C2. Equations and Inequalities:
demonstrate an understanding of variables, expressions, equalities, and inequalities, and apply this understanding in various contexts
add monomials with a degree of 1 that involve whole numbers, using tools
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 7h 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 35h:
evaluate algebraic expressions that involve whole numbers and decimal tenths
35h = 35($0.5)
2l + 2w = 2(6.5 cm) + 2(5.1 cm)
= 13 cm + 10.2 cm
= 23.2 cm
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.
solve equations that involve multiple terms and whole numbers in various contexts, and verify solutions
Provide students with equations to solve that require adding monomials of degree 1, such as 5m + 3m = 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 5m + 3m = 16:
Provide students with equations of the form 3m + 4 = 16 + 3, and have them simplify the right side of the equation, solve using a flow chart, and then check their answer.
solve inequalities that involve two operations and whole numbers up to 100 and verify and graph the solutions
Have students solve a variety of problems involving inequalities. For example:
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.