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# Preservation of Equality

Strand: Patterns and Relations (Variables and Equations)
Outcome: 3

## 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?
Strand: Patterns and Relations (Variables and Equations)

 Specific Outcomes 5. Demonstrate and explain the meaning of preservation of equality, concretely and pictorially.

 Specific Outcomes 3. Demonstrate an understanding of preservation of equality by: modelling preservation of equality, concretely, pictorially and symbolically applying preservation of equality to solve equations.

 Specific Outcomes 2. Model and solve problems concretely, pictorially and symbolically, using linear equations of the form: ax = b x/a = b, a≠0 ax + b = c x/a + b = c, a≠0 a(x + b) = c where a, b and c are integers.

### Big Ideas

Equality and inequality express relationships between quantities. When the quantities balance, there is equality. The equal sign is a symbol that indicates the quantity on the left side of the sign is the same as the quantity on the right. When there is an imbalance, there is inequality. The symbols on either side of the equality or inequality represent a quantity; e.g., 2 + 3 and 2n + 4 are both expressions for numbers.

The National Council of Teachers of Mathematics states, "In understanding equality, one of the first things students must realize is that equality is a relationship, not an operation" (2000–2007). Students often think of the equal sign as a symbol that tells them to do something or find the answer, but "they should come to view the equals sign as a symbol of equivalence and balance" (NCTM 2000, p. 39). For example, 5 + 2 = ? means that adding 2 to 5 results in 7. When students have this "operator view of equality," they have difficulty making sense out of a number sentence like 8 + 4 = ? + 5.

Liping Ma views the equal sign as "the soul of mathematical operations. In fact, changing one or both sides of an equal sign for certain purposes while preserving the 'equals' relationship is the 'secret' of mathematical operations" (1999, p. 111).

Patterns and Pre-algebra, Grades 4–6 discusses the various relationships that can be shown using equality.

Equality and inequality between quantities can be considered as:

• whole to whole relationships (five red chips = five blue chips or 5 = 5)
• part-part to whole relationships (3 + 5 = 8)
• whole to part-part relationships (8 = 3 + 5)
• part-part to part-part relationships (4 + 4 = 3 + 5).

Reproduced from Alberta Education, Patterns and Pre-Algebra, Grades 4–6 (unpublished workshop handout) (Edmonton, AB: Alberta Education, 2007), p. 54.

Compensation is used in deciding if the expressions on both sides of the equal sign represent the same quantity; e.g., 38 + 72 = 40 + 70 is a true statement because the 2 that is added to 38 to make 40 is compensated by subtracting 2 from 72 to make 70. Similarly, 85 – 28 = 87 – 30 is a true statement because the difference is constant; i.e., 2 is added to each number on the left side of the equal sign to make the expression on the right side so the difference between the two numbers remains the same.

Solving equations requires that the balance of the equation is maintained so that the expressions on either side of the equal sign represent the same quantity. Relations between the expressions on either side of the equal sign are examined and used to simplify the process; e.g., if a quantity is added to one side of the equation then, to maintain equality, the same quantity must be added to the other side of the equation. Similarly, the equality or balance must be maintained when a number is subtracted, multiplied or divided.