Working with Expressions and Equations Part 2

Part of my class routine is a do now at the beginning of every class.  Students walk into class and pick up the packet for the day.  They get to work quickly on the problems.  Often, I create do nows that have problems that connect to the task that students will be working on that day.  For this lesson I want students to practice identifying the independent and dependent variables.

I ask for volunteers to define the independent and dependent variables in their own words.  We refine the definitions.  I call on students to identify the variables in #1 and #2.  I want students to be able to articulate that the number of miles Dan drives depends on the number of hours he drives.  The longer he drives the further he’ll go.  I want students to be able to articulate that the amount of money you spend on a data plan depends on the number of months you use the plan.  The longer you use the plan, the more money you will spend on the data plan.  If students struggle to articulate these patterns I will set up a table and have students generate values as the hours/months increase.

After the Do Now, I have a student read the objectives for the day.  I tell students that they will be creating expressions and equations to model situations.   These expressions and equations will be more complicated than the ones that we created previously.

On page 2 we review the equation for the amount of money collected and identify the independent variables.  I introduce the concepts of income, costs, and profit.  Students share out ideas for costs.  Students will likely identify cost of paying employees, cost of tools and materials (tractor, gas, clippers, signs ),  advertising (tv/radio ad, website, etc), and utilities (water, electricity).  These are all things that cost money!  Profit is the amount of money a business still has after paying its costs.

I have students brainstorm for a minute to create an expression for #1 on page 3.  Students share out their ideas.  Some students may struggle to connect 8n (the amount of money collected) and 75 (the cost of one day of running the maze).  Other students may struggle with whether the expression should be 8n – 75 or 75 – 8n.  I will revisit the idea of profit being the difference between the amount of money the maze takes in and the costs of running the maze. 

Once we have agreed on the expression for the profit, students will work in partners to answer question 2.  I am looking to see that students are correctly using substitution.  A common mistake is that a student will plug in a value for n, but forget to multiply that value by 8. For example, for 2a a student may substitute 80 for n and get 880 – 75, instead of multiplying 8 times 80 and getting 640 – 75. 

We move on to part C on page 4.  I ask students what they think each line graph represents.  I am looking for students to connect the top graph with the graph that they created in the last lesson.  Students should notice that the bottom graph represents the profit that the business makes.  I am looking for students to observe that (0,-75) means that if there are no visitors, the profit is -$75, or the maze loses $75.  How could that be?  That means that the business spent more money than it brought it. 

Before moving on to the next section, I will ask students to use the graph and their equation to predict what the smallest number of visitors it would take for the maze to make a profit.  Some students may look at the graph and see that when there are 10 visitors, the amount of profit is above $0.  Other students may use substitution to show that 8(10) – 75 = 5, so when there are 10 visitors there is a profit of $5.  Other students may create a table or chart showing that with 9 visitors the maze has a loss of $3 and that with 10 visitors the maze has a profit of $5.

I have students work on the Practice section (pages 5-6) in homogeneous partner pairs.  See my video Creating Homogeneous Groups in my Strategy Folder.  I walk around and monitor student progress. 

If students are struggling, I may ask them the following questions:

• What is the expression that we used to represent the amount of money collected by the maze in one day?  How can we use this expression to help us show the amount of money collected on Saturday and Sunday?
• What does the variable r represent?  What does the variable u represent?
• What does it mean when two things are equivalent?
• What operation is occurring when the 8 is outside of the parentheses? 
• How can you test to see if your expression is equivalent to 8 (r + u)?
• If the cost for running the maze is $75 for one day, what is the cost of running the maze for 2 days?
• How do we find profit?  Look back to our notes on page 2.

If students are correctly working through the examples, they can move on to the extra practice on pages 7 and 8. 

I have students work on the Practice section (pages 5-6) in partners.  I walk around and monitor student progress. 

If students are struggling, I may ask them the following questions:

• What is the expression that we used to represent the amount of money collected by the maze in one day?  How can we use this expression to help us show the amount of money collected on Saturday and Sunday?
• What does the variable r represent?  What does the variable u represent?
• What does it mean when two things are equivalent?
• What operation is occurring when the 8 is outside of the parentheses? 
• How can you test to see if your expression is equivalent to 8 (r + u)?
• If the cost for running the maze is $75 for one day, what is the cost of running the maze for 2 days?
• How do we find profit?  Look back to our notes on page 2.

If students are correctly working through the examples, they can move on to the extra practice on pages 7 and 8.