Fractions: Number Meaning And Number Relationships Grade 3

Fractions: Number Meaning andNumber RelationshipsGrade 3Orcutt Union School DistrictJanuary 28, 2020

Today’s Agenda and OutcomesObjective: Participants will unwrap the fraction standards todetermine the concepts, procedures and type of problemsstudents are expected to master. Participants will engagein learning activities that build meaning of fractions andfraction relationships using various tools and strategies. Agenda: Unwrap standardsBuilding meaning of fractions using areas, number lines, and setsMake fraction equivalencies using areas and number linesCompare fractions using the same whole by comparing samenumerators, same denominators, or benchmark values. Instructional PlanningMD School Solutions, Inc. 2016

Introductions and Outcomes Nametags Name (write this big) Site Yrs. in education Introduction What is your favorite number? Write it as a multiplication fact. Talking Chips Take 4 coffee filters and put 3 chips in each one.MD School Solutions, Inc. 2016

Multiplication and DivisionMD School Solutions, Inc. 2014

Understanding Multiplication and Divisionx (# ofgroups)MD School Solutions, Inc. 2016(how manyitems ineachgroup)(Total # ofitems)

Multiplication What models are we asking students touse? How often?MD School Solutions, Inc. 2016

4 Ways to Represent Fold your paper into 2 equal parts horizontallyand then vertically to create 4 equal sections. You are given a fact to display in 4 differentways. Write your fact in the centerRepeated Addition Number Line1) repeated addition equation2) number line6 x 9 3)an arrayArrayEqual groups4) equal groupsMD School Solutions, Inc. 2016

MD School Solutions, Inc. 2016

What is Division?How does it relate to multiplication?What are fair shares? What does it mean to partitionequally? (3.OA.2)MD School Solutions, Inc. 2014

Division as an Operation6 2 Partitive – Finding the size ofthe groupThe question: How do youdivide or partition 6 thingsinto 2 groups?There are 3 in each group.Measurement – Finding the numberof groups of a specific size0MD School Solutions, Inc. 20166The question: How manygroups of 2 units are in 6?There are 3 groups of 2 in6.

Strategies Counting on Doubling Combining twoknown factsMD School Solutions, Inc. 2016

MD School Solutions, Inc. 2016

Reflection What have you implemented to supportstudents build the concepts ofmultiplication and division? What is working? Where are students still getting stuck? With whatspecifically? What are you still doing consistently tobuild fluency and to measure fluency?MD School Solutions, Inc. 2016

MD School Solutions, Inc. 2014

MD School Solutions, Inc. 2014

MD School Solutions, Inc. 2014

MD School Solutions, Inc. 2014

Critical AreasMD School Solutions, Inc. 2014

MD School Solutions, Inc. 2016 Problem Types ProblemStructures InstructionalConsiderationsfor SolvingProblemFluency NumberMeaning NumberRelationships NumberMagnitude Operations ofNumbersProblem SolvingNumber SenseCritical Areas Identified UnderstandingFluency BuildingFlexibleStrategies MeasuredProgress

Critical Areas IdentifiedGrade 3Operations andNumbers andAlgebraic tiplyDefining a EquivalentInterpret and divide fractionfractionsproducts within 100andandcomparinginterpretfractionsquotientsMD School Solutions, Inc. 2016Measurement3.MD.2Solveproblemsof massandvolumeusing alloperations3.MD.7Conceptsof area asit relates tomultiplication anddivision.

