math institute teaching fractions
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5/22/2018 Math Institute Teaching Fractions
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Grades 3 - 5
Michele Lippens Special Educator
Cambridge PublicSchools
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About this Presentation Common Core Standards
Road Map to the CommonCore
Why Fractions? Identifying Strategies for
Comparing Fractions
Developing an
understanding of Fractionsas Numbers
Using Visual Models
Lesson Ideas and Resources
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The Common CoreAchieve equity for all
students Teachers concentrate
on a more focused setof major concepts
Engages students insolving real-world
math problems Incorporates
mathematical practicestandards
The standards are designed tobe robust and relevant to the
real world, reflecting theknowledge and skills that our
young people need for success incollege and careers. ~CommonCore State Standards Initiative(2012)
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Common Core Standards For
Mathematical Practice Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoningof others
Model with mathematics
Use appropriate tools strategically
Attend to precision Look for and make use of structure
Look for and express regularity in repeated reasoning.
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Road Map to Common CoreUnderstanding Fractions Operations with Fractions
Grade 3
Students will learn about
Unit fractions and their size.
Compare and find equivalent fractionsusing visual models 3.NF.3
Fractions on a number line 3.NF2
Grade 4
Students will learn about
Comparing two fractions with differentnumerators/denominators by usingbenchmark fractions/visual models 4.NF.1
Grade 4
Students will learn to
Add & Subtract fractions with like
denominators using visual models 4.NF3d Create equal fractions (3/8 = 1/8 + 1/8 +1/8)
Multiply whole numbers by a fraction.
Grade 5
Add & Subtract fractions with unlikedenominators using visual models (5.NF.6)
Multiply a fraction by a fraction.
Divide a whole number by a fraction and a
fraction by a whole number.
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Common Themes - Common Core Construct mental images
Use multiple fraction models
Extended time/practice with
models Real-world problem solving
Discussion of the reasoning
strategies used whencomparing fractions
Bray & Sanchez, Teaching Children Mathematics, NCTM.org Sept. 2010
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Why Fractions? Complex subject
Requires conceptualunderstanding and
reasoning Operations with fractions
are counterintuitive towhat children have learnedabout whole numbers
Fractions follow studentsthroughout their school
years and beyond
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Did this everhappen to
you?
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Comparing FractionsBIG Ideas Understand strategies
students are using in orderto support student learning
Identify what strugglinglearners are missing inorder to provideinterventions and direct
instruction Explicitly teach these
strategies using multiplemodes and models
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Strategies for Comparing Fractions
1. Same Denominator
2. Same Numerator
3. Transitive
4. Residual
Same Denominator -The same denominator
implies that one iscomparing parts of theunit that are the same
size
Same Numerator -involves understanding
that an inverserelationship exists
between the number ofparts a unit is
partitioned into and
the size of the parts.
Transitive - students
use a single outsidevalue to compare bothfractions
Residual - studentsfocus on the leftover
part in judging therelative size of the
fractions
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Same Denominator
More Same Sized PartsBig Ideas
Students use concrete materials to compare
fractions and use this knowledge to move to moreabstract thinking in the later grades.
Concrete models should first focus on area models
Pieces are the same size
Largest numerator is the largest fraction
Elementary and Middle School Mathematics: Teaching Developmentally (van de Walle 2007)
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Same DenominatorVisual ModelThis comparison focuses on the number of partsselected/shaded. Students should recognize that italways refers to the same whole.
3/10
5/10
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Same NumeratorSame Number
But Different-Sized PartsBig Ideas
Always refers to the same whole
Students become familiar with models forfractions and describing part-to-wholerelationships
With fractions, the more pieces, the smaller thesize of each piece
Students reasonabout size of pieces
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Same NumeratorVisual ModelThis model demonstrates reasoning about the size ofeach part. Students need to shift their thinking fromcomparing whole numbers.
3/8
3/10
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Transitive Strategy: Benchmark
When students use the benchmark of and onethey are using the transitive property.
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ActivityComparing Fractions
(Transitive Strategy)Which Fraction is greater:
14/24 or 17/36
Bray & Sanchez, Teaching Children Mathematics, NCTM.org Sept. 2010
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Answer: Which Fraction is Bigger?Although both fractions are close to
, 14/24 is more than because 12/24
is exactly and 14/24 is greater thanthat. Likewise, 17/36 is less than because 18/36 is exactly and 17/36 isless than that. So 14/24 is greater than17/36
Bray & Sanchez, Teaching Children Mathematics, NCTM.org Sept. 2010
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Residual Strategy: Fill the WholeWhen comparing fractions students decide on therelative size of each fraction by reflecting on theamount away from the whole. Students rely on the
same numerator strategy; they compare whats left,to determine which of the original fractions have thelargest amount away from one.
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ActivityComparing Fractions
(Residual Strategy)
Which fraction is greater?
3/4 and 7/8
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Turn and TalkStudent ThinkingWhat does the student know?
Is the student using the Residual Strategy?
Is the student thinking abstractly? Is there evidence of the student reasoning about
numbers?
What does the student need to understand in order to
use this strategy?
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Reason Abstractly and
Quantitatively
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Make Sense of Problems Teaching students
both HOW and WHYcomputational
procedures workresults in greatermath proficiency.~Tim Lamarre
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Real Word Problem Solving Ms. Lippens is making
brownies. One batchneeds 2/3 cups of
chocolate chips. But, shewants to make 5 batchesso she will have enoughto feed all her students.
