a story of units

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NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Units

A Story of UnitsGrade 4 – Module 5

Fraction Equivalence, Ordering, and Operations


N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units

Session Objectives

• Examination of the development of mathematical understanding across the module with a focus on the Concept Development within the lessons

• Introduction to mathematical models and instructional strategies to support implementation of A Story of Units



Agenda

Introduction to the ModuleConcept DevelopmentModule Review



Curriculum Overview of A Story of Units



Module Overview

• Read the Narrative, Standards, Lesson Objectives, and Vocabulary.

• What is familiar to you and your students?• What will be new learning?



Mid-Module Assessment

• Study this document, and the End Module Assessment, prior to teaching the Lessons.

• Provides a snapshot of the concepts taught over the next 21 Lessons.

• Scan the Mid-Module Assessment to get an idea of what topic we are about to cover.

Topics A - D



Agenda




Topic A: Decomposition and Fraction Equivalence • 6 lessons to show fraction equivalence

through decomposition of fractions• Tools: number bond, folding paper

strips, tape diagram, area model

4.NF.3b, 4.NF.4a



Decomposition of Fractions

Lesson 1, Concept Development

• Use a strip of paper to represent, for example, thirds and sixths.

• Show how a non-unit fraction is the sum of unit fractions.

• Extend learning to show that non-unxxit fractions can be expressed as the sum of two or more fractions (unit or non-unit).



Decomposition of Fractions• Extend learning to fractions representing

numbers greater than one.• Use tape diagrams to represent fractional parts.

• Discover that there are multiple ways to decompose fractions.





Debrief: How is decomposing a non-unit fraction similar to decomposing whole numbers in Kindergarten and Grade 1?




• Express a non-unit fraction as a whole number times a unit fraction.





• Decompose fractions into sums of smaller unit fractions.

• Draw and label to show the equivalence of fractions.





Lesson 5, Application Problem




• Show fraction equivalence using an area model.

• Model equivalence by drawing an area model.





• Create equivalent fractions by drawing an area model and then dividing it into smaller parts.





Turn and Talk to those sitting at your table: • How can decomposing fractions lead to better

understanding of the representation of non-unit fractions and to fraction equivalence?

• How do number bonds, tape diagrams, and area models aid in the understanding of the decomposition of fractions?

Topic A Debrief



Topic B: Fraction Equivalence Using Multiplication and Division• 5 lessons to show fraction equivalence

using multiplication and division.• New Terms: numerator, denominator • Tools: tape diagrams, area model,

number line

4.NF.1



Fraction Equivalence




Fraction Equivalence• Decompose the area model to show equivalence.• Create equivalent fractions by multiplying the

numerator and denominator by the same number.

Lessons 7 and 8, Concept Development




• Compose larger fractional units.

• Discover that the numerator and denominator can be divided by the same number to model equivalence.






• Use knowledge of common factors to show equivalent fractions.

• Show equivalence using the greatest common factor.




• Use a number line to show equivalence.





Turn and Talk to those sitting at your table: • How does the modeling used within this Topic help

students to conceptually understand fractional equivalence?

Topic B



Topic C: Fraction Comparison

• 4 lessons to show comparison of fractions using benchmark numbers and common units

• New Term: benchmarks• Tools: number line, tape diagrams,

area models, number bonds

4.NF.2



Fraction Comparison

• Think about fractions in relation to 0, ½ , and 1, first in isolation and then in fraction comparison.




Fraction Comparison

• Compare two fractions greater than one using benchmarks.




Fraction Addition and Subtraction

• Subtraction of a fraction from a whole using different methods




Fraction Comparison

• Common numerators

• Numerators with related units




Fraction Comparison

• Denominators with related units




Fraction Comparison• Denominators with

unrelated units• Use of the area model to

create common units for comparison

• Use of number bonds to ‘pull out’ a whole and then compare the parts




Fraction Comparison

Turn and Talk: • What methods of fraction comparison are familiar?• What methods of fraction comparison are new?• How can each of the methods be used to help

students as they compare fractions?

