1 Decimal Notations for Fractions Number and Operations Fractions /4.NF Domain: Cluster: Standard: 4.NF Number and Operations Fractions Understand decimal notation for fractions, and compare decimal fractions. 4.NF.C.5 Express a fraction with denominator as an equivalent fraction with denominator, and use this technique to add two fractions with respective denominators and. For example, express 3/ as 30/, and add 3/ + 4/ = 34/. 4.NF.C.6 Use decimal notation for fractions with denominators or. For example, rewrite 0.62 as 62/; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. The standards covered in this section fall under the cluster for understanding decimal notation for fractions and comparing decimal fractions. This cluster connects to the second critical area of focus for fourth grade which requires students to develop an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions. More specifically within this cluster, we ll be dealing with understanding fraction equivalence between fractions with the denominator and (and adding such fractions), fraction equivalence with decimals, and comparing two decimals to the hundredths. The standards within this cluster should be covered after the standards in the cluster for building fractions from unit fractions. The knowledge gleaned from understanding and generating equivalent fractions will be invaluable as students embark on working with fractions with the denominator as well as building an understanding of decimals and decimal notation. Moreover, the skills learned within this cluster will serve as a foundation in fifth grade as students perform operations with decimals to the hundredths. Before we dig in to each of the standards presented within this section, be sure to review the helpful literature and resource books that can complement your lessons and provide additional support for students.
2 Literature Books to Support Teaching the Standard(s) in this Section Fractions, Decimals, and Percents by David A. Adler Do You Know Dewey? by Brian P. Cleary Bob the Alien Discovers the Dewey Decimal System by Sandy Donovan Parting is Such Sweet Sorrow by Linda Powley Great Videos to Support What You re Teaching There are many instructional videos that you can use to supplement your lessons. Save the following links to videos for future viewing during class time, within your math center, as part of a lesson, or for students to use at home for review or supplemental learning. Benjamin Light 4.NF.5 Converting Decimals to Fractions Math Antics Converting Base Fractions Expanded Notation Decimals and Decimal Fractions Writing Tenths and Hundredths with Decimals Decimals in Standard and Word Form Decimals on a Number Line Decimals in the Tenths and Hundredths Tenths and Hundredths on a Number Line Comparing and Arranging Decimals Great Virtual Games and Resources to Support What You re Teaching Fractions to Decimals Fruit Shoot: Shoot the fruit with the decimal equivalent of the fraction.
3 The Decimal Detectives: Students are quizzed on their knowledge of a decimal s place value on a number line to help catch a crooked decimal! Matching Math Tenths: Match the decimals to the tenths with its model. Matching Math Hundredths: Match the decimals to the hundredths with its model. Comparing Decimals Fruit Shoot: Compare the two decimals and shoot the fruit with the appropriate symbol (<, >, or =). Scooter Quest Decimals: Students use their knowledge of decimal place value to deliver the newspaper to the correct house. 4.NF.C.5 / Express a fraction with denominator as an equivalent fraction with denominator, and use this technique to add two fractions with respective denominators and. For example, express 3/ as 30/, and add 3/ + 4/ = 34/. Thus far, students have been working mostly with like denominators, specifically the denominators 2, 3, 4, 5, 6, 8,, and 12. As part of this domain, students must also be familiar with fractions with the denominator. Fractions with the denominator come into play in this standard as students compare their values with fractions with the denominator. Students will use their knowledge of equivalent fractions to generate fractions with the denominator into equivalent fractions with the denominator. Additionally, they will be adding fractions with the denominators and. First and foremost, students must be taught that a fraction with the denominator and a fraction with the denominator both represent the same whole. Previously, students learned that fractions with the same denominator represent the same whole. Now, they are beginning to learn that numbers with different denominators can also represent the same whole. This, of course, varies depending on the denominators. Working with the denominators and is a great starting point to help students learn this concept. We know that fractions can be represented by a number line. Let s take, for example, a fraction with the denominator. The denominator tells us that we must divide the number line into ten equal parts; each part represents 1 /. Notice that / is the same as 1 whole because the denominator and the numerator are the same.
4 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / / Now, what if we wanted to create a number line with a fraction with the denominator? Well, we d use the same process except this time, we ll divide the number line into equal parts. Each part now represents 1 / / 20 / 30 / 40 / 50 / 60 / 70 / 80 / 90 / / Do you notice some similarities between the two number lines? Both number lines are the same length and have the endpoints 0 and 1. Notice that / is equal to 1 just like / is equal to 1. That must mean that both denominators refer to the same whole! Still not convinced? Let s compare / and / to 1 whole using models. 1 whole ( 1 / 1 ) / / Above we have three models, 1 whole, shaded parts out of equal parts, and shaded parts out of equal parts. You can see that / and / both have the same whole. In other words, they are equal to 1 whole. Keeping this idea in mind, take a look at the problem below. Is 5 / equivalent to 50 /?
