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Developing Fraction Concepts
Learner OutCOmes After reading this chapter and engaging in the embedded activities and reflections, you should be able to: 15.1 Describe and give examples for fractions constructs. 15.2 Name the types of fractions models and describe activities for each. 15.3 Explain foundational concepts of fractional parts, including iteration and partitioning, and connect these ideas to CCSS-M expectations. 15.4 Illustrate examples across fraction models for developing the concept of equivalence. 15.5 Compare fractions in a variety of ways and describe ways to teach this topic conceptually. 15.6 Synthesize how to effectively teach fraction concepts.
F
ractions are one of the most important topics students need to understand in order to be successful in algebra and beyond, yet it is an area in which U.S. students struggle. NAEP test results have consistently shown that students have a weak understanding of fraction con- cepts (Sowder & Wearne, 2006; Wearne & Kouba, 2000). This lack of understanding is then translated into difficulties with fraction computation, decimal and percent concepts, and the use of fractions in other content areas, particularly algebra (Bailey, Hoard, Nugent, & Geary, 2012; Brown & Quinn, 2007; National Mathematics Advisory Panel, 2008). Therefore, it is absolutely critical that you teach fractions well, present fractions as interesting and important, and commit to helping students understand the big ideas.
Big IDEAS
◆ ◆ (^) For students to really understand fractions, they must experience fractions across many constructs, including part of a whole, ratios, and division. ◆ ◆ (^) Three categories of models exist for working with fractions—area (e.g., 13 of a garden), length (e.g., 34 of an inch), and set or quantity (e.g., 12 of the class). ◆ ◆ (^) Partitioning and iterating are ways for students to understand the meaning of fractions, especially numerators and denominators. ◆ ◆ (^) Equal sharing is a way to build on whole-number knowledge to introduce fractional amounts. ◆ ◆ (^) Equivalent fractions are ways of describing the same amount by using different-sized fractional parts. ◆ ◆ (^) Fractions can be compared by reasoning about the relative size of the fractions. Estimation and reasoning are important in teaching understanding of fractions.
C h a p t e r
340 Chapter 15 Developing Fraction Concepts
meanings of Fractions
Fraction understanding is developmental in nature. Fraction experiences should begin as early as first grade. In the Common Core State Standards students partition shapes and refer to the fractional amounts in grades 1 and 2 as “equal shares.” In grade 3, fractions are a major emphasis, with attention to using fraction symbols, exploring unit fractions (fractions with numerator 1), and comparing fractions. Grade 4 focuses on fraction equivalence and begins work on fraction operations (Chapter 16). This emphasis over years of time is an indication of both the complexity and the importance of fraction concepts. Students need significant time and experiences to develop a deep conceptual understanding of this important topic. Understanding a fraction is much more than recognizing that 35 is three shaded parts of a shape partitioned into five sections. Fractions have numerous constructs and can be repre- sented as areas, quantities, or on a number line. This section describes these big ideas. The next sections describe how to teach the concepts of fractions.
Fraction Constructs Understanding fractions means understanding all the possible concepts that fractions can rep- resent. One of the commonly used meanings of fraction is part‐whole. But many who research fraction understanding believe students would understand fractions better with more emphasis across other meanings of fractions (Clarke, Roche, & Mitchell, 2008; Lamon, 2012; Siebert & Gaskin, 2006).
Pause (^) & Reflect Beyond shading a region of a shape, how else are fractions represented? Try to name three ideas. ●
Part-Whole. Using the part‐whole construct is an effective starting point for building mean- ing of fractions (Cramer & Whitney, 2010). Part‐whole can be shading a region, part of a group of people ( 35 of the class went on the field trip), or part of a length (we walked 3^12 miles).
measurement. Measurement involves identifying a length and then using that length as a measurement piece to determine the length of an object. For example, in the fraction 58 , you can use the unit fraction 18 as the selected length and then count or measure to show that it takes five of those to reach 58. This concept focuses on how much rather than how many parts, which is the case in part‐whole situations (Behr, Lesh, Post, & Silver, 1983; Martinie, 2007).
Division. Consider the idea of sharing $10 with 4 people. This is not a part‐whole scenario, but it still means that each person will receive one‐fourth ( 14 ) of the money, or 2^12 dollars. Divi- sion is often not connected to fractions, which is unfortunate. Students should understand and feel comfortable with the example here written as 104 , 4 10, 10 , 4, 224 , and 2^12 (Flores, Samson, & Yanik, 2006).
