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Fraction Progressions PD Module

Presented at STAR – July 31, 2013Casper Hilton Garden Inn

Laurie Hernandez, M.Ed.WDE Math ConsultantLaurie.Hernandez@wyo.gov

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Grades 2 – 6 Fraction Progressions

Agenda

•Objectives of Presentation•Fraction Overview•The Meaning of Fractions•Equivalent Fractions•Comparing Fractions•Operations with Fractions

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• 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.

OBJECTIVES

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• 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?

Key Questions to Consider Throughout the Day:

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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.)

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Fractions Progressions Overview

http://youtu.be/X9NFEZIkoH0

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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; 2nd grade – Geometry – Cluster #1). *note: gray boxes remain blank.

▫Share your findings with the group. (10 min.)

FRACTIONS PROGRESSIONSGRADES 2 & 3 GRADE 4 GRADE 5

The meaning of fractions

The number line and number line diagrams

Equivalent Fractions Equivalent Fractions

Adding and subtracting fractions Adding and subtracting fractions

Comparing fractions Comparing Fractions

Multiplication of a fraction by whole number

Multiplying and dividing fractions

Multiplication as scaling

Decimals

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FRACTIONS PROGRESSIONSGRADES 2 & 3 GRADE 4 GRADE 5

The meaning of fractions2.G.3/3.NF.1

The number line and number line diagrams3.NF.2.a, b

Equivalent Fractions3.NF.3.a, b, c

Equivalent Fractions4.NF.1

Adding and subtracting fractions4.NF.3.a, b, c/4.NF.3.d

Adding and subtracting fractions5.NF.1/5.NF.2

Comparing fractions2.MD.3/3.NF.3.d

Comparing Fractions4.NF.2

Multiplication of a fraction by whole number4.NF.4.a, b, c

Multiplying and dividing fractions5.NF.3/5.NF.4.a/5.NF.6/5.NF7.a

Multiplication as scaling5.NF.5.a/5.NF.5.b

Decimals4.NF.5/4.NF.6/4.NF.7

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Where are the Cookies?

Mrs. James left a tray of cookies on the counter early one morning. Larry walked by before lunch and decided to take of the cookies on the tray. Later that afternoon Barry came in and ate of the remaining cookies. After supper Terry saw the tray of cookies and ate of the cookies remaining at that time. The next morning Mrs. James found the tray with only 6 cookies left. How many cookies were on the tray when Mrs. James first left it on the counter?

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Unit 1: The Meaning of Unit Fractions

http://youtu.be/9Z_yLiwbEf0

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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.)

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Specifying the Whole

Explaining what is meant by

“equal” parts

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Unit 2: Equivalent Fractions

http://youtu.be/lRTeNZaaLMI

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Equivalent Fractions

Using an area model prove and justify that:

4.NF.1 Explain why a fraction is equivalent to a fraction 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.

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Equivalent FractionsUsing an area model prove and justify

that:

3–5 Number and Operations—Fractions Progressions

¿

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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.)

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Unit 3: Comparing Fractions

http://youtu.be/hm5DI_zlSD8

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Which Fraction is Larger?(The following fractions are for demonstration purposes only and are NOT grade

specific.)

1. Using number line diagrams, determine which fraction in each pair is larger.

a. b. c.

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”.

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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.)

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Unit 4: Adding Fractions

http://youtu.be/0ZblmRwktTo

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Activity

•Solve the following problem without using the traditional “common denominator” approach:

1𝑢𝑛𝑖𝑡

1𝑢𝑛𝑖𝑡34

13412

912

Demonstration of one Possible Solution

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1312

Final Solution

1𝑢𝑛𝑖𝑡

4 / 129/12

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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.)

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Unit 5: Multiplying Fractions (Part 1)

http://youtu.be/F1QtzR-0_Dc

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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.)

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Unit 6: Multiplying Fractions (Part 2)

http://youtu.be/XzkSAytyBFQ

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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.)

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• With your partner discuss how transparencies and color markers can be used to model the problem below:

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2

4

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Questions for Discussion

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Demonstration of one Possible Solution

Model for Model for

Rotate 90

Model for

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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.)

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Unit 7: Dividing Fractions

http://youtu.be/co0qfzyu4qk

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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.)

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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.)

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Where are the Cookies?

Mrs. James left a tray of cookies on the counter early one morning. Larry walked by before lunch and decided to take of the cookies on the tray. Later that afternoon Barry came in and ate of the remaining cookies. After supper Terry saw the tray of cookies and ate of the cookies remaining at that time. The next morning Mrs. James found the tray with only 6 cookies left. How many cookies were on the tray when Mrs. James first left it on the counter?

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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.)

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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.)

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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.)

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• 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.)

Reflections

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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.

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Questions?