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Making Sense of Fractions: Laying the Foundation for Success in Algebra

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How can developing fraction reasoning help students to. reason algebraically? The Big Questions ... Contact Us: nbezuk_at_mail.sdsu.edu. sklass_at_projects.sdsu.edu ... – PowerPoint PPT presentation

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Title: Making Sense of Fractions: Laying the Foundation for Success in Algebra


1
  • Making Sense of Fractions Laying the Foundation
    for Success in Algebra
  • Nadine Bezuk and Steve Klass
  • NCTM Annual Conference--Salt Lake City
  • April 10, 2008

2
The Big Questions
  • What makes fractions so difficult for students?
  • What do students need to know and be able to do
    so they can reason with fractions?
  • How can developing fraction reasoning help
    students to reason algebraically?

3
Connecting Arithmetic and Algebra
  • If students genuinely understand arithmetic at a
    level at which they can explain and justify the
    properties they are using as they carry out
    calculations, they have learned some critical
    foundations of algebra.
  • Carpenter, Franke, and Levi, 2003, p. 2

4
Laying the Foundation for Algebra
  • Encourage young students to make algebraic
    generalizations without necessarily using
    algebraic notation.
  • NCTM Algebra Research Brief

5
Foundation for Fraction Reasoning
  • Fraction concepts and number sense about
    fractions
  • Equivalence
  • Order and comparison
  • Meaning of whole number operations
  • Students need to understand these topics well
    before they can be successful in operating with
    fractions.
  • Students need to be successful with fraction
    reasoning and operations if we want them to have
    success in transitioning to algebraic thinking.

6
From the NCTM Focal Points Relating Fractions
and Algebra
  • Grade 3 - Foundational fraction concepts,
    comparing, ordering, and equivalence. . . They
    understand and use models, including the number
    line, to identify equivalent fractions.
  • Grade 4 - Decimals and fraction equivalents
  • Grade 5 - Addition and subtraction of fractions
  • Grade 6 - Multiplication and division of
    fractions
  • Grade 7 - Negative integers
  • Grade 8 - Linear functions and equations

7
Types of Models for Fractions
  • Area/region
  • Fraction circles, pattern blocks, paper folding,
    geoboards, fraction bars, fraction strips/kits
  • Set/discrete
  • Chips, counters, painted beans Length/linear
  • Linear
  • Number lines, rulers, fraction bars, fraction
    strips/kits

8
What Should Students Understand about Fraction
Concepts
  • Meaning of the denominator (number of equal-sized
    pieces into which the whole has been cut)
  • Meaning of the numerator (how many pieces are
    being considered)
  • The more pieces a whole is divided into, the
    smaller the size of the pieces

9
What is Equivalence, Anyway?
  • Equivalence means equal value
  • A fraction can have many different names
  • Understanding that 1/2 is equivalent to many
    other fractions helps learners to use that
    benchmark
  • Simplify when and why
  • (does simplify mean reduce?)

10
Ordering Fractions

Fractions with the same denominator can be
compared by their numerators.
11
Ordering Fractions

Fractions with the same numerator can be compared
by their denominators.
12
Ordering Fractions

Fractions close to a benchmark can be compared by
finding their distance from the benchmark.
13
Ordering Fractions

Fractions close to one can be compared by finding
their distance from one.
14
Strategies for Ordering Fractions
  • Same denominator
  • Same numerator
  • Benchmarks close to 0, 1, 1/2
  • Same number of missing parts from the whole
    (Residual strategy)

15
Clothesline Fractions Activity
16
Clothesline Fractions Activity
17
Clothesline Fractions Activity
18
Clothesline Fractions Activity
19
Clothesline Fractions Activity
20
Clothesline Fractions Activity
X ,
21
Clothesline Fractions Activity
(where x ?0)
22
Clothesline Fractions Activity
23
Clothesline Fractions Activity
(where x ?0)
24
Clothesline Fractions Activity
(where x ?0)
25
Clothesline Fractions Activity
(where x ?-1)
26
The Number Line Helps Develop
  • Fraction sense
  • Benchmarks
  • Relative magnitude of fractions
  • Algebraic connections

27
What Should Kids Know?
  • Fractions arent just between zero and one they
    live between all the numbers on the number line
  • A fraction can have many different names
  • There are more strategies than just finding a
    common denominator for comparing and ordering
    fractions
  • Fractions can be ordered on a number line just
    like whole numbers.
  • The thinking involved when placing fractions on a
    number line can be symbolized algebraically.

28
Contact Usnbezuk_at_mail.sdsu.edusklass_at_projects.s
dsu.eduSlides and Fraction Tents Master are
available athttp//pdc.sdsu.edu(click on PDC
Presentations)
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