How many cups ofchocolate chips does sheneed in all?
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Develop an Understanding of Fractions
as Numbers
Many experiences
withphysical
models are
needed to
overcome the
influence of
childrens wholenumber thinking.
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Beyond PizzaMultiple concrete models
Physical aids are just one componentin the acquisition of concepts
Verbal, pictorial, symbolic and real-world representations
Opportunities to talkaboutmathematical ideas
Conceptual knowledge priortoformal work with symbols andalgorithms
(Teachers Guide to the RNP: Initial FractionIdeas, p. 12).
http://www.cehd.umn.edu/ci/rationalnumberproject/RNP1-09/RNP1-09_TG.pdfhttp://www.cehd.umn.edu/ci/rationalnumberproject/RNP1-09/RNP1-09_TG.pdfhttp://www.cehd.umn.edu/ci/rationalnumberproject/RNP1-09/RNP1-09_TG.pdfhttp://www.cehd.umn.edu/ci/rationalnumberproject/RNP1-09/RNP1-09_TG.pdf -
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Lesson Idea: Area Models Developing the idea of the relative amount of the area that
has been shaded is not possible unless the size of the basicrectangle is sometimes varied within one exercise.
Central idea of relative sizes is a key idea for fractions
What is the same and what is different?Lesson Idea: Students determine relative use of space in
newspapers for ads, articles, pictures etc..
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Attend to Precision
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Fractions have MAGNITUDE Fractions are NUMBERS
Clearly illustrate the magnitude of fractions
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Lesson Idea: Measurement with
Fraction Strips To start, take a strip of card stock
or construction paper thatrepresents the initial unit ofmeasure (i.e., a whole) and use
that strip to measure objects inthe classroom (desk, chalkboard,book, etc.).
When the length of an object isnot equal to a whole number ofstrips, teachers can providestudents with strips thatrepresent fractional amounts ofthe original strip
1/2 1/4
0.5 0 .25
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Visual Models: Fraction Wall Focuses on creating equal-
sized pieces
Visualizes equivalentfractions
Visualizes that 1/7is afraction (this is not on
many commercial models) Shows that 1/7will be a bit
bigger than 1/8and a bitsmaller than 1/6, etc.
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Online Resourceswww.geogebratube.org
www.coolmath.com
The National Library ofVirtual Manipulatives,(NLVM) http://nlvm.usu.edu/en/nav/topic_t_1.htm
lwww.thinkingblocks.com
www.adaptedmind.com
www.ixl.com/Math
www.sheppardsoftware.com/math.htm
www.mathisfun.com
www.helpwithfractions.c
om
http://www.geogebratube.org/http://www.coolmath.com/http://nlvm.usu.edu/en/nav/topic_t_1.htmlhttp://nlvm.usu.edu/en/nav/topic_t_1.htmlhttp://nlvm.usu.edu/en/nav/topic_t_1.htmlhttp://www.thinkingblocks.com/http://www.adaptedmind.com/http://www.ixl.com/Mathhttp://www.sheppardsoftware.com/math.htmhttp://www.sheppardsoftware.com/math.htmhttp://www.mathisfun.com/http://www.helpwithfractions.com/http://www.helpwithfractions.com/http://www.helpwithfractions.com/http://www.helpwithfractions.com/http://www.mathisfun.com/http://www.sheppardsoftware.com/math.htmhttp://www.sheppardsoftware.com/math.htmhttp://www.ixl.com/Mathhttp://www.adaptedmind.com/http://www.thinkingblocks.com/http://nlvm.usu.edu/en/nav/topic_t_1.htmlhttp://nlvm.usu.edu/en/nav/topic_t_1.htmlhttp://nlvm.usu.edu/en/nav/topic_t_1.htmlhttp://www.coolmath.com/http://www.geogebratube.org/ -
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Online Resources:
Fractions on a Number Line http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp
/ks2/maths/fractions/level4.htm
http://www.visualfractions.com/GrampStrict/findgrampystrict.html
http://www.funbrain.com/linejump/index.html
http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks2/maths/fractions/level4.htmhttp://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks2/maths/fractions/level4.htmhttp://www.visualfractions.com/GrampStrict/findgrampystrict.htmlhttp://www.visualfractions.com/GrampStrict/findgrampystrict.htmlhttp://www.funbrain.com/linejump/index.htmlhttp://www.funbrain.com/linejump/index.htmlhttp://www.visualfractions.com/GrampStrict/findgrampystrict.htmlhttp://www.visualfractions.com/GrampStrict/findgrampystrict.htmlhttp://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks2/maths/fractions/level4.htmhttp://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks2/maths/fractions/level4.htm -
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Rational Number ProjectAn ongoing research
project investigatingstudent learning and
teacher enhancementAdvocates teaching
fractions using a modelthat emphasizes multiplerepresentations andconnectionsamongdifferent representations.
In 2009 they publishedRNP Initial FractionIdeas containingfree
lesson and materialsteachers can use in theirclassrooms.
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IESInstitute of Education SciencesMission is to provide
rigorous and relevant
evidenceon which toground educationpractice and policy and
share this informationbroadly.
www.ies.ed.gov/
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Questions and Comments
Teachers need tounderstand the reasoningbehind computations thatinvolve fractions so theycan clearly and coherentlyexplain to students why
the procedures work, notjust the sequence of stepsto take.
www ies ed gov/
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