FractionsMD School Solutions, Inc. 2014

StandardsDocumentsStandards What do theysay? How are youmaking sense ofthe expectationsof thestandards? How will thestandards guideyour instruction?CONCEPTSWhatconcepts dostudentsneed tounderstand?PROCEDURESWhatproceduresdo studentsneed tomaster?PROBLEMSOLVINGTOOLSWhat toolsshouldstudents useto makingmeaning ofthis concept?What types ofproblem solvingare studentsexpected to do?This Photo by Unknown Author is licensed under CC BY-SAMD School Solutions, Inc. 2016

“Current instruction that gives a briefintroduction through part-wholefractions and then proceeds tointroduce computation proceduresdoes not give children the time theyneed to construct important ideas andways of thinking.”(Lamon, 1999; emphasis added)MD School Solutions, Inc. 2014

Confusions and MisconceptionsConfusion results when students try to applytheir understanding of whole numbers to fractions.The textbook approach to teaching: Moves too quickly over the major concepts Rushes to have students practice procedures,resulting in simplistic, mechanical problem solving Has limited representation, weakening conceptualunderstandingMD School Solutions, Inc. 2016

Confusions and MisconceptionsFallout from Whole NumbersWhat do you see? 2 and 6 2 and 8Is there arelationship?MD School Solutions, Inc. 201628

Confusions and MisconceptionsSome Models Can Be Confusing𝟐Is this𝟖or “2 green and 6 red”?Say: “Two-eighths”not “2 over 8” or “2 out of 8.”MD School Solutions, Inc. 2016

Confusions and Misconceptions Fractions and the Concept of InfinityINFINITY0123456INFINITY1000 1001 1002MD School Solutions, Inc. 20161003 1004 1005 1006

Confusions and Misconceptions Fractions and the Concept of InfinityINFINITE NUMBER OF FRACTIONS01INFINITE NUMBER OF FRACTIONS1000MD School Solutions, Inc. 20161001

Confusions and MisconceptionsAn Infinite Number of Fractions?0MD School Solutions, Inc. 20161428382448

Confusions and MisconceptionsAn Infinite Number of Fractions?0142838616MD School Solutions, Inc. 20167162448816

Confusions and MisconceptionsAn Infinite Number of Fractions?0142838616MD School Solutions, Inc. 2016716143224488161615 3232

Confusions and Misconceptions Take the time to develop the concept offractions. Foundational to decimals, percentage, ratios, andproportions Use visual representations and classroom talk. Develop a robust understanding of part-wholerelationships. Avoid simplistic instruction Before teaching operations on fractionsMD School Solutions, Inc. 2016

Confusions and Misconceptions Students need to learn how to: Represent a fraction multiple ways. Show the value of a fraction based on differentdefinitions of the whole. Work with equivalent fractions. Understand that comparisons can only occurwhen the wholes are equivalent. Use the meaning of numerator and denominatorto compare fractions and reason about the size ofa fraction.MD School Solutions, Inc. 2016

MD School Solutions, Inc. 2016

Building Number Sense of Fractions Number Meaning Counting, Writing, Naming, and Representing Number Relationships Building a Sense of Operation Number Magnitude and SizeMD School Solutions, Inc. 2016

Building Meaning of FractionsArea ShapesWhat does 1/3 of this rectanglelook like?MD School Solutions, Inc. 2016

Building Meaning of FractionsLength ModelsNumber Line01323Cuisenaire RodsFraction BarswholehalvesthirdsfourthsfifthssixthsMD School Solutions, Inc. 20161

Building Meaning of FractionsSet ModelsMD School Solutions, Inc. 2016

Symbolic NotationThese words are Level 3 vocabulary anddon’t have meaning or any commonreference for children. The wordsthemselves will NOT help studentsunderstand their meaning.2(# of equal parts)Means enumeration 8Numeratoror countingMD School Solutions, Inc. 2014Denominator(# of equal parts in the whole)Means name of the class ortype of thing beingrepresented like adenomination of moneybeing counted.

