In this activity, students use tables to make comparisons between 2 patterns. 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 that these properties are true for any linear or exponential function.

Students may struggle to write an expression using the variable . 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 .

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 does 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 they 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 representations 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 which expressions matched each of the situations?” (To fit the first situation that is tripling, I looked for things being multiplied by 3 multiple times. For the second situation, it was growing at a constant rate of 3, so I looked for things where 3 was added multiple times.
• “How could you tell which table matched each of the situations?” (I looked for values that are being multiplied by 3 for each increase of 1 in to match the first situation. For the second situation, I looked for values that have a difference of 3 for each increase of 1 in .
• “How is the number of followers growing every 2 years?” (It is tripling twice, so they are being multiplied by 9 every 2 years.)
• “How is the amount of water in the tank increasing every 2 minutes?” (3 gallons of water are being added twice, so 6 gallons are added every 2 minutes.)