Lesson 1

The purpose of this warm-up is for students to discuss the multiplicative relationships between the place values of the digits in two numbers. This will be useful when students write multiplication and division expressions to represent place value relationships in a later activity. While students may notice and wonder many things about these numbers, the place value relationships between the digits in the numbers and the numbers themselves are the important discussion points. 

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses. 

• “How does the value of 8,200 compare to the value of 820?” (It’s 10 times as much.)
• “How does the value of 0.82 compare to the value of 0.082? How do you know?” (It’s also ten times as much since there are ten thousandths in a hundredth.)

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 ten and then use them for measurement conversions.

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

• Groups of 2
• Display the numbers: 60, 6
• “How many times the value of 6 is 60? How do you know?” (10 times because it’s 6 tens)
• Display the equation: \(60 = 10 \times 6\).
• “What division equation shows that 60 is ten times the value of 6?” (\(60 \div 6 = 10\) is another way of saying that 60 is ten 6s or ten times 6.)
• Display the equation: \(60 \div 6 = 10\). 
• “You are going to write equations like these relating different numbers.”

• 5 minutes: partner work time

MLR7 Compare and Connect

• “Create a visual display that shows your equations. You may want to include details such as notes, diagrams or drawings to help others understand your thinking.”
• Monitor for students who:
  • identify an equation that is incorrect during the gallery walk
  • notice place value patterns during the gallery walk
• 2–5 minutes: independent or group work
• 5–7 minutes: gallery walk

• Invite students to share equations they made.
• Display the equation: \(0.6 \div 10 = 0.06\)
• “How do you know this equation is true?” (When I divide tenths into ten 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. \(0.6 = 10 \times 0.06\).)
• Display the equation: \(600 \times 0.01 = 6\)
• “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. \(600 \div 100 = 6\).)
• 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 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 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 \(\frac{1}{10}\) when it shifts one place to the right (MP7).

The former idea will be further developed in the next lesson where 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.

• Groups of 2
• “We are going to continue to work with the numbers from the previous activity to explore more patterns.”

• 5 minutes: individual work time
• 5 minutes: partner work time

• “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 \(\frac{1}{10}\) or 0.1. It is divided by 10.)
• Invite students to share the numbers that they listed that come before 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.”