Download Exploring Patterns and Algebraic Thinking

Patterning and Algebra
Suggested Reading (journal articles) posted on PJ Math Blog (Teaching Resources):
o “Developing Algebra Eyes and Ears”
o “Understanding the Equals sign”
o “Integrating Math and Music”
Before formal schooling, children develop beginning concepts related to patterns, functions, and algebra. They
learn repetitive songs, rhythmic chants, and predictive poems / stories that are based on repeating and growing
patterns. Their observations and discussions of how quantities relate to one another, lead to initial experiences
with functional relationships (early number sense). Their representations of mathematical situations using
concrete objects, pictures, and symbols are the beginnings of mathematical modeling. (NCTM Principles and
Standards for School Mathematics, 2000, pg 91).
Developmentally Appropriate Teaching Sequence
Recognize
→
Copy →
Extend
- Finish
- Middle
- Begin
→
Create
Patterning expectations should be explored and developed through integration within other strands (e.g.)
patterns in numbers, geometry, measurement and data management, etc.
Explore & Model Patterns in Multiple Ways:
Numbers
Colours
Shapes
Sizes
Words/stories
Concrete Materials
Movement
Body positions/dance
Sample Prompting Questions
•
What comes next in the pattern?
•
What do you notice about the pattern?
•
How could you describe the pattern?
•
How can you repeat or extend the pattern?
•
How are these patterns alike? Different?
–
2, 4, 6, 8,… AND 3, 5, 7, 9,…
–
2, 4, 6, 8, 10,… AND 2, 4, 8, 16, 32,…
•
Model the patterning in another way.
–
RED, BLUE, YELLOW, RED, BLUE, YELLOW …
•
What is the “pattern unit”? Describe the pattern unit.
–
ABBABBABB…
–
ABCCABCCA…
–
NOTE: A “pattern core/unit” must repeat TWICE!!
Sounds / Music
What does “Algebra” look like?
 Understanding Variable
 Solving and Creating Equations
 Exploring Growing & Shrinking Patterns
The main concepts include:
1. Equality (the meaning of the equals sign) … Use a “Balance” beginning in Grade 1!
2. Variable versus a Constant
 For example, for the “table problem” the number of tables is a “variable”, yet the “+2 people” at
the end of each group of tables is a “constant”. Both are represented in the following equation:
# of People = 2 (# of tables) + 2
3. Functional versus successor / recursive relationships
4. Representing relationships (or “rules”) in tables, graphs, words, pictorially, in real life contexts, and/or
using concrete materials.
x
1
2
3
4
y
9
19
29
39
Common responses include:
• “x goes up by ones and y goes up by tens” (successor/recursive relationship identified)
• “the x tells you what the next digit is in the next y”
• “to work out y, put a zero onto x and take away 1” (functional relationship identified, yet clarification
required)
• “take 1 off the x, and write a nine after it” (place value relationship, yet clarification required)
These descriptions can be used to extend the table to the 50th or 100th term … BUT, they cannot all be used to
write an algebraic equation (generalization)!
In the Junior grades “Successor / Recursive” relationships are important, but they also begin to explore
“Functional” relationships.