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

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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

- 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?

“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

Encourage young students to make algebraic generalizations without necessarily using algebraic notation.

NCTM Algebra Research Brief

- 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.

- 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

- 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

- 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

- “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”?)

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

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

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

Fractions close to one can be compared by finding their distance from one.

- Same denominator
- Same numerator
- Benchmarks: close to 0, 1, 1/2
- Same number of missing parts from the whole (”Residual strategy”)

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- Fraction sense
- Benchmarks
- Relative magnitude of fractions
- Algebraic connections

- 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.

Contact Us:[email protected]@projects.sdsu.eduSlides and Fraction Tents Master are available at:http://pdc.sdsu.edu(click on “PDC Presentations”)