C2. Equations and Inequalities:
demonstrate an understanding of variables, expressions, equalities, and inequalities, and apply this understanding in various contexts
translate among words, algebraic expressions, and visual representations that describe equivalent relationships
s + s + s + s
a + b + c
bh ÷ 2
Individually, in pairs, or in a large group, have students match statements to algebraic expressions. A single copy of BLM: 5C2.1 Statements and Algebraic Expressions can be cut into pieces and given to individual students or to pairs, or one copy can be cut up and distributed among the whole group.
After all the paper strips have been matched, one statement and one expression will be left over. Ask students to write the missing expression for the statement and the missing statement for the expression.
To introduce students to the concept of translating words into algebraic expressions, it can be helpful to begin with a few specific numerical examples and then generalize them using a variable. For example, to translate three more than a number, have students create a table with inputs 1 to 5 in column 1. In column 2, have them write the change that produces the output – in this example, “Add 3”. Then have them write the outputs, including for the general case, in column 3.
Ask students to express, in a variety of ways, the sum of the lengths of the four equal sides of a square. They may write the expression as side + side + side + side, s + s + s + s, or 4 × s. Discuss the fact that when a variable is multiplied by a number, it can be written with no space and no multiplication sign, that is, as 4s. Depending on the situation, students may also encounter other ways to express multiplication, such as 4*s in spreadsheets and coding. Discuss the fact that these are widely understood mathematical conventions.
In a similar way, ask students to represent the formula for the following in different ways and using different conventions:
evaluate algebraic expressions that involve whole numbers
Provide students with expressions, such as the following, that underscore the idea of equivalence and encourage students to read an expression for meaning before computing:
Model student thinking about these expressions on an open number line.
Have students evaluate different ways of writing the same formula. For example:
Ask students to explain why these formulas are equivalent.
Present students with scenarios such as the following, and ask them to explain their reasoning:
Present students with problems that involving working with formulas. For example:
solve equations that involve whole numbers up to 100 in various contexts, and verify solutions
80 = 40 + 5a
40 + 40 = 40 + 5a
40 = 5a
a = 8
Introduce students to the use of flag diagrams as an input/output machine for performing operations in the order in which they appear. For example, 3 × 4 + 5 can be modelled as:
Support students in making connections between this and the use of flags to solve for an unknown value. For example, 4m + 5 = 17 can be solved using flags by first writing:
and then reversing to do the inverse operations, starting at 17:
Have students create their own flag diagrams to solve the following equations:
Have students develop their own equation, then exchange it with a partner. Partners can then solve each other’s equations using flag diagrams.
Have students solve equations that require them to determine the unknown value that makes the two expressions on either side of the equal sign equivalent (e.g., 5 + 4m = 8 + 17). Be sure to include variables in different locations, such as 10 = 3n + 1. Students can use various strategies to find the unknown, such as guess-and-check, the balance model, or flag diagrams. Note that when they use a flag diagram, they may need to manipulate the expression so that it is in a form similar to 3x + 4 = 16.
solve inequalities that involve one operation and whole numbers up to 50, and verify and graph the solutions
Ask students to rewrite inequalities in a different way, such as by moving the placement of the variable. For example:
Have students solve a variety of problems involving inequalities. For example: