This activity encourages students to use the number line to make sense of powers of 10 and to think about how to rewrite expressions for very small numbers. Students use the structure of the number line to compare numbers and extend that understanding to estimate relative sizes of other numbers when no number lines are given (MP7). 

The goal of this discussion is for students to articulate how to rewrite very small numbers written using one power of 10 with a different power of 10.

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 ?” (10 because the exponent is increasing by 1)
• “If is multiplied by 10 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 10.)
• “What is the resulting equivalent expression?” ()
• “How can be written using ? (Since is being divided by , the factor 0.0123 needs to be multiplied by , resulting in .)

In this activity, students convert small decimals into a multiple of a power of 10 with a negative exponent for the first time. The first problem leads to a product of an integer and a power of 10, and the second leads to a product of a decimal and a power of 10. Again, students build experience with scientific notation before the term is formally introduced.

If students write the mass of a proton as  but find that there are not enough tick marks on the number line, consider asking:

• “Would 1.7 fit on the given number line?” (Yes.)
• “How could the expression be written with 1.7 as one of the factors while still keeping the value of the expression the same?” (Divide 17 by 10 and multiply by 10, resulting in .

The goal of this discussion is for students to share the strategies they used when writing small decimals as a multiple of a power of 10. Some students may describe a strategy where they “move” the decimal point or count decimal places. Encourage students to develop a more precise explanation by emphasizing that each time a number is multiplied by , its value decreases by one place value. So for example, the value 0.0000003 is equal to because 3 has been multiplied by seven times.

If time allows, have students practice writing decimals as a multiple of a given power of 10. Note that students are not expected to write these values using scientific notation at this time.

• “What is 0.00014 written as a multiple of ?” ()
• “What is 0.0000008 written as a multiple of ?” ()
• “What is 0.00256 written as a multiple of ?” ()