Algebraic Thinking: Generalizations, Patterns & FunctionsVan De Walle, Karp & Bay-WilliamsChapter 14Gillian EdwardsED - 616

What was your experience of algebra in school? Why do we need to teach algebraic thinking?

What is Algebraic Thinking?Definition

Algebraic thinking or algebraic reasoning involves forming generalizations from experiences with numbers and computation, formalizing these ideas with the use of a meaningful symbol system and exploring the concepts of pattern and functions. (p. 254)

What do we want kids to be able to do?Kaput describes 5 forms of algebraic thinking (p. 254)

1.Making generalizations from arithmetic and patterns

2.Meaningful use of symbols

3.Study structure in the number system

4.Study patterns and functions

5.Mathematical modelling integrating the first 4 list items http://www.k5learning.com/math/

algebra

Generalization from Arithmetic and PatternsWhen students are able to make generalizations about solving a problem, they can apply that rule to other problems that have the same pattern.

Meaningful Use of SymbolsA strong understanding of mathematical symbols is essential for students to progress in algebra. Students need to understand

the equals sign = sign represents equivalence (not the answer is…)

Equations with variables allow for the expression of generalizations.

Variables (like x) can represent a unique unknown value OR

Variables may represent varying quantities

In the following expression, what number do you think belongs in the box? 8 + 4 = + 5

Structure in the Number System“Focus on exemplars to guide students to generalize” and make conjectures rather than presenting the properties as rules to be learned.”

Properties of odd and even numbersCan you ever get an integer by dividing an odd

number by 2?

Commutative, associative and distributive properties.

True or False? 7 x 9 = 7 x 3 x 3

True or False? 6 + 2 + 3 = 2 + 3 + 6

True or False? 8 - 4 = 4 - 8

Patterns and FunctionsUnderstanding patterns and functions builds algebraic reasoning skills (p. 267). We want students to:

Identify and extend repeating patterns beginning in kindergarten

Explore growing patterns from 4th grade - middle school

Interpret and construct graphs using real functions K - 8

Use linear functions from middle grades and understand rate of change and proportional and non-proportional situations.

Sketch a graph for the value of a lovingly maintained 1969 Volkswagen Kombi van from its time of purchase to the present

Kombi Value

Time

$

Mathematical Modelling“The process of beginning with real phenomena and attempting to mathematize them,” (p. 276). Students apply algebraic thinking to real contexts to come up with models (which we could also think of as rules or formulas.)

http://everydaymath.uchicago.edu/

If John has 9 apples and Julia gives him 3 more, how many apples does he have now?

Teaching Considerations1. Emphasize appropriate algebra vocabulary

2. Multiple representations for functions

a. context

b. table

c. verbal description

d. symbols

e. graphs

3. “Algebrafy” with measurement

4. Use real data (sports, census, class experiments) so that students know how to use “messy data.

(p. 277 - 283)