Since we have nine toothpicks, we need to split the toothpicks up evenly between the groups. Let’s start with one toothpick in each group and continue to add toothpicks until all of the toothpicks are gone. So how many toothpicks are in each group? There are three toothpicks. How many counters are in each group? There is one counter in each group. Now replace the counters and toothpicks one at a time by drawing them as you remove each.

Next create three groups with one counter and three toothpicks in each by drawing a circle around each group. What other words can you think of that can be used to describe one of something? Another word for a chapter or section in the math book is known as a unit. If we only have one counter, what can we call it? We can call it a unit. So what is the ratio of toothpicks to one counter? Let’s look at one group. Since one counter is represented in each group. In each group we have one counter and three toothpicks. So the ratio of toothpicks to one counter is three to one. What is unique or special about the ratio we just identified? The second value is one or a unit. When the second value is a one, or a unit, we call that ratio a unit rate. What is the unit rate for each group of counters and toothpicks? The unit rate is three toothpicks to one counter.

Note: That when we’re determining a unit rate, the denominator will always be one.

Let’s take a look at another example, this time using stars and moons. How many stars are pictured? There are eight stars. How many moons? There are four moons. So what is the ratio of stars to moons? We can represent this ratio as eight to four, eight colon four, or eight over four. Now how many equal groups can be created with one moon in each group? Let’s draw circles to represent each group, making sure that we only include one moon in each group. There are four groups. How many stars are in each group? There are two stars in each group. And how many moons are in each group? There is one moon in each group. So what is the unit rate of stars to moons? We need to look at one group to find the unit rate. There are two stars to every moon.

UNIT RATES – ABSTRACT

In this next activity we will continue to talk about unit rates. Although this time we will not have any pictures to help us. Let’s take a look at this question together. What is the unit rate of water to one cup of juice if the ratio of water to juice is nine to three? What is the problem asking me to find? The problem is asking me to find the unit rate of water to one cup of juice. So what is the first quantity in the ratio of water to juice? The amount of water in the ratio is nine cups. What is the second quantity in the ratio of water to juice? The second quantity for the juice is three cups. So what is the ratio of water to juice? The ratio of water to juice is nine to three. Let’s record this information in the graphic organizer. The first quantity represents the amount of water. In the ratio the amount of water is nine cups. Represent the number nine for quantity one. Quantity two represents the amount of juice in the ratio. Three represents the amount of juice in the quantity. So quantity two is three. The ratio can be written as nine to three. Why is the ratio written in that order? The water is mentioned first in the ratio statement. So we must represent the water first when we write the ratio.

In a unit rate, what must the denominator be? The denominator needs to be one when we’re talking about a unit rate. So how can we change the fraction that represents the ratio of water to juice into a fraction with a denominator of one? We can simplify by dividing both the numerator and the denominator by three. What is the equivalent fraction for nine over three with a denominator of one? If we divide the numerator and the denominator each by three, we find the equivalent fraction three over one. So what is the unit rate? There are three cups of water to one cup of juice. Let’s record this information in the graphic organizer. When we found the equivalent ratio we divided the numerator and denominator each by three, which gave us the equivalent ratio of three over one. This means that for the unit rate, three cups of water to one cup of juice.