The purpose of this activity is to remind students about the need for exponential notation when thinking about problems involving repeated multiplication. Before students begin working, they are explicitly asked to make an estimate. Making a reasonable estimate and comparing a computed value to one’s estimate is often an important aspect of making sense of problems (MP1).

If some students interpret the last question to mean “How many more cubes does the 28th tower have compared to the 25th tower,” or if students think they need to know exactly how many cubes are used in each tower to be able to answer this question, consider asking:

• “How many cubes are needed for the 5th tower?” (32 or cubes)
• “How many cubes are needed for the 6th tower?” (64 or cubes)
• “How many times bigger is 64 compared to 32? How many times bigger is compared to ?” (Either way the values are expressed, each new tower needs 2 times as many cubes as the one before it.)

The goal of this discussion is for students to understand exponential notation and use it to reason about a situation that involves repeated multiplication. Display the table for all to see.

Tell students that exponents allow us to perform operations and reason about numbers that are too large to calculate by hand. Explain that the “expanded” column shows the factors being multiplied, the “exponent” column shows how to write the repeated multiplication more succinctly with exponents, and the “value” column shows the decimal value. Discuss with students:

• “How many times larger is than ? (4 times larger)
• "How does the "expanded" column help explain this? (The “expanded” column shows that has 2 more factors of 2 compared to , and .)
• “How many times larger is than ?” (16 times larger because has 4 more factors that are 2, and .)

This activity prompts students to think about repeated division by 2 as being equivalent to repeated multiplication by . This will lay the foundation for understanding negative exponents in later lessons.

The goal of this discussion is for students to understand that repeated division by 2 corresponds to repeated multiplication by . Ask students to discuss their responses with their partner before discussing the following questions:

• “What are some different ways we can describe what is happening to the height of this tower each day?” (The height is cut in half. The height is getting divided by 2.)
• “How can division by 2 be represented with multiplication? That is, if we want to cut something in half, what value can we multiply by?” (We can multiply by or 0.5.)
• “How can repeated division by 2 be represented with exponents?” (Since dividing by 2 is the same as multiplying by , repeated division by 2 corresponds to repeated multiplication by , which can be written as or )