REPRESENTING PROPORTIONAL RELATIONSHIPS WITH EQUATIONS
The objective for this lesson on representing proportional relationships with equations is, the student will represent proportional relationships with equations.
The skills students should have in order to help them in this lesson include, recognizing patterns, writing ratios, unit rates, and finding constant of proportionality.
We will have three essential questions that will be guiding our lesson. Number 1, what is the form of any equation that shows a proportional relationship? Number 2, what does the coefficient of the x-variable represent in the equation? And number 3, where are the independent and dependent variables in the equation?
Begin by completing the warm-up on finding the constant of proportionality to prepare for representing proportional relationships with equations in this lesson.
SOLVE PROBLEM
The SOLVE problem for this lesson is. Nick is purchasing some supplies for a party. He is trying to decide how many cases of water he can afford to buy. Each case of water costs four dollars and fifty cents. What equation can he use to represent the proportional relationship between the number of cases of water he buys and the cost? We will begin by studying the problem. First we need to identify where the question is located within the problem, and underline the question. The question in this problem is, what equation can he use to represent the proportional relationship between the number of cases of water he buys and the cost? 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 equation that Nick can use to find the cost of any number of cases of water. During this lesson we will learn how to write an equation to show the relationship between two variables, in a proportional relationship. We will use this knowledge to complete this SOLVE problem at the end of the lesson.
CONCRETE
Let’s begin by looking at proportional relationships with equations using our red and yellow chips. Place three yellow chips on the work space. Next place one red chip on the work space below the yellow chips. Describe the relationship between the red and yellow chips. For every one red chip there are three yellow chips. Identify the unit rate of yellow to red chips. Remember that a unit rate is a ratio with one as the denominator. And describes the relationship between two quantities. The unit rate is three to one. What is another term we can use for the unit rate? It is also known as the constant of proportionality. What is the constant of proportionality for the number of red chips to the number of yellow chips? The constant of proportionality is one red chip to three yellow chips or one over three. Explain the meaning of one over three as the constant of proportionality. This means that for every one red chip there are three yellow chips. Discuss a variable that can be used to represent the red and yellow chips.
If you want to identify the chips with a variable, what might be a good letter to use? We can use the letter y to represent to the yellow chips and the letter r to represent the red chips. Let’s record this information in our graphic organizer. The variable for our red chips is r. And the variable for our yellow chips is y. Note: That generally in tables, we will see the independent variable on the top and the dependent variable on the bottom when the table is horizontal. When the table is vertical, the independent variable will be on the left. Remember that when forming the constant of proportionality, the dependent variable will be in the numerator and the independent variable will be in the denominator of the ratio. Be careful when working with the table. Because in the table the independent variable is on the top, where in our fraction the independent variable is on the bottom or in the denominator of the ratio.
What term can we use to represent the yellow chips? The yellow chips represent the dependent variable. Explain. The number of yellow chips is dependent on how many red chips are shown. Let’s record the yellow chips as our dependent variable in the graphic organizer. So what term can we use to represent the red chips? The red chips represent the independent variable. Explain. The number of red chips is independent from the number of yellow chips, meaning that the yellow chips depend on the number of red chips. Let’s record our red chips as the independent variable in our graphic organizer.
Now place three more yellow chips on the work space. And place one red chip below the second set of yellow chips. Did the constant of proportionality change? Why or why not? No, because there are still three yellow chips to every red chip. Let’s record the values that we know. We know that when there was one red chip, we had three yellow chips. And when we had two red chips, we had six yellow chips.
Now place three more yellow chips on the work space. And place one red chip below the third set of yellow chips. This time did the constant of proportionality change? Why or why not? No, the constant of proportionality did not change, because there are still three yellow chips to every red chip. Let’s record the values of three and nine(you show 9 and 3) in our graphic organizer. When there are three red chips, there are nine yellow chips. Is there a consistent relationship between the values of the red chips and the yellow chips in the table? Yes. What is the relationship? Define as an operation that can be used to represent the relationship. The relationship can be defined as multiplication. We multiply the number of red chips represented by the variable r, times three to determine the number of yellow chips represented by the variable y. So if we had four red chips, how many yellow chips would there be? And how do you know? There would be twelve yellow chips, because four times three is twelve. What if we had five red chips? How many yellow chips would there be? And how do you know? There would be fifteen yellow chips, because five times three is fifteen. So write an equation that can be used to describe the relationship between the number of red chips and the number of yellow chips. Remember that we will use the variable r to represent the red chips and the variable y to represent the yellow chips. The number of yellow chips is equal to three times the number of red chips. So our equation is y equals three r. Three r represents that we are multiplying three times the value for the variable r.
