C2. Equations and Inequalities
Specific Expectations
Variables
C2.1
identify when symbols are being used as variables, and describe how they are being used
- variables as 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 |
- the symbol □ is used to represent a specific unknown (placeholder) in this skip-counting pattern:
- 2, □, 6, 8
- the symbol ❤ is used to represent a quantity that varies, such as the distance between students’ homes and a particular location (e.g., the local library):
- Symbols can be used to represent quantities that change or quantities that are unknown.
- 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 find different addends for a sum no more than 100, 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, as opportunities arise, discuss quantities that are unknown or quantities that vary. For example, the lunch break is always 40 minutes, but the time each student takes to eat their lunch will vary. Another example is that the weekend is almost always two days (Saturday and Sunday), but the number of days until the weekend will vary depending on what day it is.
Have students determine what the symbol stands for in a variety of scenarios, and discuss situations where the value is variable, such as:
- 5, 10, 15, □, 25, 30:
- The square represents 20, and only 20.
- The ❤ represents the distance from home to the local library, which will vary depending on where you live.
- 2 + 5 = 🐱:
- The 🐱 represents 7, and only 7.
- △ + △ = 6:
- The △ represents 3, and only 3.
- △ + □ = 6:
- The △ and the □ will vary, depending on the value of the other symbol.
Equalities and Inequalities
C2.2
determine what needs to be added to or subtracted from addition and subtraction expressions to make them equivalent
- using a balance model:
- using hops on a number line:
- using ten frames or sectioned circles:
- 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 ($$\neq$$) is a symbol denoting that the two expressions are not equivalent.
Note
- When using a balance model, the representations of the addition or subtraction expressions are manipulated until there is an identical representation on both sides of the balance.
- When using a balance scale, the objects on the scale are manipulated until the scale is level.
Provide students with opportunities to work with a number balance to compare two expressions. Have them first predict which addition or subtraction expression has a greater value, then represent each on the balance. Support them in recognizing that the balance will tip towards the expression of greater value. Ask them to add or remove the balance masses until they have created balance, or equivalency.
Have students represent two addition expressions using a number line or a sectioned circle for each expression. Ask them to predict the amount that needs to be added for the two expressions to be equal. Have them verify their prediction by using a structured number line.
C2.3
identify and use equivalent relationships for whole numbers up to 100, in various contexts
- possible contexts:
- decomposing numbers, such as:
96 = 50 + 40 + 6 - composing numbers, such as:
50 + 40 + 6 = 96 - representing numbers in a variety of ways; for example, 96 can be represented as:
- decomposing numbers, such as:
- using the commutative property to add a smaller number onto a larger number, such as:
8 + 52 = 52 + 8 - using the associative property to make tens, such as:
(11 + 12) + 8 = 11 + (12 + 8) - identifying equivalent monetary values of coins and bills (e.g., 1 ten-dollar bill is equivalent to 2 five-dollar bills, 1 five-dollar bill and 5 loonies, 10 loonies, and so on)
- 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 and associate properties of addition are founded on equivalency.
Have students dedicate a section of their “learning log” to situations that involve equivalence. From time to time, have them record these situations. For example, students may be asked to represent a number such as 96 in more than one way, using tools. They should indicate their strategy; for example, “I swapped some of the tens for 2 fives because I know that 10 = 5 + 5.” They should then label each representation using symbols; for example, 90 + 6 and 50 + 40 + 6:
When students are adding whole numbers, have them demonstrate equivalence using the commutative property of addition, such as 8 + 52 = 52 + 8, and ask them to describe how the expressions are equivalent. In a similar way, have students demonstrate equivalence using the associative and commutative properties of addition. For example, 12 + 11 + 8 is the same as 12 + 8 + 11. Students are not expected to be able to name or define these properties, but they should be able to use them appropriately. For example, they can use the term “making tens” when they rearrange numbers using the associative and commutative properties.