1. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 What are Fractions? Part 1 Common Core Leadership in Mathematics (CCLM^2) Tuesday, July 10, 2012 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.

2. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 CCSSM and Fractions

3. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 “Difficulty with learning fractions is pervasive and is an obstacle to further progress in mathematics and other domains dependent on mathematics, including algebra. It has also been linked to difficulties in adulthood, such as failure to understand medical regimens.” (“Foundations for Success” The National Mathematics Panel Final Report, 2008-page 28)

4. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Getting Started with Fractions

5. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 We are learning to … recognize fractions as numbers using models. emphasize partitioning in learning about fractions. understand and use unit fraction reasoning. We will be successful when we can … explain the mathematical content and language in 1.G.3, 2.G.3, 3.G.2 and provide examples of the mathematics and language. Learning Intentions and Success Criteria

6. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Standards 1.G.3, 2.G.3, and 3.G.2 Individually –Read the standards. –Highlight key words and key phrases. –Use your worksheet to draw a representation for each standard. Pairs –Compare highlighted ideas. –Using your worksheet discuss how ideas of partitioning are developing across grades 1 – 3.

7. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 1.G.3. Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. Fractions in Grade 1 (Geometry Domain)

8. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 2.G.3. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. Fractions in Grade 2 (Geometry Domain)

9. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 3.G.2. Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape. Fractions in Grade 3 (Geometry Domain)

10. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Attending to precision... What specific language was used to describe the ideas students need to understand about fractions? What language was left out? Debrief of Language

11. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Examining Partitioning....“early experiences with physically partitioning objects or sets of objects may be as important to a child’s development of fraction concepts as counting is to their development of whole number concepts” (Behr and Post, 1992)

12. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Partitioning as a Foundation for Fractions

13. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Equal Shares Four children are sharing 7 candy bars. How much candy will each child get if they share the candy bars equally? In groups of four: Use paper strips to solve the problem. Use the language of 1.G.3, 2.G.3, and 3.G.2, to describe the shares.

14. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Stages of Partitioning Read pgs. 71 – 75 from Focus on Fractions (through stages of partitioning). Highlight key phrases from the reading. * Star the important ideas. ? Question mark the confusing thoughts. Table group discussion: Summarize and clarify thoughts on partitioning. Work through problem number 1 on pages 77 & 78.

15. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Fraction Models: Area, Set, and Number Line Features of Fraction Models How the whole is defined How “equal parts” are defined and What the fraction indicates

16. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Slate Work Take a slate and divide it in thirds. On each 1/3 draw a model: an area model, a set model, and a number line. Discuss the features of each model.

17. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Features of Models What is the whole? How are equal parts defined? What does the fraction indicate? Area Model The whole is determined by the area of a defined region Equal area The part covered of whole unit area Set Model The whole is determined by definition (of what is in the set) Equal number of objects The count of objects in the subset of the defined set of objects. Number Line Unit of distance or length (continuous) Equal distance The location of a point in relation to the distance from zero with regard to the defined unit.

18. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Standard 3.NF.1. 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. Read the first part of the standard. In pairs, practice using the language of this standard to describe your unit fractions.

19. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 We are learning to … recognize fractions as numbers using models. emphasize partitioning in learning about fractions. understand and use unit fraction reasoning. We will be successful when we can … explain the mathematical content and language in 1.G.3, 2.G.3, 3.G.2 and provide examples of the mathematics and language. Learning Intentions and Success Criteria