1. Quadratic Patterns
Big Ideas:
Comparing mathematical relationships algebraically, numerically, verbally, or
pictorially/concretely, helps us see that there are classes of relationships with common
characteristics and helps us describe each member of the class.
Many equivalent representations can describe the same situation or generalization. Each
representation may give a different insight into certain characteristics of the situation or
generalization.
Mathematical Processes: [C, CN, PS, R, V]
Materials:
Colour tiles, o/h colour tiles
Getting Ready
Display first 3 figures in a pattern using colour tiles. (L-shape, see below)
th
Have Ss build 4 figure and describe what they see.
Ss may see corner tile and two arms. Replace blue corner tiles with red tiles.
“There is a relationship btwn the figure number and the number of tiles.
Knowing this relationship will allow us to get information about the number of
tiles for a given figure number (without having to build it).”
“How can we organize this information?” (table of values – figure number/red
tiles/blue tiles/total)
Extend to 10, 100, f.
Write an equation that tells how to calculate the number of tiles.
Before
“Looking at the colour tiles, not just the table, can help us write this equation.”
The equation is t = 2f + 1 (b = 2n ,r = 1). Emphasize functional rather than
recursive relation.
“Patterns can be visualized in multiple ways. Can you see the pattern in a
different way?”
Ss may see row across bottom and a column on the left. Replace blue tiles in
first column with red tiles.
Organize this info in table. Extend. Generalize. The equation is t = f + 1 + f(r =
f + 1, b = f).
“What do you notice?” (Expressions are equivalent! They must be. The
number of tiles is the same, no matter how we visualize the pattern.)

3. During Let the Students Do the Math
Have Ss choose one (or more) of the other patterns and write equations
that tell how to calculate the number of tiles.
Note: patterns/equations are quadratic.
“Explain your thinking using pictures, words, and numbers.”
“How else could you visualize it?”
Observe & assess – look for different strategies/visualizations (e.g., square,
linear, constant terms; areas of rectangles; addition/subtraction of
linear/constant terms)
For Ss having difficulty, encourage use of two (or more) colours. Have
students look for patterns for one colour.
Extension: Write an equation. Create a pattern that matches this equation in
as many ways as you can.
After Bringing it Together
Show & share student solutions.
“What patterns did you discover?”
Teacher helps students make connections to big ideas.
“How do you know the equations are correct/match?”
“What do the numbers in the equations tell you about how the pattern grows?”