Lesson 18

Usemos hechos de números enteros

Warm-up: Verdadero o falso: Dinámicas de grupo (10 minutes)

Narrative

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.

Launch

• 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

Activity

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

Student Facing

Decide si cada afirmación es verdadera o falsa. Prepárate para explicar tu razonamiento.

• $$30 \times 2 \times 10 = 6 \times 10$$
• $$30 \times 2 \times 10 = 20 \times 3 \times 10$$
• $$60 \times 10 = 30 \times 20$$

Activity Synthesis

• 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.)

Activity 1: De acuerdo o en desacuerdo (15 minutes)

Narrative

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.

Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words.
Supports accessibility for: Memory, Organization

• Groups of 2

Activity

• 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 $$3 \times 0.7 = (3 \times 7) \times 0.1$$.

Student Facing

1. Decide si cada ecuación es verdadera o falsa, y explica o muestra por qué.

1. $$4 \times 0.7 = 28$$
2. $$5 \times 0.8 = 0.40$$
3. $$6 \times 0.03 = (6 \times 3) \times 0.01$$
4. $$8 \times 0.07 = (8 \times 7) \times 0.1$$
2. En cada caso, llena el espacio en blanco para que la ecuación sea verdadera.

1. $$3 \times 0.7 = 3 \times 7 \times \underline{\hspace{0.9cm}}$$
2. $$3 \times 0.07 = 3 \times 7 \times \underline{\hspace{0.9cm}}$$
3. $$5 \times \underline{\hspace{0.9cm}} = (5 \times 4) \times 0.1$$

Activity Synthesis

• 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 $$8 \times 7 \times 0.1$$.)

Activity 2: Interpretemos diagramas y expresiones (20 minutes)

Narrative

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

Activity

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

Student Facing

1. Explica o muestra cómo está representada cada expresión en el diagrama.
1. $$3 \times 0.12$$
2. $$(3 \times 12) \times 0.01$$
3. $$(3 \times 0.1) + (3 \times 0.02)$$
2. Encuentra el valor de $$(3 \times 12) \times 0.01$$. Explica o muestra cómo razonaste.
3. Encuentra el valor de $$(3 \times 0.1) + (3 \times 0.02)$$. Explica o muestra cómo razonaste.

Student Response

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.

Activity Synthesis

• 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.)

Lesson Synthesis

Lesson Synthesis

“Hoy usamos nuestra comprensión del valor posicional para multiplicar números decimales” // “Today we used our understanding of place value to multiply decimals.”

Display:

$$25 \times 0.3$$

$$25 \times 0.03$$

“Describan el proceso que usarían para encontrar el valor de estas expresiones” // “Describe the process you would use to find the value of these expressions.” (Find $$25 \times 3 = 75$$ and then multiply that by 0.1 or 0.01. The first one is 75 tenths or 7.5 and the second one is 75 hundredths or 0.75.)

“¿Cómo podemos multiplicar cualquier número entero por una cantidad de décimas o de centésimas?” // “How can we multiply any whole number by an amount of tenths or hundredths?” (Find the whole number multiplied by the number of tenths or hundredths and multiply that result by 0.1 or 0.01.)