Operator. Fractions can be used to indicate an operation, as in 45 of 20 square feet or 23 of the audience was holding banners. These situations indicate a fraction of a whole number, and students may be able to use mental math to determine the answer. This construct is not emphasized enough in school curricula (Usiskin, 2007). Just knowing how to represent frac- tions doesn’t mean students will know how to operate with fractions, which occurs in various other areas in mathematics ( Johanning, 2008).
ratio. Discussed at length in Chapter 18, the concept of ratio is yet another context in which fractions are used. For example, the fraction 14 can mean that the probability of an event is one in four. Ratios can be part‐part or part‐whole. For example, the ratio 34 could be the ratio of those wearing jackets (part) to those not wearing jackets (part), or it could be part‐whole, meaning those wearing jackets (part) to those in the class (whole).
342 Chapter 15 Developing Fraction Concepts
reach higher levels of understanding is to use multiple representations, multiple approaches, and explanation and justification (Harvey, 2012; Pantziara & Philippou, 2012). This chapter is designed to help you help students deeply understand fractions.
Complete Self-Check 15.1: Meanings of Fractions
models for Fractions
There is substantial evidence to suggest that the effective use of visuals in fraction tasks is important (Cramer & Henry, 2002; Siebert & Gaskin, 2006). Unfortunately, textbooks rarely incorporate manipulatives, and when they do, they tend to be only area models (Hodges, Cady, & Collins, 2008). This means that students often do not get to explore fractions with a variety of models and/or do not have sufficient time to connect the visuals to the related con- cepts. In fact, what appears to be critical in learning is that the use of physical tools leads to the use of mental models, which builds students’ understanding of fractions (Cramer & Whitney, 2010; Petit, Laird, & Marsden, 2010). Properly used, tools can help students clarify ideas that are often confused in a purely symbolic model. Sometimes it is useful to do the same activity with two different representa- tions and ask students to make connections between them. Different representations offer dif- ferent opportunities to learn. For example, an area model helps students visualize parts of the whole. A linear model shows that there is always another fraction to be found between any two numbers—an important concept that is underemphasized in the teaching of fractions. Some students are able to make sense of one representation, but not another. Importantly, students need to experience fractions in real-world contexts that are meaningful to them (Cramer & Whitney, 2010). These contexts may align well with one representation and not as well with another. For example, if students are being asked who walked the farthest, a linear model is more likely to support their thinking than an area model. Table 15.1 provides an at‐a‐glance explanation of three types of models—area, length, and set—defining the wholes and their related parts for each model. Using appropriate rep- resentations and different categories of models broaden and deepen students’ (and teachers’) understanding of fractions. An increasing number of Web resources are available to help represent fractions. One excellent source, though subscription based, is Conceptua Fractions (https://www.youtube .com/watch?v=7OJTjYxWCIU), developed by Conceptua Math. This site offers free tools that help students explore various fraction concepts using area, set, and length models (includ- ing the number line). The activities can be prescribed by the teacher and contain formative assessment resources.
standards for mathematical Practice mP5. Use appropriate tools strategically.
Table 15.1 mODeLs FOr FraCtiOn COnCePts anD HOW tHey COmPare
model What Defines theWhole What Defines theParts What the Fraction means
Area The area of the defined region
Equal area The part of the area covered as it relates to the whole unit Length or number line The unit of distance or length
Equal distance/length The location of a point in relation to 0 and other values on the number line Set Whatever value is determined as one set
Equal number of objects
The count of objects in the subset as it relates to the defined whole Source: Based on Petit, Laird, & Marsden (2010).
Models for Fractions 343
area models
With these visuals, fractions are based on parts of an area. See Figure 15.1 for examples. Area is a good place to begin fraction explorations because it lends itself to equal sharing and partitioning. Circular Fraction pieces are the most commonly used area model. One advantage of the circular model is that it emphasizes the part‐whole concept of fractions and the meaning of the relative size of a part to the whole (Cramer, Wyberg, & Leavitt, 2008). Other area models in Figure 15.1 demonstrate how different shapes can be the whole. Grid or Dot paper provides flexibility in selecting the size of the whole and the size of the parts (see Blackline Masters 5– for a selection). Many commercial versions of area manipulatives are available, including cir- cular and rectangular pieces, pattern blocks, geoboards, and tangrams. Activity 15.1 (adapted from Roddick & Silvas-Centeno, 2007) uses pattern blocks to help students develop concepts of partitioning and iterating.
