Chapter 12: "Exploring Fraction Concepts"
Have you ever cut a whole pizza to make individual slices? Have you ever mixed half-a-cup of chocolate chips in cookie dough? Have you ever given your students five index cards each? If you have, then you have used fractions. Although most individuals are not aware of it, we apply fraction concepts every single day. They have multiple applications in our day-to-day living. Within the classroom, you are there to guide students’ thought processes and scaffold their problem-solving strategies. But, when they are adults, you are not going to be there. Therefore, it is super important that your students understand how to apply fraction concepts! They must know how to use fractions in truly authentic ways. Unfortunately, learning fraction concepts can be quite troubling for most students. They are so familiar with whole-number thinking that the transition to fraction-based thinking is difficult. For instance, when computing with fractions, students tend to use whole-number operation rules for addition and subtraction. When adding 3/4 + 2/4, 5/8 is a common answer. According to Van de Walle et al. (2014), “Although we want students to build on prior knowledge of whole numbers, fractions become more challenging when students misapply whole-number thinking to solve fraction situations” (p. 206). Students must be explicitly taught how fractions are similar to and different from whole numbers. Yet, talking about these similarities and differences only goes so far. Students must be provided with authentic activities that exemplify fraction concepts and encourage fraction-based thinking (Van de Walle et al., 2014). I have compiled some helpful teaching tips, so you can successfully support your students' conceptual understanding of fractions.
- Fraction Purposes: Students do not learn every fraction concept in a single school year. The concepts build upon themselves over many years of mathematical instruction. Although you only teach specific concepts, it is important that you understand those that students will learn beyond your classroom. This includes the many purposes that fractions serve in our daily lives. There are five fraction purposes: part-whole relationship, measurement, division, operator, and ratio. Typically, the part-whole relationship is the starting point of fraction-based thinking. This embodies that fractions represent the relationship between a whole and its fractional parts. For example, 3/8 of a pizza represents 3 out of the 8 slices of pizza were eaten. Students must understand that a whole can be partitioned into equal-sized parts, before tackling more challenging concepts. Van de Walle et al. (2014) stated, “Measurement involves identifying an amount of a continuous unit [length, area, volume, or time], and then comparing that amount to a whole unit that is equal to 1” (p. 205). For example, 3/4 of a cup of water represents 3/4 of 1 cup of water. Division is not commonly related to fractions, but it should be! Dividing something amongst individuals is creating equal shares. For example, dividing 30 cookies amongst 15 people is creating 2 equals shares per person. Operator embodies the idea that fractions represent multiplication in certain situations. For example, 1/4 of 56 people represents 14 people out of the 56. Ratio, one of the most difficult concepts, embodies the relationship between two or more quantities (Van de Walle et al., 2014). For example, 2/3 in a recipe may represent 2 cups of milk to 3 cups of oatmeal. These fraction purposes may seem overwhelming. But, it is important that you understand them and pass that knowledge on to your students.
- Parts of a Whole: According to Van de Walle et al. (2014), “The first goal in the development of fractions should be to help students construct the idea of fractional parts of the whole- the parts that result when the whole or unit has been partitioned into equal-sized portions or equal shares” (p. 203). That seems simple enough, right? Unfortunately, learning that concept is not always simple for students. Their brains are so familiar with whole-number thinking that the idea of partitioning a whole is confusing! Therefore, they must be provided with many opportunities to physically partition a whole, compare the equal shares, and manipulate them. The best method for introducing this concept is implementing sharing activities (Van de Walle et al., 2014). You can pose sharing activities in the form of stories. For instance, you could say, “I have six candy bars to share amongst four people. How many candy bars should each person get?” Van de Walle et al. (2014) stated, “Students initially perform sharing tasks (division) by distributing items on at a time. When this process leaves leftover pieces, it is much easier to think of sharing fairly if the items can be subdivided or partitioned” (p. 203). It is super helpful for students to grasp the concept of equal shares, if they have physical objects to manipulate. Also, sharing activities are the perfect way to introduce fraction language, such as halves and thirds (Van de Walle et al., 2014). There are a ton of ways to implement sharing activities with manipulatives and fraction language. Be creative with it!
- Fraction Notation: It is important that students are taught the conventional method for properly writing fractions. They must understand that they have a top component and a bottom component that are separated by a horizontal bar. However, it is even more important that students understand what the top component and bottom component represent. Explicitly teach the proper mathematical terms! Tell students that the top component of the fraction is the numerator. Then, explain exactly what it represents. Van de Walle et al. (2014) explained the numerator, “This is the counting number. It tells how many shares or parts we have. It tells how many have been counted. It tells how many parts we are talking about. It counts the parts or shares” (p. 216). When teaching students about the numerator, follow the same process. Van de Walle et al. (2014) explained the denominator, “This tells what is being counted. It tells how big the part is. If it is a 4, it means we are counting fourths; if it is a 6, we are counting sixths” (p. 216). Those are great, simple explanations. I encourage you to use them!
