Fraction Progressions PD Module Presented at STAR – July 31, 2013 Casper Hilton Garden Inn Laurie Hernandez, M.Ed. WDE Math Consultant 1

Grades 2 – 6 Fraction Progressions Agenda Objectives of Presentation Fraction Overview The Meaning of Fractions Equivalent Fractions Comparing Fractions Operations with Fractions 2

Gain an understanding of the fraction progressions across Grades 2-6, informed by research on children’s cognitive development and the structure of mathematics. Collaborate within and across grades. Further develop professional learning using additional resources by grade level. 3 OBJECTIVES

What type of FOCUS do I need in my grade level to help a student be successful on a problem such as this? How do we work together within AND across grade levels to ensure COHERENCE? How do we maintain proper RIGOR in our instruction including: Conceptual Understanding, Fluency, and Application? 4 Key Questions to Consider Throughout the Day:

Fraction Overview Please refer to the Fraction Progressions Overview document. ▫Please read the document individually. ▫Underline the sentences which you believe are the most important in unit development. ▫Share your sentences with the group. (10 min.) 5

Fractions Progressions Overview 6

Activity Work in pairs using your CCSS-M document to, ▫Complete the Fractions Progressions Table by identifying the fraction standards in grades 2, 3, 4, or 5 that match the descriptors. (5 min.) ▫Label the grade, domain, and cluster (i.e. 2.G.1; 2 nd grade – Geometry – Cluster #1). *note: gray boxes remain blank. ▫Share your findings with the group. (10 min.) 7

FRACTIONS PROGRESSIONS GRADES 2 & 3GRADE 4GRADE 5 The meaning of fractions The number line and number line diagrams Equivalent Fractions Adding and subtracting fractions Comparing fractionsComparing Fractions Multiplication of a fraction by whole number Multiplying and dividing fractions Multiplication as scaling Decimals 8

FRACTIONS PROGRESSIONS GRADES 2 & 3GRADE 4GRADE 5 The meaning of fractions 2.G.3/3.NF.1 The number line and number line diagrams 3.NF.2.a, b Equivalent Fractions 3.NF.3.a, b, c Equivalent Fractions 4.NF.1 Adding and subtracting fractions 4.NF.3.a, b, c/4.NF.3.d Adding and subtracting fractions 5.NF.1/5.NF.2 Comparing fractions 2.MD.3/3.NF.3.d Comparing Fractions 4.NF.2 Multiplication of a fraction by whole number 4.NF.4.a, b, c Multiplying and dividing fractions 5.NF.3/5.NF.4.a/5.NF.6/5.NF7.a Multiplication as scaling 5.NF.5.a/5.NF.5.b Decimals 4.NF.5/4.NF.6/4.NF.7 9

Where are the Cookies? 10

Unit 1: The Meaning of Unit Fractions 11

Activity Read the section of the Progressions Document on development of the meaning of fractions and the number line. Work in pairs to answer the following question: ▫What are the important aspects of fractions that provide opportunities for the mathematical practice of attending to precision? (10 min.) 12

Specifying the Whole Explaining what is meant by “equal” parts 13

Unit 2: Equivalent Fractions 14

Equivalent Fractions 15

Equivalent Fractions 3–5 Number and Operations—Fractions Progressions 16

Discussion After watching the video and doing the activity, how has your perception of equivalent fractions and creating experiences for students about equivalent fractions changed? (5 min.) 17

Unit 3: Comparing Fractions 18

Which Fraction is Larger? (The following fractions are for demonstration purposes only and are NOT grade specific.) 2. What rules about the relative sizes of fractions can you state from these examples? Be as precise as you can in expressing your rules, without using the terms: “numerator”, “denominator”, “top number”, or “bottom number”. 19

Activity Read the section of the Progressions Document on Grade 4 Equivalent Fractions. Work in pairs to answer the following question: ▫How can the use of area models and number line diagrams solidify a student’s understanding of fraction comparison? (10 min.) 20

Unit 4: Adding Fractions 21

Activity 22

Demonstration of one Possible Solution 23

Final Solution 4 / 129/12 24

Activity Read the section of the Progressions Document on Grade 4 and Grade 5 Adding and Subtracting Fractions. Work in pairs to answer the following question: ▫How could a student build on their previous understanding of adding/subtracting whole numbers in order to add/subtract fractions? (5 min.) 25

Unit 5: Multiplying Fractions (Part 1) 26

Questions for Discussion Work in pairs to, ▫discuss how multiplying a fraction by a whole number is similar to/different from multiplying whole numbers. (5 min.) ▫discuss some of the misconceptions students may have when multiplying a fraction by a whole number. (5 min.) 27

Unit 6: Multiplying Fractions (Part 2) 28

Activity Work in pairs to,  discuss one advantage and one disadvantage of using an area model when multiplying two fractions.  create an area model that justifies each of your responses. (10 min.) 29

With your partner discuss how transparencies and color markers can be used to model the problem below: 30 Questions for Discussion

31 Demonstration of one Possible Solution

Activity Read the section of the Progressions Document on Grade 4 and Grade 5 Multiplying and Dividing Fractions. (10 min.) Work with a partner to respond to the following item: ▫Explain how creating a story/real-world context might assist a student in understanding fraction multiplication. (5 min.) 32

Unit 7: Dividing Fractions 33

50-Pounds of Rice If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Using a visual model, show how each person gets the same amount. (5 min.) 34

Multiplying/Dividing Fractions Discussion Questions What models are used for multiplying/dividing fractions in the videos and Progressions? What are the advantages to using different models of multiplying/dividing fractions? (10 min.) 35

Where are the Cookies? 36

Activity Work with a partner to respond to the following items: ▫How would your students approach this problem? ▫What conceptual understanding of fractions does a student need in order to solve the previous problem? ▫What instructional strategies would you use to reach students at various levels of mathematical ability?(12 min.) 37

Questions for Further Investigations What opportunities should students be given to assist with building their conceptual understanding of fractions? How do the various models of fractions build understanding? What are the consequences of a student being bound to one model (e.g. only using circles)? (7 min.) 38

Questions for Further Investigations Whole group: ▫How could various models have been used to facilitate understanding of any of the previous activities and what does the student’s choice of model tell the teacher about student understanding?(7 min.) 39

Discuss as a whole group the following:  What type of FOCUS do I need in my grade level to help a student be successful on a problems similar to those presented in today’s professional development?  How do we work together within AND across grade levels to ensure COHERENCE?  How do we maintain proper RIGOR in our instruction including: Conceptual Understanding, Fluency, and Application? (15 min.) 40 Reflections

Next Steps Work with your students, gather student work, re-visit, and share students’ understanding and misconceptions with team or PLC. ▫What worked? ▫What didn’t? ▫Evaluate if individual students are ready to move on to the next concept. 41

Questions? 42