PERCENT OF CHANGE
INTRODUCTION
The objective for this lesson on Percent of Change is, the student will use proportional reasoning to find the percent of change from one value to another.
The skills students should have in order to help them in this lesson include, writing and solving percent equations and changing percents to decimals.
We will have three essential questions that will be guiding our lesson. Number 1, explain what you are finding when you find the percent of change. Number 2, explain what is meant by a fifty percent decrease. And number 3, give an example in which the amount of increase is the same as the original amount.
Begin by completing the warm-up on percent problems, to prepare for percent of change in this lesson.
SOLVE PROBLEM – INTRODUCTION
We will begin this lesson by completing a SOLVE problem. Matthew was trying to earn money for his sister’s birthday. The first week he mowed one lawn and earned ten dollars. The second week he picked up two more lawns, mowing three lawns total and earning a total of thirty dollars that week. What is the difference in the amount of money he earned from week one to week two?
We will begin by Studying the Problem. First we need to identify where the question in located within the problem and underline the question. What is the difference in the amount of money he earned from week one to week two? 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 difference in the amount of money he earned between each week.
In Step O, we will Organize the Facts. First we need to identify the facts. Matthew was trying to earn money for this sister’s birthday, fact. The first week he mowed one lawn and earned ten dollars, fact. The second week he picked up two more lawns, mowing three lawns total and earning a total of thirty dollars that week, fact. What is the difference in the amount of money he earned from week one to week two?
Now that we have identified the facts, we are ready to eliminate the unnecessary facts. These are the facts that will not help us to find the difference in the amount of money he earned between each week. Matthew was trying to earn money for his sister’s birthday. Knowing that he was earning for his sisters birthday will not help us to find the difference in the amount of money he earned between each week. So we will eliminate this fact. The first week he mowed one lawn and earned ten dollars. Knowing what he earned the first week will help us to find the difference in the amount he earned from week one to week two. So we will keep this fact. The second week he picked up two more lawns, mowing three lawns total and earning a total of thirty dollars that week. Knowing how much he earned in week two will also help us to find the difference in the amount of money he earned between each week. So we will keep this fact as well.
Now that we have eliminated the unnecessary facts, we can list the necessary facts. He earned ten dollars week one and he earned thirty dollars week two.
In Step L, we will Line Up a Plan. First we need to write in words what your plan of action will be. We can subtract the amount of money he earned in week one from the amount of money he earned week two, to find the difference in the amount of money he earned between each week.
What operation or operations, will we use in our plan? We will use subtraction.
Now that we have lined up our plan, it is time to Verify Your Plan with Action. First let’s estimate your answer. We know that he earned thirty dollars in week two, and ten dollars in week one, so the difference should be less than what he earned the second week. Our estimate can be less than thirty dollars.
Now let’s carry out your plan. We said that we wanted to subtract the amount of money he earned in week one from the amount of money he earned in week two. We will subtract ten dollars from thirty dollars. Thirty dollars minus ten dollars equals twenty dollars.
In Step E, we will Examine Your Results.
Does your answer make sense? Here compare your answer to the question. Yes, the answer makes sense because I found the difference in the amount he earned each week.
Is your answer reasonable? Here compare your answer to the estimate. Yes, it is close to my estimate of less than thirty dollars.
And is your answer accurate? Here check your work. Yes, the answer is accurate.
We are now ready to write your answer in a complete sentence. The difference in the amount of money he earned between week one and week two is twenty dollars.
DISCOVERY ACTIVITY – PERCENT OF CHANGE – EXTEND THE SOLVE PROBLEM
We are now going to take the SOLVE problem we just completed and extend it to Percent of Change. Take a look at the questions below the SOLVE problem.
What does the word difference mean? Difference means change or subtraction. What does the word increase mean? When something goes up. And what does the word decrease mean? It means when something goes down.
Now take a look at Problem four. When amounts change, they can increase or decrease. Look at the examples below, decide if the change is an increase or a decrease, and how much of a change it is.
