1. 2010 NCTM Annual Conference–San Diego, CA
“Putting a Face on X”: Connecting Number Sense with Algebraic Reasoning
Nadine Bezuk & Steve Klass
2010 NCTM Annual Conference–San Diego, CA

2. @ 2010 SDSU Professional Development Collaborative
The Big Questions
How can we help students reason algebraically?
How can developing fraction reasoning help students reason algebraically?
What do students need to know and be able to do so they can reason with fractions?
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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
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4. Laying the Foundation for Algebra
Encourage young students to make algebraic generalizations without necessarily using algebraic notation.
NCTM Algebra Research Brief
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5. Foundation for Fraction Reasoning
Fraction concepts and number sense about fractions
Equivalence
Order and comparison
Students need to understand these topics well before they can be successful in reasoning with fractions.
Students need to be successful with fraction reasoning if we want them to have success in transitioning to algebraic thinking.
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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
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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
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8. @ 2010 SDSU Professional Development Collaborative
Considerations
Most children need to use concrete models over extended periods of time to develop mental images needed to think conceptually about fractions
Children benefit from opportunities to talk to one another and with their teacher about fraction ides as they construct their own understandings of fraction as a number
Students who don’t have mental images for fractions often resort to whole number strategies
(Post, et al. 1985, Cramer, et al. 1997)
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9. @ 2010 SDSU Professional Development Collaborative
Ordering Fractions
Fractions with the same denominator can be compared by their numerators.
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10. @ 2010 SDSU Professional Development Collaborative
Ordering Fractions
Fractions with the same numerator can be compared by their denominators.
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11. @ 2010 SDSU Professional Development Collaborative
Ordering Fractions
Fractions close to a benchmark can be compared by finding their distance from the benchmark.
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12. Ordering Fractions
The denominator is twice the value of the numerator. 12
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13. @ 2010 SDSU Professional Development Collaborative
Ordering Fractions
Fractions close to one can be compared by finding their distance from one.
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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”)
Be here by 15 minutes
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15. Introducing the “Clothesline” Activity
Task: Order fraction tents on a clothesline.
Mathematically justify the reasons for your ordering.
Materials: fraction tents and clothesline, string, yarn, etc..
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16. “Clothesline” Fractions Activity, ,
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17. “Clothesline” Fractions Activity,
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18. Free Online Resource
Conceptua Math
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19. “Clothesline” Fractions Activity, ,
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20. “Clothesline” Fractions Activity, ,
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21. Ordering Fractions =
(where x ≠ -1 or 1)
The denominator is twice the value of the numerator. 21
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22. “Clothesline” Fractions Activity,
X ,
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23. “Clothesline” Fractions Activity, ,
(where x ≠ 0)
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24. “Clothesline” Fractions Activity
(where x ≠ 0)
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25. @ 2010 SDSU Professional Development Collaborative
“Putting a Face on X”
Students often have difficulty making sense of what “X” means. These resources help students:
develop understanding of the meaning of variables, “putting a face on X”,
develop number sense about fractions, and
support algebraic reasoning.
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26. @ 2010 SDSU Professional Development Collaborative
Contact Us: FREE online fraction number line: Slides and Fraction Tents Master are available at:
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