Lesson 1 Cafeteria Actions and Reactions Develop Understanding

Jump Start

Evaluate the following expression for $x = 3$ . Show each step of your work so you can review it with a neighbor.

$\frac{4 (2 x + 3) + 4}{8}$

Learning Focus

Solve multi-step linear equations using inverse operations.

How does connecting the operations in an equation to an action in a story help me develop and justify my solution process for solving the equation?

How does the structure of an equation give me clues about how to solve it?

Open Up the Math: Launch, Explore, Discuss

Elvira, the cafeteria manager, has just received a shipment of new trays with the school logo prominently displayed in the middle of each tray. After unloading $4$ cartons of trays in the pizza line, she realizes that students are arriving for lunch and she will have to wait until lunch is over before unloading the remaining cartons. The new trays are very popular, and in just a couple of minutes, $24$ students have passed through the pizza line and are showing off the school logo on the trays. At this time, Elvira decides to divide the remaining trays in the pizza line into $3$ equal groups so she can also place some in the salad line and the sandwich line, hoping to attract students to the other lines. After doing so, she realizes that each of the $3$ serving lines has only $12$ of the new trays.

“That’s not many trays for each line. I wonder how many trays there were in each of the cartons I unloaded?”

1.

Help the cafeteria manager answer her question using the data in the story about each of the actions she took. Explain how you arrived at your solution.

Elvira is interested in collecting data about how many students use each table during lunch. She has recorded some data on sticky notes to analyze later. Here are the notes she has recorded:

Some students are sitting at the front table. (I got distracted by an incident in the back of the lunchroom, and forgot to record how many students.)
Each of the students at the front table has been joined by a friend, doubling the number of students at the table.
Four more students have just taken seats with the students at the front table.
The students at the front table separated into three equal-sized groups and then two groups left, leaving only one-third of the students at the table.
As the lunch period ends, there are still $12$ students seated at the front table.

Elvira is wondering how many students were sitting at the front table when she wrote her first note. Unfortunately, she is not sure what order the middle three sticky notes were recorded in since they got stuck together in random order. She is wondering if it matters.

2.

Does it matter which order the notes were recorded in? Determine how many students were originally sitting at the front table based on the sequence of notes that appears above. Then, rearrange the middle three notes in different orders and determine what the new order implies about the number of students seated at the front table at the beginning.

Pause and Reflect

3.

Here are three different equations that could be written based on a particular sequence of notes. Examine each equation, and then list the order of the five notes that is represented by each equation. Find the solution for each equation.

$\frac{2 (x + 4)}{3} = 12$

$2 (\frac{x}{3} + 4) = 12$

$\frac{2 x + 4}{3} = 12$

Ready for More?

1.

Create your own story context with a sequence of actions that can be represented symbolically with an equation. Then, create a set of sticky note statements for your situation. Rearrange your notes in ways that can be represented by a variety of different equations, even though the actions remain the same, then record your new equations with the corresponding stories.

2.

Are there arrangements of the notes that produce different answers? If so, list the equations.

3.

Can you find different arrangements of the notes that produce the same answer? Why might that be so? List the equations.

Takeaways

Today, we used operations of arithmetic to represent actions in a story context. We used grouping symbols and order of operations to write an equation to model a story context, and then used the story context to help us justify a process for solving the equation.

In general, we learned that when solving a linear equation, we un-do operations in the equation by .

The order for applying inverse operations is .

We can justify each step in the equation-solving process by using properties of equality and properties of operations, such as:

The Distributive Property:

The Addition Property of Equality:

The Subtraction Property of Equality:

The Multiplication Property of Equality:

The Division Property of Equality:

Vocabulary

distributive property of multiplication over addition
inverse operation
multi-step equation
properties of equality
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned how to solve multi-step equations in the context of using operations to represent actions in a story. As we observed how to “un-do” the actions of the story, we developed a strategy for solving the equation by using inverse operations. The order in which inverse operations are applied when solving an equation matters, and we learned how to pay attention to the structure of the equation for clues to the order in which we should use inverse operations.

Retrieval

1.

Graph the equation and determine if the given point is a solution.

$y = - 2 x + 5$

Point: $(1, 3)$

Indicate whether the given value is a solution to the equation. Show your work to justify your response.

2.

$n = 5$

$- 5 n + 13 = - 9$

3.

$b = - 7$

$2 (b + 5) = 3 (b - 2) + 23$