## C2. Equations and Inequalities

### Specific Expectations

#### Variables and Expressions

C2.1

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

**representing addition of monomials:**

**representing addition with pattern blocks:**

**representing subtraction using algebra tiles to compare two expressions:**- 5
*x*– 2*x*= ?

- 5

- 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*is 1. When the exponent is not shown, it is understood to be one. - 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*. - Monomials with a degree of 1 with the same variables can be subtracted; for example, 10
*y*– 8*y*= 2*y*. - Monomials can be subtracted in different ways. One way is to compare their representations and determine the missing addend (e.g., 3
*x*+ ? = 7*x*). Another way is to remove them from the expression representation (e.g., 3*x*-tiles are physically removed from the collection of 7*x*-tiles).

*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 and subtracting monomials using tools supports students in understanding which monomials can be simplified. Only monomials with the same variables (like terms) can be simplified.

Have students create a tile or a picture using two different types of pattern blocks. Have them determine how many of each type of pattern block they used. Have them write an algebraic expression for the tile or picture that they created such that each variable represents the cost of each of the types of blocks. The sample design below has 4 yellow hexagons and 24 trapezoids. An algebraic expression to represent the cost of the design is 4*h* + 24*t*, where *h* is the cost of a hexagon and *t* is the cost of a trapezoid:

Have students use pattern blocks or algebra tiles to model the addition or subtraction of a variety of monomials. It is important for students to notice that only items that are alike can be added or subtracted. For example, yellow hexagons can only be combined with other yellow hexagons, and red trapezoids can only be combined with other red trapezoids. Challenge students to come up with a design that models subtraction. For example, the skateboarder below can become a walker by removing the skateboard.

- Skateboarder: 4
*r*+ 5*g*+ 2*p*+ 11*b*+ 4*y* - Walker:

4*r* + 5*g* + 2*p *+ 11*b* + 4*y* – (7*b* + 2*y)*

= 4*r *+ 5*g* + 2*p* + 11*b* – 7*b + *4*y – *2*y*

= 4*r *+ 5*g* + 2*p* + 4*b* + 2*y*

- where
*r*= red trapezoid,*g*= green triangle,*p*= purple square,*b*= blue rhombus, and*y*= yellow hexagon

C2.2

evaluate algebraic expressions that involve whole numbers and decimal numbers

**algebraic expressions:**- 2
*l*+ 2*w* *bh**bh*÷ 2 or*lwh**πr*^{2}- 2
*πr* *πd*- 3
*m*+ 2*n*– 1

- 2
**evaluating expressions:**- 35
*h*represents the cost of the tiles:- if the cost of 1
*h*= $0.75, then

- if the cost of 1

- 35

35*h* = 35($0.75)

= $26.25

**evaluating expressions in formulas:**- Volume of a cube =
*s*^{3}:

- Volume of a cube =

if* s *= 5 cm, then

*s*^{3 }= (5 cm)^{3}

= 5 cm × 5 cm × 5 cm

= 125 cm^{3}

- 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(7.2) when*m*= 7.2. The operation between 4.5 and (7.2) 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 radius of a circle’, radiusA” is instructing the computer to define the variable “radiusA” and store whatever the user inputs into the temporary location called radiusA. The instruction: “calculate 2*radiusA, diameterA” instructs the computer to take the value that is stored in radiusA and multiply it by two, and then store that result in the temporary location, which is another variable called “diameterA”.

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, the hexagons cost $1.75 each and the trapezoids cost $1.30 each.

Formulas have algebraic expressions. For example, the area of a circle can be determined using the algebraic expression π*r*^{2}, where* r* is the radius. Have students evaluate a variety of formulas, including those used in Strand E: Spatial Sense when appropriate.

