UNIT RATES
INTRODUCTION
The objective for this lesson on Unit Ratesis, the student will understand unit rates and be able to compute unit rates associated with ratios of fractions.
The skills the students should have in order to help them with this lesson include simplifying fractions, multiplying fractions, dividing fractions and reciprocal.
We will have three essential questions that will be guiding our lesson. Number 1, what is a ratio? Number 2, how do you write a ratio as a unit rate? And number 3, what does a unit rate tell you about the relationship between two quantities?
Begin by completing the warm-up on multiplying and dividing fractions to prepare for unit rates in this lesson.
SOLVE INTRODUCTION
The Solve Problem for this lesson is, Jeremiah is mowing the yard, which is two acres. He has mowed one- fourth of an acre in one- twelfth of an hour. When he is done mowing his own yard, he also has to mow the neighbor’s yard, which is one acre. If he can continue at the same rate, what is the unit rate for the number of acres he can now in one hour?
We will begin by studying the problem. First we need to identify where the question is located within the problem and underline the question. If he can continue at the same rate, what is the unit rate for the number of acres he can mow in one hour? Now that we have identified the question, we will put this question in our own words in the form of a statement. This problem is asking me to find, the unit rate of the number of acres he can mow in one hour.
During this lesson we will learn how to find unit rates involving ratios with fractions. We will use this knowledge to complete this Solve Problem at the end of the lesson.
RATIOS – CONCRETE AND PICTORIAL
Let’s begin by placing two yellow counters on the workspace under letter A. Next place four red counters on the workspace under letter A. Compare the number of red counters to the number of yellow counters. There are four red counters and two yellow counters. Now replace the counters with drawings under letter A. Next let’s place two yellow counters on the workspace under letter B. And place six red counters on the workspace under letter B. Compare the number of red counters to the number of yellow counters. There are six red counters and two yellow counters. Let’s replace the counters with drawings under letter B. Now place three yellow counters on the workspace under letter C. Place nine red counters on the workspace under letter C. Compare the number of red counters to the number of yellow counters. There are nine red counters and three yellow counters. Replace the counters with drawings under letter C. The number of objects to be compared will be referred to as quantities. A quantity is how much there is of something. It has a number value.
What is the relationship of the red counters to the yellow counters in Box A? There are four red counters to two yellow counters. This relationship is called a ratio, which is defined as a comparison of one quantity to another quantity. What types of notation can be used to represent any ratio? We can write a ratio as a fraction. Using the word “to” in between the two quantities or separating the two quantities with a colon. We can write the comparison of four yellow counters to two red counters as follows: four over two, four to two, or four colon two. What do you notice about how the ratio is written? The first number in the ratio is the first number that is listed in words. Since the box says red to yellow, the number for the red counters has to come first. Fill in the three ways to write a ratio in the second box under A.
UNIT RATES - CONCRETE
Let’s place two yellow counters and four red counters next to Number 1. What is the ratio of red counters to yellow counters? There are four red counters to two yellow counters. We can write this as four over two, four to two, or four colon two. Discuss how you can make equal groups with one yellow counter in each group. You can separate the yellow counters into two separate groups. Then move the red counters one at a time so that they are evenly divided between the two yellow counters. How many equal groups did we create with one yellow counter in each group? We created two equal groups. How many red counters are in each group? There are two red counters in each group. Draw the two yellow counters and four red counters circling each group that was created. What do you notice about the relationship between the red and yellow counters in the first circle? There are two red counters for one yellow counter. What do you notice about the relationship between the red and yellow counters in the second circle? There are two red counters for one yellow counter.
Here’s a quick review: What vocabulary are you familiar with that refers to the number one? The number one is also known as a unit. The relationship between the red and yellow counters in the circle is called a unit rate. How can you define or explain a unit rate? A unit rate is when we compare two different quantities where the second quantity is one. What is the unit rate for the first circle? Two red counters for each yellow counter. What is the unit rate for the second circle? Two red counters for each yellow counter. What do you notice about the unit rate for each circle? The unit rate is the same for both circles! What is the unit rate for each group? Two red counters for each yellow counter.
