Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Fractions: Teaching with Understanding Part 2 This material was developed for use by participants in the Common Core Leadership in Mathematics (CCLM^2) project through the University of Wisconsin-Milwaukee. Use by school district personnel to support learning of its teachers and staff is permitted provided appropriate acknowledgement of its source. Use by others is prohibited except by prior written permission.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 We are learning to: Understand and use unit fraction reasoning. Use reasoning strategies to order and compare fractions. Read and interpret the cluster of CCSS standards related to fractions. Success Criteria: Explain the mathematical content and language in 3.NF.1, 3.NF.2 and 3.NF.3, 4.NF.2 and provide examples of the mathematics and language. Learning Intentions and Success Criteria

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Fraction Strips

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Making Fraction Strips White: whole Green:halves, fourths, eighths Yellow:thirds, sixths, ninths ?:twelfths Note relationships among the fractions as you fold. Remember – no labels.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Benefits of Fractions Strips Why is it important for students to fold their own fraction strips? How does the “cognitive demand” change when you provide prepared fraction strips? How might not labeling fraction strips with numerals support developing fraction knowledge? How this tool supports 1.G.3, 2.G.3, and 3.G.2?

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012

CCSSM Focus on Unit Fractions

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Standard 3.NF.1 Unit Fractions Fold each fraction strip to show only one “unit” of each strip. Arrange these unit fractions from largest to smallest. What are some observations you can make about unit fractions?

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Fractions Composed of Unit Fractions Fold your fraction strip to show ¾. How do you see this fraction as ‘unit fractions’?

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Looking at a Whole Arrange the open fraction strips in front of you. Look at the thirds strip. How do you see the number 1 on this strip using unit fractions? In pairs, practice stating the relationship between the whole and the number of unit fractions in that whole (e.g., 3/3 is three parts of size 1/3).

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Standard 3.NF.1. Non-unit Fractions In pairs, practice using the language of the standard to describe non-unit fractions.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute NF. 1 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. How do you make sense of the language in this standard connected to the previous activities?

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Why focus on unit fractions? 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. How will you explain the meaning of standard 3.NF.1 to colleagues in your schools? What conjectures can you make as to why the CCSSM is promoting this unit-fraction approach?

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Number Line Model

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Number Line Model What do you know about a number line that goes from 0 to 4? 0 4

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Sequential & Proportional Strategies Draw two number lines from 0 to 4. Use whole numbers & fractions to show parts on the number line. # line 1 show sequential reasoning # line 2 show proportional reasoning Is it harder when you have to mark fractions? Why ?

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 On your slate draw another number line from 0 to 2 that shows thirds. Mark 5/3 on your number line. Explain to your shoulder partner how you marked 5/3. 01

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 NF Progressions Document What are the CCSSM expectations for number lines? Read: “The Number Line and Number Line Diagrams” on page 3. Read: Standard 3.NF.2, parts a and b. With a partner, explain this standard to each other while referring to your drawing.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Standard 3.NF.2 3.NF.2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. a.Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. b.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.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Explain Ken’s thinking?

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Explain Judy’s thinking?

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 On your slate, draw a number line from 0 to 1. Use proportional thinking to place and on the number line.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Equivalency

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Equivalency Place the whole fraction strip that represents 0 to 1 on a sheet of paper. Draw a line labeling 0 and 1. Lay out your fraction strips, one at a time, and make a tally mark on the line you drew. Write the fractions below the tally mark. Look for patterns to help you decide if two fractions are equivalent.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Which fractions are equivalent? How do you know?

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 NF Progressions Document Number off by twos: ones study Grade 3, twos study Grade 4. Grade 3 Equivalent Fractions Read pp. 3-4; study margin notes and diagrams. Study standard 3.NF.3. Grade 4: Equivalent Fractions Read p. 5; study margin notes and diagrams. With your shoulder partner, identify what distinguishes student learning at each grade.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute NF.3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. a.Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. b.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. c.Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Standard 3.NF.3, Parts a, b, & c

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. Standard 4.NF.1

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Comparing Fractions

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Compare Fractions by Reasoning about their Size More of the same-size parts. Same number of parts but different sizes. More or less than one-half or one whole. Distance from one-half or one whole (residual strategy–What’s missing?)

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Standards 3.NF.3d and 4.NF.2 3.NF.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. 4.NF.2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Standard 3NF3d & 4NF2 On your slate, provide an example of comparing fractions as described in these standards. What is the difference between the two standards? Share with your partner.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Ordering Fractions #1 1.1/4, 1/2, 1/9, 1/5, 1/ /15, 3/9, 3/4, 3/5, 3/ /25, 7/18, 8/15, 7/8

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Ordering Fractions #2 1)Write each fraction on a post it note. 2)Write 0, ½, 1, and 1 ½ on a post it note and place them on the number line as benchmark fractions. 3)Taking turns, each person: Places one fraction on the number line and explains their reasoning about the size of the fraction.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Ordering Fractions 3/83/106/5 7/477/10025/26 7/1513/2417/12 8/3 16/17 5/3

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Extension of Unit Fraction Reasoning Jason hiked 3/7 of the way around Devil’s Lake. Jenny hiked 3/5 of the way around the lake. Who hiked the farthest?  Use fraction strips and reasoning to explain your answer to this question.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 The Garden Problem Jim and Sarah each have a garden. The gardens are the same size. 5/6 of Jim’s garden is planted with corn. 7/8 of Sarah’s garden is planted with corn. Who has planted more corn in their garden?  Use fraction strips and reasoning to explain your answer to this question.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Reflect Summarize how you used reasoning strategies to compare and order fractions based on their size.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Translating the Standards to Classroom Practice Discuss the progression of the standards we did today. Is the progression logical? Discuss how the standards effect classroom practice. What will need to change?

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Let’s Rethink the Day We know we are successful when we can…  Explain the mathematical content and language in 3.NF.1, 3.NF.2 and 3.NF.3, 4.NF.2 and provide examples of the mathematics and language.