PICTORIAL
Now let’s use a pictorial representation to talk about proportional relationships with equations. Take a look at the diagram here with stars and hearts. If there is only one heart, how many stars are there? When there is one heart, there are two stars. If there are two hearts, how many stars are there? For two hearts, there are four stars. And if there are three hearts, how many stars are there? For three hearts, there are six stars. So this means that for every one heart, you must add how many stars? You must add two stars for every one heart. What is the constant of proportionality for the number of stars to hearts? For every two stars, there is one heart. The constant of proportionality is two over one. Explain how many stars you would have with four hearts. If you had four hearts, you would need two stars for each of the four hearts. So you would have a total of eight stars. Every time we add a heart, we must add two stars to continue the pattern. There are always twice as many stars as hearts. The number of stars that are in the diagram depends on the number of hearts in the diagram. So, if s represents the number of stars, the dependent variable, and h represent the number of hearts, the independent variable, we can write an equation the shows the relationship between the hearts represented by the variable h and stars represented by the variable s. Remember that the value of the dependent variable depends on the value of the independent variable. How did you determine the number of stars represented by the variable s there would be for each number of hearts represented by the variable h? You need to multiply the number of hearts by two! Because for every heart there are two stars. How could we show that in an equation? The number of stars equals two times the number of hearts. So the equation can be written as s equals two h. The number of hearts represented by the variable h multiplied by two equals the number of stars represented by the variable s.
TABLES
Now let’s use a table to talk about proportional relationships with equations. The table seen here shows how many minutes of commercials are within television shows of certain lengths. How can we find the constant of proportionality? We can find the unit rate. So how do we find the unit rate for the first ratio in the table, which is fifteen over sixty? Remember that when we’re finding the unit rate we need to have a denominator of one. In order to turn this fraction into a fraction with a denominator of one we need to divide the numerator and denominator by sixty. Find the constant of proportionality for all rows of the table. For our first ratio fifteen over sixty we will divide both the numerator and denominator by sixty. This gives us twenty five-hundredths in the numerator and one in the denominator. Our constant of proportionality is twenty five-hundredths. Our second ratio in the table is twenty two and five tenths over ninety. We will need to divide our numerator and denominator each by ninety. When we do so we find that the constant of proportionality for this ratio is also twenty five-hundredths. Our third ratio is thirty over one hundred twenty. Again we will divideour numerator and denominator both by the value of the denominator which is one hundred twenty. When we do so we find that the constant of proportionality is twenty five-hundredths. And our fourth ratio in the table is sixty over two hundred forty. We will divide the numerator and denominator each by two hundred forty, which give us a constant of proportionality again of twenty five-hundredths.
Looking at the table if there are one hundred twenty minutes in a show, how many minutes of commercials will there be? There will be thirty minutes of commercials.
So if there are one hundred fifty minutes in a show, how many minutes of commercials will there be? Since one hundred fifty minutes is not represented in our table we need to use a different method to find the number of minutes of commercials. Knowing that our constant of proportionality is twenty five-hundredths we can multiply one hundred fifty minutes by twenty five-hundredths, which will tell us that there will be thirty seven and five-tenths minutes of commercials, when there are one hundred fifty minutes in a show.
What if there are two hundred minutesin a show, how many minutes of commercials will there be? Again, this is not represented in our table, but we can use the constant of proportionality to help us to find the answer. Two hundred times twenty five-hundredths will tell us that we will have fifty minutes of commercials when there are two hundred minutes in a show. How did you determine the number of minutes of commercials there would be for the number of minutes in a television show? We multiply the number of minutes in the television show by the constant of proportionality, which is twenty five-hundredths.
