Lesson 23

The purpose of this warm-up is for students to analyze statements about quotients of whole numbers by 0.1 or 0.01. Students compare the value of quotients by 0.1 and 0.01. They can analyze the statements either by calculating the value of the expressions or reasoning about the relationship between 0.1 and 0.01, namely that there are ten hundredths in a tenth. 

• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

• Share and record answers and strategy.
• Repeat with each statement.

• Display second statement.

• “How can you prove this statement is true without finding the value of the expressions on both sides of the equal sign?” (I can think about the size of the quotients. They both show 6 being divided into groups, but the size of the groups is bigger in \(6\div0.1\) so there will be fewer groups.)

The purpose of this activity is for students to find quotients of a whole number by tenths or hundredths. This activity focuses on divisors of 0.2 and 0.02 to build on students' previous work dividing by 0.1 and 0.01. The strategies, diagrams and place value reasoning used in the previous lessons still apply. Students may also think about the problems using multiplication. For example \(1 \div 0.2 = 5\) because \(5 \times 0.2 = 1\).

• Groups of 2
• Give students access to blackline masters with grids.

• 1–2 minutes: quiet think time
• 8–10 minutes: partner work time
• Monitor for students who represent their reasoning with:
  • diagrams
  • expressions or words using multiplication

If a student is not able to use the value of the first expression in each set to find the value of the later expressions, ask them to describe what they notice and wonder about each set of expressions. Use numbers and symbols when possible to record the relationships they describe.

• Ask selected students to share the representations listed above.
• “How are the representations the same? How are they different?” (They both figured out how many tenths or hundredths are in one whole. Then they multiplied that by the number of wholes. One of them used a picture and one of them reasoned about the size of tenths and hundredths to find how many 0.2's and 0.02's are in one whole.)
• Display a student's completed work showing the values of the quotients.
• “What patterns do you see across the sets of equations?” (The quotients with a divisor of 0.01 are 10 times larger than those with a divisor of 0.1. The quotients with a divisor of 0.1 are a tenth of the value of those with a divisor of 0.01.)
• Record the patterns as students describe them.

The purpose of this activity is for students to find the value of quotients where the divisor is less than 1 and where the dividend is large enough that drawing a complete diagram is cumbersome. Instead, students are encouraged to use a diagram to find how many divisor sized groups are in 1 whole and then multiply the dividend by that number to find the value of the quotient. When students use the value of \(1 \div 0.2\) and multiplication to find the value of \(12 \div 0.2\) they are using regularity in reasoning (MP8).

This activity uses MLR1 Stronger and Clearer Each Time. Advances: Conversing.

• Groups of 2

• Give students access to blackline masters with hundredths grids.

• 3–5 minutes: independent work time
• Monitor for students who use the following strategies for problem 1:
  • reason about how many divisor-sized groups are in 1 whole, possibly with a diagram, and then multiply by the dividend
  • use multiplication equations to find the value of the division expressions

If a student is not sure how to explain Tyler’s reasoning in problem 2, ask them to read the expression to you and record the words they use. Then ask them how those words relate to the words Tyler used.

• Ask previously identified students to share their solutions and strategies for evaluating the expression in the first problem.
• “How was multiplication useful to find the value of the expression in the first problem?” (I didn't have to draw 12 wholes because I knew there were five 0.2s in each one so that would be \(12 \times 5\) total.)

MLR1 Stronger and Clearer Each Time

• “Share your response to problem 2 with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 3-5 minutes: structured partner discussion
• Repeat with 2 different partners.
• “Revise your initial draft based on the feedback you got from your partners.”
• 2-3 minutes: independent work time