Circular “pie” pieces (^) Any piece can be selected as the whole. Rectangular regions Fourths on a geoboard
Drawings on grids or dot paper Pattern blocks Paper folding
One-third, — 155 One-fifth or two-tenths
Figure 15.1 Area models for fractions.
CCSSM: 1.G.a.3; 2.G.a.3; 3.NF.a.
Playground Fractions Create this “playground” with your pattern blocks (see pattern Block playground activity page). It is the whole. For each fraction below, find the pieces of the playground and draw it on your paper. For grades 1 and 2 use words, not fraction symbols (e.g., half of, one-half, or four-thirds). 1 2 playground^ 1 3 playground 112 playgrounds 23 playground 2 playgrounds (^43) playground
Activity 15.
Models for Fractions 345
Like with whole numbers, the number line is used to compare the relative size of num- bers. Importantly, the number line reinforces that fact that there is always one more fraction to be found between two fractions. The following activity (based on Bay‐Williams & Martinie,
- is a fun way to use a real‐world context to engage students in thinking about fractions through a linear model.
CCSSM: 3.NF.a.2a, b; 3.NF.a.3a, b, d
Who Is Winning? Use Who Is Winning? activity page and give students paper strips or ask them to draw a number line. this activity can be done two ways (depending on your lesson goals). First, ask students to use reasoning to answer the question “Who is winning?” Students can use reasoning strategies to compare and decide. Second, students can locate each person’s position on a number line. explain that the friends below are playing “red Light, Green Light.” the fractions tell how much of the distance they have already moved. Can you place these friends on a line to show where they are between the start and finish? Second, rather than place them, ask students to use reasoning to answer the question “Who is winning?”
Mary: 34 harry: 12 Larry: (^56) han: 58 Miguel: 59 angela: (^23)
this game can be differentiated by changing the value of the fractions or the number of friends (fractions). the game of “red Light, Green Light” may not be familiar to eLLs. Modeling the game with people in the class and using estimation are good ways to build background and support students with disabilities.
Activity 15.
engLisH Language Learners
stuDents with sPeCiaL neeDs
set models
In set models, the whole is understood to be a set of objects, and subsets of the whole make up fractional parts. For example, 3 objects are one‐fourth of a set of 12 objects. The set of 12 in this example represents the unit, the whole or 1. The idea of referring to a collection of counters as a single entity makes set models difficult for some students. Putting a piece of yarn in a loop around the objects in the set to help students “see” the whole. Figure 15.3 illustrates several set models for fractions. A common misconception with set models is to focus on the size of a subset rather than the number of equal sets in the whole. For example, if 12 counters make a whole, then a sub- set of 4 counters is one‐ third, not one‐fourth, because 3 equal sets make the whole. However, the set model helps establish important connections with many real‐world uses of fractions and with ratio concepts. Two color counters are an effective set manipulative. Counters can be flipped to change their color to model vari- ous fractional parts of a whole set. Any countable objects (e.g., a box of crayons) can be a set model (with one box being the unit or whole). The following activity uses your students as the whole set. It can be done as an energizer, warm-up, or full lesson.
3 5
9 15
Two-color counters in arrays. Rows and columns help show parts. Each array makes a whole. Here or are yellow.
Two-color counters in sets showing red. The whole must be clearly indicated.
1 (^13)
12 makes 1 whole
2 3
6 Objects. Shows or 9 are cars.
Figure 15.3 Set models for fractions.
346 Chapter 15 Developing Fraction Concepts
Students must be able to explore fractions across the three area, length, and set models. As a teacher, you will not know whether they really understand the meaning of a fraction such as 1 4 unless you have seen a student represent^ one-fourth using all three models.
