In this activity, students use their understanding of decimal place value and base-ten diagrams to practice working with the structure of scientific notation before it is formally introduced. They express numbers as sums of terms, with each term being a multiple of a power of 10. For example, 254 can be written as . 

The goal of this discussion is to make sure students see the connection between decimal place value and sums of terms that are multiples of powers of 10.

Begin by inviting students to share their responses to the first matching problem. If not brought up in students’ responses, bring attention to the fact that for expressions a and d, both diagrams F and G could be used depending on the choice of the value of the small square. Discuss the following questions:

• “How are the diagrams related to our base-ten numbers and place value system?” (In base-ten numbers, each place value is ten times larger than the one to its right; so every 1 unit of a place value can be composed of 10 units of the next place value to its right. The diagrams work the same way: Each shape representing a base-ten unit can be composed of 10 that represent another unit that is one tenth of its value.)
• “How are the diagrams related to numbers written using powers of 10?” (We can think of each place value as a power of 10. So a ten would be , a hundred would be , a tenth would be , and so on.)
• “If each large square represents , what do 2 large squares and 4 long rectangles represent?” (A long rectangle is a tenth of the large square, so we know the long rectangle represents . This means that 2 large squares and 4 long rectangles represent .)
• “Why is it possible for one base-ten diagram to represent many different numbers?” (Because the structure of our place value system—where every group of 10 of a base-ten unit composes 1 of the next higher unit—is consistent across all place values.)

This activity motivates students to find easier ways to communicate about very large and very small numbers by using powers of 10, in preparation for working with scientific notation. In describing quantities that have many digits, students must use language precisely (MP6). 

The purpose of the discussion is for students to hear different strategies for communicating very small and very large numbers. Direct students’ attention to the reference created using Collect and Display. Ask students to share how they described the very large numbers to their partner, or how the very large numbers were described to them. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional equivalent phrases. For example, 150,000,000,000 could be described as “one hundred fifty billion,” as “15 followed by 10 zeros,” as , or as .

Display the first 2 rows of the table for all to see. If necessary, explain how 150,000,000,000 and are equivalent. If time allows, ask students to come up with additional equivalent expressions for 150,000,000,000 that use powers of 10 (, ).

Next, ask students to share how they described very small numbers to their partner, or how they were described to them. Display the remaining rows of the table for all to see. Tell students to consider the third row, and discuss the idea that multiplying by means multiplying by , and results in a number that is one place value smaller. That means that 0.0000000000048 is 48 multiplied by 0.1 or 13 times, or . Another way to write this is . If time allows, ask students to come up with additional equivalent expressions that use powers of 10 (, ).