Level E Lesson 6
Compare and Round Decimals to Thousandths
In lesson 6 the objective is, the student will compare and round decimals to the thousandths.
The skills students should have in order to help them in this lesson include, comparing decimals to the hundredths, and rounding whole numbers.
We will have three essential questions that will be guiding our lesson. Number 1, what is the first step for comparing decimals? Number 2, when comparing decimals, in what place value should you start comparing? And number 3, when is it useful to round decimal values?
The SOLVE problem for this lesson is, Chad is arranging the batting order for an upcoming baseball game. Michael has a batting average of 0.634, and Andrea has a batting average of 0.643. For his fourth batter, Chad would like to pick the player with the higher batting average. Between Michael and Andrea, who should Chad pick to be the fourth batter?
We will start this lesson by Studying the Problem. We need to identify where the question is located within the problem, and underline the question. Between Michael and Andrea, who should Chad pick to be the fourth batter? Now we want to take this question and put it in our own words in the form of a statement. This problem is asking me to find the fourth batter Chad should pick, who has the higher batting average.
During this lesson we will learn how to compare decimals to the thousandths. We will use this knowledge to complete this SOLVE problem at the end of the lesson.
Throughout this lesson students will be working together in cooperative pairs. All students should know their role as either partner A or partner B before beginning this lesson.
We will begin this lesson by comparing decimals in a Place Value Chart. We will start by placing the numbers the problem seen here into the Place Value Chart. We are looking to find which of these numbers has the greater value. On the first line on our Place Value Chart we will place the first number from the problem 25 and 452 thousandths. On the second line we will place the second number from our problem 25 and 453 thousandths. To find which number is larger we want to compare the digits in each place value. We will begin at the far left. If the digits are the same continue moving to the right. In the tens place are digits are the same. So let’s move to the ones place. These are also the same, so we will move to the tenths place. These are also the same, so let’s move to the hundredths place, same. So we’ll move to the thousandths place. These digits are different. When the digits are different, circle them. Look at the circled digits and identify which digit is larger. Because 3 is larger than 2, the bottom decimal 25 and 453 thousandths is the larger decimal. Let’s relay that information to the statement for Question 1. We found that the second number is larger, so the first number is smaller. Because 25 and 452 thousandths is smaller, we can say that it is less than 25 and 453 thousandths. Place a “less than” symbol (<) inside the circle.
Now you will play the game Decimal Duels with a partner. Start by cutting up the 10 decimal cards. Next you will place them face down in a pile. Each partner then pulls a card off the pile and writes his or her decimal on one side of the circle. For this example partner A pulls the card 1 and 601 thousandths. And partner B pulls the card 1 and 113 thousandths. Each partner then writes their number in the Place Value Chart. They will then compare their number and complete the statement at the top with a less than, greater than, or equal to sign to make it true. We will start at the left on our Place Value Charts comparing the digits one at a time. In the ones place both partners have the same digit, so we’ll move to the tenths place. In the tenths place the digits are different, circle those digits and compare, 6 is greater than 1. So partner A’s number is greater than partner B. The winner of this round would be partner A. As partner A has the greater decimal number. The student who wins the most rounds will be the overall winner. If there is a tie play one additional round to break the tie.
In this next example we will be comparing the decimals that are shown. This example is a little bit different from the ones that we have just completed as there is no Place Value Chart to help us. We will write our first decimal number first, 61 and 215 thousandths. Next write the second number underneath the first number, making sure to line up the decimals one under the other. Our second number is 61 and 251 thousandths. Now to compare our decimal numbers we will look at the digits in each place value beginning at the far left. If the digits are the same keep moving to the right. We will begin in the tens place. Our tens digits are the same, so we will move to the ones digit. Our ones digits are the same so we will move to the tenths place. Our tenths digits are the same, so we will move to the hundredths place. Our hundredths digits are different. When the digits are different circle them. Look at the circled digits and identify which digit is larger. Because 5 is larger than 1, the bottom decimal 61 and 251 thousandths is the larger decimal. Let’s relay that information to the statement for Question 1. We said that our second decimal number is larger, so our first decimal number is the smaller number. Because 61 and 215 is smaller, we can say that it is less than 61 and 251 thousandths. Place a “less than” symbol (<) inside the circle. In this example we were still able to compare our numbers even without our Place Value Chart. You can always line up the decimals and compare from left to right in any situation. You can even draw your own columns if that will help to line up the place values.
Now that we know how to compare decimals let’s review the steps for rounding whole numbers. First, identify the number. Second, decide to what place you will round. Third, decide what two numbers the number is between. Fourth, circle the digit to be rounded. Fifth underline the digit that tells “which way”. In step 6, the statement is going to help us to tell which way to go, if we are going to round up or if we are going to round down. In step 6, we think, if it’s 4 or below, it’s way to low, which means we will be rounding down. If it 5 or above, then up you go, which would mean we would round up. And our last step is to write the rounded number.