FormaTive assessmenT Notes. A straightforward way to assess students’ knowledge of a fractional amount is to give them a piece of paper folded into thirds. Write area, length, and set at the top of each section and give them a fractional value (e.g., 34 ). Observe as they (1) draw a picture and (2) write a sentence describing a context or example for the selected fraction. This can be done exactly for commonly used fractions or can be an estimation activity with fractions like 3158. ■
Technology Note. Virtual manipulatives are available for all three models. Virtual manipulatives have been found to positively affect student achievement, especially when they are paired with using the actual manipulatives (Moyer-Packenham, Ulmer, & Anderson, 2012). Recommended sites include: Cyberchase (PBS): Cyberchase is a popular television series. Their website offers videos that model fractions with real‐world connections and activities such as “Thirteen Ways of Looking at a Half ” (fractions of geometric shapes) and “Make a Match” (concept of equivalent fractions). Illuminations (NCTM) Fractions Model: Explore length, area, region, and set models of fractions, including fractions greater than one, mixed numbers, decimals, and percentages. Math Playground Fraction Bars: On this site you can explore fractional parts, the concepts of numerator and denominator, and equivalence. National Library of Virtual Manipulatives: This site offers numerous models for exploring fractions, including fraction bars and fraction pieces. There is also an applet for comparing and visualizing fractions. ■
Complete Self-Check 15.2: Models for Fractions
Fractional Parts
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 fair shares. (Recall that Table 15.1 describes the meanings of parts and wholes across each type of model.) Students understand the idea of separating a quantity into two or more parts to be shared fairly among friends. This is the beginning of understanding fractions and in the CCSS-M
CCSSM: 3.NF.a.1; 3.NF.a.3b
Class Fractions Use a group of students as the whole—for example, six students if you want to work on halves, thirds, and sixths. Invite them to come to the front of the room. Say to the group, “If you [are wearing tennis shoes, have brown hair, etc.], move to the right. If not, move to the left.” ask everyone, “What fraction of our friends [are wearing tennis shoes]?” Invite them to whisper to a partner, then share with the class. Change the number of selected students in the whole. You can also ask, “how many friends in one-half of this group? One- fourth of this group?” Connect to the symbols for fractions (e.g., 12 , 13 , and 16 ), asking students to write the fraction that tells how many students [are wearing tennis shoes]. also, if you have six students in your group, and you have three in your subgroup, then students are likely to say “three-sixths” and “ one-half.” Discuss that these values are equivalent.
Activity 15.
348 Chapter 15 Developing Fraction Concepts
part is one-fourth of the unit). It is important for students to understand, however, that some- times visuals do not show all the partitions. For example, consider the following picture:
Referring back to the two criteria, a student might think, “If I partitioned this so that all pieces were the same size, then there will be four parts; therefore, the smaller partitioned region represents one‐fourth”—not one‐third, as many students without a conceptual under- standing of fractional parts might suggest. Some manipulatives, like fraction bars or fraction circles, can mislead students to believe that fractional parts must be the same shape as well as the same size. Color tiles can be used to create rectangles that address this misconception. Ask students to describe the fractional parts in a rectangle, such as the one illustrated here:
Students who recognize that each color represents thirds understand that fractional parts must be the same size, and that the shape of the thirds may be different. Area models are the first types of models to use in teaching fractional parts. Young stu- dents, in particular, tend to focus on shape, when the focus should be on equal‐ sized parts. Activity 15.1 is an example of how you can use pattern blocks to focus on partitioning into equal‐sized parts. You can build on this activity by building other shapes that use different pattern block pieces and then have students figure out how much each piece is of the whole. View how a classroom teacher helps his students understand parts of the whole using pattern blocks. Activity 15.4 uses partitioned drawings to develop the concept of fractional parts.
CCSSM: 1.G.a.3; 2.G.a.3; 3.NF.a.
Partitioning: Fourths or Not Fourths? Use Fourths or Not Fourths activity page showing examples and nonexamples (which are very important to use with students with disabilities) of fourths (see Figure 15.5). ask students to identify the wholes that are correctly divided into fourths (equal shares) and those that are not. For each response, have students explain their reasoning. repeat with other fractional parts, such as thirds or eighths. to challenge students, ask them to draw shapes that fit each of the four categories listed on the next page for other fractional parts, such as sixths. (See Sixths or Not Sixths activity page.)
Activity 15.
stuDents with sPeCiaL neeDs
Fractional Parts 349
In the preceding activity, the shapes fall in each of the following categories:
1. Same shape, same size: (a) and (f ) [equivalent] 2. Different shape, same size: (e) and (g) [equivalent] 3. Different shape, different size: (b) and (c) [not equivalent] 4. Same shape, different size: (d) [not equivalent]
FormaTive assessmenT Notes. Activity 15.4 is a good diagnostic interview to assess whether students un- derstand that it is the size that matters, not the shape. If students get all correct except (e) and (g), they hold the misconception that parts should be the same shape. Future tasks are needed that focus on equivalence. For example, you can ask students to take a square and subdivide a picture themselves, as in Activity 15.5. ■
(a) (b) (c)
(e)
(d)
(f) (g)
Figure 15.5 Given a whole, find fractional parts.
