- Determine dependent and independent variables in real-world situations.
- Write algebraic expressions and equations to represent real-world situations.
- Solve algebraic equations given the value of a variable.
- Solve expressions and equations containing positive and negative rational numbers.
See standards below.
- Digital Interactive Tool: “Launching Into Expressions and Equations” (optional)
- Interactive whiteboard OR computer/projector hookup (optional)
- Internet connection (optional)
- Computers for small groups and/or all students (optional)
- Planning a Trip to the Amazon: Using Variables to Represent Numbers and Write Expressions printable
- Exploring the Amazon: Translating a Real-World Situation From a Word Problem Into an Equation printable
- Answer Key for Adventures in Expressions and Equations
Note: The lesson includes both online and printable components but was designed to be a meaningful learning experience whether or not the online components are used.
INTRODUCTION TO NEW MATERIAL
Step 1: Introduce the lesson by telling students that they will be describing the world’s largest habitat, the ocean. Ask students: What are some of the threats to this habitat’s life-forms? What are some solutions for combating these threats?
Step 2: Tell students that Ireland implemented a solution in 2002 to combat the amount of litter present in the ocean: they began to charge a €0.15 tax for each plastic shopping bag that customers used at stores. (A euro is worth about the same amount as an American dollar, and its sign is € rather than $.)
Ask: If a shopper requires 1 plastic shopping bag, how much tax will they pay for the plastic bag? (€0.15) 2 plastic bags? (€0.30) 3 plastic bags? (€0.45) Ask students if there is some general way that they can calculate the tax a customer would pay for any number of plastic bags that they use? Have pairs discuss and then share their thoughts with the class. At this point, verbal descriptions, rather than expressions or equations, can and should be suggested (example: multiply the number of bags they use by €0.15).
Step 3: Define variable, independent variable, and dependent variable for students.
- variable: an unknown or changing value
- independent variable: a variable whose value does not depend on another variable’s value; a freely chosen value (often represented by x)
- dependent variable: a variable whose value relies on the value of the independent variable (often represented by y)
In the situation described above, ask students to identify the independent variable and the dependent variable. Then have them discuss in pairs why the number of bags used is the independent variable and why the total tax charged is the dependent variable. Have students share their thoughts with the class.
Step 4: Define expression, equation, and algebraic equation:
- expression: a mathematical phrase including numbers, operators, and/or variables (examples: 7; b + 2; 40xy)
- equation: a statement that shows two equal expressions (examples: 23 + 7 = 30; 9 = 9)
- algebraic equation: an equation that includes variables (examples: 0.8 + c = 40; 6h = g)
Tell students that often independent variables are assigned the letter x, and dependent variables are assigned the letter y. If, however, they would like to represent values with other letters, such as b for bags and t for tax, that is often done as well. Ask students to discuss the relationship between the number of bags used and the tax charged, and write an equation relating the two values.
Step 5: Model for students how to solve the equation (y = 0.15x) for a given number of bags. Then have students practice this skill with other values of x.
Step 6: Tell the students that Ireland has had to spend money in order to implement and enforce the new tax. Each year, Ireland has to pay €350,000 to administer the plan. What algebraic equation could represent the amount of money spent to administer the plan for a given number of years? (y = €350,000x, where x is the number of years the plan has run) When the tax began, it cost a fixed €1,200,000 to set up the plan. Tell students to modify their equation to include the fixed setup cost of the plan. (y = €350,000x + €1,200,000, where x is the number of years the plan has run) Then have students practice solving the equation with different values of x.
Step 7: Tell students that Ireland increased its plastic bag tax to €0.22 in 2007. Suppose a family has a budget of €100 to spend on groceries each week. What equation can represent the amount of money a family can spend on groceries in relation to the number of plastic bags they use for the groceries? (y = €100 – €0.22x) Have students solve this equation for a variety of numbers of plastic bags.
GUIDED PRACTICE and INDEPENDENT PRACTICE
Step 8: Mix and match from among these digital and printable materials below depending on your class’s needs and technological capabilities.
These engaging materials situate students in real-world scenarios of exploration—from an astronaut blasting off into space to a biologist swimming the Amazon River. This highlights the value of math in real-world situations and careers, in which expressions and equations are necessary tools for solving problems.
- Module 1 of the Digital Interactive Tool: Variables in Expressions and Equations.
- Planning a Trip to the Amazon: Using Variables to Represent Numbers and Write Expressions
- Exploring the Amazon: Translating a Real-World Situation From a Word Problem Into an Equation
Tell students that another factor concerning scientists is sea level trends. Individually, or as a class, they can visit tides and current website in order to identify and write equations for sea level changes in various areas of the world over a certain number of years. They can then solve the equations to determine the sea level change for a specified number of years in those areas.
(CCSS and NCTM):
- Grade 6: Algebraic equation, dependent variable, independent variable, solving equations (CCSS 6.EE.B.6 & 7)
- Grade 7: Rational numbers, converting between units of measure (CCSS 7.EE.B.3)
- Grades 6–8: (CCSS MP1, 2, 4, 6 & 7); NCTM Algebra