Algebra Forum IV San Jose, CA May 22, 2012 Compiled and Presented by April CherringtonJoan Easterday Region 5Region 1 Susie W. Hakansson, Ph.D. California Mathematics Project Teaching and Learning Fractions with Conceptual Understanding

2 Fractions from a number line approach represents a shift in thinking about fractions, moving beyond part-whole representations to thinking of a fraction as a point on the number line. Included in this session will be the following: rationale, comparing and ordering, and justification. Description

3 (20 minutes) Introduction (90 minutes) Breakout session (20 minutes) Reflection Outline for Today

4 CaCCSS-M Task Force Conceptual understanding Order problems Cognitive level Language issues Why number line? Fraction progressions Standards for Mathematical Practice Challenges students face Overview of break out session Introduction

5 Greisy Winicki-Landman, Chair Nadine Bezuk April Cherrington Pat Duckhorn Joan Easterday Doreen Heath Lance Fractions Task Force Pam Hutchison Natalie Mejia Gregorio Ponce Debbie Stetson Kathlan Latimer

6 “… almost all teachers are placing a lower priority on student understanding in recent years, ….” “… the sort of high quality PD that an really affect teachers in their ability to produce students who understand is very, very difficult to do, and very few people have much clue about how to do it.” Scott Farrand Demands of CaCCSS-M

7 8 / 15 > 1 / 2 (?) 7 / 22 > 1 / 3 (?) 6 / 11 > 7 / 15 (?) 7 / 8 > 8 / 9 (?) Fraction Sense: Comparing Solve these problems mentally without using algorithms. Justify your thinking.

Cognitive Demand Spectrum MemorizationProcedures Without Connections to understanding, meaning, or concepts Procedures With Connections to understanding, meaning, or concepts Doing Mathematics Tasks that require memorized procedures in routine ways Tasks that require engagement with concepts, and stimulate students to make connections to meaning, representation, and other mathematical ideas

The soldier decided to desert his dessert in the desert. Upon seeing the tear in the painting, I shed a tear. After a number of injections, my jaw got number. A minute is a minute part of a day. 9 Why Is English So Hard?

There is no egg in eggplant and no ham in hamburger. How can a slim chance and a fat chance be the same, while a wise man and a wise guy are opposites? Did you say thirty or thirteen? Did you say two hundred or two hundredths? Did you say fifty or sixty? 10 Why Is English So Hard?

Math WordMeaning 1 (Math)Meaning 2 SolutionThe answerTwo or more substances mixed together ExerciseMath problems to solvePhysical movement to stay fit ProductThe answer when you multiply two or more numbers Something made by humans or machines ExpressionMath statement with numbers and/or variables To show emotion Dual Meaning Words

12 9 th and 10 th graders’ responses Tom had 5 apples. He ate 2 of them. How many apples were left? A. 10 B. 7 C. 5 D. 3 ( 100% ) Guinevere had 5 pomegranates. She ate 2 of them. How many pomegranates were left? A. 10 ( 22% ) B. 7 ( 24% ) C. 5 ( 23% ) D. 3 ( 31% ) The Guinevere Effect

13 Access prior knowledge Frontload language Build on background knowledge Extend language Be aware of multiple meanings of words Have students Think, Ink, Pair, Share (TIPS) Key Strategies for English Learners

14 Teachers learn to amplify and enrich--rather than simplify--the language of the classroom, giving students more opportunities to learn the concepts involved. Aída Walqui, Teacher Quality Initiative

15 “ Hung-Hsi Wu attempts to bring coherence to the teaching and learning of fractions by beginning with the definition of a fraction as the length on the number line (1998). This approach eliminates the ‘conceptual discontinuity’ (2002) encountered moving from work with whole numbers to fractions; it also brings coherence to the various meanings of fractions and allows for both conceptual work to operations on fractions (2008). Wu asserted that ‘The number line is to fractions what one’s fingers are to whole numbers...” Why Number Line?

16 Basic Assumptions about the Number Line and Its Use Numbers go on infinitely in both directions. On a conventional horizontal number line, the numbers increase from left to right. The numbers to the right of zero are the positive numbers, and those to the left are negative numbers. 0 is not positive nor negative.

17 Basic Assumptions about the Number Line and Its Use Using the number line, there are basically two types of tasks: Given a point on the number line, assign a number to it (its coordinate) Given a number, place it as a point on the number line. The length of the interval from 0 to 1 is called the unit and it determines the distance between every pair of consecutive integers on the line.

18 Basic Assumptions about the Number Line and Its Use Fractions can be placed on the number line by partitioning the length from 0 to 1 into d equal parts. One of these parts has the length 1/ d ; n of those parts has the length n/d. The fraction n/d is the point at the end of the segment of length n/d. Given the unit interval, each point on the number line can be associated to infinitely many fractions: the name will depend on the partition chosen. All of these fractions are equivalent.

19 WHY THE Number Line? It serves as a visual/physical model to represent the counting numbers and constitutes an effective tool to develop estimation techniques, as well as a helping instrument when solving word problems. It constitutes a unifying and coherent representation for the different sets of numbers (N, Z, Q, R), which the other models cannot do.

20 WHY THE Number Line? It is an appropriate model to make sense of each set of numbers as an expansion of other and to build the operations in a coherent mathematical way. It enables to present the fractions as numbers and to explore the notion of equivalent fractions in a meaningful way.