CCSSM: 1.G.a.3; 2.G.a.3; 3.NF.a.
Finding (All the) Fair Shares Give students dot paper and ask them to enclose a region that lends itself to partitioning with a particular fractional part. For example, they might enclose a 3-by-6 rectangle if they are going to partition into thirds. ask students to find a way to partition the rectangle into thirds. then redraw another rectangle that is the same size whole and partition it a different way to show thirds. ask students to find a way to show thirds where the thirds are different shapes. See how many ways they can find. For eLLs, fraction parts sound like whole numbers (e.g., fourths and fours). Be sure to emphasize the th on the end and explicitly discuss the difference between four areas and a fourth of an area.
Activity 15.
engLisH Language Learners
Partitioning with Length models. The explanation of partitioning in the CCSS-M may be difficult to interpret. For example:
3.NF.A.2b: Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. (CCSSO, 2010, p. 24)
Put more simply, students need to be able to partition a number line into fourths and realize that each section is one-fourth:
1 4
0 4 2 4
3 4
Size of interval:^34
4 4 5 4
6 4
7 4
8 4 9 4
10 4
11 4
Number lines are difficult for students. Students may ignore the size of the interval (McNamara & Shaughnessy, 2010; Petit et al., 2010). Students can develop an understanding of the number line by folding paper strips. Provide examples where the shaded sections are in
Fractional Parts 351
Locating a fractional value on a number line is particularly challenging but very import- ant for students to be able to do. Shaughnessy (2011) found four common errors students make in working with the number line: They use incorrect notation, change the unit (whole), count the tick marks rather than the space between the marks, and count the ticks marks that appear without noticing any missing ones. This is evidence that we must use number lines more extensively in exploring fractions (most real-life contexts for fractions are measurement related). Partitioning is a strategy commonly used in Singapore (a high-performing country on international mathematics assessments) as a way to solve story problems. Consider the follow- ing story problem (Englard, 2010):
a nurse has 54 bandages. Of those, 29 are white and the rest are brown. how many of them are brown?
A bar diagram can be used as a tool for solving the problem. A student first partitions a strip into nine parts and then figures out the equal shares of bandages for each partition:
6 6 6 6 6 6 6 6 6
Did you notice that this is an example of fraction as operator? These types of partitioning tasks are good building blocks for multiplying with fractions.
Partitioning with set models. Students can partition sets of objects such as coins, counters, or baseball cards. When partitioning sets, students may confuse the number of counters in a share with the name of the share. In the example in Figure 15.4, the 12 counters are partitioned into 6 sets— sixths. Each share or part has two counters, but it is the number of shares that makes the partition show sixths. As with the other models, when the equal parts are not already figured out, then students may not see how to partition. Students seeing a picture of two cats and four dogs might think 24 are cats (Bamberger, Oberdorf, & Schultz‐Ferrell, 2010). Consider the following problem:
eloise has 6 trading cards, andre has 4 trading cards, and Lu has 2 trading cards. What fraction of the trading cards does Lu have?
A student who answers “one-third” is not thinking about equal shares but about the number of people with trading cards. Understanding that parts of a whole must be partitioned into equal‐sized parts across dif- ferent models is an important step in conceptualizing fractions and provides a foundation for exploring sharing and equivalence tasks, all of which are prerequisites to performing fraction operations (Cramer & Whitney, 2010).
sharing tasks
An important recommendation by the IES research team on ways to help students learn frac- tions states, “Build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts” (Siegler et al., 2010, p. 1). In particular, they suggest using equal-sharing activities to develop the concepts of fraction, equivalence, and ordering of frac- tions. See equal Sharing Stories expanded Lesson for a lesson designed for grades 1 or 2. Students in the early grades partition by thinking about fair shares (division). Sharing tasks are generally posed in the form of a simple story problem. Four friends are sharing two cookies. How many cookies will each friend get? Then problems become slightly more difficult: Suppose there are four cookies to be shared fairly among three children. How much will each child get? See how eduardo reasons about this sharing situation. Students initially perform sharing tasks by
352 Chapter 15 Developing Fraction Concepts
distributing items one at a time. When this process leaves leftover pieces, students must figure out how to subdivide so that every group (or person) gets a fair share. Contexts that lend to subdividing an area include cookies, brownies, sandwiches, pizzas, and so on. Pattern blocks are a good tool to focus on equal shares because each piece is not an equal share, so creating shapes with pattern blocks and asking about equal shares helps students focus on the important idea of fair (equal) shares. Ask students to create a “cookie” using the six different pattern block shapes and ask, “Can this cookie be shared fairly with 6 people?” (Ellington & Whit- enack, 2010). The answer is “no.” Then, ask students to build a cookie that can be shared fairly. Sharing brownies is a classic activity that focuses on partitioning to make equal shares (see, for example, Empson, 2002). Using concrete tools such as dough can make sharing accessible even to kindergartners (Cwikla, 2014).
