This activity encourages students to use the number line to make sense of powers of 10 and to think about how to rewrite expressions in the form , where is between 1 and 10 (as in the case of scientific notation). Students use the structure of the number line to compare numbers, and to extend their use to estimate relative sizes of other numbers when no number lines are given (MP7).

In the digital version of the activity, students use an applet to place points on a number line. The applet allows students to quickly check and revise their placement if necessary. The digital version may be helpful for students who rush to place the values on the number line and would benefit from immediate feedback.

The goal of this discussion is for students to see how rewriting numbers to have the same power of 10 makes it easier to compare their relative sizes. Here are some questions for discussion:

• “How can 4,000,000, , and all be written as a multiple of ?” ()
• “Why are numbers easier to compare when they are all multiplied by the same power of 10?” (Since they are all multiplied by the same power of 10, we needed to compare only the first number of each expression.)
• “If the population of the United States is roughly people and the global population is roughly people, about how many times larger is the global population than the U.S. population?” ( is roughly 20 times larger than , because 8 is roughly twice as large as 3.5, and is 10 times larger than .)

This activity promotes a deeper understanding of expressions written as the product of a single digit and a power of 10. Students investigate a table of the speed of light waves through different materials. To distinguish more easily between the different speeds of light through various materials, the interval between and is magnified on the number line. This illustrates the base-ten structure and allows for a more accurate placement of numbers with more decimal places. 

Once a number line is labeled with powers of 10 and its structure is understood, numbers written as a multiple of the same power of 10 can be placed on the number line fairly straightforwardly (MP7).

In the digital version of the activity, students use an applet to place points on a number line. The applet allows students to magnify an interval of the number line and easily change the placement of points. The digital version may be helpful for illustrating that the second number line is really a zoomed in version of an interval in the first number line.

The goal of this discussion is to check that students understand how to rewrite very large numbers written using one power of 10 with a different power of 10. 

Begin by asking students to explain how they were able to compare the speeds of light through the different materials even though they were each written in different ways. (Rewriting the numbers so they had the same power of 10 made it easier to compare them.) While students do not yet know the formal definition of scientific notation, this discussion should help illustrate how expressing values in this format allows for an easier comparison.

Invite students to share how they rewrote using . If not brought up in students’ explanations, consider asking the following series of questions:

• “What can be multiplied by to get ?” (100 or )
• “If is multiplied by to get , the value of the expression will change. What can be done to keep the value of the expression the same?” (Divide the other factor by .)
• “What is the resulting equivalent expression?” ()