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Making Sense of Fractions: Laying the Foundation for Success in Algebra
Nadine Bezuk and Steve KlassNCTM Annual Conference--Salt Lake City
April 10, 2008
2
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?
The Big Questions
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
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.
Foundation for Fraction Reasoning
6
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
From the NCTM Focal Points: Relating Fractions and Algebra
7
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
Types of Models for Fractions
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
5
7 ,
3
7 ,
6
7Fractions with the same denominator can
be compared by their numerators.
11
Ordering Fractions
4
10 ,
4
37 ,
4
8Fractions with the same numerator can
be compared by their denominators.
12
Ordering Fractions
4
9 ,
8
15 ,
1
2Fractions close to a benchmark can be compared by finding their distance from
the benchmark.
13
Ordering Fractions
99
100 ,
6
7 ,
15
16Fractions 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
1
1
2,
3
4,
16
“Clothesline” Fractions Activity
7
4 2
3 1 ,
17
“Clothesline” Fractions Activity
15
29
8
15,
4
9,
18
“Clothesline” Fractions Activity
6
27
1
4,
3
13,
19
“Clothesline” Fractions Activity
15
16
1
8,
5
6,
20
“Clothesline” Fractions Activity
−xX , 2x ,
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“Clothesline” Fractions Activity
1
2x
x
2,
(where x ≠0)
1
2x ,
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“Clothesline” Fractions Activity
1 - x
x +2 , x - 1,
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“Clothesline” Fractions Activity
3
x
1
x,
2
x,
(where x ≠0)
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“Clothesline” Fractions Activity
(where x ≠0)
x + 1
x,x −1
x,x
x
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“Clothesline” Fractions Activity
x - 1
x +1(where x ≠-1)
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The Number Line Helps Develop
Fraction sense
Benchmarks
Relative magnitude of fractions
Algebraic connections
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Fractions aren’t 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.
What Should “Kids” Know?
28
Contact Us:[email protected]@projects.sdsu.edu
Slides and Fraction Tents Master are available at:http://pdc.sdsu.edu
(click on “PDC Presentations”)