C2. Equations and Inequalities
Specific Expectations
Variables and Expressions
C2.1
translate among words, algebraic expressions, and visual representations that describe equivalent relationships
- algebraic expressions:
- s + s + s + s
- 2l + 2w
- bh
- 3m + 2n − 1
- x − y
- translation among words, algebraic expressions, and visual representations of formulas:
| Words | Algebraic Expression | Visual Representation |
| The perimeter of a square is the sum of the lengths of the four equal sides. |
Perimeter: s + s + s + s or 4s |
|
| The perimeter of a triangle is the sum of the lengths of the three sides. |
Perimeter: a + b + c |
 |
| The area of a rectangle is equal to the length of its base times its height. |
Area: bh |
 |
| The area of a triangle is equal to the length of its base times its height divided by two. |
Area: bh ÷ 2 or $$A=\frac{b h}{2}$$ |
 |
- translation among words, algebraic expressions, and visual representations of relationships:
| Words | Algebraic Expression | Visual Representation |
| sum of two numbers | n + m |  |
| difference between two numbers | n − m |  |
| difference as distance | m − n |  |
| product of two numbers | xy |  |
| three more than a number | n + 3 |  |
| three less than a number | n − 3 |  |
| three more than double a number | 2n + 3 |  |
- Algebraic expressions are a combination of variables, operations, and numbers, such as 3a and a + b.
- Algebraic expressions are used to generalize relationships. For example, the perimeter of a square is four times its side length (s), which can be expressed as 4s.
- For expressions like 3a, it is understood that the operation between the number, 3, and the variable, a, is multiplication.
- When two expressions are set with an equal sign, it is called an equation.
Note
- The letter x is often used as a variable. It is important for students to know when it is being used as a variable.
- The letters used as a symbol are often representative of the words they represent. For example, the letters l and w are often used to represent the length and width of a rectangle, and also the formula for the area of a rectangle, A = lw.
- Many forms of technology require expressions like 3a to be entered as 3*a, where the asterisk is used to denote multiplication. The expression a ÷ 2 is entered as a/2.
- Words and abbreviated words are used in a variety of coding languages to represent variables and expressions. For example, in the instruction: “input ‘the side length of a square’, sideA”, the computer is defining the variable sideA and stores whatever the user inputs into its temporary location.
Individually, in pairs, or in a large group, have students match statements to algebraic expressions. A single copy of BLM: 5C2.1 Statements and Algebraic Expressions can be cut into pieces and given to individual students or to pairs, or one copy can be cut up and distributed among the whole group.
After all the paper strips have been matched, one statement and one expression will be left over. Ask students to write the missing expression for the statement and the missing statement for the expression.
| Statements | Algebraic Expressions |
| Multiply n by 4, and divide by 2. | $$\frac{4n}{2}$$ |
| Multiply n by 2, and add 13. | 2n + 13 |
| Subtract n from 12, and multiply by 4. | (12 – n) × 4 |
| Add n to 13, and divide by 2. | $$\frac{13 + n}{2}$$ |
| Multiply n by 2, and subtract from 13. | 13 – 2n |
| Add 3 to n, and divide by 2. | $$\frac{n + 3}{2}$$ |
| Multiply n by 4. | 4n |
| Subtract 2 from n, and multiply by 4. | 4(n – 2) |
| Divide n by 2, and subtract 4. | $$\frac{n}{2}-4$$ |
| Divide 12 by n, and add 2. | $$\frac{12}{n}+2$$ |
| Divide 12 by n, and multiply by 3. | 3 – $$\frac{12}{n}$$ |
| Multiply n by 2, and add 7. | 2n + 7 |
To introduce students to the concept of translating words into algebraic expressions, it can be helpful to begin with a few specific numerical examples and then generalize them using a variable. For example, to translate three more than a number, have students create a table with inputs 1 to 5 in column 1. In column 2, have them write the change that produces the output – in this example, “Add 3”. Then have them write the outputs, including for the general case, in column 3.