#### Equalities and Inequalities

C2.3

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

**equations involving multiple terms and one variable:**- 5
*m*+ 3*m*= 10 + 6 - 10 – 6 = 5
*x*– 3*x* - 3.2
*d*= 7.3 + 5.5 - 14.5 – 1.5 = 6.5
*h*

- 5

- 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 – 2.1 = 10.4. The first diagram shows the flow of operations performed on the variable
*m*to produce the result 10.4. 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,
*A*=*lw*, if*l*= 10.5, and*w*= 3.5, then*A*= (10.5)(3.5) = 36.75. If*A*= 36.75 and*l*= 10.5, then 36.75 = 10.5*w*, and this will require dividing both sides by 10.5 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 them to subtract monomials of degree 1 involving whole numbers, such as 5*m* – 3*m* = 16. Once they have simplified an equation, they can use a variety of methods to solve for the unknown value. It is important to have them check their solutions by substituting the value into the equation to verify that both sides of the equation are equal. For example, they might use the structure of an LS/RS (left side/right side) check to substitute their solution into the original equation, then evaluate 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 *m* = 8 is correct:

Have students solve equations involving decimal numbers, such as finding the length of a rectangle to the nearest tenth, given an area of 42.5 cm^{2} and a width of 3.2 cm.

C2.4

solve inequalities that involve multiple terms and whole numbers, and verify and graph the solutions

**inequalities and solutions:****5****x****+ 3****x****≥ 10 + 6:**

- “What variable am I solving for?“

*x*

- “How many
*x*’s are there?”

8*x* (5*x *+ 3*x*)

- “What else do I know?”

8x ≥ 16

- “If 8
*x*= 16, then how much is each*x*worth?”

“I know that 8 × 1 = 8, so that doesn’t work.”

“If *x* = 2, then 8 × 2 = 16. 16 is equal to 16, so that works.”

“If *x* = 3, then 8 × 3 = 24 and 24 is greater than 16, so that also works.”

- “So I know that the solution must be 2 or greater.”

*x* ≥ 2:

**14 – 6 > 5****m****– 3****m****:**

- “What variable am I solving for?”

*m*

- “How many
*m*’s are there?”

2*m* (5*m* – 3*m*)

- “What else do I know?”

8 > 2*m*

- “If I know that 8 > 2
*m*, I also know that 2*m*< 8. I find that easier to work with.” - “If 2
*m*< 8, then how much is each*m*worth?”

“I know that 2 × 4 = 8, so I will test numbers greater and less than 4.”

“If *m* = 3, 2 × 3 = 6, which is less than 8, so that works.”

“If *m* = 5, 2 × 5 = 10, which is not less than 8, so that doesn’t work.”

- “So the solution must be less than, and not include, 4.”

*m* < 4

- An inequality 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 when an inequality involves “less than” or “greater than”; if the inequality includes the equal sign (=), then a closed dot is used.

*Note*

- Inequalities that involve multiple terms may need to be simplified before they can be solved.
- The solution for an inequality that has one variable, such as 2
*x*+ 3*x*< 10, can be graphed on a number line.

Have students simplify algebraic expressions, such as:

- 3
*p*+ 2 + 5*p* - 10
*h*−*h*− 7 - 9 + 8
*t*+ 2 - 15 − 9
*m*+ 2 − 3*m*

Ask students to come up with their own algebraic expressions to simplify and exchange in a small group, or alternatively to use as a warm-up activity for other lessons.

Have students solve inequalities that can be solved by simplifying like terms and graph their solutions on a number line. This might include inequalities such as:

- 5
*x*+ 3*x*≥ 10 + 6 - 14 – 6 > 5
*m*– 3*m* - 5
*x*+ 3 > 3*x*+ 5 - 12 − 3
*x*< 4*x*− 2

A possible strategy in solving inequalities is to first solve the inequality as an equality, then test numbers greater than and less than the solution in order to determine the range of numbers for which the inequality holds true. Reinforce the proper use of closed or open circles to represent solutions. For example:

*m*< 4:

Ask students to come up with their own inequalities to solve and exchange in a small group or use as a warm-up activity for other lessons.