Lesson 6

The purpose of this activity is for students to choose how they can apply math concepts and strategies to a problem arising in a real-world context: predicting whether a restaurant will make a profit.

After students have estimated the monthly cost of their ongoing expenses on their list, poll the class on their total estimated monthly costs.

If students have completed the previous lesson about using spreadsheets to calculate the cost of ingredients for one serving of their recipe, they can use that as the basis for calculating the percentage of the markup. If students have not calculated the cost of ingredients in a previous lesson, then tell students that the ingredients for one serving of a meal typically cost about:

• $2 for a fast-food restaurant
• $5 for a casual sit-down restaurant
• $10 to $25 for a formal sit-down restaurant

Point out that restaurants have many more expenses than just the cost of the food. Give students quiet work time followed by partner discussion.

Consider having students pause their work after the first question so that you can record their answers displayed for all to see, and all students will have access to this same information for the rest of the activity.

Poll the class on whether they think each of the restaurants A, B, and C made money for the month. Select students to share their reasoning.

Note: The way the problem is written, there is not just one correct answer to the question. Whether or not each restaurant made money depends on how much the students estimated for the total monthly expenses (excluding food).

Next, ask students to trade with a partner, and check their work for calculating the percentage of the mark up.

Ask students to discuss:

• “How does the amount you plan to charge for each meal compare to your partner’s amount?”
• “How does your mark up percentage compare to your partner’s percentage?”

Consider telling students that many restaurant owners use 300% as an estimate of a good percentage for the mark up on their meals to be able to make a profit. That means the price of the meal is 4 times what the cost of the ingredients were.

The purpose of this activity is for students to write and solve equations as a strategy to compare the projected costs of using reusable versus disposable plates and forks. First, students examine dot plots representing the average number of customers served per day at a sample of restaurants to make a prediction about how many customers they might serve per day. Then, students see that the cost of buying disposable plates and forks can be modeled with a proportional relationship, while the cost of buying and washing reusable plates and forks can be modeled with an equation in the form \(px+q=r\).

Display the dot plots about the number of customers served per day. Invite students to share what they notice and wonder.

Some things students may notice:

• Many of the fast food restaurants serve more customers per day than the full service restaurants.
• There is a lot of overlap between the two distributions, from 300 to 600 customers.

Some things students may wonder:

• Were the restaurants included in the samples selected at random?
• Is there a meaningful difference between the average number of customers served at these two types of restaurants?
• About how many customers would my restaurant serve per day?

Give students quiet work time followed by whole-class discussion.

Poll the class on how many days they can you use the reusable plates and forks for the same cost as using disposable plates and forks for 1 year. Select students to share what this tells us about the situation. (If their answer is greater than the number of days they planned for their restaurant to be open during the year, then this means that buying and washing reusable plates and forks is cheaper than using disposable plates and forks.)

Select students to share their reasoning about whether there is a proportional relationship between the cost of using disposable or reusable and the number of days.

Students might share the following ideas:

• The relationship for the cost of using disposable looks like it is proportional because we wrote it in the form \(y=kx\).
  • The relationship for the cost of using disposable would be close, but not exactly proportional, because the equation is assuming an average number of customers per day and in real life the restaurant serves a different number of people each day.
• The relationship for the cost of using reusable is not proportional because:
  • there is a start-up cost of buying the reusable plates and forks.
  • the equation cannot be written in the form \(y=kx\). There has to be a term that is added that represents the start-up costs.
  • if we graphed the relationship, it would not go through the origin, but would cross the y-axis at a point that represents the start-up costs.