## C2. Equations and Inequalities

### Specific Expectations

#### Variables

C2.1

identify quantities that can change and quantities that always remain the same in real-life contexts

**quantities that remain the same and quantities that change:**

Quantities that Remain the Same (constants) |
Quantities that Change (variables) |

24 hours in a day |
number of hours spent playing |

12 months in a year |
number of months until a special event |

number of cents in a dollar |
number of cents to buy something |

number of wheels on a tricycle |
number of tricycles |

- Quantities that can change are also referred to as “variables”.
- Quantities that remain the same are also referred to as “constants”.

*Note*

- Identifying quantities in real life that stay the same and those that can change will help students understand the concept of variability.
- Identifying what is constant and what changes is one aspect of mathematical modelling.
- When students create models of 10 by adding numbers (terms), they are implicitly working with variables. These terms are variables that can change (e.g., in coding, a student’s code could be TotalSteps = FirstSteps + SecondSteps).
- In mathematics notation, variables are only expressed as letters or symbols. When coding, variables may be represented as words, abbreviated words, symbols, or letters.
- Students are also implicitly working with variables as they are working with attributes (e.g., length, mass, colour, number of buttons), as the value of those attributes can vary.

Throughout the year, where possible, address the concept of variability by identifying quantities that stay the same and quantities that can change. For example, when students are working with time, they can identify the elements of time that are always the same, such as 24 hours in a day. Then discuss how this is different from the number of hours in a day they might play, because the number of hours they might play can change from one day to another.

#### Equalities and Inequalities

C2.2

determine whether given pairs of addition and subtraction expressions are equivalent or not

**showing****equivalence****using a****balance model****:**

**showing equivalence using a number line:**

**showing equivalence using****ten frames****or sectioned circles:**- “8 is always 2 less than 10”:

**showing equivalence using a tape diagram (****part-whole****model):**

- Numerical expressions are equivalent when they produce the same result, and an equal sign is a symbol denoting that the two expressions are equivalent.
- Numerical expressions are not equivalent when they do not produce the same result, and an equal sign with a slash through it () is a symbol denoting that the two expressions are not equivalent.

*Note*

- The equal sign should not be interpreted as the "answer", but rather, that both parts on either side of the equal sign are equal, therefore creating balance.

Provide students with a variety of addition and subtraction expressions, and have them represent the expressions using a number balance. Have them track which expressions are equivalent and which are not.

C2.3

identify and use equivalent relationships for whole numbers up to 50, in various contexts

- decomposing numbers, such as:
- 36 = 30 + 6

- composing numbers, such as:
- 30 + 6 = 36

- representing numbers in a variety of ways, such as representing 6 as:

- developing math facts, such as:
- 2 + 8 = 10, so 8 + 2 = 10 (commutative property):

- identifying equivalent monetary values of coins (e.g., 1 quarter is equivalent to 2 dimes and 1 nickel, or 5 nickels, or 1 dime and 3 nickels)

- When numbers are decomposed, the parts are equivalent to the whole.
- The same whole can result from different parts.

*Note*

- Many mathematical concepts are based on an underlying principle of equivalency.
- The commutative property is an example of an equivalent relationship.

Ask students to represent numbers in a variety of ways using manipulatives (e.g., buttons, beads, sticks, relational rods, tiles), and ask them to explain how their representations of the numbers are equivalent.

When students are developing a math fact, such as 3 + 5 = 8, ask them to show that when the addends are reversed, the sum is the same, that is, 5 + 3 = 8.