21 WHY THE Number Line? The number line, in some way, looks like a ruler, fostering the use of the metric system and the decimal numbers. It fosters the discovery of the density property of rational numbers. It provides an opportunity to consider numbers that are not fractions.

Grades 3, 4 and 5 Number and Operations - Fractions Common Core Standards Mathematics Grade 3 Develop understanding of fractions as numbers Develop understanding of fractions as numbers Grade 4 Extend understanding of fraction equivalence and ordering Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. Understand decimal notation for fractions, and compare decimal fractions. Grade 5 Use equivalent fractions as a strategy to add and subtract fractions. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Grades 6 and 7 Number and Operations - Fractions Common Core Standards Mathematics Grade 6 Apply and extend previous understandings of multiplication and division to divide fractions by fractions. Apply and extend previous understandings of multiplication and division to divide fractions by fractions. Grade 7 Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

24 We will focus two of the Standards for Mathematical Practice: Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others CaCCSS-M : Mathematical Practice

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26 Reason Abstractly and Quantitatively DO STUDENTS: Explain a problem to themselves, determine what it

27 Construct Viable Arguments and Critique the Reasoning of Others Use multiple representations (verbal descriptions,? Standards for Mathematical Practice

28 What are some of the challenges that students have with fractions? Question

29 Overview of Breakout Sessions Appropriate grade level problem Twelve (12) cards Videos of students working with 12 cards Human Number Line activity Reflection

Slides for Breakout 30

Breakout Reflection What mathematics did you use in the activities? How did your reasoning and explaining support and expand your understanding of the mathematics? How did these activities provide access to all students? Give specific examples. 31

Return to main room 32

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34 Use the structure of the number line and benchmarks to determine value of the “?” of each number line strip. With a partner agree on the value of the “?” and share your strategies. Part IB: Equivalent Fractions, Comparing and Ordering Fractions

CCSS-M Task Force: CAMTE, CDE, CISC, CMC, CMP Ordering 12 Cards Order Number Lines Explain Reasoning Hardest Easiest M2 A3

Human Number Line: Ordering Divide into groups of 10. Each of you will be given a card. One person will place himself/herself on a line to establish a point. Each subsequent person is to place himself/herself to the left, right, or in between the existing numbers to maintain proper order. This task focuses on order and not proportionality. 36

Reflection: Think, Ink, Pair, Share How has this experience with these activities expanded your concept of fractions? How will this inform your instruction in the classroom? How will you amplify and enrich the language to support English learners? 37

38 SOLVE THIS WITHOUT USING ALGEBRA OR a/b = c/d What is the ratio of men to women in a town where two-thirds (2/3) of the men are married to three-fourths (3/4) of the women?

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45 Support English learners in mathematics with high cognitive demand tasks. Become familiar with the CaCCSS-M, particularly the Standards for Mathematics Practice. Gain a conceptual understanding of the number line as a big idea in the CaCCSS-M. Use the number line in working with fractions. Goals for Institute

46 How will we support English learners in mathematics with high cognitive demand tasks? What did we learn about the CaCCSS-M, particularly the Standards for Mathematics Practice? What understanding of the number line as a big idea in the CaCCSS-M did we gain? How will we use the number line in working with fractions? Reflection

47 Delivery of Instruction Engage students in high cognitive demand tasks. Assess by walking around (ABWA). Provide access to the language of mathematics. Allow for discourse (Think, Ink, Pair, Share--TIPS) Set high expectations and increase expectations Allow students to explore why—metacognition

48 Acquiring the Knowledge Become familiar with the content and academic language of your lesson and possible misinterpretations. Frontload the academic/mathematics/English language of the mathematics content. Amplify and enrich the language. English language learners are trying to catch a moving target. Be aware of how your students interpret the academic and mathematics language.

49 When you design instruction, you start with the cognitively demanding mathematics you want students to learn. You gain an in-depth understanding of the content. Then you incorporate the most effective instructional practices to meet the needs of your students. You access prior knowledge, build background knowledge, and extend language. Putting This Together

50 You instinctively know to frontload the language, to ask questions of students, to have students think, ink, pair, share, to increase discourse, to increase student engagement, to assess, etc. This process becomes second nature to you. You are addressing the needs of ALL students, particularly English learners. Putting This Together

51 Equity and Quality How do we provide access to ALL students? We want students to make sense of rigorous, high quality, and cognitively demanding mathematics. We want them to approach the zone of proximal development, not the zone of minimal effort (e.g., assigning only lower level problems, not requiring homework, expecting less).

52 Equity and Quality We want equity and quality. Equity without quality is meaningless. Quality without equity is unjust. We must always ask ourselves, what can we do to incorporate both?

53 It is up to us to provide greater access and opportunity to high cognitive level mathematics by enriching the language and by using instruction that supports the learning needs of ALL students. It is up to us to acquire the understanding of fractions from a number line approach. What will you do to acquire the knowledge so that the understanding of the content and the integration/infusion of mathematics and language becomes second nature to you? Final Thoughts

Assessment What did I learn? What do I plan to implement in my classroom? Questions I have remaining. I suggest... 54

55 April Cherrington: Patricia Duckhorn: Susie W. Hakansson: Greisy Winicki-Landman: Contact Information

56 CaCCSS-M Task Force Materials: TODOS: Mathematics for ALL: California Mathematics Project: Links