Unit 4, Lesson 1
Planning a Party
Integrated Math 1
1.1
Warm-up
Standards Alignment
Building On
Addressing
HSA-CED.A.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Building Toward
Instructional Routines
NoneMaterials
NoneActivity Narrative
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.
Activity
NoneHere are some letters and what they represent. All costs are in dollars.
- represents the cost of a main dish.
- represents the number of side dishes.
- represents the cost of a side dish.
- represents the total cost of a meal.
- Discuss with a partner: What does each equation mean in this situation?
- Write a new equation that could be true in this situation.
Student Response
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Building on Student Thinking
Activity Synthesis
Invite students to share their interpretations of the given equations and the new equations that 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: ? How do you know?" (It could be. Example: If is 7.50 and is 12.50, then the equation is true.)
- "When might the equation be false?" (Example: If is 7.50 and is anything but 12.50, or if the value of is not 5 more than , 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 . 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
Activity
Materials
NoneActivity Narrative
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).
1.3
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
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 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. Make connections between representations visible. Display or provide charts with symbols and meanings. To encourage critical thinking and application to the expressions created in this activity, invite students to name additional examples of other variables that might be considered constraints. Examples of additional constraints might include to represent how long the party could last, or to say that each student will get less than 3 beverages.
Supports accessibility for: Language, Conceptual Processing
Supports accessibility for: Language, Conceptual Processing
Lesson Synthesis
Some key takeaways from this lesson are the ideas that real-life situations often involve constraints, and 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 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?" (food preferences, amount of food per person, possible prices of food)
- "Suppose we had gathered information differently, for instance, by asking every student the exact food and amount they prefer. 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 loves cheese and would like only 1 slice. How would that have changed the cost estimate?" (If the assumption was a slice of cheese per person, it would probably cost less, but many students might not be able to enjoy the cheese 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 that affect the models. If the information or assumptions change, the model would also change.
Student Lesson Summary
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 to represent the number of people at the party and the letter for the total cost in dollars.
We can also write expressions using these numbers and letters. For instance, the expression is a concise way to express the overall cost of food if it costs \$5.50 per guest and there are 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
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, and entertainment. We can represent this relationship with:
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 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.