TOP OR BOTTOM: WHICH ONE MATTERS?

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1 Chapter TOP OR BOTTOM: WHICH ONE MATTERS? Helping Students Reason About Generalizations Regarding Numerators and Denominators Strategy # Provide opportunities for students to investigate, assess, and refine mathematical rules and generalizations. What s the Math?... What s the Research?... Classroom Activities. Number Line Activities with Cuisenaire Rods...3. Is It Always True?... From Principles and Standards for School Mathematics Number and Operations Standard: Grades 3 5: Through the study of various meanings and models of fractions how fractions are related to each other and to the unit whole and how they are representedstudents can gain facility in comparing fractions, often by using benchmarks such as or. 5 Beyond Pizzas & Pies Copyright by Math Solutions

3 students to discuss the question with their table groups and to try to come up with a generalization about fraction sizes in general. Well, Dominique began, I think you could say that if the number of pieces the thing is cut into is smaller, the fraction is larger. And if the number of pieces the thing is cut into is larger, the fraction is smaller. Thank you, Dominique. Can someone restate Dominique s idea using the math word that means the number of pieces the thing is cut into? asked Ms. Alvarez. Noticing Noah s hand, Ms. Alvarez called on him. Here goes. The smaller the denominator, the larger the fraction. The larger the denominator, the smaller the fraction. What do you think about that? Ms. Alvarez asked the class. Several students nodded enthusiastically and many gave Noah a thumbs-up. Let s all say Noah s idea together: The smaller the denominator, the larger the fraction. The larger the denominator, the smaller the fraction. Ms. Alvarez recorded Noah s conjecture while chanting along with the class, happy that they seemed to understand this important idea. A few days later Ms. Alvarez was looking over the students homework and was dismayed to see that in response to the prompt Circle the larger fraction: 3 many of them had indicated that was greater than 3. When Ms. Alvarez asked Dominique to explain her answer during homework check, Dominique replied, Remember you taught us last weekthe smaller the denominator, the larger the fraction. The larger the denominator, the smaller the fraction. Two is smaller than four, so one-half has got to be bigger than three-fourths. As several students nodded their agreement with Dominique, Ms. Alvarez realized they had overgeneralized this important idea. She also realized her students were thinking only of the denominators of the fractions they were comparing, not realizing that in order to understand the value of a fraction they needed to consider both the numerator and denominator as well as their relationship to each other. Ms. Alvarez decided she would need to review this with her students and help them understand that both parts of a fraction are equally important. Top or Bottom: Which One Matters? Beyond Pizzas & Pies Copyright by Math Solutions

4 What s the Math? When comparing fractions, students need to consider the size of the wholes and interpret each fraction as a single number defined by the relationship between the numerator and the denominator. Fractions can be compared in several different ways. Finding a common denominator or cross-multiplying to compare are two common approaches that students learn in school, but these approaches do not explicitly require a consideration of the size of the fractions. As Van de Walle, Karp, and Bay-Williams (9) note, other strategies include: More of the same-sized parts (same denominators); Same number of parts but parts of different sizes (same numerators); More and less than one-half or one whole; and Closeness to one-half or one whole. These strategies may support reasoning about the size of the fractions; however, they may not work with all fractions. Here are examples of these and other comparison strategies: Finding a Common Denominator , so 5 6 Cross-Multiplication , so 5 6 More of the Same-Size Parts 3 3 Beyond Pizzas and Pies: Essential Strategies for Supporting Fraction Sense Beyond Pizzas & Pies Copyright by Math Solutions

5 Same Number of Parts But Parts of Different Sizes Fourths are larger then eighths, so 3 3 Closeness to One Whole is less than. 5 6 is 6 less than. 6 is greater than, so is closer to is away from 5 6 is away from Distance from Zero One could locate both numbers on a number line to determine distance from (alternatively, a single number line could be used): is further away from 5 is closer to Size of the Shaded Area 5 6 Top or Bottom: Which One Matters? 9 Beyond Pizzas & Pies Copyright by Math Solutions