A local neighborhood watch program reported that since they began the program, car break-ins have gone from five per month to one per month in their neighborhood. Was there an increase or a decrease in the number of car break-ins? There was a Decrease. How do you know? First there were five then there was one, which is less. How much did the break-ins decrease by? The break-ins decreased by four per month. How do you know? I subtracted, five minus one equals four.
Let’s take a look at another situation. The population of Mt. Holly, North Carolina in the year two thousand was about nine thousand six hundred residents. In the year two thousand eight, the population was about ten thousand one hundred. Was there an increase or a decrease? There was an increase. How do your know? Well, first there was nine thousand six hundred and now there are ten thousand one hundred. How much did the population increase by? It increased by five hundred residents. How do you know? I subtracted ten thousand one hundred minus nine thousand six hundred equals five hundred.
Let’s look at another situation. A pair of boots normally sells for eighty nine dollars. They are on sale for fifty dollars. Was there an increase or a decrease in the price of the boots? There was a decrease. How do you know? First they were eighty nine dollars, now they are fifty dollars. How much did the price decrease by? The price decreased by thirty nine dollars. How do you know? I subtracted eighty nine dollars minus fifty dollars equals thirty nine dollars.
And let’s look at one more situation together. In the year nineteen ninety six the minimum wage was four dollars and seventy five cents. In the year two thousand nine, the minimum wage was seven dollars and twenty five cents. Was there an increase or decrease in the minimum wage? There was an increase. How do you know? First it was four dollars and seventy five cents and now it is seven dollars and twenty five cents. How much did the wage increase by? It increased by two dollars and fifty cents. How do you know? I subtracted seven dollars and twenty five cents minus four dollars and seventy five cents equals two dollars and fifty cents.
Now let’s refer back to the SOLVE problem you completed about Matthew earning money by mowing lawns. We will use this problem to complete the graphic organizer in Problem five.
How much money was earned during week one? Matthew earned ten dollars during week one. And how much money was earned during week two? Matthew earned thirty dollars in week two. Let’s include this information in our graphic organizer. Money earned week one was ten dollars. And money earned week two was thirty dollars. What is the difference in the amount that was earned between each week? When we completed the problem we found that the difference was twenty dollars. Is that an increase or a decrease? Matthew’s income increased between week one and week two. Let’s include this information in our graphic organizer. The difference between the weeks was twenty dollars. It was an increase in the amount of money Matthew made. Describe what a ratio is. A ratio is a comparison of one quantity to another quantity. What is the ratio of the difference of what Matthew earned between each week to the amount he earned the first week? The ratio is twenty to ten. Let’s include this information in our graphic organizer. The ratio of the difference to what happened is twenty to ten. Can we express the ratio as a decimal? Yes, twenty to ten as a decimal is two and zero hundredths. Let’s change your decimal to a percent. The whole number two written as a percent is two hundred percent. Let’s include this information in the graphic organizer. We can express the ratio as a decimal as two and zero hundredths. We can write the ratio as a percent two and zero hundredths can be written as the percent two hundred percent. What does the ratio you created tell us? The ratio tells the comparison of the change to the initial amount earned. How did you find that change as a percent? Calculate the ratio as a decimal and convert it to a percent. What you just found was the ratio of the change to the original amount and then converted it to a percent. This is called percent of change. What does two hundred percent represent? It represents the percent of change from the amount of money earned from week one to week two.
PERCENT OF CHANGE – SOLVE PROBLEM
Let’s complete the SOLVE problem together. A local neighborhood watch program reported that since they began the program, car break-ins have gone from five per month to one per month in their neighborhood. What is the percent of change in the number of break-ins per month?
We will begin by, Studying to Problem. First we need to identify where the question is located within the problem and underline the question. What is the percent of change in the number of break-ins per month? 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 percent of change in the number of break-ins.
In Step O, we will Organize the Facts. First we need to identify the facts. A local neighborhood watch program reported that since they began the program, fact, car break-ins have gone from five per month, fact, to one per month in their neighborhood, fact. What is the percent of change in the number of break-ins per month?
