Patterns

BIG IDEAS:

(taken from “Big Ideas by Dr. Small”):

1. Patterns represent identified regularities. There is always an element of repetition.

2. Patterns can be represented in a variety of ways (i.e. pictures, words, graphs, sequences, tables).

3. Some ways of displaying data highlight patterns.

STUDENT LEARNING GOALS:

GOAL #1: I can represent a pattern as a sequence, table of values, picture, graph and in words.

• VIDEO: Introduction to Linear (Arithmetic) Sequences (Source: Khan Academy)

• VIDEO: Linear versus Non-Linear Functions (Source: Khan Academy)

• VIDEO: Extending Linear Patterns (Source: Khan Academy)

• VIDEO: Organizing Patterns in T-Tables (Source: Khan Academy)

• VIDEO: Modelling Patterns using Pictures (Source: Khan Academy)

• VIDEO: Equations, Table and Graphs (Source: BettsMath)

• GIZMO: Function Machine 2 (Source: ExploreLearning)

• PRACTICE: Math Patterns (Source: Khan Academy)

• PRACTICE: Extending Linear Sequences (Source: Khan Academy)

GOAL #2: I can determine the explicit pattern rule (algebraic equation) for a linear pattern.

• VIDEO: How to Find the nth term

• VIDEO: Developing the nth term (Source: Khan Academy)

• VIDEO: Determining Explicit Formulas (Source: Khan Academy)

• PRACTICE: Finding the nth term (Source: FlashMaths)

• PRACTICE: Finding Equation from a Graph – click on Formulas (Source: ThatQuiz)

• GIZMO: Arithmetic and Geometric Sequences (Source: ExploreLearning)

• GIZMO: Points, Lines and Equations (Source: ExploreLearning)

GOAL #3: I can make predictions from patterns.

• VIDEO: Graphing Linear Equations (Source: Khan Academy)

• PRACTICE: Plotting Equations – click on “Graphing” (Source: ThatQuiz)

CURRICULUM EXPECTATIONS:

• represent, through investigation with concrete materials, the general term of a linear pattern, using one or more algebraic expressions (e.g.,“Using toothpicks, I noticed that 1 square needs 4 toothpicks, 2 connected squares need 7 toothpicks, and 3 connected squares need 10 toothpicks. I think that for n connected squares I will need 4 + 3(n – 1) toothpicks, because the number of toothpicks keeps going up by 3 and I started with 4 toothpicks. Or, if I think of starting with 1 toothpick and adding 3 toothpicks at a time, the pattern can be represented as 1 + 3n.”)

• represent linear patterns graphically (i.e., make a table of values that shows the term number and the term, and plot the coordinates on a graph), using a variety of tools (e.g., graph paper, calculators, dynamic statistical software)

• determine a term, given its term number, in a linear pattern that is represented by a graph or an algebraic equation (Sample problem: Given the graph that represents the pattern 1, 3, 5, 7,…, find the 10th term. Given the algebraic equation that represents the pattern, t = 2n – 1, find the 100th term.).

• model linear relationships using tables of values, graphs, and equations (e.g., the sequence 2, 3, 4, 5, 6,… can be represented by the equation t = n + 1, where n represents the term number and t represents the term), through investigation using a variety of tools (e.g., algebra tiles, pattern blocks, connecting cubes, base ten materials) (Sample problem: Leah put $350 in a bank certificate that pays 4% simple interest each year. Make a table of values to show how much the bank certificate is worth after five years, using base ten materials to help you. Represent the relationship using an equation.)

• make connections between solving equations and determining the term number in a pattern, using the general term (e.g., for the pattern with the general term 2n + 1, solving the equation 2n + 1 = 17 tells you the term number when the term is 17)