| Input | Change Rule | Output |
| 1 | Add 3 | 4 |
| 2 | 5 | |
| 3 | 6 | |
| 4 | 7 | |
| 5 | 8 | |
| n | n + 3 |
Ask students to express, in a variety of ways, the sum of the lengths of the four equal sides of a square. They may write the expression as side + side + side + side, s + s + s + s, or 4 × s. Discuss the fact that when a variable is multiplied by a number, it can be written with no space and no multiplication sign, that is, as 4s. Depending on the situation, students may also encounter other ways to express multiplication, such as 4*s in spreadsheets and coding. Discuss the fact that these are widely understood mathematical conventions.
In a similar way, ask students to represent the formula for the following in different ways and using different conventions:
- perimeter of a rectangle
- perimeter of a parallelogram
- perimeter of a rhombus
- area of a rectangle
- area of a square
- area of a rhombus
- area of a triangle
C2.2
evaluate algebraic expressions that involve whole numbers
- algebraic expressions:
- s + s + s + s
- 2l + 2w
- bh
- bh ÷ 2 or $$\frac{b h}{2}$$
- 3m + 2n − 1
- evaluating expressions:
- using substitutions:
| 2l + 2w | if l = 6 and w = 5: = 2(6) + 2(5) = 12 + 10 = 22 |
| $$\frac{b h}{2}$$ | if b = 4 and h = 3: = $$\frac{(4)(3)}{2}$$ = $$\frac{12}{2}$$= 6 |
- To evaluate an algebraic expression, the variables are replaced with numerical values and calculations are performed based on the order of operations.
Note
- When students are working with formulas, they are evaluating expressions.
- Replacing the variables with numerical values often requires the use of brackets. For example, the expression 4s becomes 4(s) and then 4(5) when s = 5. The operation between 4 and (5) is understood to be multiplication.
- Many coding applications involve algebraic expressions being evaluated, and this may be carried out in several steps. For example, the instruction: “input ‘the side of a square’, sideA” is instructing the computer to define the variable sideA and store whatever the user inputs into the temporary location called sideA. The instruction: “calculate 4*sideA, perimeterA” instructs the computer to take the value that is stored in “sideA” and multiply it by 4, and then store that result in the temporary location, which is another variable, called perimeterA.
Provide students with expressions, such as the following, that underscore the idea of equivalence and encourage students to read an expression for meaning before computing:
- 14 + 14 + 14 =
- 214 + 214 + 214 =
- (24 + 24 + 24 + 24) ÷ 4 =
- (14 + 28) ÷ 3 =
Model student thinking about these expressions on an open number line.
Have students evaluate different ways of writing the same formula. For example:
- P = l + l + w + w, where l represents the length and w represents the width
- P = 2l + 2w
- P = 2(l + w)
Ask students to explain why these formulas are equivalent.
Present students with scenarios such as the following, and ask them to explain their reasoning:
- If g – 227 = 543, what does g – 230 equal?
- If m + 7 = 12, what does m + 12 equal?
- If (a + 3) × 2 = 20, what does (a + 3) × 5 equal?
Present students with problems that involving working with formulas. For example:
- A teacher asked their students to find the formula for the perimeter of this parallelogram:
- Maude says that the perimeter is a + a + b + b.
- Jean says that the perimeter is 2a + 2b.
- Esteban says that the perimeter is 2(a + b).
- Who is right? Justify your thinking.
Equalities and Inequalities
C2.3
solve equations that involve whole numbers up to 100 in various contexts, and verify solutions
- equations:
- x + 68 = 95
- 56 = x − 34
- 1 + x + 4 = 32 + 45
- 7n = 56
- 2n + 10 = 30
- 42 − 5 = 3n − 2
- solving equations using number sense and reasoning:
- 80 = 40 + 5a
80 = 40 + 5a
40 + 40 = 40 + 5a
40 = 5a
a = 8
- solving equations using a tape diagram:
- 3x − 4 = 20:
- solving equations using flag diagrams:
- 4m + 5 = 17:
- Equations are mathematical statements such that the expressions on both sides of the equal sign are equivalent.