5 How can we determine if the two fractions are equivalent? Start by looking at the denominators. We have a denominator of and a denominator of. We now know that the denominators and refer to the same whole. That means we can create identical models to represent each fraction. That is, the models can be of the same shape and size. For this example, we chose a square. 5 / 50 / In the first model representing 5 /, the denominator tells us to divide the model into ten equal parts. The numerator tells us to shade in five parts. In the second model representing 50 /, the denominator tells us to break the model into equal parts and the numerator 50 tells us to shade in 50 parts. Compare the two models. Do they look the same? Yes! We already know they have the same whole. We can now see from the shaded portions that they are equivalent. 5 / and 50 / are equivalent fractions. Now, try the problem below. Find a fraction with a denominator that is equivalent to 40 /.
6 In the problem above, we have the fraction 40 /. Write down the fraction. We want to find a fraction that is equivalent, so we ll write an equal sign. Our equivalent fraction must have the denominator. Write a fraction with a denominator. What we don t know is the numerator. So, write a question mark to represent the unknown numerator. 40 =? Next, we ll draw two identical models because we know the denominators and refer to the same whole. To represent the denominator, we ll divide the first model into equal pieces. The second model is divided into ten equal parts to represent the denominator. We know the numerator for the first fraction is 40, so we ll shade in 40 parts in the hundreds model. To create an equivalent fraction in the second model, we ll shade in the same amount. 40 /? / The models now represent equivalent fractions. Now, we must simply determine what the denominator is for the second model to find the answer. We know that the numerator
7 represents the number of parts shaded. So, let s count the number of shaded parts. We have four shaded parts so we know the numerator is 4! The answer is 4 /. 4 / is equivalent to 40 /. 40 = 4 To check to make sure the answer is correct, we can use number lines. We ll create two identical number lines for the two fractions / 20 / 30 / 40 / 50 / 60 / 70 / 80 / 90 / / / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / / We can see that both number lines have the same points and the same shaded portions which tells us that the fractions are equivalent. The answer is correct! Once students have fully grasped the idea that a fraction with a denominator has the same whole as a fraction with the denominator, it s time to get them started on adding fractions with the denominators and. We ll continue to use models as tools for finding the answer. Take a look at the problem below
8 Adding fractions with unlike denominators is a completely new concept for students. Luckily, using models to add fractions is not! So, we ll use modeling to solve this problem. We know that the denominators and refer to the same whole. Again, we ll create identical models to represent the fractions. + 3 / 50 / In the first fraction model, we know to divide the model into ten equal parts per the denominator and shade in three parts per the numerator. The same goes for the second fraction model with equal parts and 50 shaded parts. Now, we know that adding requires us to combine numbers. So, let s try combining the shaded parts from the first model into the second model. + = 3 / 50 / We have the model for the answer. To determine the fraction, we ll need to look at the number of equal parts and the number of shaded parts. We already know that the second model in the addition problem had equal parts so the denominator is. What about
9 the numerator? This is quite a tedious process, but once the shaded parts are counted, we know that we have a numerator of 80. The answer is 80 /! Is there an easier way to write 80 /? How about 8 /? How do we know for sure that 80 / is equivalent to 8 /? Well, suppose we did the reverse and instead of adding the shaded portion of the first model into the second model, we combined the shaded portion of the second model into the first model. We would have equal parts and 8 shaded parts, giving us a fraction of 8 /. = 8 / 80 / Using modeling is an excellent way to help students make the connection between equivalent fractions with the denominators ten and one hundred. Models allow students to visually see that although two fractions may appear different, they both represent the same quantity. The goal of this standard is for students to understand fractions beyond the digits they consist of, and acquire a deeper understanding of fractions as parts of a whole. 4.NF.C.6 / Use decimal notation for fractions with denominators or. For example, rewrite 0.62 as 62/; describe a length as 0.62 meters; locate 0.62 on a number line diagram. This standard introduces students to the concept of decimals and is directly connected to the aforementioned standard on fractions with the denominators and. In order for students to master this standard, they need a deep understanding of place value. Previously, students learned to work with place value charts with numbers up to one million. It is important that students fully understand the concept of place value in whole numbers as that concept will now be extended to fractions and their decimal equivalents. In this section, we teach students how to convert decimals to fractions and fractions to decimals using place value charts, number lines, and division.