Let’s apply these steps to an example. We want to round 841 to the nearest hundred. First, we identify the number, 841. Second, decide to what place you will round. This problem tells us we want to round to the nearest hundred. Third, we decide what two numbers the number is between. 841 is between 800 and 900. Next, we will circle the digit to be rounded. The digit in the hundreds place is the 8, so this is the number that we circle, and then we underline the digit that tells which way. The digit that tells which way is to the right of the number to be rounded. So we will underline the 4. Now let’s use our statement to help us. If it’s 4 or below, its way to low, if it’s 5 or above, then up you go. Since our digit is a 4, it’s way to low, meaning that we will round down. 841 rounded to the nearest hundred, is 800.
Now that we have reviewed how to round whole numbers, we will be learning how to round numbers to decimals in any place value. Each pair of students will need 2 beans, in order to help them in rounding our decimals. This first example has the number 2 and 4 tenths in our Place Value Chart. We want to round this number to the nearest ones place. The digit in the ones place is a 2. Our number line goes from 2 and 0 tenths to 3 and 0 tenths. This is because 2 and 4 tenths falls between 2 and 0 tenths and 3 and 0 tenths. We want to place a bean on 2 and 4 tenths on our number line. Is 2 and 4 tenths closer to 2 and 0 tenths or closer to 3 and 0 tenths? 2 and 4 tenths is closer to 2 and 0 tenths than 3 and 0 tenths. So we will place a second bean on 2 and 0 tenths. The 4 in the tenths place puts the number closer to 2 and 0 tenths. 2 and 4 tenths rounded to the nearest ones place is 2 and 0 tenths. Because the number was rounded to the nearest one the tenths digit becomes a 0 as a place holder. Because it’s a 0 that is in the tenths place we can also write this number simply as 2.
Let’s look at another example together. The number in this Place Value Chart is 2 and 9 tenths. We are rounding to the nearest ones place. The value of the digit in the ones place is 2. Our number line starts at 2 and 0 tenths and goes to 3 and 0 tenths, because 2 and 9 tenths falls between 2 and 0 tenths and 3 and 0 tenths. We will place a bean on 2 and 9 tenths on our number line. Is 2 and 9 tenths closer to 3 and 0 tenths, or closer to 2 and 0 tenths? 2 and 9 tenths is closer to 3 and 0 tenths than 2 and 0 tenths. So we will place another bean on 3 and 0 tenths. The place value to the right of the number to be rounded is a 9. That 9 tells us that we’re going to round up. So 2 and 9 tenths rounded to the nearest ones place is 3 and 0 tenths. The tenths digit becomes a 0 because we are rounding to the nearest one. We rounded up to 3 because 2 and 9 tenths is closer to 3 than 2 on the number line. Because there is a 0 in the tenths place and that is the only digit after our decimal point, we can write this number simply as the whole number 3.
Remember: when rounding, the number that decides if we should round to the higher or lower value is the number to the right of the number to be rounded. If we round to the ones place, the place value to the right which is the tenths place will decide if we round up or down.
In our next example we want to round to the nearest tenths place. Our number is 6 and 14 hundredths, and we are rounding to the nearest tenths place. The digit in the tenths place is a 1, so our number line starts at 6 and 10 hundredths and ends at 6 and 20 hundredths, as 6 and 14 hundredths falls between these two values. The scale is divided into hundredths. We will place a bean on the number 6 and 14 hundredths on our number line. Is our bean closer to 6 and 10 hundredths or closer to 6 and 20 hundredths? We can use the digit in the hundredths place to help us in answering this question. The digit in the hundredths place will help us to decide if we round our number up or down. As the digit in the hundredths place is a 4 we know if it’s 4 or below, it’s way to low, so we will be rounding down. 6 and 14 hundredths is closer to 6 and 10 hundredths than 6 and 20 hundredths. So we will place a bean on 6 and 10 hundredths. 6 and 14 hundredths rounded to the nearest tenths place is 6 and 10 hundredths. The hundredths digit becomes zero, because we are rounding to the nearest tenths value. As the zero is our last digit after the decimal point in our number we can write this as 6 and 1 tenth.
Our next example is 6 and 18 hundredths, and we are rounding to the nearest tenths place. The digit in the tenths place is a 1. Our number line starts at 6 and 10 hundredths and goes to 6 and 20 hundredths, because 6 and 18 hundredths falls between these two values. Our scale is again broken down into hundredths. We want to place a bean on 6 and 18 hundredths on our number line. Now let’s look at the digit in the hundredths place. This decides if we will round our number up or down. Since the digit in the hundredths place is an 8, and thinking back to our statement, if it’s 4 or below, it’s way to low, if it’s 5 or above then up you go, having an 8 in the hundredths place tells us that we will round up. We can also see from our number line that 6 and 18 hundredths is closer to 6 and 20 hundredths than it is to 6 and 10 hundredths, so we can place a bean on 6 and 20 hundredths. 6 and 18 hundredths rounded to the nearest tenths place is 6 and 20 hundredths. The hundredths digit becomes a 0 and the tenths digit becomes a 2, as we are rounding to the nearest tenths value. We can also write this number as 6 and 2 tenths. As the 0 is the last digit after the decimal in our number.