Now place two yellow counters and six red counters next to the Number 2. What is the ratio of red counters to yellow counters? The ratio is six to two. We can write this as six over two, six to two, or six colon two. Now move the counters to create groups so that every group has only one yellow counter. Since we have two yellow counters we will create two groups. We will place our red counters in each of these two groups. How many groups can you create to show one yellow counter in each? We said that we created two groups. How many red counters are in each group? There are three red counters in each group. Now draw the two yellow counters and six red counters with one yellow counter in each group. We will circle each group. What is the unit rate for each group? Three red counters for each yellow counter.
UNIT RATES – PICTORIAL AND ABSTRACT
Take a look at Question 1. What do you notice about the unit rates in Problems 1 – 3 on the previous page? All of the unit rates tell how many red counters there are for one yellow counter. Using this information, what is a possible definition for unit rate? A unit rate compares two different quantities where one of the quantities is one. How did you determine how many red counters there were for each yellow counter? The red counters were split evenly among the yellow counters. What operation does that action represent? Division. Were the number of groups always the same? No, there were as many groups as there were yellow counters. How do you think we should decide what number to divide by when finding a unit rate? Divide by how many yellow counters there are.
Let’s take a look at question 3 and the graphic organizer below it. What is the comparison for the first row of the graphic organizer? The comparison is Moons to Trapezoids. Which value should come first in the ratio? And why? Moons should come first, because it is the first word in the comparison. Identify the number of the moons. There are twelve moons. Identify the number of trapezoids. There are four trapezoids. How can that relationship be written as a ratio? With the word “to”, with a colon or as a fraction. Which method would be the easiest to use if we needed to simplify to the unit rate? Write the ratio as a fraction in the second column. The ratio is twelve over four. Remember, in order to be a unit rate, one of the numbers in the ratio must be a one. What operation did we use to find the unit rate? We used division. What number could we divide both the denominator and numerator by in order to determine the unit rate? We can divide by four. Why? If we divide both the numerator and denominator by four, we will have a unit rate with a value of one in the denominator. Let’s complete this in the graphic organizer under, divide to get a unit rate. Twelve over four divided by four over four equals three over one. This means that there are three moons for every one trapezoid. Complete the graphic organizer.
PICTORIAL AND ABSTRACT – UNIT RATES WITH FRACTIONS
Here you see the completed graphic organizer. What was different with the last two unit rates? They have a mixed number and a fraction. What do you think this means when we work with unit rates? We do not always have a whole number value to work with.
Let’s read Problem number 1 together. Marie is painting her bedroom. One-half gallon of paint will cover one-sixth of her wall. We can use pictures and unit rates to find the number of gallons of paint it will take to paint one wall. The rectangle below represents Marie’s wall. Into how many sections should we divide the wall? We will divide the wall into six sections. How do you know? In Problem number 1, they talk about one-sixth of the wall. Let’s split the wall into six equal parts. Each of you will divide the rectangle on your paper into six equal parts. How many gallons of paint will cover each section of the wall? One-half gallon will cover each section of the wall. Let’s label in our drawing one-half gallon of paint is needed to cover each section. How many half gallons of paint will it take to paint Marie’s wall? We can see it will take six-half gallons. How many gallons of paint will it take to paint Marie’s wall? It will take three gallons. Since there are six sections and each section takes one-half gallon we can multiply six times one-half. This gives us six over two which when simplified equals three. Do you think it will always be possible to draw the situation? It may be possible, but it certainly won’t always be the most efficient way to solve the problem, and in some cases it may not be possible to draw. How could we determine the unit rate without the picture? We can use the same strategy we used in the graphic organizer. What will we write in the second column of the graphic organizer? We will write the ration with a fraction in the numerator and a fraction in the denominator. How do we write the ratio of paint tothe area of the wall as a ratio? One-half gallon of paint to one-sixth of the all. Can we write that ratio as a fraction? Explain. Yes, the one-half would be the numerator and the one-sixth would be the denominator, because in the ratio paint comes before the area of the wall. Let’s write this ratio in our graphic organizer. One-half over one-sixth. What operation is represented by the fraction bar between one-half and one-sixth? Division. Is it possible the divide fractions? Yes. What is the next step to finding the unit rate? Divide both the numerator and denominator by one-sixth, because we want to have a one in the denominator and any number divided by itself is equal to one. Let’s set up the ratio division in the third column. We have one-half over one-sixth and we are going to divide both the numerator and the denominator by one-sixth. In the past, when we divided fractions, what operation did we use? We multiplied by the reciprocal of the second value. Let’s show this in our graphic organizer. We will multiply one-half times six over one and one-sixth times six over one. The question was then, how many groups of one-sixth are in one-half? When we multiply the reciprocal we find that there are three groups of one-sixth in one-half. Three gallons of paint to one wall.
