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
2The 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?
3Connecting 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
4Laying the Foundation for Algebra
- Encourage young students to make algebraic
generalizations without necessarily using
algebraic notation. - NCTM Algebra Research Brief
5Foundation 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.
6From 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
7Types 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
8What 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
9What 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?)
10Ordering Fractions
Fractions with the same denominator can be
compared by their numerators.
11Ordering Fractions
Fractions with the same numerator can be compared
by their denominators.
12Ordering Fractions
Fractions close to a benchmark can be compared by
finding their distance from the benchmark.
13Ordering Fractions
Fractions close to one can be compared by finding
their distance from one.
14Strategies for Ordering Fractions
- Same denominator
- Same numerator
- Benchmarks close to 0, 1, 1/2
- Same number of missing parts from the whole
(Residual strategy)
15Clothesline Fractions Activity
16Clothesline Fractions Activity
17Clothesline Fractions Activity
18Clothesline Fractions Activity
19Clothesline Fractions Activity
20Clothesline Fractions Activity
X ,
21Clothesline Fractions Activity
(where x ?0)
22Clothesline Fractions Activity
23Clothesline Fractions Activity
(where x ?0)
24Clothesline Fractions Activity
(where x ?0)
25Clothesline Fractions Activity
(where x ?-1)
26The Number Line Helps Develop
- Fraction sense
- Benchmarks
- Relative magnitude of fractions
- Algebraic connections
27What 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.
28Contact Usnbezuk_at_mail.sdsu.edusklass_at_projects.s
dsu.eduSlides and Fraction Tents Master are
available athttp//pdc.sdsu.edu(click on PDC
Presentations)