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 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. Making internet-enabled devices available gives students an opportunity to choose appropriate tools strategically (MP5). 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 if planning to serve pizza, 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).

If students do not understand what is meant by “quantities that might change,” ask them if it is more likely that the cost of party items increases on the day of the party, or the number of students changes. In a model that incorporates both of these quantities, they may wish to use a number for the cost of each item and a letter for the number of students present that day.

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 food items per person
• The cost of delivery

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. 

If students have trouble thinking of constraints for a chosen variable, ask about extreme values. For instance, ask: "Do you think a large package of 150 stickers 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 . Ask these students to explain the meaning of the “>” symbol. Ask: "If 29 students come to class, can we write ?"

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: A large package of stickers comes with exactly 150 individual stickers, or 50 balloons are needed to make an arch.
• Other times, a constraint involves a boundary or a limit. For example: The party must cost less than $90, or at least 25 students must be in class to have the party.