Figure 15.6 Ten brownies shared with four children.
CCSSM: 1.G.a.3; 2.G.a.
Cookie Dough: Cut Me a Fair Share! Give students a ball of dough and a plastic knife. explain that they are going to be finding a way to share each group of cookies fairly with a group of students. Start with an example that is not too difficult. For example: Four friends want to share ten brownies so that each friend gets the same amount of brownies. how much will each friend get? Invite students to shape their dough into squares for brownies and then show how to share them fairly with four friends, using a paper knife if necessary. encourage students to share their ways of thinking about this problem. a strategy many students will use for this problem is to deal out two brownies to each child and then halve each of the remaining brownies (see Figure 15.6). then, offer a selection of other sharing tasks with different numbers of brownies and different number of sharers (see addi- tional examples below).
Activity 15.
“Kids and Cookies” is an excellent online tool for sharing cookies (both round and rectan- gular). Display the situations on an interactive whiteboard and ask for different ways to share fairly (you can begin with whole numbers and increase in difficulty) (Center for Technology and Teacher Education, n.d.). The relationship between the number of things to be shared and the number of sharers determines problem difficulty. Students’ initial strategies for sharing involve halving, so a good place to begin is with two, four, or even eight sharers. Here are some examples: 5 brownies shared with 2 children 5 brownies shared with 4 children 7 brownies shared with 4 children 2 brownies shared with 4 children 4 brownies shared with 8 children 3 brownies shared with 4 children These can be prepared as Brownie Sharing Cards. The last example, three brownies shared with four children, is more challenging because there are more sharers than items, and it involves more than just finding halves. One strategy is to partition each brownie into four parts and give each child one-fourth from each brownie—a total of three-fourths. Stu- dents (even adults) are surprised at the relationship between the problem and the answer. Felisha explains the fractional amount each of 5 children get when sharing 2 cookies, but loses track of what the whole is in determining each person’s share.
354 Chapter 15 Developing Fraction Concepts
Students need to understand that 34 , for example, can be thought of as a count of three parts called fourths (Post, Wachsmuth, Lesh, & Behr, 1985; Siebert & Gaskin, 2006; Tzur, 1999). If you know the kind of part you are counting, you can tell when you get to one, when you get to two, and so on. Students should be able to answer the question, “How many fifths are in one whole?” just as they know how many ones are in ten. However, the 2008 National Assessment of Education Progress (NAEP) results indicated that only 44 percent of students answered this question correctly (Rampey, Dion, & Donahue, 2009). This is the focus of Activities 15.9 through 15.11.
CCSSM: 3.NF.a.1; 3.NF.a.2a, b
More, Less, or Equal to One Whole Give students a collection of fractional parts (all the same-size pieces) and indicate the kind of fractional part they have. For example, if done with Cuisenaire rods, the collection might have seven light green rods/strips with a caption or note indicating “each piece is 18 .” the task is to decide if the collection is less than one whole, equal to one whole, or more than one whole. ask students to draw pictures or use symbols to explain their answer. as students count each collection of parts, discuss the relationship to one whole. ask questions that help students focus on the meaning of the numerator and denominator, such as “Why did we get almost two wholes with seven-fourths, and yet we don’t even have one whole with ten-twelfths?”
Activity 15.
After exploring Activity 15.9 with same-sized pieces, try Activity 15.10, which returns to using the pattern blocks to help students focus on the size of the parts, not the number of pieces or partitions (Champion & Wheeler, 2014; Ellington & Whitenack, 2010).
