The purpose of this Number Talk is for students to reason about place-value relationships and the properties of multiplication. The elicited understandings and strategies will be helpful in later lessons and units when students multiply large numbers. In this unit, students produce and interpret multiplication expressions in terms of volume.
Launch
Display one expression.
“Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron // “Give me a signal when you have an answer and can explain how you got it.”
1 minute: quiet think time
Activity
Record answers and strategies.
Keep expressions and work displayed.
Repeat with each expression.
Student Task Statement
Encuentra mentalmente el valor de cada expresión.
Student Response
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Advancing Student Thinking
Activity Synthesis
“¿Qué patrones observaron en los problemas que resolvimos?” // “What patterns do you notice in the problems we solved?” (There is a 6 and a 2 in each product. There are also factors of 10, and the 6 and the 2 are sometimes multiplied by a factor of 10.)
Consider asking:
“¿Alguien puede expresar el razonamiento de _____ de otra forma?” // “Who can restate _____’s reasoning in a different way?”
“¿Alguien usó la misma estrategia, pero la explicaría de otra forma?” // “Did anyone have the same strategy but would explain it differently?”
“¿Alguien pensó en el problema de otra forma?” // “Did anyone approach the problem in a different way?”
Activity 1
20 mins
Encontremos volúmenes de figuras
Standards Alignment
Building On
Addressing
5.MD.C.5.c
Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
The purpose of this activity is for students to find the volumes of figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts. There are different ways to decompose the figures. Monitor for students who break apart the figures differently and find the same total volume. To reinforce earlier work with cubic units of measure, ask students for the unit of measure in their response if they state the volume as only a number (MP2). If students finish early, give them isometric grid paper to draw a figure composed of two rectangular prisms for their partner to find the volume.
Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were needed in order to solve the problem. Display the sentence frame: “La próxima vez que necesite encontrar el volumen de una figura compuesta por dos prismas rectangulares, voy a . . .” // “The next time I need to find the volume of a figure composed of two rectangular prisms, I will . . . .“ Supports accessibility for: Conceptual Processing, Organization, Memory
Launch
Groups of 2
Display the image from the student book, with the unknown side lengths.
“¿Cómo podemos encontrar las longitudes de lado desconocidas?” // “How can we find the missing side lengths?”
1 minute: quiet think time
2 minutes: partner discussion
Share and record responses on the image of the figure. (We can subtract 5 from 7 in both cases.)
Activity
“Ahora cada uno va a encontrar el volumen de una de las figuras y después va a intercambiar su figura con la de su compañero. Luego, deben encontrar el volumen de su nueva figura usando una estrategia de descomposición diferente a la de su compañero” // “Now each partner will find the volume of one of the figures, and then switch papers. Your job is to then find the volume, using a different decomposition strategy than your partner.”
If there is time, the groups of 2 students can make groups of 4 students and share responses and strategies.
10 minutes: independent work time, with partner discussion
Give students access to isometric dot paper to draw the figures if they finish early.
Student Task Statement
6-sided rectangular prism. Straight sides. All side lengths meet at right angles. Side lengths. Bottom, 7 feet. Right side rises 5 feet, then goes left 5 feet, goes up an unlabeled length, then goes left an unlabeled length, then goes down 7 feet. Width shown as 2 feet.
Compañero A: encuentra el volumen de la figura 1.
Compañero B: encuentra el volumen de la figura 2.
Figura 1
Figura 2
Activity Synthesis
Display Figure 2.
“¿Cómo partieron esta figura para encontrar el volumen?” // “How did you break up this shape to find the volume?” (I cut off the overhanging piece vertically to make two rectangular prisms. I cut off the bottom piece that the shape is resting on, horizontally.)
“¿Cómo encontraron las longitudes de los lados de los prismas rectangulares?” // “How did you find the side lengths of the rectangular prisms?” (Some of the lengths are provided, and I found the others by subtracting.)
“¿Obtuvieron el mismo volumen cuando partieron la figura de manera diferente? ¿Por qué?” // “Did you get the same volume when you broke up the figure differently? Why?” (Yes. The calculations were different, but both tell me how many cubic inches it takes to fill the shape.)
Activity 2
15 mins
Expresiones para representar el volumen de una figura
Standards Alignment
Building On
Addressing
5.MD.C.5.c
Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add and , then multiply by ” as . Recognize that is three times as large as , without having to calculate the indicated sum or product.
