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## Fractions 3-6

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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 ... – PowerPoint PPT presentation

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Title: Fractions 3-6

1
Fractions 3-6
• Central Maine Inclusive Schools
• October 18, 2007
• Jim Cook

2
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?

3
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.

4
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

5
MLR goals for students
• Read, write model, and compare simple fractions
with denominators of 2, 3, 4
• 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

6
MLR goals for students
• Read, compare, order, classify, and explain
simple fractions through tenths
• Solve real-life problems involving addition and
subtraction of simple fractions
• 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

7
MLR goals for students
• 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
and subtraction with simple fractions with common
denominators

8
MLR goals for students
• 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

9
MLR goals for students
• 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

10
MLR goals for students
• 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

11
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

12
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

13
Fractions as parts of regions
• NCTM lesson Fun with Fractions
• http//illuminations.nctm.org/LessonDetail.aspx?id
U113

14
Wipe-Out
• Version I
• Version II

15
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

16
Number Line modelfractions as locations on a
number line
• Find the number line master in your packet

17
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

18
Fraction Representations
• Ask students to make connections
• How are these different representations alike?

19
Fraction Representations
• Ask students to consider negative examples.
• Why is this not 1/3?
• Why is this not ¼?

20
Fraction Representations
• Have students generate their own fractional parts
• Make a fraction kit.

21
Fraction Kit Activities
• Cover Up
• Uncover
• Version I
• Version II
• Whats missing
• Comparing pairs

22
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

23
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.

24
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.

25
Fractions as Quotients
• Try sharing 12 cookies between 4 people.
• Try sharing 3 cookies between 4 people.
• 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.

26
Lesson Ideas for mixed numbers
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.

27
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

28
• 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.

29
• Help students make the connection between the
procedures for adding fractions and their
experience with manipulatives. They might even

30
Multiplying Fractions
• Use physical objects and drawings to develop
meaning.
• Use whole number meaning for multiplication.
• Use the commutative property and of
• Use rectangles
• Help students develop the rules for themselves

31
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

32
Dividing Fractions
• Use fraction pieces
• ¾ ½
• how many ½s are in ¾?

33
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.