Lesson 1

Planning a Pizza Party

1.1: A Main Dish and Some Side Dishes (5 minutes)

Warm-up

This warm-up elicits the idea that an equation can contain only letters, with each letter representing a value. It also reminds students that an equation is a statement that two expressions are equal, and that different expressions could be used to represent a quantity. Later in this lesson and throughout the unit, students will create, interpret, and reason about equations with letters representing quantities.

Launch

Arrange students in groups of 2. For the last question, ask each partner to come up with a new equation.

Student Facing

Here are some letters and what they represent. All costs are in dollars.

• $$m$$ represents the cost of a main dish.
• $$n$$ represents the number of side dishes.
• $$s$$ represents the cost of a side dish.
• $$t$$ represents the total cost of a meal.
1. Discuss with a partner: What does each equation mean in this situation?
1. $$m=7.50$$
2. $$m = s + 4.50$$
3. $$ns = 6$$
4. $$m + ns = t$$
2. Write a new equation that could be true in this situation.

Activity Synthesis

Invite students to share their interpretations of the given equations and the new equations they wrote. Then, discuss with students:

• "What is an equation? What does it tell us?" (An equation is a statement that an expression has the same value as another expression.)
• "Can equations contain only numbers?" (Yes.) "Only letters?" (Yes.) "A mix of numbers and letters?" (Yes.)
• "The last question asked you to write an equation that could be true. Could this equation be true: $$m + 5 = t$$? How do you know?" (It could be. If $$m$$ is 7.50 and $$t$$ is 12.50, then the equation is true.)
• "When might the equation be false?" (If $$m$$ is 7.50 and $$t$$ is anything but 12.50, or if the value of $$t$$ is not 5 more than $$m$$, then it is false.)
• "One equation tells us that a main dish is $7.50. Another equation tells us that it is equal to the expression $$s + 4.50$$. Could both be true? Are they both appropriate for expressing the cost of a main dish?" (Yes. The first one tells us the price in dollars. The second tells us how the price compares to a side dish.) 1.2: How Much Will It Cost? (20 minutes) Activity This activity prompts students to create expressions to represent the quantities and relationships in a situation and engages them in mathematical modeling. Students plan a pizza party and present a cost estimate. To do so, they need to consider relevant variables, make assumptions and estimates, perform calculations, and adjust their thinking along the way (MP4). Some students may choose to perform research and revise their models as they gather new information. There are many possible solutions to the task. As students discuss their ideas, monitor for those who: • find and use actual data or exact values (for example, count the number of students in the class, research the cost of a large pizza at a nearby shop, or quickly survey the class for topping preferences) • estimate quantities based on prior knowledge (for example, the cost of a large pizza in a recent purchase, or the number of slices they and their friends generally consume at lunch time) • make assumptions about behaviors, preferences, or quantities (for instance, assume that a certain percentage of the class prefers a certain topping) Launch Ask students if they have ever been in charge of planning a party. Solicit a few ideas of what party planners need to consider. Ask students to imagine being in charge of a class pizza party. Explain that their job is to present a plan and a cost estimate for the party. Arrange students in groups of 4. Provide access to calculators and, if feasible and desired, access to the internet for researching pizza prices. Students can also make estimates based on prior experience, refer to printed ads, or use their personal device to look up pricing information. Limit the time spent on the first question to 7–8 minutes and pause the class before students move on to subsequent questions. Give groups of students 1–2 minutes to share their proposals with another group. Then, select a few groups who used contrasting strategies (such as those outlined in the Narrative) to briefly share their plans with the class. Record or display their plans for all to see. Next, ask students to complete the remaining questions. If needed, give an example of an expression that can be written to represent a cost calculation. Student Facing Imagine your class is having a pizza party. Work with your group to plan what to order and to estimate what the party would cost. 1. Record your group’s plan and cost estimate. What would it take to convince the class to go with your group’s plan? Be prepared to explain your reasoning. 2. Write down one or more expressions that show how your group’s cost estimate was calculated. 1. In your expression(s), are there quantities that might change on the day of the party? Which ones? 2. Rewrite your expression(s), replacing the quantities that might change with letters. Be sure to specify what the letters represent. Student Response Teachers with a valid work email address can click here to register or sign in for free access to Student Response. Student Facing Are you ready for more? Find a pizza place near you and ask about the diameter and cost of at least two sizes of pizza. Compare the cost per square inch of the sizes. Student Response Teachers with a valid work email address can click here to register or sign in for free access to Extension Student Response. Anticipated Misconceptions If students do not understand what is meant by “quantities that might change,” ask them if it is more likely that the cost of a pepperoni pizza increases on the day of the party, or that some students are absent that day. In a model that incorporates both of these quantities, they may wish to use a number for the cost of each type of pizza and a letter for the number of students present that day. Activity Synthesis Invite groups who did not previously share their plans to share the expressions they wrote and explain what the expressions represent. After each group shares, ask if others calculated the costs the same way but wrote different expressions. As students present their expressions, record the quantities that they mention and display them for all to see. Some examples: • the number of students in the class • the number of pizza slices per person • the cost of delivery • the price per topping Briefly discuss the quantities that students anticipate would change (and therefore would replace with letters). Explain to students that the expressions they have written are examples of mathematical models. They are mathematical representations of a situation in life that can be used to make sense of problems and solve them. We will look more closely at how expressions could represent the quantities in a situation like party planning, which involves certain conditions or requirements. Representing, Conversing: