# Lesson 8. Decimals. Objectives

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1 Student Name: Date: Contact Person Name: Phone Number: Lesson 8 Decimals Objectives Understand conceptually what decimal notation means Be able to convert from decimals to mixed numbers and fractions Round decimals to a given place value

2 Authors: Jason March, B.A. Tim Wilson, B.A. Editor: Graphics: Linda Shanks Tim Wilson Jason March Eva McKendry National PASS Center BOCES Geneseo Migrant Center 27 Lackawanna Avenue Mount Morris, NY (585) (585) (fax) Developed by the National PASS Center under the leadership of the National PASS Coordinating Committee with funding from the Region 20 Education Service Center, San Antonio, Texas, as part of the Mathematics Achievement = Success (MAS) Migrant Education Program Consortium Incentive project. In addition, program support from the Opportunities for Success for Out-of-School Youth (OSY) Migrant Education Program Consortium Incentive project under the leadership of the Kansas Migrant Education Program.

3 After work, you go to the store to buy some produce for a fruit salad you want to make. You see a deal for bananas on a sign, Ten bananas for one dollar. From what we know about fractions, one banana costs = of a dollar. 10 From your knowledge of money, you also know that one tenth th of one dollar is a dime, or \$0.10 So, = Here is yet another way to represent fractions! 0.10 is an example of a decimal. A decimal is a number that can represent a whole part and a fractional part. A decimal point,, written as a period (.), is written to separate the whole part from the fractional part. For example, 3.5 is a decimal, and so is A great way to begin an understanding of decimals is to think of them using money.. With decimals, the first number after the decimal point is in the tenths place. This makes sense because one dime is \$0.1, or one-tenth of a dollar. The second number to the right of a decimal point is in the hundredths place.. With money, this number tells you the number of pennies you have. This makes 1 sense since one penny is 1 one-hundredth ( ) of a dollar. Here is a diagram to help you better 100 understand whole number and decimal places. hundred thousands ten thousands thousands, hundreds tens ones Math On the Move Lesson 8 1 tenths hundredths thousandths ten thousandths hundred thousandths millionths

4 Example Write the place value of each digit in the number Solution We see that 1 is in the tenths place, 2 is in the hundredths place, 3 is in the thousandths place, 4 is in the ten-thousandths thousandths place, 5 is in the hundred-thousandths place, and zero is in the millionths place. We can also say that there is 1 tenth, 2 hundredths, 3 thousandths, 4 ten thousandths, 5 hundred thousandths, and zero millionths. Now you give it a try. 1. Write each digit in the correct place value in the chart below. Then write the place p value of the digit that is farthest to the right. a) 3.1 b) 2.03 c) d) e) a) b) c) d) e).. Hundreds Tens Ones. Tenths Hundredths Thousandths Ten Thousandths Hundred Thousandths Math On the Move 2

5 So how do we write and say the entire decimal? Example Write out using words. Solution This is a mixture of whole numbers and a fractional part. By counting the number of decimal places, we see this number goes to the thousandths place. We say it like this: three and four hundred thirteen thousandths Let s notice a few things here. When we say and, we really mean to say, there is a decimal point here. The next part we say is the fraction. We read it as if it s a new number, and then say the place value of the right-most digit. Four hundred thirteen is the number. Thousandths is the last place value. This is also exactly how we say mixed numbers. Without much work, we can also write decimals as mixed numbers = Once again, the number is read as three and four hundred thirteen thousandths From mixed number form, we know how to change this into an improper fraction as well! = + = so 3 = Math On the Move Lesson 8 3

6 Algorithm To write a decimal with words: 1. Write the number to the left of the decimal as you would any whole number. 2. In place of the decimal point, p use the word and. 3. Write the number to the right of the decimal point, as you would any whole number. 4. At the end, write the final digit s place value. It should end in ths. (tenths, hundredths, thousandths, ) Algorithm To write a decimal as a mixed number: 1. Rewrite all the digits to the left of the decimal point; this is the whole number part. 2. Write all the digits to the right of the decimal point as a s the numerator of the fraction. 3. For the denominator, write the place value of the right-most digit. (10, 100, 1000, 10000, , ) For example, = Now you give it a try. Math On the Move 4