7 Circle the larger fraction: Explain your answer. Figure The smaller the number the bigger the pieces. provided a more generalized rationale, that the smaller the number the bigger the pieces. (See Figure.) This finding indicates that students may be responding to only one aspect of the written fraction, in this case the denominator, instead of attending to both the numerator and denominator as necessary components of the number. This is like looking at a number such as 3, where each of the digits provides crucial information about the value of the number, but the digits have to be considered in relation to each other to provide the whole story. If students focus only on the numerator or denominator, they are not considering the value of the fraction. Even among students who circled the correct answer ( ) some indicated that their reason for choosing was based on the value of the numerator only. (See Figure 5.) Still other students paid attention to both the numerators and denominators when comparing the fractions, but did not consider the relationship between them. For instance, a subset of students indicated that is the larger fraction because is greater than 5 and is greater than 6. (See Figure 6.) These types of responses present compelling evidence that many Circle the larger fraction: Explain your answer. Figure 5 This one is bigger because there are more pieces. Top or Bottom: Which One Matters? Beyond Pizzas & Pies Copyright by Math Solutions

8 Circle the larger fraction: Explain your answer. Figure 6 Since is bigger than 6 and is bigger than five, is bigger. students are not attending to the relationship between numerators and denominators when asked to reason about fractional values and make comparisons between two or more fractions. This provides further confirmation of the claims made by Nancy Mack, a professor at Grand Valley State University, that students often rely on whole number strategies when solving fraction problems (). When asked to compare two fractions such as and 3, many of Ms. Alvarez s students had overgeneralized the important mathematical idea that fractions with smaller denominators are larger than those with larger denominators. What her students failed to realize was that this generalization or rule had developed from a situation where the numerators of both fractions were the same ( versus 6 ). When comparing and 3, many students did not realize or remember that both the numerators and denominators of the fractions needed to be considered when comparing their values. When students overgeneralize in this manner they may not be thinking of fractions as numbers, but only as groups of items or parts of shaded areas. The following classroom activities will help students increase their understanding of fractions as numbers and increase their ability to develop and apply appropriate generalizations. Beyond Pizzas and Pies: Essential Strategies for Supporting Fraction Sense Beyond Pizzas & Pies Copyright by Math Solutions

9 Classroom Activities Activity. Number Line Activities with Cuisenaire Rods Overview In these activities students use the number line to discover that the rule the smaller the denominator, the larger the fraction holds true only when the numerators are the same because is indeed larger than, and is larger than. In addition, is larger than, and is larger than. Students can also see, however, that when the numerators are different, the comparison of any two fractions is more complex. Knowing the denominator of a fraction is a very important aspect of understanding its value, but it is only part of the information that is necessary. In much the same way, knowing only the numerator of a fraction is not sufficient when it comes to understanding the value.. Pass out the -cm Number Lines recording sheets (Reproducible a). Using -cm as the unit interval allows students to use the rods to partition the unit into twelfths, sixths, fourths, thirds, and halves. Materials -cm Number Lines recording sheet, copy per student (Reproducible a) Cuisenaire rods, set per pair of students Manipulative Note Cuisenaire rods are wooden or plastic blocks that range in length from to centimeters. Each rod of a given length is the same color. That is, all of the -cm rods are white, all of the -cm rods are red, all of the 3-cm rods are light green, and so on. Figure -cm Number Lines (Reproducible a) Top or Bottom: Which One Matters? 3 Beyond Pizzas & Pies Copyright by Math Solutions