CCSSM: 3.NF.a.1; 3.NF.a.2a, b
Pattern Block Creatures ask students to build a pattern Block Creature that fits with a set of rules (a creature represents one-whole). these rules can begin with just stating a fractional quantity for a color, such as “the red trapezoid is one-fourth of the creature.” But, more constraints can be added to the rules. For example: the blue parallelogram is one-sixth of the creature. Use at least two colors to build your creature. the yellow hexagon is one-half of the creature. Use three colors to build your creature. Green triangles are one-third of your creature. Use four different colors to build your creature. after a student creates their creature, they can sketch the creature on paper and write the rule below it. Other stu- dents can critique the creature to see if it follows the rules it was given. alternatively, the student can write their rule as “the red trapezoid is ______ of my creature” and trade it with another student to see if they can figure out the fractional amount.
Activity 15.
CCSSM: 3.NF.a.1; 3.NF.a.3a, c
Calculator Fraction Counting Many calculators, like the tI-15, display fractions in correct fraction format and offer a choice of showing results as mixed numbers or simple fractions. ask students to type in a fraction (e.g., 14 ) and then + and the fraction again. to count, press 0 , , , repeating to get the number of fourths wanted. the display will show the counts by fourths and also the number of times that the key has been pressed. ask students questions such as the following: “how many fourths to get to 3?” “how many fifths to get to 2?” these can get increasingly more challenging: “how many fourths to get to 4^12 ?” “how many two-thirds to get to 6? estimate and then count by two-thirds on the calculator.” Students, particularly students with disabilities, should coordinate their counts with fraction models, adding a new fourths piece to the pile with each count.
Activity 15.
stuDents with sPeCiaL neeDs
Fractional Parts 355
The TI-15 display can be switched back and forth from mixed number to fractions, rein- forcing the equivalence of values such as 1^23 and 53. A variation on Activity 15.11 is to show students a mixed number such as 3^18 and ask how many counts of 18 on the calculator it will take to count that high. The students should try to stop at the correct number, 258 , before pressing the mixed‐number key. Iterating applies to all models but is particularly connected with length models because iteration is much like measuring. Consider that you have 2^12 yards of ribbon and are trying to figure out how many fourths you can cut. You can draw a strip and start counting (iterating) the fourths:
1 ft 1 ft ft (^1) – 2
Using a ribbon that is 14 of a yard long as a measuring tool, a student marks off ten fourths:
Students can participate in many tasks that involve iterating lengths, including ones where they are asked to find what the whole or unit is.
CCSSM: 3.NF.a.1; 3.NF.a.2a, b
A Whole Lot of Fun Use a Whole Lot of Fun activity page and a strip of paper like the one here:
tell students that this strip is three-fourths of one whole (unit). ask students to sketch strips of the other lengths on their paper (e.g., 52 ). You can repeat this activity by selecting other values for the starting amount and selecting different fractional values to sketch. a context, such as walking, is effective in helping students make sense of the situation. Be sure to use fractions less than and greater than 1 and mixed numbers.
Activity 15.
Notice that to solve the task in Activity 15.12, students first partition the piece into three sec- tions to find 14 and then iterate the 14 to find the other lengths. Iterating can be done with area models. Display some circular fractional pieces in groups as shown in Figure 15.9. For each collection, tell students what type of piece is being shown and simply count them together: “ One ‐fourth, two ‐fourths, three ‐fourths, four ‐fourths, five ‐ fourths.” Ask, “If we have five‐fourths, is that more than one whole, less than one whole, or the same as one whole?” To reinforce the piece size even more, you can slightly alter your language to say, “One one-fourth, two one-fourths, three one-fourths,” and so on. Iteration can also be done with set models. For example, show a collection of two‐color counters and ask questions such as, “If 5 counters is one‐fourth of the whole, how much of the whole is 15 counters?” These problems can be framed as engaging puzzles for students.
Fractional Parts 357
- The numerator is the counting number. It tells how many shares or parts we have. It tells how many have been counted.
- The denominator tells what size piece is being counted. For example, if there are four parts in a whole, then we are counting fourths. Making sense of symbols requires connections to visuals. Illustrating what 54 looks like in terms of pizzas (area), on a number line (length), or connected to filling bags with objects (set) will help students make sense of this value. One of the best things we can do for students is to empha- size equivalence and different ways to write fractional amounts.
