Fractions 3-6

Central Maine Inclusive SchoolsOctober 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 they’re 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, 1:3, 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 model—fractions 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 model—fractions 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

What’s 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. Don’t 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 2’s are in 6?” Check by multiplying ÷ means “into groups of.”

6 ÷ ½ =

Dividing Fractions

Use fraction pieces ¾ ÷ ½ = “how many ½’s 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.”