We can use variables to represent the situation. This time we are going to use the variables x and y because those are the most common variables used in writing equations. If we use y to represent the number of minutes of commercials, the dependent variable, and x represents the number of minutes in the television show, the independent variable, we can write an equation that shows the relationship between x and y. Identify the equation that could be used for the relationship. Remember that the constant of proportionality is twenty five-hundredths. Y or the number of minutes of commercials is equal to twenty five-hundredths x, which represents the length of the TV show. Our equation is y equals twenty five-hundredths x. The number of television show minutes, which is represented by the variable x, multiplied by twenty five-hundredths equals the number of commercial minutes, which is represented by the variable y.
So if there were one hundred minutes in the television show, how many minutes of commercials would there be? We can use our equation y equals twenty five-hundredths x to help us to solve the problem. We will multiply twenty five-hundredths times one hundred, which equals twenty five minutes. When a television show is one hundred minutes in length there will be twenty five minutes of commercials.
What if there were three hundred minutes in the television show, how many minutes of commercials would there be? Again we can use our equation y equals twenty five-hundredths x to help us to find the answer. We will multiply twenty five-hundredths times the number of minutes in the television show which is three hundred. Twenty five-hundredths times three hundred equals seventy five minutes. There will be seventy five minutes of commercials when the television show is three hundred minutes in length.
GRAPHS
Now let’s use a graph to talk about proportional relationships with equations. The graph seen here shows the total number of pushups that Jesse has done this week. How can we find the constant of proportionality? We can find the unit rate. How do we find the unit rate in a graph? We need to look for the number of pushups in one day. What is the unit rate? One day one Jesse did twenty five pushups. So the unit rate is twenty five. What is the number of pushups for two days? The number of pushups for two days is fifty. What is the number of pushups for three days? The number of pushups for three days is seventy five. And what is the number of pushups for four days? The number of pushups for four days is one hundred. How did you determine the number of pushups Jesse would do, based on the number of days? We can multiply the number of days by twenty five, which is our constant of proportionality. If y represents the number of pushups the dependent variable, and x represents the number of days, the independent variable, we can write an equation that shows the relationship between x and y. Identify the equation that could be used for the relationship. Since we know the constant of proportionality is twenty five we can set up the equation so that the number pushups, is equal to twenty five times the number of days. Y equals twenty five x. The number of days represented by the variable x, multiplied by twenty five equals the number of pushups, represented by the variable y.
So let’s look at the graph again. If Jesse does pushups for five days, how many pushups will he do? We can use our equation y equals twenty five x. We will multiply twenty five by five since we are talking about five days. Twenty five times five equals one hundred twenty five pushups.
What if Jesse does pushups for ten days, how many pushups will he do? Again, let’s use our equation to help us to solve this problem. The equation is y equals twenty five x. We will multiply twenty five times ten, which equals two hundred fifty pushups. If Jesse does pushups for ten days he will do a total of two hundred fifty pushups.
ABSTRACT
Up to this point we have worked with proportional relationships with equations using chips, diagrams, tables and graphs. We will now look at a word problem to discuss proportional relationships with equations. The word problem is, the veterinarian says that for every pound your dog weighs, the dog should get one-tenth cc of medicine. What is the relationship between the weight of the dog and the amount of medicine? For every one pound that the dog weighs it should be administered one-tenth cc of medicine. If the number of cc’s of medicine depends on the number of pounds the dog weighs, what is the dependent variable? The dependent variable is the cc’s of medicine, because it depends on the amount the dog weighs. What does the variable y represent? It represents to dependent variable, which as we said is the cc’s of medicine.
Now take a look at the graphic organizer. Let’s complete the information that we know so far. We know that y our dependent variable is the cc’s of medicine. So what is the independent variable? Since we know that the dependent variable is the cc’s of medicine, the independent variable is the weight of the dog. The weight of the dog does not depend on the cc’s of medicine. So it is the independent variable. What does the variable x represent? It represents the independent variable. Let’s add this information to our graphic organizer. X the independent variable is the weight of the dog.