• symbols being used to represent quantities in a relationship:
  • in equations, such as N = 🚗 × 4:

• in formulas, such as A = b × h or P = b + b + h + h, where
  • h represents the height of a rectangle
  • b represents the base of a rectangle

• symbols being used as a variable to represent an unknown quantity:
  • N = 🚗 × 4, where N is the number of wheels and 🚗 is the number of cars:
    • if there are 3 cars, there must be 12 wheels
    • if there are 8 wheels, then there must be 2 cars
  • 🚗 =

• Symbols can be used to represent quantities that change or quantities that are unknown.
• An expression is a mathematical statement that involves numbers, letters, and/or operations, for example, a + 3.
• An equation is a statement of equality between two expressions, for example, 1a + 3 = 5 + 10.
• Formulas are a type of equation, for example, A = b × h.
• 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 of the aspects of mathematical modelling.
• In mathematics notation, variables are only expressed as letters or symbols. When coding, variables may be represented as words, abbreviated words, symbols, or letters.
• In an expression like 4a, it is understood that the operation between the 4 and the a is multiplication. In working with some technologies, 4a would need to be inputted as 4*a, in which the asterisk denotes multiplication. The forward slash (/) is used for division.

• equations:
  • m + 34 = 50
  • 21 + t = 28
  • h − 7 = 13
  • 16 = b − 8
  • 1 + w + 4 = 2 + 5
  • 10 + 3 = 4 + b + 7
  • 6k = 54
  • 2 × m + 1 = 19
  • 24 − 5 = 3 × L − 2
  • a ÷ 8 = 56
• solving equations using number sense and reasoning:

• solving equations using guess-and-check:
  • 2 × m + 1 = 19:
    • guess of 6: 2 × 6 + 1 = 12 + 1 = 13 (too low)
    • guess of 8: 2 × 8 + 1 = 16 + 1 = 17 (still too low)
    • guess of 9: 2 × 9 + 1 = 18 + 1 = 19 (right on)
• solving equations using a balance model:
  • 3 × n + 4 = 28:
    • Step 1:
      • Represent the equation using 3 n’s plus 4 (circles) on the left side and 28 (circles) on the right side:

• Step 2:
  • Remove 4 circles from the left side to leave only n’s and remove 4 circles from the right side to maintain the balance:

• Step 3:
  • Rearrange the boxes and circles so that each box is opposite an equal amount of circles:

• Step 4:
  • Isolate one box to determine the number of circles for one n:

• Equations are mathematical statements such that the expressions on both sides of an equal sign are equivalent.
• In equations, symbols are used to represent unknown quantities.

Note

• To solve an equation using guess-and-check, the process is iterative. The unknown value is estimated and then tested. Based on the result of the test, the guess is refined to get closer to the actual value.
• To solve an equation using a balance model, the expressions are visually represented and are manipulated until they are equivalent.

• inequalities and solutions:
  • m ≤ 7:

• a + 3 ≤ 10:

• a is the set of numbers {0, 1, 2, 3, 4, 5, 6, 7}:

• 20 − m > 15:

• m = {0, 1, 2, 3, 4}:
  

• Inequalities can be solved as equations, but the values that result must be tested to determine if they hold true for the inequality.
• A number line shows the range of values that hold true for an inequality. An open dot on a number line is used when an inequality involves “less than” or “greater than”, and a closed dot is used when it also includes “equal to”.
• Number lines help students notice the range of values that hold true for inequalities.