Nadine Bezuk & Steve Klass. How can we help students reason algebraically? How can developing fraction reasoning help students reason algebraically? What

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Nadine Bezuk & Steve KlassNadine Bezuk & Steve Klass

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?

The Big Questions

@ 2010 SDSU Professional Development Collaborative

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


Laying the Foundation for Algebra

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 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.

Foundation for Fraction Reasoning


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


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


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)

Considerations


Ordering Fractions

5

7 ,

3

7 ,

6

7Fractions with the same denominator can

be compared by their numerators.@ 2010 SDSU Professional Development Collaborative

Ordering Fractions

4

10 ,

4

37 ,

4

8Fractions with the same numerator can

be compared by their denominators.


Ordering Fractions

4

9 ,

8

15 ,

1

2Fractions close to a benchmark can be compared by finding their distance from

the benchmark.@ 2010 SDSU Professional Development Collaborative

12

Ordering Fractions

41

29

=7

1

21 5

=1

2The denominator is twice the value of the

numerator.@2009 SDSU Professional Development Collaborative

Ordering Fractions

99

100 ,

6

7 ,

15

16Fractions close to one can be compared

by finding their distance from one.


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”)


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..


“Clothesline” Fractions Activity

1

1

2,

3

4,



7

4 2

3 1 ,

@2009 SDSU Professional Development Collaborative

Free Online Resource

Conceptua Math

@ 2010 Conceptua Math


15

29

8

15,

4

9,



6

27

1

4,

3

13,


21

Ordering Fractions

41

29

The denominator is twice the value of the numerator.


x 2 −1

2x 2 −1

=

(where x ≠ -1 or 1)


−xX , 2x ,



1

2x

x

2,

(where x ≠ 0)

1

2x ,



(where x ≠ 0)

x + 1

x,

x −1

x,

x

x


“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.


Contact Us:[email protected]

[email protected]

FREE online fraction number line:

www.conceptuamath.com.

Slides and Fraction Tents

Master are available at:

www.sdsu-pdc.org


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Nadine Bezuk & Steve Klass. How can we help students reason algebraically? How can developing fraction reasoning help students reason algebraically? What