Lesson 2

This warm-up prompts students to compare four tables. It gives students a reason to use language precisely (MP6) and gives the opportunity to hear how they use terminology and talk about characteristics of the items in comparison to one another. To allow all students to access the activity, each item has one obvious reason it does not belong. Encourage students to move past the obvious reasons and find reasons based on mathematical properties.

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. To encourage more participation, after each person shares, ask if anyone chose the same table for the same reason, or if they have alternative reasons for claiming a particular table as the one that does not belong. It may be useful to introduce the language of common differences and common factors at this time. For example, in Table B, for every increase of 2 in \(x\), the difference between \(y\)-values is 16. However, in Table C, for every increase of 1 in \(x\), the \(y\)-values are related by a factor of 4. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

In the previous lesson, students compared two patterns presented via a description and a graph. In this activity, they make the same kind of comparisons using a table. A table is particularly helpful for illustrating that:

• linear functions grow by equal differences over equal intervals
• exponential functions grow by equal factors over equal intervals

At this stage, students simply observe these properties for a few particular functions and in the context of concrete problems. Toward the end of the unit, they will prove these properties are true for any linear or exponential function.

Tell students that they will now look at a couple of possible ways a company can expand its business. Give students access to a spreadsheet tool or a calculator. Arrange students in groups of 2. If time is limited, ask one partner to complete the questions for Part A and the other to do so for Part B.

Students may struggle to write an expression using the variable \(n\). Ask them to tell you any patterns they notice in the tables and then encourage them to use those patterns to write an expression for \(n\).

To help students compare and contrast the two tables, ask:

• “How are the tables alike? Or how was the process of completing the two tables the same?” (The tables both follow a simple pattern: each table row is calculated by applying the same single arithmetic operation to the previous row. This pattern also holds for every other row or every third row, and so on. The entries in both tables increase as we go down the table.)
• “How are the tables different? Or how was the process of completing the two tables different?” (In Plan A, 20 new stores are added each year; the repeated arithmetic operation is addition. In Plan B, the number of stores is doubled each year; the repeated arithmetic operation is multiplication. Plan B grows much more quickly than Plan A.)

Students may notice in the table that the difference from the number of stores the previous year is the same as the number of stores the previous year. This is because of the doubling pattern: to find the number of stores in the next year, the number from the previous year is added to itself (or doubled).

Students study two stories characterized by different growth patterns, and match expressions and tables with the two situations. The numbers in the two scenarios are identical so students need to focus on the mathematical relationships and the operations in the expressions to successfully complete the activity. If students are evaluating the expressions, encourage them to reason about the operations without calculating anything.

Focus the discussion on how students went about matching the cards and the situations. Record students' reasoning for all to see. In particular, highlight observations about common differences and common factors. Ask questions such as:

• “Besides evaluating the expression, how could you tell (a certain expression) matched one of the situations?”
• “How could you tell (a certain table) matched one of the situations?”
• “How is the number of followers growing every 2 years?”
• “How is the amount of water in the tank increasing every 2 minutes?”