C2. Equations and Inequalities:
demonstrate an understanding of variables, expressions, equalities, and inequalities, and apply this understanding in various contexts
add and subtract monomials with a degree of 1 that involve whole numbers, using tools
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 4h + 24t, 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.
4r + 5g + 2p + 11b + 4y – (7b + 2y)
= 4r + 5g + 2p + 11b – 7b + 4y – 2y
= 4r + 5g + 2p + 4b + 2y
evaluate algebraic expressions that involve whole numbers and decimal numbers
35h = 35($0.75)
if s = 5 cm, then
s3 = (5 cm)3
= 5 cm × 5 cm × 5 cm
= 125 cm3
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 πr2, where r is the radius. Have students evaluate a variety of formulas, including those used in Strand E: Spatial Sense when appropriate.
solve equations that involve multiple terms, whole numbers, and decimal numbers in various contexts, and verify solutions
Provide students with equations to solve that require them to subtract monomials of degree 1 involving whole numbers, such as 5m – 3m = 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 cm2 and a width of 3.2 cm.
solve inequalities that involve multiple terms and whole numbers, and verify and graph the solutions
8x (5x + 3x)
8x ≥ 16
“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.”
x ≥ 2:
2m (5m – 3m)
8 > 2m
“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.”
m < 4
Have students simplify algebraic expressions, such as:
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:
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:
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.