MLR7 Compare and Connect. As students share their plans with the class, call students’ attention to the different ways the quantities are represented in expressions, and to the different expressions that represent the same quantity. Take a close look at the contrasting strategies used within the context of the situation. Wherever possible, amplify student words and actions that students use to revise their models by estimating quantities and making assumptions about the situation. Design Principle(s): Maximize meta-awareness; Support sense-making 1.3: What are the Constraints? (10 minutes) Activity Writing and solving equations often revolves around the idea of representing and satisfying constraints. This activity introduces the term “constraints” and begins to develop the idea that expressions and equations can help us describe constraints on quantities. It prompts students to recognize that quantities are sometimes constrained in terms of the values they could take or in terms of how they relate to another quantity. Launch Tell students that they will now look at some constraints of the pizza party. Explain that a constraint is something that limits what is possible or what is reasonable in a situation. For example, one constraint a teacher has to work with is the amount of time in a class period or the number of school days in a year (both of these might be a fixed number). Another constraint might be the number of students in a class (which may vary by class, but is usually no more than a certain number). Consider keeping students in groups of 4. Give students 2 minutes of quiet work time and then time to briefly share their responses with their group. Follow with a whole-class discussion. Representation: Develop Language and Symbols. Display or provide charts with symbols and meanings. Invite students to name additional examples of other variables that might be considered constraints to encourage critical thinking and application to the expressions created in this activity. Examples of additional constraints might include $$t < 2$$ to represent how long the party could last, or $$b < 3$$ to say that each student will get less than 3 beverages. Supports accessibility for: Conceptual processing; Memory Student Facing A constraint is something that limits what is possible or reasonable in a situation. For example, one constraint in a pizza party might be the number of slices of pizza each person could have, $$s$$. We can write $$s < 4$$ to say that each person gets fewer than 4 slices. 1. Look at the expressions you wrote when planning the pizza party earlier. 1. Choose an expression that uses one or more letters. 2. For each letter, determine what values would be reasonable. (For instance, could the value be a non-whole number? A number greater than 50? A negative number? Exactly 2?) 2. Write equations or inequalities that represent some constraints in your pizza party plan. If a quantity must be an exact value, use the $$=$$ symbol. If it must be greater or less than a certain value to be reasonable, use the $$<$$ or $$>$$ symbol. Student Response Teachers with a valid work email address can click here to register or sign in for free access to Student Response. Anticipated Misconceptions If students have trouble thinking of constraints for a chosen variable, ask about extreme values. For instance, ask: "Do you think a large pizza might cost \$100? How about \$3?" Some students may struggle with translating the written descriptions of the constraints into inequalities. For example, “the greatest number of students in the class would be 30” might be mistakenly written as $$s > 30$$. Ask these students to explain the meaning of the “>” symbol. Ask: "If 29 students come to class, can we write $$29 > 30$$?" Activity Synthesis Invite students to share the equations or inequalities that represent the constraints in their party plan. Emphasize the following points: • Sometimes a constraint is an exact value. For example: Ordering pizza online involves a fee of \$2.50, or the price of a large cheese pizza is \$9. • Other times, a constraint involves a boundary or a limit. For example: An order must be at least \$25 in value to qualify for free delivery, or the party must cost no more than \$90. Lesson Synthesis Lesson Synthesis Some key takeaways from this lesson are the ideas that real-life situations often involve constraints, that we can use expressions, equations, and inequalities to represent these constraints. To help students see these points, discuss questions such as: • "In planning a pizza party, what were some ways we gathered information to estimate the cost?" (counting the number of students, researching actual prices) • "What were some assumptions we made?" (pizza preferences, number of slices per person, possible prices of pizza) • "Suppose we had gathered information differently, for instance, by asking every student the exact pizza toppings and number of slices. Would that have been a reasonable approach? How would that have changed the cost estimate? " (It would be inefficient to take exact orders, cost much more, and likely mean a lot of leftovers.) • "Suppose we had made a different set of assumptions, for instance, assuming that everyone loved pepperoni and would like only 1 slice. How would that have changed the cost estimate?" (If the assumption was a slice of pepperoni pizza per person, it would probably cost less, but many students might not be able to enjoy the pizza or might not have enough.) • "In planning the party, we saw some examples of constraints. Can you think of some constraints in other situations? What might be some constraints in, say, planning a field trip, or in organizing a community service event?" Tell students that expressions, equations, and inequalities are mathematical models. A model is a mathematical representation of a real-life situation. When people create models, they rely on the information they have, but they also make assumptions and decisions which affect the models. If the information or assumptions change, the model would also change. 1.4: Cool-down - Ice Cream Party (5 minutes) Cool-Down Teachers with a valid work email address can click here to register or sign in for free access to Cool-Downs. Student Lesson Summary Student Facing Expressions, equations, and inequalities are mathematical models. They are mathematical representations used to describe quantities and their relationships in a real-life situation. Often, what we want to describe are constraints. A constraint is something that limits what is possible or what is reasonable in a situation. For example, when planning a birthday party, we might be dealing with these quantities and constraints: quantities • the number of guests • the cost of food and drinks • the cost of birthday cake • the cost of entertainment • the total cost constraints • 20 people maximum • \$5.50 per person
• \$40 for a large cake • \$15 for music and \$27 for games • ​​​​no more than \$180 total cost