- In equations, variables are used to represent unknown quantities.
- There are many strategies for solving equations, including guess-and-check, the balance model, and the reverse flow chart.
- Formulas are equations in which any of the variables can be solved for. When solving for a variable in a formula, values for the variables are substituted in, and then further calculations may be needed depending on which variable is being solved for. For example, for A = lw, if l = 10 and w = 3, then A = (10)(3) = 30. If A = 50 and l = 10, then 50 = 10w, and solving this will require either using known multiplication facts or dividing both sides by 10 to solve for w.
Note
- The strategy of using a reverse flow chart can be used to solve equations like $$\frac{m}{4}-2=10;$$ for example:
- The flow chart used in coding is different from the reverse flow chart that can be used to solve equations.
- Many coding applications involve formulas and solving equations.
Introduce students to the use of flag diagrams as an input/output machine for performing operations in the order in which they appear. For example, 3 × 4 + 5 can be modelled as:
Support students in making connections between this and the use of flags to solve for an unknown value. For example, 4m + 5 = 17 can be solved using flags by first writing:
and then reversing to do the inverse operations, starting at 17:
Have students create their own flag diagrams to solve the following equations:
- 7n – 8 = 20
- 9m + 10 = 55
Have students develop their own equation, then exchange it with a partner. Partners can then solve each other’s equations using flag diagrams.
Have students solve equations that require them to determine the unknown value that makes the two expressions on either side of the equal sign equivalent (e.g., 5 + 4m = 8 + 17). Be sure to include variables in different locations, such as 10 = 3n + 1. Students can use various strategies to find the unknown, such as guess-and-check, the balance model, or flag diagrams. Note that when they use a flag diagram, they may need to manipulate the expression so that it is in a form similar to 3x + 4 = 16.
Equalities and Inequalities
C2.4
solve inequalities that involve one operation and whole numbers up to 50, and verify and graph the solutions
- inequalities:
- s + 2 < 14:
- s is all numbers less than, but not equal, to 12
- when this inequality is expressed as a number line, the open circle indicates that 12 is not included in the solution:
- s + 2 < 14:
- 5m ≤ 50
| m | 5m | ≤ 50 |
| 0 | 0 | Yes |
| 1 | 5 | Yes |
| 2 | 10 | Yes |
| 3 | 15 | Yes |
| 4 | 20 | Yes |
| 5 | 25 | Yes |
| 6 | 30 | Yes |
| 7 | 35 | Yes |
| 8 | 40 | Yes |
| 9 | 45 | Yes |
| 10 | 50 | Yes |
| 11 | 55 | No |
- m = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
- Inequalities can be solved like an equation, and then values need to be tested to identify those that 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”.
Note
- The solution for an inequality that has one variable, such as x + 3 < 4, can be graphed on a number line.
Ask students to rewrite inequalities in a different way, such as by moving the placement of the variable. For example:
- 50 ≥ 5m could also be expressed as 5m ≤ 50
- 20 – b > 15 could also be expressed as 15 < 20 – b
Have students solve a variety of problems involving inequalities. For example:
- At an amusement park, a sign says that to ride the roller coaster, you must be between 36 inches (91 cm) and 40 inches (102 cm) tall. On a number line, represent the possible heights for those that can go on this ride.
- Trish committed to reading at least 20 minutes every day during her school’s Read-A-Thon. On Tuesday she met her goal by reading 12 minutes at school and then reading some more at home. On a number line, show how many minutes Trish could have read at home to meet her goal.
- A school bus seats at most 36 students. At each of the first two stops, 7 students get on. What are the possible numbers of students the school bus can pick up after these two stops? Represent your solution on a number line.
- The cost to park at a parking garage is $3.50 per half hour. The maximum fee per day is $21.00. How long can you park a car in the garage before you reach the maximum fee?
