# Lesson 16

Estimemos productos (optional)

## Warm-up: Observa y pregúntate: Tamaño del jardín (10 minutes)

### Narrative

The purpose of this warm-up is for students to reason about the side lengths of a garden with a given area in preparation for an upcoming activity. If students do not name any fractional side lengths, ask the synthesis question to prompt that discussion.

### Launch

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.
• “Si el jardín mide entre 30 y 40 pies cuadrados, ¿cuáles podrían ser las longitudes de sus lados? Piensen en varias opciones” //  “If the garden is between 30 square feet and 40 square feet, what are some possible side lengths?” (Sample responses: 8 feet by 4 feet, 7 feet by 5 feet)
• Share and record responses.

### Student Facing

¿Qué observas? ¿Qué te preguntas?

### Student Response

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### Activity Synthesis

• “Teniendo en cuenta el área, ¿10 pies y $$3 \frac{1}{8}$$ pies podrían ser las longitudes de los lados?” // “Would side lengths of 10 feet and $$3 \frac{1}{8}$$ feet be possible based on the area?” (Yes because 10 times 3 is 30 and since $$10 \times \frac{1}{8}$$ is more than 1 but less than 2, so the area would be more than 30 but less than 40.)

## Activity 1: El jardín de Priya (20 minutes)

### Narrative

The purpose of this activity is for students to notice and use the structure in multiplication expressions to represent the area of rectangles. Students estimate products to see if they are greater than or less than a given amount to solve a problem. Remind students they can draw a diagram if it is helpful.

MLR5 Co-Craft Questions. After students discuss what they know about gardens, read the problem statement and ask, “¿Qué preguntas matemáticas se pueden hacer sobre esta situación?” // “What mathematical questions could be asked about this situation?” Give students 2–3 minutes to write a list of mathematical questions, before comparing their questions with a partner. Invite students to make comparisons about the subject and language of their questions.
Advances: Reading, Writing
Representation: Develop Language and Symbols. Synthesis: Invite students to explain their thinking orally, using a picture or diagram to show possible side lengths.
Supports accessibility for: Conceptual Processing, Organization

### Launch

• Groups of 2
• Display image from student book:
• “¿Qué saben sobre jardines?” // “What do you know about gardens?”

### Activity

• 2–3 minutes: independent work time
• 6–7 minutes: partner discussion
• Monitor for students who:
• draw diagrams.
• estimate the approximate area.
• reason which gardens are less than 36 square feet and which ones are bigger than 36 square feet.

### Student Facing

Priya tiene suficientes materiales para construir un jardín que mida 36 pies cuadrados.

Escoge todas las longitudes que son razonables para los lados del jardín de Priya. Prepárate para explicarle a tu compañero cómo pensaste.

1. 9 pies por $$4 \frac{2}{3}$$ pies
2. 9 pies por $$3 \frac{8}{9}$$ pies
3. 12 pies por $$2 \frac{11}{12}$$ pies
4. 9 pies por $$2 \frac{2}{3}$$ pies

### Student Response

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### Advancing Student Thinking

If students don’t estimate, ask them to think of some whole-number side lengths that would work for Priya’s garden and explain why they would work. Then, ask them to consider the side lengths in the problem.

### Activity Synthesis

• Ask previously selected students to share how they know that two expressions will have values a little bit less than 36. (I know that $$9 \times 4 = 36$$ and $$3\frac{8}{9}$$ is a little less than 4. I know that $$12 \times 3 = 36$$ and $$2 \frac{11}{12}$$ is a little less than 3.)

## Activity 2: Muy alto, muy bajo, aproximadamente igual (15 minutes)

### Narrative

In the previous activity, students reasoned about the value of each product by thinking about the decomposition of the mixed number factor, and how close the mixed number is to the nearest whole number. The purpose of this activity is for students to reason about the value of products by rounding either the whole number or mixed number factors and multiplying.

