Algebraic thinking: generalizations, patterns and functions
Generalizations, Patterns &
Van De Walle, Karp & Bay-Williams
ED - 616
What was your experience of algebra in school?
Why do we need to teach algebraic thinking?
What is Algebraic Thinking?
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
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
Generalization from Arithmetic and Patterns
When 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 Symbols
A strong understanding of mathematical symbols is
essential for students to progress in algebra. Students need
the equals sign = sign represents equivalence (not the
Equations with variables allow for the expression of
Variables (like x) can represent a unique unknown value
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 numbers
Can you ever get an integer by dividing an odd number by 2?
Commutative, associative and distributive
True or False? 7 x 9 = 7 x 3 x 3
True or False? 6 + 2 + 3 = 2 + 3 + 6
Patterns and Functions
Understanding patterns and functions builds algebraic
reasoning skills (p. 267). We want students to:
Identify and extend repeating patterns beginning in
Explore growing patterns from 4th grade - middle
Interpret and construct graphs using real functions K -
Use linear functions from middle grades and
understand rate of change and proportional and
Sketch a graph for the value of a lovingly
maintained 1969 Volkswagen Kombi van
from its time of purchase to the present
“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.)
If John has 9 apples and Julia gives him 3 more,
how many apples does he have now?
1. Emphasize appropriate algebra vocabulary
2. Multiple representations for functions
c. verbal description
3. “Algebrafy” with measurement