10 . Instruct students to find the rod that allows them to partition the units on the first number line into halves. Have students use the 6-cm dark green rod to mark on the number line: Dark Green Dark Green 3. Next, have students find the rod that will enable them to partition the next number line in thirds: 3 3 Purple Purple Purple. Continue in this manner until students have partitioned and labeled the final three number lines into fourths, sixths, and twelfths: Teaching Note See Chapter 3 for further discussion of helping students understand fractions as numbers. 5. After students have partitioned and labeled their number lines, ask for observations about the numbers they have written. It is important that you begin to refer to fractions as numbers, so that students become comfortable with this language as well. 6. Pose one or more of the following questions to help students reason about fractions as numbers. Beyond Pizzas and Pies: Essential Strategies for Supporting Fraction Sense Beyond Pizzas & Pies Copyright by Math Solutions

11 Questions to Help Students Reason About Fractions as Numbers What number is halfway between zero and one? What number is halfway between zero and one-half? What other numbers are the same as one-half? What number is one-fourth more than one-half? One-sixth more than one-half? What number is one-sixth less than one? What number is one-third more than one? What number is halfway between one-twelfth and three-twelfths? Some students may initially be surprised that there are numbers between and. Realizing that lies between and on the number line reinforces the relationship between halves and fourths. Students may initially say there are several numbers here:, 3 6, and 6. This is an excellent opportunity to introduce the idea that although these look like different numbers, they are actually different ways to name the same number, much like one hundred can also be called ten tens. This is also an opportunity to discuss what names for the same number have in common. This question can help students begin to reason about relative value of different fractions and compute without the need for converting to numbers with common denominators. This question encourages students to compare fractions to the unit. This question exposes students to fractions greater than one and can support their understanding that 3 is the same as 3. This question provides another chance for students to encounter equivalents. They can also begin to reason why there is no sixth equivalent to or 3 (or 5,, 9, or ). (continued) Top or Bottom: Which One Matters? 5 Beyond Pizzas & Pies Copyright by Math Solutions

12 Which number is closest to zero? Which number is closest to one? What would you call a number halfway between zero and one-twelfth? This provides another example of when the larger the denominator, the smaller the fraction is true. This can help students see that knowing both the numerator and the denominator is necessary to understanding a fraction s value. It can also provide a very reliable and frequently sufficient way to compare fractions, without needing to find common denominators and create equivalent fractions. This question asks students to extend their understanding and provides a foundation for helping them reason about fraction multiplication, that is, Why does? Since the materials students used for this activity do not include rods that enable partitioning the number line into twenty-fourths, this last question may initially seem like a trick question to some students. By posing this question you can encourage students to look deeply at the relationships between the other fractions they have been working with. The ensuing discussion can provide many opportunities for students limited understanding to surface and for students to refine their thinking about fraction relationships. Any of these questions can be posed during whole-class discussions, given to groups to investigate and report back, or used for assessment purposes. Allow students to discuss their thinking and provide a rationale for their responses. These questions are not designed to be answered with oneword responses and no discussion. Any one of the questions can serve as a springboard for an entire lesson. 6 Beyond Pizzas and Pies: Essential Strategies for Supporting Fraction Sense Beyond Pizzas & Pies Copyright by Math Solutions

13 Assessment Opportunity As your students are working with the two-unit number line, don t be surprised if some of them initially identify the 6-cm dark green rods as fourths. This is very common! In fact, these students are correct, in that one 6-cm rod is of the distance from to ( cm). What they are not considering, however, is that the unit they are partitioning is the distance between and, and in that case each 6-cm rod is equal to. Activity. Extension Working with the Two-Unit Number Line Overview To provide opportunities for your students to work with fractions between and, use the Two-Unit Number Lines recording sheets that are numbered from to. (See Figure [Reproducibles b c].). Pass out the Two-Unit Number Lines (Reproducibles b c). Have your students use the Cuisenaire rods as in Activity.: Number Line Activities with Cuisenaire Rods to partition the number lines into halves, thirds, fourths, sixths, and twelfths.. Ask students questions as before, this time providing opportunities for them to reason about fractions greater than one as well as those less than one. Figure Two-Unit Number Lines (Reproducibles b c) Top or Bottom: Which One Matters? Beyond Pizzas & Pies Copyright by Math Solutions

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