Lesson 17

The purpose of this True or False is to consider relationships between place values. This recalls work that students did earlier in the unit and seeing these relationships expressed using multiplication naturally prepares students for the work of the next several lessons where they will learn to find products of decimals. Students will revisit multiplicative relationships of place values in a later unit.

• Display one statement.
• “Hagan una señal cuando sepan si la afirmación es verdadera o no, y puedan explicar cómo lo saben” // “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: \(100 \times 0.1\), \(10 \times 0.1\), \(10 \times 0.01\)
• “¿En qué se parecen y en qué son diferentes estas expresiones?” // “What is the same and different about these expressions?” (They all have multiples of 10 in them. They all have decimals. The values are all different.)

The purpose of this activity is for students to find decimal products in a way that makes sense to them. Many approaches are possible including:

• thinking about the meaning of place value and multiplying by place value
• using the hundredths grids
• using fractions or a number line

For the last problem, students may use their understanding of arithmetic, the distributive property, and their work on the first two problems (MP7) or they may make a new calculation. The goal of the synthesis is to share and connect different strategies for finding the values of the products.

• Groups of 2
• Make copies of hundredths grid available to students.
• Display the image from student workbook.
• “¿Cuántos ven?” // “How many do you see?” (2 big squares, 60 small shaded squares, 6 shaded rows)
• 1 minute: partner discussion
• Display expression: \(2 \times 0.3\)
• “¿Cómo está representada la expresión en el diagrama?” // “How does the diagram represent the expression?” (There are 2 groups of 0.3 shaded.)

• 5–7 minutes: partner work time
• Monitor for students who:
  • use diagrams to find the products
  • use multiplication of whole numbers and place value understanding to find the products

• Invite students to share how they found the value of \(2 \times 0.7\).
• Display student work or image from solution.
• “¿Cómo se muestra \(2 \times 0.7\) en el diagrama?” // “How does the diagram show \(2 \times 0.7\)?” (There are two groups of 0.7 shaded.)
• “¿Cómo se puede usar el diagrama para encontrar el valor de \(2 \times 0.7\)?” // “How can you use the diagram to find the value of \(2 \times 0.7\)?” (I can see that I have \(2 \times 7\) or 14 tenths and that’s 1 whole and 4 more tenths.)
• “¿En qué se parecen encontrar el valor de \(2 \times 0.08\) y encontrar el valor de \(2 \times 0.7\)? ¿En qué son diferentes?” // “How is finding the value of \(2 \times 0.08\) the same as finding the value of \(2 \times 0.7\)? How is it different?” (I could shade part of each of the large squares and then see the total. This time I shaded individual squares rather than rows of squares. In both cases, I can use multiplication to find the product.)

In the previous activity students found products of a whole number and some tenths or hundredths using hundredths grids or a strategy that made sense to them. The goal of this activity is to find these products with a greater focus on place value and the associative property of multiplication (MP7). For example, \(5 \times 0.07\) means 5 groups of 7 hundredths. That means that its value is 35 hundredths or 0.35. This way of thinking about products allows students to use what they know about finding whole number products in order to find products of a whole number and a decimal number (MP8).

This activity uses MLR3 Clarify, Critique, and Correct. Advances: Reading, Writing, Representing.

• Groups of 2
• Make hundredths grids available for students. 

• “Tómense unos minutos para encontrar el valor de las expresiones del primer problema” // “Take a few minutes to find the value of the expressions in the first problem.”
• 1–2 minutes: quiet think time
• 5 minutes: partner work time

• Read Kiran’s explanation aloud.
• “¿Qué creen que Kiran quiere decir? ¿Qué no es claro?” // “What do you think Kiran means? What is unclear?”
• 1 minute: quiet think time
• 2 minutes: partner discussion
• “Con su compañero, escriban juntos una explicación ajustada” // “With your partner, work together to write a revised explanation.”
• Display and review the following criteria: 
  • Write an explanation for each step.
  • Use specific words and phrases such as equal or groups of.
  • Use complete sentences.
  • Write expressions or equations as examples.
• 3–5 minutes: partner work time
• Select 1–2 groups to share their revised explanation with the class. Record responses as students share.
• “¿En qué se parecen y en qué son diferentes las explicaciones?” // “What is the same and different about the explanations?” (Kiran did not use any numbers or equations. He did not explain why his strategy works.)

If a student does not find the correct value of the expressions, show them \(3 \times 5\), \(5 \times3\), \(7 \times2\) and ask, ”¿En qué se parecen estas expresiones y las expresiones que se muestran en el problema? ¿En qué son diferentes?” // “How are these expressions the same as and different from the expressions in the problem?”

• “Con su compañero, adapten el razonamiento de Kiran para encontrar el valor de \(6 \times 0.07\)” // “Work with your partner to adapt Kiran’s reasoning to find the value of \(6 \times 0.07.\)”
• 2-3 partner work time
• Invite students to share their responses and reasoning for the value of \(6 \times 0.07\).
• “¿En qué se parece esto a encontrar el valor de \(6 \times 7\)? ¿En qué es diferente?” // “How is this the same as finding the value of \(6 \times 7\)? How is it different?” (I found \(6 \times 7\) but needed to remember that it’s hundredths so the product is 42 hundredths.)