10 Before we begin, let s do a brief review. As you recall, fractions can be represented in a number of ways. Parts of a group Parts of a whole Number line 0 ¼ 1 Today, we ll introduce yet another way to represent fractions: as decimals! Reviewing a number s place value is very important as well. We know that a digit s location in a number is significant. Why? Because the location tells us the value of the digit. For example, the value of 5 is different than the value of 50 which is also different from 500. We worked with a place value chart such as the one below to help us understand the values of digits in a number. Periods Hundred Millions Millions Thousands Ones Ten Millions Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones Sections At the top, we have the periods and below we have the sections. Remember that when writing a number in numerical form, we separated the periods with a comma. Now, we ll be adding another period to the place value chart. This time, the period will represent numbers that are less than one. In other words, they are fractions or decimals. The period is called thousandths.
11 Mil. Millions Thousands Ones Thousandths Mil. Mil. Thous. Thous. Thous. s s,,. 1s Tenths Hundredths Thousandths Within the thousandths period, we have the sections representing the tenths place, hundredths place, and thousandths place. Notice that although there is a comma separating the whole number periods, there is a period separating the ones period from the thousandths period. This is because fractions (with a numerator that is smaller than the denominator) are less than one whole, which means they are not whole numbers; they are parts of a whole. Therefore, the fractions represented in decimal form must go after a period. To reiterate, whole numbers go to the left of the period and decimal fractions go to the right of the period. Let s try an example to help bring this concept home. Convert 0.4 to a fraction. Above we have the decimal 0.4. The zero in front of the period tells us there are no wholes in the number. The 4 after the period tells us that this number is a decimal. To convert the decimal to a fraction, we can use a place value chart. Since we are working with decimals, we ll simplify the chart to include only the periods ones and thousandths. Ones Thousandths Hundreds Tens Ones Tenths Hundredths Thousandths 0 4. With consideration to the location of the period in the decimal, we entered the numbers into the place value chart. We can now see more clearly that there are no whole numbers and 4 tenths. 4 tenths sounds a lot like a fraction, doesn t it? That s because it IS a fraction! Since 4 is in the tenths place, we know that the fraction would have 4 as the numerator and (TENths) as the denominator. 0.4 = 4
12 There we have it, 4 / is the fraction form of 0.4! To gain a better understanding of the value of 0.4, let s represent the decimal fraction as a model and plot 4 / on a number line. The denominator tells us to divide the model into ten equal parts and the numerator tells us to shade in four. Take a look at how 0.4 looks as a model. 0.4 ( 4 / ) Next, we ll represent 0.4 on a number line. We know that a decimal is less than one whole so we ll create a number line between 0 and 1. The denominator tells us to divide the number line into ten equal parts and the numerator 4 tells us to plot a point on the fourth dash and shade up to that point / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / / Now, let s try another problem. Put 0.27 into fraction form. Once again, we are given a decimal and asked to turn it into a fraction. As we ve done previously, we ll use a place value chart to help us better understand the value of 0.27 and convert it to a fraction.
13 Ones Thousandths Hundreds Tens Ones Tenths Hundredths Thousandths Following the location of the period, we have zero ones (or wholes) in front of the period. After the period, we can see that we have two tenths and seven hundredths. Or, we can simply say that we have twenty-seven hundredths. To create a fraction, we can use 27 as the numerator and (HUNDREDths) as the denominator is equivalent to 27 /! 0.27 = 27 What if we wanted to do the reverse and turn a fraction into a decimal? Well, that s a little trickier but previous practice with expanded form and division can help us master this skill! Rewrite 52 / as a decimal. As you can see, we are now given a fraction and asked to convert the fraction to decimal form. One strategy we can use is to put the fraction into expanded form then input it into a place value chart. Looking at the numerator, we know that the expanded form of 52 is The denominator stays the same so we have 50 / + 2 /. We know that 50 / is the same as 5 /, so the expanded form of 52 / is 5 / + 2 /. 52 (50 + 2)
14 Now that we have the fraction in expanded form, we can easily input the numbers into a place value chart. Since the 5 has a denominator of, we know that 5 goes into the tenths place. 2 has a denominator of which puts it in the hundredths place. Ones Thousandths Hundreds Tens Ones Tenths Hundredths Thousandths A period goes before the thousandths period and a zero goes before the period to represent no whole numbers. Together, we have a decimal of Be sure to say out loud, point-fifty-two and fifty-two tenths to students so they make the connection between decimals and fractions. 52 = 0.52 Another strategy students can use to convert fractions to decimals is to use division. You may have noticed from a previous section that although we ve taught students to divide multi-digit numbers, we have not yet introduced long division. The CCSS does not officially introduce dividing multi-digit numbers in the standard algorithm until the sixth grade. In the interim, students have learned to divide by using their knowledge about multiplication which, in many cases, is the more efficient method for finding multi-digit quotients. We bring this up because the strategy that we introduce below will utilize a bit of long division method so it may be too advanced for some students. However, the long division is quite basic and may be useful to many students when converting fractions to decimals. Using the next method, it is important that students see fractions as a division problem. For example, instead of seeing 52 / as 52 parts out of, they should recognize that 52 is being divided by. 52
