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Fractions 3-6

- Central Maine Inclusive Schools
- October 18, 2007
- Jim Cook

Workshop Goals

- What should students know and be able to do?
- What common difficulties do students have with

fractions? - What instructional practices can help students

understand fractions?

Understanding the meaning of fractions

- Too much school instruction on fractions is with

computation. When instruction focuses on the

meaning of fractions, it is often too brief and

superficial. As a result, students are forced to

learn rules and procedures for computations

without a sound understanding of what theyre

operating on.

What are appropriate goals for fraction

instruction?

- Name fractional parts of regions and sets
- Find fractions on a number line
- Represent fractional parts using standard

notation (proper and improper fractions, mixed

numbers) and also with concrete and pictorial

representations - Understand equivalence
- Compare and order fractions
- Make reasonable estimates with fractions
- Compute with fractions
- Solve problems with fractions

MLR goals for students

- 1997 MLR Grade 3
- Read, write model, and compare simple fractions

with denominators of 2, 3, 4 - 2007 MLR Grade 3
- Students recognize, name, illustrate, and use

simple fractions - Recognize, name, and illustrate fractions with

denominators from two to ten - Recognize, name, and illustrate parts of a whole
- Compare and order fractions with like numerators

or with like denominators

MLR goals for students

- 1997 MLR Grade 4
- Read, compare, order, classify, and explain

simple fractions through tenths - Solve real-life problems involving addition and

subtraction of simple fractions - 2007 MLR Grade 4
- Students understand, name, illustrate, combine,

and use fractions - Add and subtract fractions with like denominators

and use repeated addition to multiply a unit

fraction by a whole number - List equivalent fractions
- Represent fractions greater than one as mixed

numbers and mixed numbers as fractions - Connect equivalent decimals and fractions for

tenths, fourths, and halves in meaningful

contexts

MLR goals for students

- 1997 MLR Grade 5
- Read, compare, order, use, and represent simple

fractions (halves, thirds, fourths, fifths, and

tenths with all numerators) - Compute and model addition and subtraction with

simple fractions with common denominators - Create, solve, and justify the solution for

multi-step, real-life problems involving addition

and subtraction with simple fractions with common

denominators

MLR goals for students

- 2007 MLR Grade 5
- Students understand, name, compare, illustrate,

compute with, and use fractions - Add and subtract fractions with like and unlike

denominators - Multiply a fraction by a whole number
- Develop the concept of a fraction as division

through expressing fractions with denominators of

two, four, five, and 10 as decimals and decimals

as fractions

MLR goals for students

- 1997 MLR Grade 6
- Read, compare, order, use and represent fractions

(halves, thirds, fourths, fifths, sixths, eighths

and tenths with all numerators) - Compute and model all four operations with common

fractions - Create, solve, and justify the solution for

multi-step, real-life problems with common

fractions

MLR goals for students

- 2007 MLR Grade 6
- Students add, subtract, multiply, and divide

numbers expressed as fractions, including mixed

numbers - Students understand how to express relative

quantities as percentages and as decimals and

fractions - Use ratios to describe relationships between

quantities - Use decimals, fractions, and percentages to

express relative quantities - Interpret relative quantities expressed as

decimals, fractions, and percentages

Developing the meaning of fractions

- Fractions have four basic interpretations
- Measure
- Part of a region
- Part of a set
- Location on a number line
- Quotient
- 1/3 is what you get when you divide 1 pizza

between three people - Ratio
- The ration of pizzas to people is 1 to 3, 13,

1/3 - Operator
- There are 1/3 as many pizzas as people

Developing the meaning of fractions

- Students should understand all 4 interpretations

and how they are interrelated. Present fractions

using all four interpretations. - Using manipulatives is important
- Understanding fractions as parts of regions may

be easiest

Fractions as parts of regions

- NCTM lesson Fun with Fractions
- http//illuminations.nctm.org/LessonDetail.aspx?id

U113

Wipe-Out

- Version I
- Version II

Set modelfractions as parts of sets

- Get six green triangles
- Put them into two equal groups
- Put them into three equal groups
- Fraction Line-Up

Number Line modelfractions as locations on a

number line

- Find the number line master in your packet

Fraction Representations

- Using a variety of representations and asking

students to switch between them enhances

understanding. - Real objects
- Manipulatives
- Fraction circles
- Fraction rectangles
- Pattern blocks
- Drawings
- Words
- Symbols

Fraction Representations

- Ask students to make connections
- How are these different representations alike?

Fraction Representations

- Ask students to consider negative examples.
- Why is this not 1/3?
- Why is this not ¼?

Fraction Representations

- Have students generate their own fractional parts
- Make a fraction kit.

Fraction Kit Activities

- Cover Up
- Uncover
- Version I
- Version II
- Whats missing
- Comparing pairs

Fraction Equivalence

- Students should have strong conceptual

understanding of equivalent fractions based on

lots of experience. They can then relate that

understanding to numerical methods for generating

equivalent fractions. - Simplifying fractions
- Generating fractions with common denominators

Comparing and Ordering Fractions

- Strong concepts of the relative size of

fractions, based on experience with physical

objects and drawings, supports students number

sense. - Estimating with fractions depends on ideas about

the relative size of fractions. - Without sufficient experience with physical

objects, students make errors, often using

whole-number thinking when working with fractions.

Comparing and Ordering Fractions

- Students should consider these cases
- Same denominator
- Same numerator
- Fractions with different numerators and

denominators - Students might relate fractions to a benchmark

like ½.

Fractions as Quotients

- Try sharing 12 cookies between 4 people.
- Try sharing 3 cookies between 4 people.
- Use the cookie masters in your materials.
- Students have more success making thirds, fifths,

etc. if they use toothpics. - Try sharing 7 cookies between 3 people.
- Help students connect mixed numbers and improper

fractions.

Lesson Ideas for mixed numbers

- Ask students to share cookies between different

numbers of people. - Use cookie cutouts and glue.
- Dont use a numbers that are multiples.
- Do several examples.
- Share between three, four, and six people.

Operating on Fractions

- NCTM recommends using simple denominators that

can be visualized concretely or pictorially and

are apt to occur in real-world settings. - Emphasis in instruction must shift from learning

rules for operations to understanding fraction

concepts. - Begin by asking students to use fraction pieces

to add fractions.

Adding Fractions

- Pick 2
- Make a train with two pieces on your whole strip

that are not the same color. - Build another train the same length using pieces

that are all the same color. - Record.
- Try to build other one-color trains the same

length. For each, record.

Adding Fractions

- Help students make the connection between the

procedures for adding fractions and their

experience with manipulatives. They might even

invent rules for adding fractions!

Multiplying Fractions

- Use physical objects and drawings to develop

meaning. - Use whole number meaning for multiplication.
- Repeated addition
- Use the commutative property and of
- Use rectangles
- Help students develop the rules for themselves

Dividing Fractions

- Relate dividing fractions to dividing whole

numbers - 6 2 3
- How many times can I subtract 2 from 6?
- How many 2s are in 6?
- Check by multiplying
- means into groups of.
- 6 ½

Dividing Fractions

- Use fraction pieces
- ¾ ½
- how many ½s are in ¾?

Dividing Fractions

- True or False?
- You can multiply the dividend and divisor both

by the same number, and the answer stays the

same. - If you divide a number by one, you get the same

number.