Now that we have identified the facts, we will eliminate the unnecessary facts. These are the facts that will not help us to find the percent of change in the number of break-ins. A local neighborhood watch program reported that since they began the program. Knowing that we are looking at a local neighborhood watch program is not going help us to find the percent of change in the number of break-ins. So we will eliminate this fact. Car break-ins have gone from five per month. Knowing the original number of break-ins was five per month will help us to find the percent of change in the number break-ins. So we will keep this fact. To one per month in their neighborhood. Knowing what the new number of break-ins is in the neighborhood is will also help us to find the percent of change in the number of break-ins. So we will keep this fact as well.
Now that we have eliminated the unnecessary facts, it is time to list the necessary facts. The original number of break-ins was five per month. The reduced number of break-ins is one per month.
In Step L, we will Line Up a Plan. First we will write in words what your plan of action will be. We need to create a ratio comparing the difference to the original amount. We can subtract the reduced number of break-ins from the original number of break-ins. And record this number in the numerator of our ratio. We will record the original number of break-ins in the denominator of this ratio. We can then convert the ratio to a percent to determine the percent of change. What operation or operations will we use in our plan? We need to create a ratio, we will also use subtraction and division.
In Step V, we will Verify Your Plan with Action. First let’s estimate your answer. The number of car break-ins went from five per month to one per month. So we can estimate about a seventy five percent decrease in the number of break-ins. Now let’s carry out your plan. We said that we wanted to create a ratio comparing the difference to the original amount. To find the difference we will subtract, five take away one. Our original number of break-ins was five and the reduced number of break-ins was one. Five minus one equals four. We put this number over the original number of break-ins which is five. So our ratio is four to five. When we convert this ratio into a decimal, the decimal equivalent to four over five is eight tenths. We can then convert the ratio to a percent. Eight tenths written as a percent is eighty percent. The decrease in the number of break-ins was eighty percent.
In Step E, we will Examine Your Results.
Does your answer make sense? Here compare your answer to the question. Yes, because I found the percent of change between the number of break-ins.
Is your answer reasonable? Here compare your answer to the estimate. Yes, it is close to my estimate of about seventy five percent.
And is your answer accurate? Here check your work. Yes the answer is accurate.
We are now ready to write your answer in a complete sentence. The percent of change in the number of break-ins is eighty percent.
USING PERCENT OF CHANGE TO FIND ACTUAL CHANGE
Now let’s look at some situations and determine whether they show a decrease or increase and what the percentage is.
A video game system usually costs two hundred dollars. It is on sale for thirty percent off. What is the sale price of the system? Is there an increase or decrease? The system is going on sale this represents a decrease in the price of the system. What is the percent? It is on sale for thirty percent off.
Let’s look at another situation. The toy store buys a robot action figure for fifty five dollars each from the toy company and marks the price up by seventy percent to sell to the public. Is there an increase or decrease? The toy store marks the price up. This represents an increase. What is the percent? They are marking the toy up by seventy percent.
Let’s look at another situation. In nineteen eighty five the average person watched ten hours of television per week. In nineteen ninety, another study indicated that amount had increased by forty five percent. Is there an increase or decrease? There is an increase in the amount of hours watched. What is the percent? 45 percent.
And let’s look at one more situation. Over a ten year period, the number of students enrolled at Martin Middle School decreased by fifteen percent from its opening enrollment of eight hundred. Is there an increase or decrease? There is a decrease in enrollment. What is the percent? Fifteen percent.
How are these problems different from the percent of change problems we completed? The original value and percent are given, but the new cost is not.
Use the ratio we have been using for Percent of Change, the ratio is difference over original, and determine what we will put in the place of what is missing. We can represent the difference by the variable x. When we don’t know the value, we can use a variable. What will need to be done when we determine what our variable equals? We will need to go back and “S” the problem or study the problem to determine if there was an increase or decrease, then add or subtract that from the original amount.
Let’s take a look at the first situation together and use, SOLVE to help us. We will begin by Studying the Problem. A video game system usually costs two hundred dollars. It is on sale for thirty percent off. What is the sale price of the system?