Lesson 15

The purpose of this Estimation Exploration is to recall the concept of area. Students need to think strategically because the one point of reference for the size of the grassy area in the image is the car and the road. In order to facilitate mental calculation, expect students to choose multiples of ten for the length and width of the rectangle. 

• Groups of 2
• Display the image.
• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “¿Qué usaron de la imagen para estimar el área del jardín?” // “What did you use in the image to make an estimate for the area of the garden?” (The car and the person gave me an idea of how big it is.)

The purpose of this activity is to use multiplication and division to solve area problems. In most cases the area and one side length are given and students can use division to find the missing side length. In one case the two side lengths are given and students find their product which is the area. 

• Display garden from warm-up.
• “El área de una de las partes rectangulares grandes es 9,175 pies cuadrados y la longitud de uno de los lados es 75 pies” // “The area of one of the large rectangular pieces is 9,175 square feet and the length is 75 feet.”
• Display:
  Area: 9,175 square feet
  One side length: 75 feet
• “¿Cuál sería una estimación razonable del ancho?” // “What is a reasonable estimate for the width?” (100 because and \(100 \times 75 = 7,\!500\) and \(7,\!500 \div 75 = 100\).)
• 2 minutes: partner discussion
• “¿Cómo podemos encontrar el ancho exacto del jardín?” // “How can we find the exact width of the garden?” (Divide the area by the length.)

• 1–2 minutes: quiet think time
• 5–8 minutes: partner work time
• Monitor for students who use a partial quotients algorithm to divide to share during the synthesis.
• If students try to divide to find the missing area, consider asking, “¿Cómo podemos encontrar el área de un rectángulo si conocemos el largo y el ancho?” // “How do we find the area of a rectangle given the length and width?”

• Ask selected students to share their work and explain their steps.
• “¿Cómo podemos asegurarnos de que encontramos las longitudes correctas de los lados?” // “How can we make sure that we found the correct missing side lengths?” (Multiply the length and width together to make sure it gives the area or divide the area by one of the side lengths and make sure it gives the other side length.)
• “Encontraron el área del rectángulo usando el largo y el ancho. También encontraron la longitud de uno de los lados usando el área y la longitud del otro lado. ¿Qué diferencia notaron entre estos dos procesos?” // “How was finding the area of the rectangle using the length and width different than finding one of the side lengths using the area and the other side length?” (I had to multiply to find the area from the length and width. I used division to get one side length from the area and the other side length.)

The purpose of this activity is for students to apply what they learned about dividing multi-digit numbers to find the missing side length(s) of rectangular prisms given the volume and at least one other side length. As in the previous activity, both multiplication and division are important to solve the problems. This is true both because finding partial quotients uses multiplication and because they have different choices how to find a missing side length when two side lengths and the volume are given. Monitor for students who

• find the product of the two given side lengths and then use division to find the third side length. 
• divide successively by the two given side lengths.

• Groups of 2

• 5-8 minutes: independent work time
• 5 minutes: partner work time
• Monitor for students who:
  • use a partial quotients algorithm to divide.
  • find the missing side length in the third table by dividing twice, similar to Clare’s steps in the next problem.
  • find the missing length in the third table by first multiplying the given side lengths and then dividing the volume by the product.
  • notice there are multiple possible lengths and widths for the last rectangular prism in the third table

• Ask students to share their work and reasoning.
• Highlight multiple approaches to find the missing side length in the second and third tables.
• Display second row of third table.
• “¿De qué maneras diferentes pueden encontrar el largo de este prisma rectangular?” // “What are some different ways you can find the length of this rectangular prism?” (I can first divide the volume by 2 and then divide by 48 or I can multiply 2 and 48 and then divide the volume by that product.)
• “¿Cuál fue su método preferido? ¿Por qué?” // “Which method did you prefer? Why?” (I divided the volume by 2 and then by 48 as that was quick and kept the numbers smaller.)