# Lesson 20

Productos cuyos resultados son centésimas

## Warm-up: ¿Qué sabes sobre $1\times 0.1$ y $0.1 \times 0.1$? (10 minutes)

### Narrative

The purpose of this What Do You Know About ____? is for students to think about $$0.1 \times 0.1$$ before working with this kind of expression more formally in the lesson. Students know that 0.1 is the same as $$\frac{1}{10}$$ and they know how to find products of fractions. The goal of the synthesis is to highlight this before students find products of decimals in the lesson.

### Launch

• Display the expressions.
• “¿Qué saben sobre $$1 \times 0.1$$ y $$0.1 \times 0.1$$?” // “What do you know about $$1 \times 0.1$$ and $$0.1 \times 0.1$$?”
• 1 minute: quiet think time

### Activity

• Record responses.
• “¿Cómo podríamos representar estas expresiones?” // “How could we represent these expressions?” (I could use a hundredths grid or area diagram.)

### Student Facing

¿Qué sabes sobre estas expresiones?

• $$1 \times 0.1$$
• $$0.1 \times 0.1$$

### Student Response

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### Activity Synthesis

• “¿Pueden encontrar el valor de $$0.1 \times 0.1$$?” // “Can you find the value of $$0.1 \times 0.1$$?” (Yes, 0.1 is $$\frac{1}{10}$$ so that’s $$\frac{1}{10} \times \frac{1}{10}$$ and I know that’s $$\frac{1}{100}$$.)

## Activity 1: Productos de décimas (15 minutes)

### Narrative

The purpose of this activity is for students to find products of a number of tenths and a number of tenths written as decimals. Students can think of find these products in many ways including

• using a diagram
• using whole number arithmetic and place value reasoning or properties of operations (MP7)
The goal of the synthesis is to relate these different ways of finding the product.

MLR8 Discussion Supports. Synthesis: During group work, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “Te escuché decir . . .” // “I heard you say . . .” Original speakers can agree or clarify for their partner.
Advances: Listening, Speaking
Representation: Internalize Comprehension. Provide students with a graphic organizer, such as a two-column table, to record the multiplication expression using fractions and the corresponding multiplication expression using decimal numbers to show the connection between fractions and decimal numbers and why $$0.1 \times 0.1 = 0.01$$.
Supports accessibility for: Conceptual Processing, Memory

### Required Materials

Materials to Copy

• Small Grids

• Groups of 2

### Activity

• 1–2 minutes: quiet think time
• 6–8 minutes: partner work time
• Monitor for students who:
• use grids
• use whole number facts and place value reasoning

### Student Facing

1. Encuentra el valor de cada expresión. Explica o muestra cómo razonaste. Usa las cuadrículas si te ayuda.

1. $$2 \times 0.3$$

2. $$0.2 \times 0.3$$

2. Kiran dice que $$0.2 \times 0.4 = 0.8$$. ¿Estás de acuerdo con Kiran? Explica o muestra cómo razonaste.

### Student Response

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### Activity Synthesis

• Invite students to share their reasoning for $$0.2 \times 0.4$$.
• Display a student generated diagram of $$0.2 \times 0.4$$ or the diagram from the student solution.
• “¿Cómo se muestra $$0.2 \times 0.4$$ en el diagrama?” // “How does the diagram show $$0.2 \times 0.4$$?” (There is 2 tenths of 4 tenths of the rectangle shaded.)
• “¿Cómo supieron que la región sombreada tiene un área de 0.08 unidades cuadradas?” // “How did you know that the shaded region has area 0.08 square units?” (There are $$2 \times 4$$ shaded pieces and each one is $$\frac{1}{100}$$ of the full square.)
• Display equation $$0.2 \times 0.4 = 2 \times 4 \times (0.1 \times 0.1)$$.
• “¿Cómo se muestra esta ecuación en el diagrama?” // “How does the diagram show this equation?” (The shaded part is 2 tenths of 4 tenths of the rectangle so that's $$0.2 \times 0.4$$. It’s $$2 \times 4 \times (0.1 \times 0.1)$$ because there are $$2 \times 4$$ pieces and each one has area $$0.1 \times 0.1$$ or one hundredth of a square unit.)

## Activity 2: Multipliquemos décimas (20 minutes)

### Narrative

The purpose of this activity is for students to multiply decimals by decimals, building on the strategies they saw in the previous activity. Monitor for these strategies:

• using a diagram
• using whole number products and place value understanding
• using expressions to show their thinking

### Required Materials

Materials to Copy

• Small Grids

### Launch

• Groups of 2
• Make copies of hundredths grid blackline master available.

### Activity

• 5 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for students who:
• use the grids
• multiply two whole numbers and then multiply their product by $$0.01$$

### Student Facing

1. Encuentra el valor de cada expresión. Explica o muestra tu razonamiento.

1. $$1.8 \times 0.4$$
2. $$2.5 \times 0.6$$
3. $$3.8 \times 0.7$$
2. ¿En qué se parecen estos productos? ¿En qué son diferentes?
• $$74 \times 6$$
• $$7.4 \times 6$$
• $$7.4 \times 0.6$$

### Student Response

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### Activity Synthesis

• Invite students to share their responses and reasoning for the product $$1.8 \times 0.4$$.
• Display student generated diagram or diagram in student solution.
• “¿Cómo se muestra $$1.8 \times 0.4$$ en el diagrama?” // “How does the diagram show $$1.8 \times 0.4$$ ?” (There is a full group of 0.4 and then there is 8 tenths of another group of 0.4.)
• “¿Cómo se muestra $$18 \times 4 \times 0.01$$ en el diagrama?” // “How does the diagram show $$18 \times 4 \times 0.01$$?” (There is an 18 by 4 array of pieces and each piece is a hundredth of the whole.)
• Display: $$1.8 \times 0.4 = (18 \times 4) \times 0.01$$
• Invite students to share their responses about the products $$74 \times 6$$, $$7.4 \times 6$$ and $$7.4 \times 0.6$$.
• “¿Cómo pueden usar un producto de números enteros para encontrar un producto de números decimales?” // “How can you use the whole number product to find decimal products?” (I just think about how many tenths or hundredths I have.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy encontramos productos de números decimales usando diagramas y pensando en el valor posicional” // “Today we found products of decimals using diagrams and thinking about place value.”

Display:
$$4.5 \times 8.1 = 45 \times 0.1 \times 81 \times 0.1$$

“¿Cómo sabemos que esto es verdadero?” // “How do we know this is true?” ($$4.5 = 45 \times 0.1$$ and $$8.1 = 81 \times 0.1$$ so $$4.5 \times 8.1 = 45 \times 0.1 \times 81 \times 0.1$$)

Display:
$$4.5 \times 8.1 = 45 \times 81 \times 0.01$$

“¿Cómo sabemos que esto es verdadero?” // “How do we know this is true?” (If we change the order of factors in the expression $$45 \times 0.1 \times 81 \times 0.1$$, we get $$45 \times 81 \times 0.1 \times 0.1$$ and that is equal to $$45 \times 81 \times 0.01$$.)

“¿Cómo nos ayuda esto a encontrar el valor de $$4.5 \times 8.1$$?” // “How is this helpful for finding the value of $$4.5 \times 8.1$$?” (I can just find the whole number product and then say I have that many hundredths.)

## Cool-down: Décimas (5 minutes)

### Cool-Down

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