The goal of this activity is to represent expressions as decompositions of a figure made of two non-overlapping right rectangular prisms. This gives students an opportunity to interpret parentheses in expressions while also checking their understanding of different ways to represent the volume of a rectangular prism; namely, length times width times height, and area of a base times the corresponding height.
Students work abstractly and quantitatively in this problem (MP2) as they relate abstract expressions to decompositions of figures composed of two rectangular prisms.
MLR7 Compare and Connect. Invite students to think about details they can include that will help others understand their thinking. For example, using specific language, different colors, shading, arrows, labels, notes, diagrams, or drawings. Give students time to investigate each others’ work. During the whole-class discussion, ask: “¿Qué tipo de detalles o de lenguaje les ayudaron a entender el trabajo de su compañero?” // “What kinds of details or language helped you understand your partner’s work?” Advances: Representing, Conversing
Launch
Groups of 2
“Observen la primera figura y piensen cómo la podrían descomponer en 2 prismas” // “Look at the first figure and think about how you might decompose it into 2 prisms.”
1 minute: quiet think time
“Ahora van a considerar cómo varias expresiones pueden representar el volumen de una figura” // “Now you will consider how different expressions can represent the volumes of figures.”
Activity
5 minutes: independent work time
5 minutes: partner discussion time
Monitor for students who:
Use the numbers in the expressions to determine how to break up the figure.
Break up the figure and use this to identify the expressions.
Student Task Statement
Explica cómo cada expresión representa el volumen de la figura. Muestra tu razonamiento. Organiza tu trabajo para que los demás puedan entenderlo.
Figure composed of two rectangular prisms. Front prism: 3 cubes by 3 cubes by 3 cubes. Back prism: 2 cubes by 4 cubes by 3 cubes.
¿Cómo representa cada expresión el volumen del prisma? Explica o muestra tu razonamiento. Organiza tu trabajo para que los demás puedan entenderlo.
pulgadas cúbicas
pulgadas cúbicas
8-sided figure. Straight sides. All side lengths meet at right angles. Side lengths. Bottom, unlabeled length. Right side rises 6 inches, then goes left 8 inches, then goes up 3 inches, and then goes left 4 inches. Left side, unlabeled length. Width shown as 5 inches.
Student Response
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Advancing Student Thinking
Activity Synthesis
Invite students to share how the expressions for the first prism represent its volume.
“¿Por qué hay tres factores en la expresión , pero solo hay dos factores en la expresión ?“ // “Why are there three factors in the expression but only two factors in the expression ?” (The 2, the 3, and the 4 are the length, the width, and the height of the prism in the back. The 5 is the height of the prism on the bottom, the base area of which is 6.)
“¿Qué les dicen los paréntesis en la expresión ?“ // “In the expression , what do the parentheses tell you?” (They tell me to first multiply 3 times 3, which gives me the area of the base, and then I multiply that by 2 to get the volume of the prism.)
Lesson Synthesis
“Hoy usamos expresiones para representar volúmenes de figuras hechas de prismas rectangulares” // “Today we represented the volumes of figures made of rectangular prisms with expressions.”
Display the image from the first activity:
“Esta es una de las figuras con las que trabajamos hoy” // “Here is one of the figures we worked with today.”
Display the expression .
“¿Qué parte de la figura está representada por esta expresión? ¿Cómo lo saben?” // “Which part of the figure is represented by this expression? How do you know?” (The rectangular prism at the top of the shape. It is 3 inches tall, 9 inches wide, and 7 inches deep, so its volume is .)
Draw a line to show the prism.
“¿Cuál es el volumen del otro prisma rectangular? ¿Cómo lo saben?” // “What is the volume of the other rectangular prism? How do you know?” ( cubic inches, since it is 2 inches tall, 5 inches wide, and 7 inches deep.)
Display the expression + .
“¿Cómo representa esta expresión el volumen de la figura?” // “How does this expression represent the volume of the figure?” (It shows the addition of the volumes of the two rectangular prisms that make up the figure.)
Standards Alignment
Building On
Addressing
Building Toward
5.NBT.A.2
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
If students find a volume that does not represent the volume of Figure 1 or Figure 2, consider asking,
“¿En dónde ves prismas rectangulares en esta figura?” // “Where do you see rectangular prisms in this figure?”
Show an expression that represents the volume of the figure. “¿Cómo representa esta expresión el volumen de la figura?” // “How does this expression represent the volume of the figure?”
