## C2. Equations and Inequalities

### Specific Expectations

#### Variables and Expressions

C2.1

add monomials with a degree of 1 that involve whole numbers, using tools

**representing adding monomials on a number line:**

**adding using****pattern blocks****:**- Sample 1:

- Sample 2:
- The price of one hexagon is represented by
*h*. - So the price of one tile is
*h*+*h*+*h*+*h*+*h*+*h*+*h*= 7*h*.

- The price of one hexagon is represented by

- If we have five tiles, the cost is 7
*h*+ 7*h*+ 7*h*+ 7*h*+ 7*h*= 35*h*:

**adding using****algebra tiles****:**

- A monomial with a degree of 1 has a variable with an exponent of one. For example, the exponent of
*m*for the monomial 2*m* - Monomials with a degree of 1 with the same variables can be added together; for example, 2
*m*and 3*m*can be combined as 5*m*.

*Note*

- Examples of monomials with a degree of 2 are
*x*^{2}and*xy*. The reason that*xy*has a degree of 2 is because both*x*and*y*have an exponent of 1. The degree of the monomial is determined by the sum of all the exponents of its variables. - Adding monomials using tools supports students in understanding which monomials can be combined. Only monomials with the same variables (like terms) can be combined.

Have students create a tile or a picture using only one type of pattern block. Have them identify the number of pattern blocks they used to create their tile or picture. For example, seven hexagons are used to create the tile below:

Set the context by explaining that if each hexagon is worth $1, then the entire tile is worth $7. If cost can vary, then the expression 7*h* represents the cost of a seven-hexagon tile. Have students write an algebraic expression that represents the cost of the tile or picture that they created.

Have students modify their tile from Sample Task 1 to model monomials being combined or added together. It is important for them to understand that only items that are alike can be added together. For example, yellow hexagons can only be combined with other yellow hexagons, and red trapezoids can only be combined with other red trapezoids. The sample design below shows how the tile from Sample Task 1 is used to create a bigger design, with a cost of 35*h*:

C2.2

evaluate algebraic expressions that involve whole numbers and decimal tenths

**algebraic expressions****:**- 2
*l*+ 2*w* *bh**bh*÷ 2 or- 3
*m*+ 2*n*− 1

- 2
**evaluating****expressions:**- using substitution:
- 35
*h*represents the cost of the tiles:- if the price of one
*h*is 50¢,- I know that 50¢ is the same as half of a dollar, or $0.5"

- if the price of one

- 35

- using substitution:

35*h* = 35($0.5)

= $17.50

- The perimeter of a rectangle can be determined using the formula
*P*= 2*l*+ 2*w*:- If a rectangle has a length,
*l*, of 6.5 cm and a width,*w*, of 5.1 cm, then

- If a rectangle has a length,

2*l* + 2*w *= 2(6.5 cm) + 2(5.1 cm)

= 13 cm + 10.2 cm

= 23.2 cm

- 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 4.5
*m*becomes 4.5(*m*) and then 4.5(7) when*m*= 7. The operation between 4.5 and (7) is understood to be multiplication. - Many coding applications involve algebraic expressions being evaluated. 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 sideA*sideA, areaA” instructs the computer to take the value that is stored in “sideA” and multiply it by itself, and then store that result in the temporary location, which is another variable called “areaA”.

Have students evaluate the algebraic expressions for the tile or picture they created for C2.1, Sample Task 1, when given values for the shapes. For example, ask students to determine the cost of their tile if hexagons cost 80¢ each and trapezoids cost 30¢ each. Be sure to provide costs for all pattern blocks.

Explain that formulas involve algebraic expressions. For example, the area of a parallelogram can be determined using the algebraic expression *b *×* h*, where *b* represents the base and *h* represents the height. Have students evaluate a variety of other formulas, including those used in Strand E: Spatial Sense when appropriate.