"Paper Plate Fractions: The Counting Connections"
McCoy, Barnett, and Stine (2016) stated, “Although the misapplication of whole-number thinking can certainly be a challenge, connecting early fraction work to ways of working with whole numbers can also support children’s developing understanding” (p. 246). Whole-number thinking is a solid foundation to build fraction concepts upon. However, the transition from whole-number thinking to fraction-based thinking must be properly scaffolded. The process may have some rough patches. But, students’ understanding of fraction concepts determines their mathematical success in middle school and high school. You must help your students persevere through the challenges! There are two fraction concepts that will help students transition into fraction-based thinking. The first concept is partitioning, or separating a whole into equal shares. This helps students understand the concept of fraction equivalencies. The second concept is iteration, or copying a unit and repeating it to create even larger quantities. This helps students understand the concept of counting fractional parts and its similarities to counting whole numbers. Also, iteration helps students understand the numerator’s purpose and the denominator’s purpose (McCoy et al., 2016). Luckily, these education professionals created an effective activity, Paper Plate Fractions, for teaching partitioning and iteration. I am going to share the details with you, and I hope you implement this activity in your own classroom!
- Activity Description: To begin Paper Plate Fractions, students are given eight colored circles. McCoy et al. (2016) used pink, orange, yellow, lime green, light blue, dark blue, dark green, and purple paper circles. The students are instructed to independently partition each paper circle into a specific amount of fractional parts. However, each color corresponds with the fractional parts. For instance, the pink circle is partitioned into halves, and the purple circle is partitioned into twelfths. After the students partition their paper circles, they glue them on a placemat and label them with the fractional part name. Secondly, the students are asked to count a various number of objects. McCoy et al. (2016) explained, “Our intent was to provide a foundation that would allow us to connect counting fractions to their well-established ability to count with whole numbers” (p. 247). This step scaffolds students’ understanding of iteration, because it directly relates to their prior knowledge. Thirdly, students are grouped together. Each group is given a colored paper plate that has previously been partitioned into fractional parts. For instance, the pink plate has already been partitioned into halves. No two groups have the same fractional parts. McCoy et al. (2016) matched the colors of the paper plates to the paper circles. It is the students’ responsibility to determine which fractional part they are given. Once they figure it out, they stick the paper plate pieces on the whiteboard underneath the proper fractional part name. But, they cannot form a whole with the fractional parts. They must place the paper plate pieces in rows (McCoy et al., 2016).
- Post Activity Discussion: After the students complete each step of the activity, the teacher leads a whole group discussion. She has the students count the paper plate pieces underneath each fractional part name. For instance, under her guidance, they chorally count, “one-fifth, two-fifths, and three-fifths.” Then, the teacher writes 3/5 underneath the paper plate pieces. This demonstrates the concept of iteration for the students. They can see how each fractional part is a single quantity. But, each single quantity can be repeated to create a larger quantity. McCoy et al. (2016) explained, “To encourage them [students] to see the fraction as a single quantity, we avoided using the words numerator or denominator so that the focus remained on the connection to the familiar concept of counting” (p. 248). By avoiding the words numerator and denominator, the students are not actively thinking about parts that make up a whole. Instead, they are thinking in terms of fractional parts that can be combined to create an even larger fraction. This sparks the transition between whole-number thinking and fractional-thinking. I strongly encourage you to implement Paper Plate Fractions!
Classroom Activity: Waffle Fraction Word Problem
By the end of fourth-grade, Common Core State Standard (2010) 4.NF.B.3.d expects students to solve word problems that involve addition and subtraction of fractions referring to the same whole and having like denominators. Students should be able to solve these word problems using visual fraction models and equations to represent the problems. In the above video, I explain how to create fraction word problems that relate to students' everyday lives. Meaningful context, such as waffles, helps students create a mental representation of the mathematical component. Additionally, I demonstrate how to provide visuals of everyday objects, such as waffles slices, that students can manipulate. These visual manipulatives assist students with creating their mental representations of the fractions, as they work hard to solve the word problems.
Independent Practice: Compare Fractions
According to Common Core State Standard (2010) 4.NF.A.2, fourth-graders should be able to compare fractions with different numerators and different denominators. To successfully do this, students must understand that comparisons can only be made between two fractions that refer to the same whole. Also, they must know how to create common denominators or compare to benchmark fractions. When comparing two fractions, fourth-graders use the less than (<), greater than (>), or equal to (=) symbols. There are many individual skills that combine to create this mathematical concept! Your students will need a lot of practice, before they can successfully compare fractions. That is why “Compare Fractions” is so valuable. This game instructs students to choose the sign that correctly compares the two fractions. If they choose the correct sign, they receive immediate praise. If they choose the incorrect sign, they receive immediate feedback with a positive message. For instance, this message appears on the screen, “Try again, practice makes perfect!” The students realize that they chose the incorrect symbol, yet their effort is still praised. That will motivate them to keep trying! The most valuable part of this game is the visual aids. The circles reinforce that comparisons can only be made between two fractions that refer to the same whole. Yet, they can see that the same whole has been partitioned into a different number of sections. Click on “Independent Practice” to access this awesome resource.