• representing addition of monomials:

• representing addition with pattern blocks:

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

• A monomial with a degree of 1 has a variable with an exponent of one. For example, the exponent of m for the monomial 2m 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, 2m and 3m can be combined as 5m.
• Monomials with a degree of 1 with the same variables can be subtracted; for example, 10y – 8y = 2y.
• Monomials can be subtracted in different ways. One way is to compare their representations and determine the missing addend (e.g., 3x + ? = 7x). 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 x² 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.

• algebraic expressions:
  • 2l + 2w
  • bh
  • bh ÷ 2 or
  • 
  • lwh
  • πr²
  • 2πr
  • πd
  • 3m + 2n – 1
• evaluating expressions:
  • 35h represents the cost of the tiles:
    • if the cost of 1h = $0.75, then

• evaluating expressions in formulas:
  • Volume of a cube = s³:

• 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 4.5m 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”.

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

• 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 = 10.5w, 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.

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

• 14 – 6 > 5m – 3m:

• 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 2x + 3x < 10, can be graphed on a number line.