SOLVE PROBLEM – COMPLETION - UNIT RATES
We arenow going to go back to the Solve Problem from the beginning of the lesson. Jeremiah is mowing the yard, which is two acres. He has mowed one-fourth ofan acre in one-twelfth of an hour. When he is done mowing his own yard, he also has to mow the neighbor’s yard which is one acre. If he can continue at the same rate, what is the unit rate for the number of acres he can mow in one hour?
At the beginning of the lesson we Studied the Problem. We underlined the question and put the question in our own words in the form of a statement. This problem is asking me to find the unit rate of the number of acres he can mow in one hour.
In Step O we will Organize the Facts.
First, we will identify the facts. Jeremiah is mowing the yard, which is two acres. Fact. He has mowed one-fourth of an acre, fact, in one-twelfth of an hour, fact. When he is done mowing his own yard, fact, he also has to mow the neighbor’s yard which is one acre, fact. If he can continue at the same rate, what is the unit rate for the number of acres he can mow in one hour? Now that we have identified the facts, we will eliminate the unnecessary facts. These are facts that will not help us to find the unit rate of the number of acres he can mow in one hour. Jeremiah is mowing the yard, which is two acres. Knowing how large the yard is will not help us to find the unit rate of what he can mow in one hour. So we will eliminate this fact. He has moved one-fourth of an acre, knowing how much of the yard he has moved will help us to find the unit rate, so we will keep this fact, in one-twelfth of an hour. Knowing the amount of time that it took Jeremiah to mow this portion of the yard, will help us to find the unit rate, so we will keep this fact as well. When he is done mowing his own yard, knowing what happens when he is done mowing his yard is not going to help us to find the unit rate, so we will eliminate this fact. He also has to mow the neighbor’s yard which is one acre. Knowing how large the neighbor’s yard is will not help us to find the unit rate, so we will eliminate this fact as well. Now that we have eliminated the unnecessary facts, we will list the necessary facts. Jeremiah mowed one-fourth of an acre. It took him one-twelfth of an hour.
In Step L we will Line Up a Plan.
First let’s write in words what your plan of action will be. We will begin by determining the ratio. Divide the fraction of the acre by the fraction of an hour to determine the unit rate for one hour. What operation or operations will we use in our plan? We will use division.
In Step V, we Verify Your Plan with Action.
First let’s estimate your answer. We can estimate that about three acres can be mowed in an hour. Now let’s carry out your plan. First we said that we would determine your ratio. The ratio is one-fourth of an acre to one-twelfth of an hour. We can write this as a fraction, one-fourth over one-twelfth. Now we need to divide the fraction of the acres by the fraction of an hour, to determine the unit rate. We will divide both the nominator and the denominator by one-twelfth. When we divide fractions it is the same as multiplying by the reciprocal of the second fraction. Let’s rewrite our problems to do so. One-fourth times twelve over one and one-twelfth times twelve over one. One-fourth times twelve over one equals twelve over four, which can be simplified to three. And one-twelfth times twelve over one equals twelve over twelve, which can be simplified to one. Our answer is three over one. This means that Jeremiah can mow three acres in one hour.
In Step E, we will Examine your Results.
Does your answer make sense? Here compare your answer to the question. Yes, because we are looking for the number of acres he can mow in one hour.
Is your answer reasonable? Here compare your answer to the estimate. Yes, because it matches my estimate of about three.
And is your answer accurate? Here check your work. Yes, the answer is accurate.