Pause (^) & Reflect What fraction notation might you use for the visual here (the large square represents one unit)? ●
There are (at least) three ways to notate this quantity:
5 4
Do you think that students would be able to describe this quantity in all three ways? In the fourth National Assessment of Educational Progress (NAEP), fewer than half of the seventh graders assessed knew that 5^14 was the same as 5 + 14 (Kouba et al., 1988). Throughout this chapter we have been including fractions less than 1 and fractions greater than 1. This helps students develop understanding of fractions as values that come between whole numbers (or can be equivalent to whole numbers). Too often, students aren’t exposed to numbers equal to or greater than 1 (e.g., 66 , 52 or 4^14 ), so when these values are added into the mix (no pun intended!), students find them confusing. The term improper fraction is used to describe fractions that are greater than one, such as 52. This term can be a source of confusion as the word improper implies that this representation is not acceptable, which is not the case at all—in fact, it is often the preferred representation in algebra. Instead, try not to use this phrase and instead use “fraction” or “fraction greater than 1.” Note that the word improper is not used in the CCSS-M content standards. If you have counted fractional parts beyond a whole, as discussed in the previous section, your students already know how to write 136 or 135. Ask students to use a model to illustrate these values and find equivalent representations using wholes and fractions (mixed numbers). Using connecting cubes was the most effective way to help students see both forms for recording fractions greater than 1 (Neumer, 2007) (see Figure 15.12). Students identify one cube as the unit fraction ( 15 ) for the problem ( 125 ). They count out 12 fifths and build wholes. Conversely, they could start with the mixed number, build it, and find out how many total cubes (or fifths) were used. Repeated experiences in building and solving these tasks will help students to notice a pattern that actually explains the algorithm for moving between mixed numbers and fractions greater than 1. Help students move from physical models to mental images. Challenge students to figure out the two equivalent forms by just picturing the stacks in their heads. A good explanation for
If this rectangle is one-third, what could the whole look like? If this rectangle is three-fourths, draw a shape that could be the whole. If this rectangle is four-thirds, what rectangle could be the whole?
If purple is one-third, what rods are the whole? If dark green is two-thirds, what rod is the whole? If yellow is five-fourths, what rod is one whole?
If 4 counters are one-half of a set, how big is the set?
If 12 counters are three-fourths of a set, how many counters are in the full set? If 10 counters are five-halves of a set, how many counters are in one set?
Figure 15.11 Given the part and the fraction, find the whole.
standards for mathematical Practice mP1. Make sense of problems and persevere in solving them.
standards for mathematical Practice mP8. Look for and express regularity in repeated reasoning.
358 Chapter 15 Developing Fraction Concepts
314 might be that there are 4 fourths in one whole, so there are 8 fourths in two wholes and 12 fourths in three wholes. The extra fourth makes 13 fourths in all, or 134. (Note the iteration concept playing a role.) Do not push the standard algorithm (multiply the bot- tom by the whole number and add the top), as it can interfere with students making sense of the relationship between the two and their equivalence.
Complete Self-Check 15.3: Fractional parts
equivalent Fractions
As discussed in Chapter 14, equivalence is a critical but often poorly understood concept. This is particularly true with fraction equivalence. In the CCSS-M, fraction equivalence and com- parisons are emphasized in grade 3 and applied in grade 4 (and beyond) as students engage in computation with fractions. Students cannot be successful in fraction computation without a strong understanding of fraction equivalence.
Conceptual Focus on equivalence
Pause (^) & Reflect How do you know that 46 = 23? Before reading further, think of at least two different explanations. ●
Here are some possible answers to the preceding question:
1. They are the same because you can simplify 46 and get 23. 2. If you have a set of 6 items and you take 4 of them, that would be 46. But you can make the 6 into 3 groups, and the 4 would be 2 groups out of the 3 groups. That means it’s 23. 3. If you start with 23 , you can multiply the top and the bottom numbers by 2, and that will give you 46 , so they are equal. 4. If you had a square cut into 3 parts and you shaded 2, that would be 23 shaded. If you cut all 3 of these parts in half, that would be 4 parts shaded and 6 parts in all. That’s 46 , and it would be the same amount.
All of these answers are correct. But let’s think about what they tell us. Responses 2 and 4 are conceptual, although not as efficient. The procedural responses, 1 and 3, are efficient but do not indicate conceptual understanding. All students should eventually be able to write an equivalent fraction for a given fraction. At the same time, the procedures should never be taught or used until the students understand what the result means. Consider how different the procedure and the concept appear to be:
Whole (5 cubes) or 2 wholes and
(^5) – 5
(^5) – 5
(^2) – 5 (^12) — 5
(^2) – 5
Figure 15.12 Connecting cubes are used to represent the equivalence of 125 and 2^25.