#### Equalities and Inequalities

C2.3

solve equations that involve multiple terms and whole numbers in various contexts, and verify solutions

**equations****:**- 5
*m*+ 3*m*= 10 + 6 - 5
*m*− 3*m*= 10 − 6 - 5
*x*+ 3*x*= 16 - 24 =
*d*+ 2d - 7
*n*= 56 − 14

- 5
**solving equations using a tape diagram to solve 3****x****– 2 = 20:**

- 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 to solve equations including guess-and-check, the balance model, and the reverse flow chart.
- The strategy of using a reverse flow chart can be used to solve equations like – 2 = 10. The first diagram shows the flow of operations performed on the variable
*m*to produce the result 10. The second diagram shows the reverse flow chart, or flow of the reverse operations, in order to identify the value of the variable*m*.

- 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 = 10*w*, and solving this will require either using known multiplication facts or dividing both sides by 10 to solve for*w*.

*Note*

- Some equations may require monomials to be added together before they can be solved using the reverse flow chart method.
- The flow chart used in coding is different from the reverse flow chart that can be used to solve equations.

Provide students with equations to solve that require adding monomials of degree 1, such as 5*m* + 3*m* = 16. Once they have simplified the equation, they can use a variety of methods to solve for the unknown value. It is important to have students check their solutions by substituting the value into the equation and verifying that both sides of the equation remain equal. For example, they might use the structure of an LS/RS (left side/right side) check by substituting their solution into the original equation and then evaluating each side independently. If LS = RS, the solution is correct. If LS ≠ RS, then the solution is incorrect. In the example below, the student has determined that the solution is *m* = 2. Check that *m* = 2 is the solution for 5*m* + 3*m* = 16:

Provide students with equations of the form 3*m* + 4 = 16 + 3, and have them simplify the right side of the equation, solve using a flow chart, and then check their answer.

C2.4

solve inequalities that involve two operations and whole numbers up to 100 and verify and graph the solutions

**inequalities with solutions:**- 5
*x*− 20 ≥ 80:- Note: For scenarios involving integers, students should have access to a number line, and the examples should use friendly numbers.

- 5

x |
5x − 20 |
≥ 80 |

0 | –20 | No |

1 | –15 | No |

2 | –10 | No |

3 | –5 | No |

4 | 0 | No |

5 | 5 | No |

6 | 10 | No |

10 | 30 | No |

15 | 55 | No |

20 | 80 | Yes |

30 | 130 | Yes |

40 | 180 | Yes |

50 | 230 | Yes |

- Solution:
*x*≥ 20:

- 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 by placing a dot at the greatest or least possible value. An open dot is used if the inequality involves “less than” or “greater than”; if the inequality includes the equal sign (=), then a closed dot is used.

*Note*

- The solution for an inequality that has one variable, such as 2
*x*+ 3 < 9, can be graphed on a number line.

Have students solve a variety of problems involving inequalities. For example:

- Erykah has found three pairs of running shoes that she likes, costing $50, $59, and $79. She has saved $31 already, and she has a dog-walking job where she earns $15 per hour. How many hours will she have to work to afford any of these shoes? Represent your solution on a number line.
- Bobby is buying a jewellery box for his mother for Mother’s Day. He wants to have it engraved with a special message. The store charges 10¢ for each letter engraved. Bobby plans to spend no more than $5 to engrave the jewellery box. How many letters can be in Bobby’s message? Represent your solution on a number line.
- Michael wants to bike more than 50 km this week. He rides 5 km on Monday and then wants to evenly split up the remaining kilometres over three more days. What distance must he ride on each of those three days? Represent your solution on a number line.
- An 18-wheeler truck stops at a weigh station before passing over a bridge. The mass limit on the bridge is 25 tonnes. The cab (front) of the truck has a mass of 8.9 tonnes, and the trailer (back) of the truck has a mass of 5.3 tonnes when empty. In tonnes, how much cargo can the truck carry and still be allowed to cross the bridge?

Ask students to write an inequality that has two operations and has a solution greater than 8. Then ask them to write a scenario that represents the inequality.