The purpose of this activity is for students to express place-value relationships, using multiplication and division. Students examined decimal place values in depth in the previous unit and used the relationships between the values when they performed arithmetic with decimals. Here they focus on expressing these relationships, using multiplication and division. This will be helpful throughout the next several lessons as students examine powers of 10 and then use them for measurement conversions.

This activity uses MLR7 Compare and Connect. Advances: representing, conversing.

• Invite students to share the equations they made.
• Display the equation:
• “How do you know this equation is true?” (When I divide tenths into 10 equal pieces, I get hundredths, so if I divide 6 tenths into 10 equal pieces, that's 6 hundredths.)
• “Can you express the relationship between 0.6 and 0.06, using multiplication?” (Yes, )
• Display the equation:
• “How do you know this equation is true?” (I know 100 hundredths is 1, so 600 hundredths is 6.)
• “Can you express the relationship between 600 and 6, using division?” (Yes, )
• Invite students to describe any patterns they noticed.

In the previous activity, students wrote multiplication and division equations, relating numbers with a single non-zero digit. The purpose of this activity is for students to focus on the same set of numbers and describe how the value of the non-zero digit changes when it moves one place to the left or to the right. This serves to highlight two important patterns that came out in some of the equations of the previous activity:

• The value of a digit is multiplied by 10 when it shifts one place to the left (MP7).
• The value of a digit is multiplied by 0.1 or when it shifts one place to the right (MP7).

The former idea will be further developed in the next lesson, in which students examine large numbers and exponential notation, and the latter idea will be developed when students examine conversions from a smaller metric unit to a larger metric unit.

• “What happens to the value of the 6 when it shifts one place to the left?” (It is multiplied by 10.)
• “What happens to the value of the 6 when it shifts one place to the right?” (It is multiplied by or 0.1. It is divided by 10.)
• Invite students to share the numbers that they listed that come above 600 on the list.
• “Do you think you can keep listing bigger and bigger numbers, with more and more zeros?” (Yes, I can always add more zeros. I don’t know. After 600,000, I don’t know if I can keep going.)
• “In the next lesson, we will look at some really big numbers and how they relate to multiplying over and over by 10.”