The Meaning of a/bWe have established the meaning of 1/3 A whole is partitioned into three equal sizedpieces. When we want to reference only 1 of the3 pieces, we call the size of 1/3 of the whole.MD School Solutions, Inc. 2016

The Meaning of a/bWhat does 2/3 really mean? There are 2 pieces in blue. They are each 1/3 ofthe whole. So two 1/3 pieces is the same as 2/3of the whole.MD School Solutions, Inc. 2016

Building Meaning of Fractions Partitioning is a central mathematical idea, andit is central to equal shares Begins in 1st grade Equal shares are not easy for students to draw,but they are essential for understanding(especially for addition and subtraction) The idea of relational is essentialMD School Solutions, Inc. 2016

Building Meaning of FractionsArea Models are Particularly Useful for Showing Fair SharesFair Share ExamplesMD School Solutions, Inc. 2016Non-Examples of Fair Shares

Building Meaning of FractionsArea Models are Particularly Useful for Showing Fair SharesFair Share ExamplesMD School Solutions, Inc. 2016Non-Examples of Fair Shares

Building Meaning of FractionsLength Models and Fair SharesYesFair sharesNon-examples of fair sharesMD School Solutions, Inc. 2016

Building Meaning of FractionsEach segment is equal in size so breaking up the number line thisway shows fair shares.Why are these called fourths?MD School Solutions, Inc. 2016

Building Meaning of FractionsWhat does equal shares look like in a set model?Equal SharesMD School Solutions, Inc. 2016Non-Equal Shares

Building Meaning of Fractions A beaker is considered full when the liquid reaches the fillline shown near the top. Estimate the amount of water inthe beaker by shading the drawing to the indicatedamount. The first one is done for you. Each circle represents 1 whole pie. Estimate to show howyou would cut the pie into the fractional unit that is given.MD School Solutions, Inc. 2016

Building Meaning of Fractions Circle the figures that show fair shares. Each shape is a whole divided into equal parts.Name the fractional unit. Then determines howmany of those units are shaded.MD School Solutions, Inc. 2016

Building Meaning of Fractions the whole What is the value ofArea Model?12It takes two of the trapezoids to make the wholeso one of them would be one-half of the whole.MD School Solutions, Inc. 2016

Building Meaning of Fractions the wholeWhat is the value ofMD School Solutions, Inc. 2016?16

Building Meaning of Fractions the whole What is the value ofMD School Solutions, Inc. 2016?13

Building Meaning of Fractions the wholeWhat is the value ofMD School Solutions, Inc. 2016?14

Building Meaning of Fractions the wholeWhat is the value ofWhat is the value ofWhat is the value ofMD School Solutions, Inc. 2016? 1121?621? 63

Building Meaning of Fractions the wholeWhat is the value of? 16What shape(s) will show ½ of this unit?MD School Solutions, Inc. 2016

Building Meaning of Fractions Partitioning is a central mathematical idea,and it is central to equal shares Equal shares are not easy for students todraw, but they are essential forunderstanding (especially for addition andsubtraction) Let’s stop to work on how to build the ideaof equal shares with areas using differentshapes and questions.Page 3-4MD School Solutions, Inc. 2016

The Meaning of a/b Lengths or numbers lines can also be used.Think about the red pieces. They showunits of ¼ of the whole. Use these pieces as a reference to show alength that is 2/4 or 3/4. Why can they becalled these lengths?MD School Solutions, Inc. 2016

The Meaning of a/b ¾ is the same an 3 units of ¼ in length. ¾ ¼ ¼ ¼MD School Solutions, Inc. 2016

Building Meaning of FractionsMD School Solutions, Inc. 2016

Building Meaning of Fractions Creating Paper Strips How do we use them to name fractions? How do they help students to make sense of partto whole relationships?MD School Solutions, Inc. 2016

Building Meaning of Fractions Use Paper Strips to build number lines White board Draw 5-8 lines of different lengths. Choose one line. Fold your write board to match theendpoints of your line. Where is the middle? Open it upand mark this point on the line. It shows two equalsegments or ½ of the line. Repeat with two other lines. Now “eye-spy” on the next line where you think will be thepoint to represent ½ of the line. Fold to see how close youare. REPEAT this until you find you are very close to findingthe ½ point for a given line.MD School Solutions, Inc. 2016

Building Meaning of FractionsPage 5MD School Solutions, Inc. 2016

Building Meaning of FractionsMD School Solutions, Inc. 2016

Building Meaning of FractionsMD School Solutions, Inc. 2016