DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)

Patty Coleman CCLM Project 2

July 15, 2011

CCSSM Interpretation Guide

Part 1. Standard 5.NF.1: Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 5/12 = 23/12. (in general, a/b + c/d = (ad + bc)/bd.) Domain: Number and Operations - Fractions Cluster: Addition of Fractions Grade Level: 5 Part 2. Interpretation of the standard Students will be able to find an equivalent fraction before adding and subtracting fractions (including mixed numbers) with unlike denominators. Students will need to build upon their understanding of equivalence from third and fourth grades. Students will understand that saying two fractions are equivalent is saying that the two fractions are different names for the same number. Using repeated practice with examples, students will see the pattern that leads to the strategy of multiplying or dividing both the numerator and the denominator by the same number to find equivalent fractions.

DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)

DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)

Part 3: School Mathematics Textbook Program Textbook Development: Our elementary textbook program is Everyday Math. Students begin to see equivalent fractions in third grade. The program uses equivalents for the fraction one half. ½ = 2/4 = 3/6 = 4/8 = 5/10 Students use fraction cards to see equivalent fractions in pictures and symbols. The students use the fraction cards in games to become familiar with them. There are enrichment activities that stretch the thinking to see the pattern, “find the rule” for equivalent fractions. Beginning in fourth grade, students develop a rule for finding equivalent fractions. The text uses area models to show that when parts are partitioned the whole is broken into more pieces but the shaded section does not change. Students can then see that there are different names – equivalent names – for the same number. These skills are supported through the use of games. The fifth grade textbooks spend more time on the concept of equivalent fractions. In chapter 5, they use fraction stick charts, fraction stick models and partitioning area models to review equivalent fractions. Students are prompted to study the patterns to make a rule for finding an equivalent fraction. In Chapter 6 they go on to review and find equivalent fractions in a number of ways. Using the clock model (1/4 of an hour = 15/60). Using number strip fractions: Numerator 7 14 21 28 35 42 49 56 63 70 Denominator8 16 24 32 40 48 56 64 72 80 The text then presents addition and subtraction problems in which students need to find a common denominator before solving. The text encourages students to share their strategies for finding the common denominator. There are also a number of games that reinforce the concept of finding and using common denominators. In Unit 8, common denominators are again

DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)

reviewed. Students are asked to share strategies for finding common denominators. Students then are given problems to rename fractions with common denominators before solving the addition or subtraction. The sixth grade uses the Connected Math 2 program. This program uses investigations, models, real life applications, and a lot of discussion. It does not give the algorithm for finding equivalent fractions. It prepares students to figure out how to add and subtract fractions by emphasizing flexibility in finding equivalent fractions. Class discussions make strategies explicit and efficient so students can create a general algorithm. They stress that students need to understand why it is necessary to rename fractions when adding and subtraction and the role that equivalent fractions play in that process. Conclusions: Fraction addition and subtraction concepts build from, and are dependent upon, foundational part-to-whole, equivalence and magnitude ideas (A Focus on Fractions p. 145). The Everyday Math program does begin to develop the skills needed to understand equivalent fractions beginning in third grade with simple fractions (standard 3.NF.3). They do expand on this idea in fourth grade with finding equivalent fractions (4.NF.1 and 4.NF.2). Where I think Everyday Math veers away from aligning with the standards is in fifth and sixth grade. The material in both of those grades spends a lot of time with understanding equivalent fractions which should be a skill learned in the previous grades. The fifth graders should know how to find equivalent fractions from their work in fourth grade and should be able to move on to the process of adding and subtracting fractions with unlike denominators by first finding equivalent fractions. It looks to me like this whole process is then repeated in sixth grade Connected Math 2. There is not a standard that deals with adding and subtracting fractions for sixth grade. Another weakness I see in the Everyday Math program is that they do not give enough time for exploration of the visual models. I like that they do show various models and use many different tools to show how fractions are equivalent, but it is done quickly over a short period of time. It does not provide enough time for students to really use and understand the models at a deeper level. Suggestions: Teachers need to become comfortable knowing that a standard has been taught in a previous grade so as not to repeat the learning in other

DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)

grades. We also need to be sure to develop the math in each grade level that will be needed to reach the standards in the upcoming grades. Third grade teachers need to continue to expose students to the idea of equivalence with simple fractions (3NF.3) using varied visual models (number lines, area models, set models). Fourth grade teachers need to build upon that to continue to recognize and generate equivalent fractions in more complex forms. They will also start the process of adding and subtracting fractions with like denominators in fourth grade. These are all the standards 4.NF. 1, 2 and 3. Fifth grade teachers will then build on those two skills of finding common denominators and adding/subtracting fractions to solve the problems with unlike denominators. Part 4. Formative Assessment Task The task should reveal student thinking on finding an equivalent fraction before solving. Asking for words and pictures gets the student to show their understanding of the underlying idea of equivalence and also their reasoning and math vocabulary. Anthony’s shopping list Grapes yogurt Bread eggs ½ pound ham ¾ pound roast beef How much lunchmeat does Anthony plan to buy? Use pictures and words to describe your thinking.

DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-Milwaukee. (07.15.2011)