Eddy Ringsby
CCLM^2 Project
Summer 2012
DRAFT DOCUMENT. This material was developed as part of the
Leadership for the Common Core in Mathematics (CCLM^2) project at the University of Wisconsin-Milwaukee.
Part I. Standard Grade: 5 Domain: NF – Number and Operations – Fractions Cluster: Use equivalent fractions as a strategy to add and subtract fractions. Standard: Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. In general, a/b + c/d = (ab + bc) / bd. Part II. Explanation and Examples of the Standard Standard Add and subtract fractions with unlike denominators Add fractions with unlike denominators and Subtract fractions with unlike denominators What are the students expected to be able to do? Prior Knowledge – 3.NF.1 – Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. 3.NF.3 – Explain equivalence of fractions in special cases. 3.NF.3b – Recognize and generate simple equivalent fractions, e.g. 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. 4.NF.3a – Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 4.NF.3b – Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, eg., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 2/8 + 1/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. Replace given fractions with equivalent fractions Create equivalent fractions with like denominators to allow for adding and/or subtracting 4.NF.3c – Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. Determine common denominator of two fractions by finding the product of both numbers Create equivalent fractions to add or subtract easily Example (2/3 + 3/5) You would multiply 3 X 5 and get a product of 15. Then you would need to determine equivalent fractions by using visual fraction models, sets of fractions, number lines, or area models. Finally, when students are visually comprehending the process, the use of an algorithm can be introduced and taught as another efficient and effective method. 2/3 is equivalent to 10/15, and 3/5 is equivalent to 9/15, so 2/3 + 3/5 = 10/15 + 9/15 Algorithm that can be used: (a/b + c/d = (ad + bc) / bd. To produce an equivalent sum or difference of fractions with like denominators. Add fractions with like denominators. Add or subtract fractions with like denominators. Replace equivalent sum or difference with simplified equivalent fraction through the use of visual fraction models, sets of fractions, number lines, area models, and/or an understood process of simplifying created sum or difference. Do not utilize disconnected process without understood meaning of visual representation of 19/15 is understood as being equivalent to 1 4/15. Then simplify, if applicable to situation. 10/15 + 9/15 = 19/15 Part III. School Mathematics Textbook Program (a) Textbook Development a. Grade 3 – an introduction to identifying and representing fractions and equivalent fractions and mixed numbers using visual models, sets of models, and number lines. There is very little practice with adding and subtracting fractions, and there’s no practice with adding or subtracting fractions with unlike denominators. b. Grade 4, Chapter 9 -­‐ There is a review of what’s covered in grade 3, along with an introduction of creating own visuals to compare fractions. In addition, they show how to create equivalent fractions when using fraction strips, but there are very little other visual representations used. Quickly afterwards, there’s an expectation that students use the algorithm of creating equivalent fractions. In practice, there’s very little meaning behind the exercises, and it’s not clear whether or not students would make the visual connections necessary for future understanding. A day later, students are then shown how to simply fractions by the use of dividing, with little to no connection to visual meaning. When simplifying mixed numbers, once again, students are asked to visualize the process through the use of fraction strips until they can learn how to duplicate the process using division and multiplication. c. Grade 4, Chapter 10 – There is one lesson visualizing the addition of fractions with unlike denominators using fraction strips. There is no discussion of the meaning behind the unit fraction or the non-­‐unit fraction, but it’s more of a way of somewhat understanding the proximity of the fraction to 0, 1/2, or 1 whole. There is another lesson on adding fractions with like denominators: first, visualizing through the use of fraction strips; second, through addition of the numerators; and lastly, through simplification of fractions not in simplest form. At the end of the unit, there is one lesson on adding fractions and one with subtracting fractions with unlike denominators, and again, students only use models with fraction strips for a bit and then the rest with paper and pencil changing of fractions into equivalent fractions. All along, there’s no connection to situations or real-­‐life problems. It’s simply rote practice if students understand the process. d. Grade 5, Chapter 8 – There is very little review of basic fraction understanding, and the chapter gets right into adding and subtracting fractions with like denominators and then unlike denominators. Like grades 3 and 4, there is no other strategy taught besides fraction strips, and doing rote steps to answers that aren’t connected to any contextual situations appears to be the dominant way students are expected to learn about how to add and subtract fractions. However, now in grade 5, finding the Least Common Denominator (LCD) becomes more difficult, and students are given two options to find the LCD. Either choose to list the multiples of each denominator and seek the LCD, or simply multiply the two denominators to use a common denominator in order to add or subtract the two fractions. (b) Conclusions and Suggestions a. Conclusions – The Scott Foresman textbook does appear to have ample exposures to learning about fractions, how to add and subtract fractions with like denominators, and lastly with adding fractions with unlike denominators. At the same time, the introduction and practicing of when to learn and master each of these skills mostly matches up to when the Common Core suggests that a student should become able to do these skills. However, with each lesson, there appears to be little variety in how students are to view and learn about fractions. In fact, other than a couple of instances, fractions are always taught and discussed using fractions strips, always of the same size and never created are manipulated by the students for internalization. Fraction strips are commercially created, and students never think for themselves about what size unit fractions are or how they compare to the whole, outside of this simple manipulative. There is a great deal of practice with all operations, but there is little, if any, connection to real-­‐world or contextual situations to internalize why fractions are even important. All in all, it feels that fractions are mathematics, but they’re disconnected from the remainder of what’s taught or known, and I feel it’s quite likely that students (without teacher scaffolding) may not truly appreciate fractions for what they are: a constant comparison to a whole or to another fractions. b. Suggestions – This textbook series does give one way of looking at fractions, through fraction strips. I suggest that students experience fractions in various formats. Yes, fraction strips could be used, but fractions could be learned in so many other formats: in sets, through drawings, with the area model, through the number line, in role play, as well as other applicable experiences. In addition, the learning of fractions needs to have some contextual connectedness to its practice, and this textbook series is lacking greatly in this. Fractions are simply fractions, and that way of thinking cannot allow students to internalize and apply fractions to how they think. Why would a student want to use fractions if it meant that they could simply know what fractions are? There needs to be some contextual situations in all learning experiences throughout all the grade levels, so that students begin to view the knowing, comparing, adding, and subtracting of fractions is truly something they see relevance in and may begin to find some reasoning behind its knowledge. Rather than learning about fraction strips, learning how to do something, students need to see how fractions are just another aspect of how we think about all objects, sets of objects, changes in what we already know, or in relation to sizes and amounts of those things we can already visualize. In retrospect, the designers of this textbook have provided numerous practice opportunities. What they’re missing is the connection to why one must learn fractions and the variety in how we should learn about why and how to add fractions, and later when adding and subtracting fractions with unlike denominators. If practiced in context over a few years with relevance to what they know, students would begin to see the importance of why and how to use fractions. They’d also begin to start thinking about the world in fractions terms. As this practice is repeated, adding fractions with unlike denominators just becomes another aspect of this real-­‐life necessity. So, as a teacher using this series, we must add these other opportunities and connect the learning to real situational applicable experiences. With these additions, students can also realize the importance and applicability to the learning of adding and subtracting fractions, even with unlike denominators.