math institute teaching fractions

5/22/2018 Math Institute Teaching Fractions

1/35

Grades 3 - 5

Michele Lippens Special Educator

Cambridge PublicSchools


2/35

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


3/35

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)


4/35

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.


5/35

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.


6/35

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


7/35

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


8/35

Did this everhappen to

you?


9/35

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


10/35

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


11/35

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)


12/35

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


13/35

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


14/35

Same NumeratorVisual ModelThis model demonstrates reasoning about the size ofeach part. Students need to shift their thinking fromcomparing whole numbers.

3/8

3/10


15/35

Transitive Strategy: Benchmark

When students use the benchmark of and onethey are using the transitive property.


16/35

ActivityComparing Fractions

(Transitive Strategy)Which Fraction is greater:

14/24 or 17/36



17/35

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



18/35

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.


19/35

ActivityComparing Fractions

(Residual Strategy)

Which fraction is greater?

3/4 and 7/8


20/35

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?


21/35

Reason Abstractly and

Quantitatively


22/35

Make Sense of Problems Teaching students

both HOW and WHYcomputational

procedures workresults in greatermath proficiency.~Tim Lamarre


23/35

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?


24/35

Develop an Understanding of Fractions

as Numbers

Many experiences

withphysical

models are

needed to

overcome the

influence of

childrens wholenumber thinking.


25/35

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


26/35

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..


27/35

Attend to Precision


28/35

Fractions have MAGNITUDE Fractions are NUMBERS

Clearly illustrate the magnitude of fractions


29/35

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


30/35

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.


31/35

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/


32/35

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


33/35

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.


34/35

IESInstitute of Education SciencesMission is to provide

rigorous and relevant

evidenceon which toground educationpractice and policy and

share this informationbroadly.

www.ies.ed.gov/


35/35

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/

math institute teaching fractions

Documents