Now let’s go back and talk about these four examples we just completed together. In problems 1 and 2, which number value helped you determine which way to go on the number line? In problems 1 and 2 we were rounding to the nearest ones place. And the number that helped us to determine which way to go on the number line was the number in the tenths place. In problems 3 and 4, which number value helped you determine which way to go on the number line? In problem 3 and 4, we were rounding to the nearest tenths place. So the number value that helped us to determine which way to go on the number line was the hundredths place.
Knowing this, let’s look at another example. In this example we have the number 7 and 2 tenths, and we are rounding to the nearest ones place. 7 and 2 tenths is going to be between 7 and 0 tenths and 8 and 0 tenths. So we will circle the digit to be rounded, which is our 7. Next let’s underline the digit that tells us which way to go. Now let’s place a bean on our number line at 7 and 2 tenths. Since we don’t always have a bean available to us to place on our number line we can also use a point to represent where this number is located. We will replace our bean with a point at 7 and 2 tenths. Is 7 and 2 tenths closer to 7 and 0 tenths or 8 and 0 tenths? It is closer to 7 and 0 tenths than 8 and 0 tenths because a 2 is in the tenths place. 7 and 2 tenths when rounded to the nearest ones place is 7 and 0 tenths. The 2 becomes a 0 because we are rounding to the ones place. Since the 0 is holding the place value after the decimal we can write this number simply as the whole number 7.
Let’s record the steps that we followed in rounding the numbers 7 and 2 tenths using the graphic organizer. First we rounded to the nearest tenth. The number is between 7 and 0 tenths and 8 and 0 tenths. Then we circle the digit to be rounded. The 7 is in the ones place, so we put a circle around the 7. Next, we underline the digit that tells which way to go. So we will underline the 2, as it is to the right of the ones place. Remember our says, if it’s 4 or below, it’s way to low. If it’s 5 or above, then up you go. Since our number is 4 or below, we will round down, our rounded number is 7 and 0 tenths or 7.
This next example does not have the place value chart or the number line to help us, so we will rely on the steps that we have developed throughout this lesson. First, we want to know to what place value are we rounding. In this example we are going to round to the nearest hundredths place. The number is between 5.410 thousandths and 5.420 thousandths. We will circle the digit to be rounded. The digit in the hundredths place is a 1. So we circle the 1 in the hundredths place. Next, we underline the digit that tells which way to go. This digit is always directly to the right of the digit that we are rounding. So we will underline the 2 in the thousandths place. Now we’ll use our saying, if it’s 4 or below, it’s way to low. If it’s 5 or above, then up you go. Since the 2 in the thousandths place is 4 or below, we know that we will need to round down. Our rounded number is 5.410 thousandths. We can also write this as 5.41 hundredths, since the 0 was the last digit after the decimal point in our answer.
We are now going to go back to the SOLVE problem from the beginning of the lesson. Chad is arranging the batting order for an upcoming baseball game. Michael has a batting average of 0.634, and Andrea has a batting average of 0.643. For his fourth batter, Chad would like to pick the player with the higher batting average. Between Michael and Andrea, who should Chad pick to be the fourth batter?
At the beginning of the lesson we Studied the Problem. We underlined the question. Between Michael and Andrea, who should Chad pick to be the fourth batter? And then put this question in our own words in the form of a statement. This problem is asking me to find the fourth batter Chad should pick, who has the higher batting average.
In Step O, we will Organize the Facts. We will start by identifying the facts. Chad is arranging the batting order for an upcoming baseball game, fact. Michael has a batting average of 0.634, fact, and Andrea has a batting average of 0.643, fact. For his fourth batter, Chad would like to pick the player with the higher batting average, fact. Between Michael and Andrea, who should Chad pick to be the fourth batter? Next, we will eliminate the unnecessary facts, or those that will not help us to choose which batter should go fourth based on a higher batting average. Knowing that Chad is arranging the batting order for an upcoming baseball game is not going to help us to know which batter should be chosen to go fourth, so we will eliminate this fact. Michael has a batting average of 0.634, this is important to choosing a fourth batter, and Andrea has a batting average of 0.643. This is also important to choosing a fourth batter. For his fourth batter, Chad would like to pick the player with the higher batting average, knowing that we are looking for the higher batting average, is also going to be important, so we will keep these facts. Now let’s list the necessary facts. Michaels average is 0.634, Andrea’s average is 0.643, And Chad wants the higher batting average.