C2. Equations and Inequalities
Specific Expectations
Variables
C2.1
identify and use symbols as variables in expressions and equations
- symbols being used to represent quantities in a relationship:
- in equations, such as N = 🚗 × 4:
Number of Cars (🚗) |
Number of Wheels (N) |
1 |
4 |
2 |
8 |
3 |
12 |
4 |
16 |
- 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
- 🚗 =
- N = 🚗 × 4, where N is the number of wheels and 🚗 is the number of 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.
Have students dedicate a section of their “learning log” to variables and symbols. Throughout the year, have them enter situations they encounter where variables or symbols are used in equations to represent quantities that are unknown or in expressions to represent quantities that vary. Examples may include a missing element in a pattern that is represented with a symbol, an unknown value needed to make a statement true, the use of letters to label the sides of shapes that can have any value, or their use in formulas such as A = bh.
Throughout the year, as opportunities arise, discuss the ways symbols are used as variables. For example, a symbol in a pattern might be used to represent a missing element. Another example is when a symbol is used to represent a specific unknown, such as 3 × 7 = 30 − □.
Equalities and Inequalities
C2.2
solve equations that involve whole numbers up to 50 in various contexts, and verify solutions
- 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:
25 + 5a = 50
25 + 5a = 25 + 25
5a = 25
a = 5
- 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)
- 2 × m + 1 = 19:
- 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 1:
- 3 × n + 4 = 28:
- 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.
Provide students with relevant contexts that require them to determine the unknown value that makes the two expressions on either side of the equal sign equivalent. For example, there are 2 boxes of muffins, each containing the same number of muffins. One box contains 12 carrot muffins and 8 blueberry muffins. The other box contains oatmeal muffins and 4 bran muffins. How many oatmeal muffins are there? [12 + 8 = n + 4]
Students can use various strategies to find the unknown. If they use a guess-and-check strategy, encourage them to be strategic in their choices and continue to adapt their choices as they check their answer.
They may also use a balance model, in which they represent each expression and manipulate the expressions until they isolate the variable (see the example that highlights this process). This process helps to build understanding of algebraic manipulations, which is the eventual goal. Ensure that students make a statement about what the variable represents, for example, n = 16.
Provide students with opportunities to solve a range of equations that have the variable in different locations, for example, m + 4 = 17, 4 + m = 17, 17 = 4 + m.
C2.3
solve inequalities that involve addition and subtraction of whole numbers up to 20, and verify and graph the solutions
- inequalities and solutions:
- m ≤ 7:
- a + 3 ≤ 10:
a |
+ 3 |
≤ 10 |
0 | 3 | Yes |
1 | 4 | Yes |
2 | 5 | Yes |
3 | 6 | Yes |
4 | 7 | Yes |
5 | 8 | Yes |
6 | 9 | Yes |
7 | 10 | Yes |
8 | 11 | No |
9 | 12 | No |
- a is the set of numbers {0, 1, 2, 3, 4, 5, 6, 7}:
- 20 − m > 15:
20 | − m | > 15 | |
20 | − 0 | = 20 | Yes |
20 | − 1 | = 19 | Yes |
20 | − 2 | = 18 | Yes |
20 | − 3 | = 17 | Yes |
20 | − 4 | = 16 | Yes |
20 | − 5 | = 15 | No |
20 | − 6 | = 14 | No |
- 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.
Provide each student with a portable whiteboard or large paper. Present a variety of scenarios where there might be more than one answer, such as those below. Ask students to record their solutions on their whiteboards.
As you work through each scenario with the class, invite students to come to the front of the class and show their answers. Support students in recognizing that there is more than one right answer for each of these scenarios.
- If you had to dribble the ball no more than 10 times before you could throw it, how many times could you dribble the ball?
- If you had to find an object in the school that is shorter than 5 m but longer than 1 m, how long could your object be?
- If you were to draw a line that is longer than 15 cm but shorter than 20 cm, how long could that line be?
- A parking garage has a speed limit of 10 km per hour. What speed could someone drive without breaking the speed limit?
- If you were to walk partway to the door and the door is 20 tiles away, which tile could you walk to?
Have students solve a variety of problems that involve inequalities. For example:
- At a grocery store, the express line can be used by shoppers who have 10 or fewer items. What number of items can you have in order to use that line? Represent your solution on a number line.
- There are swimmers in the pool. 4 more swimmers jump in. Now there are fewer than 10 swimmers in the pool. How many swimmers were in the pool to begin with? Represent your solution on a number line.
- Zheng borrowed 5 books from the library and can sign out as many as 13 books at a time. On a number line, show how many more books Zheng can borrow. Represent your solution on a number line.
- Alex has $9 and wants to buy an eraser and some pencils. The eraser costs $3 and the pencils cost $2 each. How many pencils can Alex buy? Represent your solution on a number line.
Ask students to create their own problems that involve solving an inequality. Have them exchange their problems with a partner or a group, or do this as a whole-class activity.