7 2. Write each decimal using words, then as a mixed number, then as a fraction in lowest terms. a) 2.6 Words: Mixed Number: Fraction: b).43 Words: Mixed Number: Fraction: c) Words: Mixed Number: Fraction: Math On the Move Lesson 8 5

8 Consider two whole numbers, 340 and Believe it or not,, 340 = The number looks a little strange, doesn t it? We don t usually see numbers written this way, since the first zeros before the 3 are meaningless. However, we do need the zero after the 4. If we drop the zero at the end of 340, it changes its meaning. Similar things can be done with decimals. The following decimals are all equal = = = = = They re the same because the place value of the 4 and 3 never change. Any number of zeros may be added at the end of a decimal without changing the value of the decimal. 3.. True/False a) 07 = 7 b) 4 = 40 c) = d) 3.4 = 03.4 e) = f) = Knowing this helps us put decimals in order. Math On the Move 6

9 Example Which is larger,.2, or.19? Solution You might be tempted to say that.19 is larger than.2, since 19 > 2. But let s think about this first. We know that.2 =.20 Thinking in terms of money, we also know that \$0.20 is more money than \$0.19. Therefore,.2 >.19 What about the next two decimals? Example Which is larger, 0.2 or ? Solution Let s line up these two numbers according to their place values Notice that the top number has 2 tenths, and the bottom only has 1 tenth plus something that is less than one tenth, so.2 > We see that decimals are nice to work with when looking at the size of two numbers. This is why we use decimals instead of fractions for money. What we have observed here will help us use the next method of comparing the size of two decimals. Math On the Move Lesson 8 7

10 Algorithm To compare the size of decimals: 1. Line up the two decimals according to place value. An easy way to do this is to make sure the decimal points are on top t of each other. 2. Compare place values until a difference is found. First check the whole number parts. If those are the same, check the tenths place of each. If they are the same, check the hundredths, then the thousandths, etc. until you find a place value v where the digits aren t the same. 3. Determine which is larger. In the place value where you find the difference, the larger digit tells you which number is larger. Example Compare and Solution Step 1: Line up the two numbers by their decimal points Step 2: Compare place values until a difference is found. We see that t the first difference comes in the hundred thousandths place. Step 3: Determine which is larger. We circled the digits 6 and 5. 6 > 5, so > Math On the Move 8

11 4.. Compare the following decimals d using <, >, or =. a).12 and.13 b).102 and.13 c) 1.35 and.999 d) and Math On the Move Lesson 8 9

12 You continue walking through the grocery store and see another sign that reads, Tuna, half off of \$1. You understand that half of \$1.00 \$ is \$0.50. This means that, You remember that 1 2 = 0R1? = 0.5 also means Therefore, 1 2 = But wait, shouldn t The answer is, that before, we were only dividing with whole numbers. Now, we may use decimals instead of remainders. Here s how: 2) 1 Make 1 into 1.0, and put a decimal point directly above the division bar too ) 1.0 2) Now just divide, as if it s the whole number ) ) Example Write 5 8 as a decimal. Math On the Move 10

13 Solution Remember, 5 8 = 5 8 8) ) 5.0 ) 5.0 8) 5.0 8) If there is a remainder, er, create a decimal 2 20 and keep adding zeros to the dividend until there is no remainder ) Walking through the aisles at the grocery store,, you see a sign that reads, Markers, 3 for \$1 or 1 for \$.35, and you wonder which one is the better deal. In the first deal, 3 markers for one dollar, the price of one marker is represented as 1 3 of a dollar. How much is this? Let s remember that fractions mean division, so = Math On the Move Lesson 8 11

14 =, and = No matter how long we keep dividing, this decimal will never end, and we will keep adding on another 3 forever! To show that a decimal will never end and goes on in the same pattern, we write a bar over the part that repeats. So, is written as.3. Because we are dealing with money, we will round this decimal to the hundredths place (Rounding will be explained later in this lesson). Thus, one marker from the first deal will cost \$.33. That is better than the \$.35 of the other deal. Another example of a decimal that never r ends is and this is written as Notice that we write the bar only over the numbers that repeat, which makes the decimal easier to read. These are examples of repeating decimals. A repeating decimal is a decimal that has an infinite number of digits,, and the digits continue in a set pattern. For example, =.3, and =.473 are repeating decimals. A decimal that ends is called a terminating decimal. For instance, i.173 and 33.2 are terminating decimals. Any fraction can be made into either a terminating or a repeating decimal! Math On the Move 12