Topic C



Topic D: Fraction Addition and Subtraction

• 6 lessons to show addition and subtraction of fractions with an emphasis on common units

• New Terms: common denominators, mixed units

• Tools: number line, tape diagram, number bond

4.NF.3a, 4.NF.4d




• Unit language is used to relate addition and subtraction of fractions to addition and subtraction of whole numbers.





• Conversion from a fraction greater than one to a mixed number using a number bond





Students work in groups to find multiple ways of solving






measuring inches

cooking a recipe

telling time buying deli meat

money

measuring turns

gas gauge

measuring spoons and cups





• Students use multiple solution strategies to solve.• Use RDW to solve. Try solving 2 ways.




• Adding with related units





Turn and Talk:• How is subtracting fractions similar to subtracting

whole numbers?• Discuss the progression and the importance of the

number bond model across grades.

Topic D



Topic E: Extending Fraction Equivalence to Fractions Greater Than One • 7 lessons working with fractions

greater than one to rename, order, and compare.

• New Terms: line plot• Tools: number line, tape diagram,

number bond, line plot

4.NF.1, 4.NF.2, 4.NF.3



Extending Fraction Equivalence to Fractions >1





• Rename fractions as mixed units using decomposition

• Convert using multiplication




Extending Fraction Equivalence to Fractions >1Use RDW to solve:





• Convert a mixed number to a fraction greater than one.




Extending Fraction Equivalence to Fractions >1• Compare fractions greater than one by reasoning

using benchmark numbers.





• Comparison with related denominators

• Comparison with unrelated denominators

•





Plot fractions on a line plot and then use the line plot to help solve word problems.





Turn and Talk:• How does the content of this Topic build on the

learning of Topics A-D? How does the content of A-D help students realize success with this Topic?

• Which of the strategies from this module have you previously taught?

Topic E



Topic F: Addition and Subtraction of Fractions by Decomposition• 6 lessons working with fractions

greater than one to estimate and find sums and differences

• Tools: number line, number bond

4.NF.3c, 4.NF.3d, 4.MD.4



Addition & Subtraction of Fractions by Decomposition


• Rounding is used to establish an estimated sum or difference.



Addition & Subtraction of Fractions by Decomposition• Use different strategies to add fractions





Addition of mixed numbers using different strategies






Subtraction of a fraction from a mixed number using different strategies




Subtract a mixed number from a mixed number using multiple strategies





Subtract with mixed numbers





Turn and Talk:• What from this Topic was new learning?• What advantages can you see in introducing multiple

strategies to students?

Topic F



Topic G: Repeated Addition of Fractions as Multiplication

• 6 lessons focusing on addition of fractions as multiplication

• Tools: number line, tape diagram, number bond, line plot

4.NF.4abc, 4.MD.4



Repeated Addition of Fractions as Multiplication

Represent the multiplication of n times a/b as (n×a)/b using the associative property and visual models.




Repeated Addition of Fractions as MultiplicationPatti needs 6 pieces of yarn that are each 3 eighths

meter long. How much yarn does Patti need?





Lesson 37 and 38, Concept Development




Solve multiplicative comparison word problems.




Repeated Addition of Fractions as MultiplicationSolve word problems involving the multiplication of a whole

number and a fraction including those involving line plots.





Turn and Talk:• How do the strategies taught within this Topic help

students to be successful when solving problems involving a whole number times a mixed number?

Topic G



Topic H: ExplorationFind and use a pattern to calculate the sum of all fractional parts

between zero and one.

Students work together to solve and critique the thinking of others.

4.OA.5



Complete the End-of-Module Assessment.

• Now with all of the mathematical knowledge and understanding of the models, review the assessment.

• You may work alone or with your table to discuss challenges or successes students and teachers may face.

• How do the Topics in this Module help to prepare students for success?

End-of-Module Assessment



Agenda




Biggest Takeaway

Turn and Talk:• What questions were answered for you?• What new questions have surfaced?



Key Points• The learning of fractions follows the same instructional

sequence as the learning of whole numbers.• Fractional units behave just as all other units do, and can be

manipulated like whole numbers.• Decomposing fractions strengthens the part-whole

relationship.• Fraction equivalence and comparison are supported by visual

models.

a story of units

Documents

story of unitsgrade

network team institutegrade

minute materials

mathematical models

decimal fractions

instructional strategies

concept developments

lessons introduction