C2. Equations and Inequalities:
demonstrate an understanding of variables, expressions, equations, and inequalities, and apply this understanding in various contexts
add and subtract monomials with a degree of 1, and add binomials with a degree of 1 that involve integers, using tools
Arrange students in pairs. Have each student in a pair create a tile using the same type of pattern block. Have each student write an algebraic expression to represent the cost of their tile. Next, have the students in each pair determine the difference between the costs of their tiles. For example, tile A can be represented by 7h and tile B can be represented by 14h, where h represents the cost of each hexagon:
Have students use pattern blocks or algebra tiles to model adding a variety of binomials. It is important for students to see that only like terms can be simplified. For example:
evaluate algebraic expressions that involve rational numbers
if 1h = $0.75 and 1t = $0.25, then
7h + 12t = 7($0.75) + 12($0.25)
= $5.25 + $3.00
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 $10.75 each and the trapezoids cost $8.50 each.
Formulas have algebraic expressions. Have students evaluate a variety of formulas, including those introduced in Strand E: Spatial Sense and the Science and Technology curriculum, when appropriate.
solve equations that involve multiple terms, integers, and decimal numbers in various contexts, and verify solutions
Provide students with equations to solve that require them to collect like terms, such as 5x – 10 = –3x + 6.
Once they have simplified an equation, they can use a variety of methods to solve for the unknown value. It is important for them to 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 both sides of 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 case below, the student has determined that the solution is x = 2:
Have students solve equations involving decimal numbers, such as finding the length of one side of a right-angle triangle to the nearest tenth when given the length of the other two sides, or finding side lengths of a shape when given some information about it:
solve inequalities that involve integers, and verify and graph the solutions
5x − 10 + 10 < −3x +6 + 10
5x < −3x + 16
5x + 3x < −3x + 3x + 16
8x < 16
10 + 6 ≤ −5x – 3x
16 ≤ −8x
−8x ≥ 16
−2 ≥ x, which can also be expressed as x ≤ −2
Have students solve various inequalities that involve simplifying like terms. For example,
A possible strategy for solving inequalities is to first solve the inequality as an equality, then test numbers greater than and less than the solution to the equality in the original inequality to determine the range of numbers for which it holds. Reinforce the proper use of closed or open circles to represent solutions.