We can use both numbers and letters to represent the quantities. For example, we can write 42 to represent the cost of entertainment, but we might use the letter $$n$$ to represent the number of people at the party and the letter $$C$$ for the total cost in dollars.

We can also write expressions using these numbers and letters. For instance, the expression $$5.50n$$ is a concise way to express the overall cost of food if it costs \$5.50 per guest and there are $$n$$ guests. Sometimes a constraint is an exact value. For instance, the cost of music is \$15. Other times, a constraint is a boundary or a limit. For instance, the total cost must be no more than \\$180. Symbols such as <, >, and = can help us express these constraints.

quantities

• the number of guests
• the cost of food and drinks
• the cost of birthday cake
• the cost of entertainment
• the total cost

constraints

• $$n \leq 20$$
• $$5.50n$$
• $$40$$
• $$15 + 27$$
• $$C \leq 180$$

Equations can show the relationship between different quantities and constraints. For example, the total cost of the party is the sum of the costs of food, cake, entertainment. We can represent this relationship with:

$$C = 5.50n + 40 + 15 + 27 \qquad \text {or} \qquad C=5.50n+82$$

Deciding how to use numbers and letters to represent quantities, relationships, and constraints is an important part of mathematical modeling. Making assumptions—about the cost of food per person, for example—is also important in modeling.

A model such as $$C = 5.50n + 82$$ can be an efficient way to make estimates or predictions. When a quantity or a constraint changes, or when we want to know something else, we can adjust the model and perform a simple calculation, instead of repeating a series of calculations.