Linear Relations

Strand: Patterns and Relations (Patterns)
Outcomes: 1, 2

Step 1: Identify Outcomes to Address

Guiding Questions

What do I want my students to learn?
What can my students currently understand and do?
What do I want my students to understand and be able to do, based on the Big Ideas and specific outcomes in the program of studies?

See Sequence of Outcomes from the Program of Studies

Strand: Patterns and Relations (Patterns)

Grade 6

Grade 7

Grade 8

Specific Outcomes
1.	Represent and describe patterns and relationships, using graphs and tables.
2.	Demonstrate an understanding of the relationships within tables of values to solve problems.

Specific Outcomes
1.	Demonstrate an understanding of oral and written patterns and their equivalent linear relations.
2.	Create a table of values from a linear relation, graph the table of values, and analyze the graph to draw conclusions and solve problems.

Specific Outcomes
1.	Graph and analyze two-variable linear relations.

Big Ideas

Mathematics is often referred to as the science of patterns. Patterns permeate every mathematical concept and are found in everyday contexts. The various representations of patterns, including symbols and variables, provide valuable tools in making generalizations of mathematical relationships. Some characteristics of patterns include the following.

There are different types of patterns that can be modelled in a variety of ways.
Patterns include repetitive patterns and growth patterns.
Patterns using concrete and pictorial representations can be translated into patterns using numbers to represent the quantity in each step of the pattern. The steps in a pattern are often translated as the sequence of items in the pattern.
Growth patterns are evident in a wide variety of contexts, including arithmetic and geometric situations. Arithmetic patterns are formed by adding or subtracting the same number each time. Geometric patterns are formed by multiplying or dividing by the same number each time.
Patterns are used to generalize relationships: recursive and functional.
The description that represents how a pattern changes from one step to another step is a recursive relationship and describes the evolution of the pattern; i.e., an expression that explains what you do to the previous number in the pattern to get the next one (Van de Walle and Lovin 2006).
The description that determines the number of elements in a step by using the number of the step is a functional relationship and is known as a pattern rule; i.e., an expression that explains what you do to the step number to get the value of the pattern for that step (Van de Walle and Lovin 2006).
Variables are used to describe generalized relationships in the form of an expression or an equation (formula) (Van de Walle and Lovin 2006).

Algebraic reasoning is directly related to patterns because this reasoning focuses on making generalizations based on mathematical experiences and recording these generalizations by using symbols or variables (Van de Walle and Lovin 2006).

"A variable is a symbol that can stand for any one of a set of numbers or objects" (Van de Walle and Lovin 2006, p. 274). Variables are used in different ways as mathematical literacy is developed. They can be used:

in equations as unknown numbers; e.g., 4 + x = 6
to describe mathematical properties; e.g., a + b + b + a
to describe functions; e.g., input (n): 1, 2, 3, 4 . . . output (3n): 3, 6, 9, 12 …
in formulas to show relationships; e.g., C = πd.

Adapted from W. George Cathcart, Yvonne M. Pothier and James H. Vance, Learning Mathematics in Elementary and Middle Schools (2nd ed.) (Scarborough, ON: Prentice-Hall Canada, 1997), pp. 352–353. Adapted with permission from Pearson Education Canada.