When students try to make a product close to 20 using the given digits, they will rely on number sense but also may need to experiment and refine their choices and strategy after finding the value of the product (MP1).

• Groups of 2

### Activity

• 5–8 minutes: independent work time
• 1–2 minutes: partner discussion
• Monitor for students who:
• Multiply the whole numbers in problem 1 to find their product that is too low.
• Round the mixed number factor to the nearest whole number before multiplying to find the just right estimate.

### Student Facing

1. Escribe un producto de números enteros que sea un poco menor, otro un poco mayor y otro aproximadamente igual al valor de $$7 \times12\frac89$$.
1. un poco menor:
2. un poco mayor:
3. aproximadamente igual:
2. Escribe un producto de números enteros que sea un poco menor, otro un poco mayor y otro aproximadamente igual al valor de $$9 \times4\frac{2}{29}$$.
1. un poco menor:
2. un poco mayor:
3. aproximadamente igual:
3. Sin calcular, usa los números 2, 3, 5, 6 y 7 para completar la expresión de forma que tenga un valor cercano a 20.
$$\underline{\hspace{1 cm}}\times\underline{\hspace{1 cm}}\frac{\boxed{\phantom{\frac{aaai}{aaai}}}}{\boxed{\phantom{\frac{aaai}{aaai}}}}$$
4. Explica cómo sabes que tu expresión tiene un valor cercano a 20.

### Student Response

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### Activity Synthesis

• Ask selected students to share.
• Consider asking:
• “¿Qué estrategias usaron para determinar el producto aproximadamente igual?” // “What strategies did you determine the just right product?” (I rounded the mixed number to a whole number based on the size of the fraction and multiplied.)
• “En el tercer problema, ¿su producto es mayor, menor o igual que 20? ¿Cómo lo saben?” // “In the third problem, is your product more, less, or equal to 20? How do you know?”

## Lesson Synthesis

### Lesson Synthesis

“Hoy hicimos estimaciones razonables del valor de expresiones de multiplicación. También usamos lo que sabemos sobre las propiedades de las operaciones para encontrar el valor de las expresiones” // “Today we made reasonable estimates for the value of multiplication expressions and used what we know about the properties of operations to find the value of the expressions.”

Display:
$$(5 \times 4) - (5 \times \frac {1}{4})$$

• 20
• 19
• 16

“¿Cuál es una estimación razonable del valor de esta expresión?” // “What is a reasonable estimate for the value of this expression?” (19 because we are subtracting a little bit more than one from 20. This means 20 is too high and 16 is too low.)

Display:
$$(5 \times 4) - (5 \times \frac {1}{4}) = (5 \times 3) + (5 \times \frac {3}{4})$$

“¿Cómo sabemos que esta ecuación es verdadera?” // “How do we know this equation is true?”
(Both expressions are equivalent to $$18 \frac{3}{4}$$. Some will notice both can be rewritten as $$5 \times 3 \frac {3}{4}$$.)

## Cool-down: Estima y resuelve (5 minutes)

### Cool-Down

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## Student Section Summary

### Student Facing

En esta sección, aprendimos cómo encontrar el área de un rectángulo que tiene un lado de longitud fraccionaria. La región sombreada tiene $$4 \times \frac {2}{3}$$ de área porque están sombreados 4 grupos de $$\frac{2}{3}$$ de unidad cuadrada. El área es $$\frac {8}{3}$$$$2 \frac{2}{3}$$ porque hay 8 partes sombreadas y cada una mide $$\frac{1}{3}$$ de unidad cuadrada.

También aprendimos a multiplicar un número mixto por un número entero. Usamos diagramas de área y expresiones para entender por qué funcionan nuestras estrategias. Por ejemplo, para resolver $$3 \frac {3}{4} \times 2$$, podemos usar la expresión $$(3 \times 2) + ( \frac {3}{4} \times 2)$$. Podemos ver ambas expresiones en el diagrama.