15 52 To solve for this division problem, we ll put the dividend (52) inside the house and the divisor () on the outside. To start, we ll ask ourselves how many times the dividend can go into the divisor. That is, how many times can go into 52? Zero! Because is larger than 52. So, we ll write a zero above 52. Then, we ll multiply zero by. We know anything multiplied by zero is zero so we ll write a zero below 52. Subtract the zero from the divisor. 52 minus 0 is still is the number leftover after we divided so, essentially, we can treat 52 as the remainder (part of a whole). Previously, we wrote the remainder as r 52. However, that doesn t work in this case because we have no whole number. Also, the goal is to create a decimal. Instead, we ll add a period to the answer and bring 52 up as a decimal: NF.C.7 / Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. In this standard, students must compare two decimals to the hundredths. To properly compare two decimals, students must understand the relationship between decimals and fractions with the denominators and. They must also grasp the concept of place value for decimals. Students have already worked with comparing multi-digit whole
16 numbers and fractions with different numerators and denominators so the process of comparing quantities should be very familiar to them. The tricky part now is that they are dealing with decimals and there are some extra steps involved in finding the answer. Nonetheless, we continue to use modeling and number lines as familiar tools that students can use to compare decimals To compare these two decimals, we begin by converting the decimals to fractions. We can use place value charts to do the conversions. Ones Thousandths Hundreds Tens Ones Tenths Hundredths Thousandths Both numbers are in the tenths place which tells us that the denominator is. Therefore, we have 7 / and 4 / as our fractions. 7 4 Now, we ll draw models to represent each fraction. Since the fractions have the same denominator, we know we are referring to the same whole. Consequently, the models should be identical and divided into ten equal parts. In the first model, we shade in 7 parts; in the second model, we shade in 4 parts. 7 /
17 4 / We can see from the models that 7 / is greater than 4 /. In other words, 0.7 is greater than 0.4. We have the answer! > Now, we ll try a problem that s a bit trickier! This problem is trickier because upon first glance, students may think that 0.16 is greater than 0.9 because 16 is greater than 9. But is that really the case? Again, we ll use a place value chart to convert the decimals into fractions. Ones Thousandths Hundreds Tens Ones Tenths Hundredths Thousandths
18 Looking at the chart, we can see that we have 16 hundredths which tells us the first fraction has a denominator of : 16/. The second decimal is 9 tenths which means we have a denominator of : 9/ Again, we ll draw models to represent each fraction. Since the fractions have the denominators and, we know we are referring to the same whole. Consequently, the models should be identical. The first model represents 16 / which means it must be divided into equal parts with 16 parts shaded. In the second model representing 9 /, we divide the model into ten equal parts and shade in 9 parts. 16 / 9 / The models clearly show that 16 / is less than 9 /. Thus, 0.16 is less than 0.9!
19 < To make sure the answer is correct, we can use a number line to compare the decimals as well. Since the decimals represent fractions that are less than one whole, we ll draw a number line of 0 to is equivalent to 16 / which tells us to divide the first number line into equal parts and shade to the 16 th dash. 0.9 is equivalent to 9 / so we create a second number line with ten equal parts and shade up to the ninth dash / 20 / 30 / 40 / 50 / 60 / 70 / 80 / 90 / / / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / / The number lines confirm that 16 / is indeed less than 9 /. Perfect! The standards covered within this section serve as great milestones for students as they journey through the CCSSM. The standards require students to use skills acquired from other standards and grade levels on the place value system and make connections to build on their prior knowledge to develop new skills. Moreover, these standards also prepare students for their work in fifth grade as they will be required to perform operations with decimals to the hundredths. In addition to the strategies provided above, allowing students to physically work with base ten blocks is always a good idea when dealing with fractions. You should also encourage students to pronounce decimals and fractions out loud. For example, 0.32 as point thirty-two, thirty-two hundredths, or even the sum of three tenths and two hundredths. As students master the skills presented within these standards, the method of connecting fractions and decimals to their word form will help students develop a greater understanding of decimal value without having to refer to the place value chart.
20 For practice with the skills taught in this section of the course, provide students with the Denominators & Decimals worksheet in the Course Guides folder.