C2. Equations and Inequalities
Specific Expectations
Variables
C2.1
describe how variables are used, and use them in various contexts as appropriate
- variables used to represent quantities in a relationship:
- in properties:
- a + b = b + a (commutative property)
- in formulas:
- perimeter of a square = s + s + s + s:
- in properties:
- in tables of values:
Number of Bicycles (b) | Number of Wheels (Wheels = 2 × b) |
1 | 2 |
2 | 4 |
3 | 6 |
4 | 8 |
- variables (could be a symbol or letter) used to represent an unknown quantity:
- 8 + □ = 16
- 🔺 × 3 = 15
- n + 6 = 20 − 5
- determining whether two multiplication expressions are equivalent:
- determining whether an addition and a subtraction expression are equivalent, using code:
- These expressions are equivalent if both turtles end up the same distance from zero. Both turtles end up at 72, which is the same distance from zero:
- Variables are used in formulas (e.g., the perimeter of a square can be determined by four times its side length (s), which can be expressed as 4s).
- Variables are used in coding so that the code can be run more than once with different numbers.
- Variables are defined when doing a mathematical modelling task.
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 find different addends for a sum no more than 200, they are implicitly working with variables. These numbers are like 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, as opportunities arise, discuss the ways variables are used. For example, when finding the perimeter of a square, students could label its sides using the variable s. Another example is using a variable for a specific unknown, such as n + 2 = 7.
Equalities and Inequalities
C2.2
determine whether given sets of addition, subtraction, multiplication, and division expressions are equivalent or not
- determining whether two multiplication expressions are equivalent:
- determining whether an addition and a subtraction expression are equivalent, using code:
- These expressions are equivalent if both turtles end up the same distance from zero. Both turtles end up at 72, which is the same distance from zero:
- Numerical expressions are equivalent when they produce the same result, and an equal sign is the symbol denoting two equivalent expressions.
- Numerical expressions are not equivalent when they do not produce the same result, and an equal sign with a slash through it ($$\neq$$), is a symbol denoting that the two expressions are not equivalent.
- Various strategies can be used to determine whether expressions are equivalent. Visual representations of the expressions can be manipulated until they look the same or close to the same.
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.
Ask students to compare pairs of expressions involving the same operation; for example, 3 × 4 and 2 × 6 or 4 × 2 and 3 × 3. Have them use relational rods to represent the expressions and determine whether they are equivalent. If they are not, ask students to create two expressions that are equivalent and represent them with relational rods and with an equation.
Ask students to compare pairs of expressions that do not involve the same operation. For example, to compare 16 + 32 and 50 − 2, have them use code to model two dogs moving horizontally across the screen to determine whether they both end up the same distance away from zero.
C2.3
identify and use equivalent relationships for whole numbers up to 1000, in various contexts
- various contexts:
- recognizing equivalent expressions:
2 × 6 = 6 + 6
6 + 6 = 2 + 2 + 2 + 2 + 2 + 2 - representing numbers in a variety of ways, such as the following different ways to make 12:
- recognizing equivalent expressions:
- adding numbers using the commutative property:
- recognizing that the addends can be flipped to make the calculation easier; for example, 8 + 352 = 352 + 8
- multiplying numbers using the commutative property:
- using known facts to determine unknown facts; for example, 5 × 2 = 2 × 5
- adding using the associative and commutative properties:
- recognizing that addends can be rearranged to make the calculation easier, such as by making tens; for example, 263 + 364 + 7 = 263 + 7 + 364
- When numbers are decomposed, the sum of the parts is 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 properties of addition and multiplication are founded on equivalency.
Have students dedicate a section of their “learning log” to equivalence. Throughout the year, have them enter situations they encounter that involve equivalence. For example, they may represent a number such as 12 as the product of two numbers, using a visual. Have them describe that visual in words so that they notice the numbers that are being combined to create the equivalent representations, for example, three sets of 4-rods and four sets of 3-rods. When students are adding or multiplying whole numbers, have them show the equivalence using the commutative property and ask them to describe how the expressions are equivalent.
Ask students to represent a number such as 12 with relational rods. Then, ask them to represent that same number using multiplication, in more than one way, and label each representation using symbols.
When students are adding whole numbers, have them demonstrate equivalence using the commutative property of addition, for example 8 + 352 = 352 + 8, and ask them to describe how the expressions are equivalent. They should also be encouraged to use the commutative property of multiplication, for example 5 × 2 = 2 × 5. In a similar way, have them demonstrate equivalence using the associative and commutative properties of addition. For example, 263 + 364 + 7 is the same as 263 + 7 + 364. Students are not expected to be able to name or define these properties, but they should be able to use them appropriately.