Lesson 18

The purpose of this True or False is for students to demonstrate strategies and understandings they have for using the associative property of multiplication. The numbers in this warm-up are whole numbers. In this lesson, students will use whole number products to find the value of the product of a whole number and a decimal and this requires using the associative property of multiplication.

• Display one equation.
• “Hagan una señal cuando sepan si la ecuación es verdadera o no, y puedan explicar cómo lo saben” // “Give me a signal when you know whether the equation is true and can explain how you know.”
• 1 minute: quiet think time

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

• Display first equation.
• “¿Cómo pueden mostrar que esto es falso sin encontrar el valor de ambos lados?” // “How can you show this is false without finding the value of both sides?” (I know \(30 \times 2\) is not 6 and then multiplying both sides by 10 will not make them equal.)
• Display second equation.
• “¿Cómo pueden mostrar que esto es verdadero sin encontrar el valor de ambos lados?” // “How can you show this is true without finding the value of both sides?” (I know \(30 \times 2\) and \(20 \times 3\) are equal and then they are both multiplied by 10.)

The purpose of this activity is for students to use place value reasoning and properties of operations to relate products of a whole number and a decimal to products of a whole number and either 0.1 or 0.01 (MP7). Students may decide that an equation is false without finding the value of both sides. For example, in the first problem, they may determine that 28 is unreasonable because 0.7 is less than one whole so the answer will be less than 4. When students find the value that makes equations true, they think about place value and the associative property of multiplication.

• Groups of 2

• 4 minutes: independent work time
• 4 minutes: partner discussion
• Monitor for students who:
  • compare the size of the product to the size of the factors to determine reasonableness. For example, explain that \(4 \times 0.7\) is not equal to 28 because the product should be less than 4.
  • use the associative property to represent the product of a whole number and a decimal as a product of two whole numbers and a decimal such as \(3 \times 0.7 = (3 \times 7) \times 0.1\).

• Display the equation \(8 \times 0.07 = (8 \times 7) \times 0.1\) from the first problem.
• “¿La ecuación es verdadera o falsa?” // “Is the equation true or false?” (false)
• Display expression: \( (8 \times 7) \times0.1\)
• “¿Cómo podríamos ajustar esta expresión para que la ecuación sea verdadera?” // “How could we revise this expression to make the equation true?” (Change it to read \(56\times0.01\), \(8\times7\times0.01\), or \(0.56\).)
• Display equation: \(8 \times \underline{\hspace{0.9cm}} = (8 \times 7) \times 0.1\)
• “¿Qué puedo escribir en el espacio en blanco para que la ecuación sea verdadera?” // “What can I write in the blank to make the equation true?” (0.7 since that’s 8 groups of 7 tenths or \(8 \times 7 \times 0.1\).)

The purpose of this activity is for students to use a diagram to support understanding two different ways to calculate the product of a whole number and a decimal number. The first strategy is one that students saw in the previous lesson, namely using whole number arithmetic to find the number of hundredths in the product and then multiplying that number by 0.01. The second strategy is useful specifically when the decimal has both tenths and hundredths. Using the distributive property, students can find the product of the whole number and the tenths and then the product of the whole number and the hundredths and combine these (MP7). In the next several activities, students will use both of these strategies as they build their understanding of how to find the product of a whole number and a decimal. 

• Groups of 2

• 2 minutes: quiet think time
• 10 minutes: partner work time

If students don’t explain how the diagram represents an expression, ask, “¿Cómo se muestra la multiplicación de números decimales en el diagrama?” // “How does the diagram show multiplication of decimals?” Then, connect the student’s response to the expressions.

• Invite students to share their calculations of the value of \(3 \times 0.12\) using the different expressions.
• “¿Cómo les ayudó la expresión \((3 \times 12) \times 0.01\) a encontrar el valor de \(3 \times 0.12\)?” // “How did the expression \((3 \times 12) \times 0.01\) help to find the value of \(3 \times 0.12\)?” (I was able to just multiply whole numbers and then notice that the product is that many hundredths.)
• “¿Cómo les ayudó la expresión \((3 \times 0.1) + (3 \times 0.02)\) a encontrar el valor de \(3 \times 0.12\)?” // “How did the expression \((3 \times 0.1) + (3 \times 0.02)\) help to find the value of \(3 \times 0.12\)?” (I multiplied the tenths and then the hundredths and added them together.)
• “¿Cuál estrategia prefieren?” // “Which strategy do you prefer?” (I like the first strategy because I can just use what I know about whole number products and it will always work.)