15 Example Write 4 5 as a decimal. Solution We will use long division so 4 = 0.8, a terminating decimal. 5 Example A sign in the store reads Paper towels, 11 rolls for \$3.00. How much will 1 roll of paper towels cost? Solution We must divide 3.00 into 11 equal groups, so we must solve As soon as you see the same remainder twice, you can tell that the decimal is a repeating decimal, and you may stop dividing! Our answer is3 11 =.27 Math On the Move Lesson 8 13

16 Convert the following fractions to decimals. The decimals may either terminate or repeat. 5) ) ) 5 6 In the previous example, we found that one roll of paper towels costs \$0.27. This seems strange. Have you ever seen the price of something shown as a repeating decimal? Stores don t say that something costs \$ Math On the Move 14

17 Let s think about why this is. Look at a picture of.27, and of In order to count the extra.0027 on the right, we split one cent into 100 equal pieces p ieces, and we took 27 of them. One cent is worth so little. If we break it into 100 pieces, the value of one is one ten- thousandth of a cent. This value is so small that it is not needed when dealing with money. We don t care about it! Because of this,, stores round to the nearest cent. Therefore, our o answer to the previous example is that one roll of paper towels costs \$0.27 Let s learn how to round! Example Round to the nearest tenth. Solution Step 1: Look at the number to the right of o f the tenths place Step 2: Compare the number to 5. Notice that 3 < 5, so we must round down and leave the tenths place the same. Our answer is Here is an algorithm to help you round Math On the Move Lesson 8 15

18 Algorithm To round a number to a given place value: 1. Look at the number to the right of the place value you are asked to round to. 2. Compare that number to 5. a. If that number is less than 5, round down and leave the given place value the same b. If that number is greater than or equal to 5, round up and increase the given place value by 1 i. If the number in the given place value is a 9, make it a 0 and increase the value of the number to the left of our given place value Round to the nearest hundredth = 5 Round up rounds to 1.90 Convert the following fractions to decimals. The decimals may either terminate or repeat. Then, round each decimal to the nearest hundredth Math On the Move 16

19 Review 1. Highlight the following definitions: a. decimal b. decimal point c. repeating decimal d. terminating decimal 2. Write one question you would like to ask your mentor, or one new thing you learned in this lesson. Math On the Move Lesson 8 17

20 Practice Problems Math On the Move Lesson 8 Directions: Write your answers in your math journal. Label this exercise Math On the Move Lesson 8,, Set A and Set B. Set A 1. Write the decimal with words, then as a mixed number, then as an improper fraction in simplest form Use an inequality sign (> or <) to compare each pair of decimals. a) and b) and 1.1 c) and 0.01 d) and Round each decimal to the nearest hundredth. a) b) c) d) e) f) 13.8 Set B 1. Write the following decimals so that their place values are lined up and Write the amount as a decimal part of a dollar. (Hint: think of how many cents each equals.) a) 1 quarter b) 4 nickels c) 89 pennies d) 14 dimes 3. A number is between 0 and 1. The place value farthest to the right is the thousandths place. The number contains the digits 0, 2, and 5. Using each digit, what is the smallest number that these digits can make? What is the greatest? Explain your thinking. Math On the Move 18

21 1) a) Tenths b) Hundredths c) Thousandths d) Ten Thousandths e) Hundred Thousandths 2) a) 2.6 is two and six tenths, = 2 = b).43 is forty three hundredths, c) is one and six thousand, five hundred twenty-four ten-thousandths, thousandths, = 1 = ) a) True b) False c) False d) True e) True f) True 4) a).12 <.13 b).102 <.13 c) 1.35 >.999 d) < ) ) ) ) ) ) Math On the Move Lesson 8 19

22 NOTES End of Lesson 8 Math On the Move 20

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