Lesson 6

The purpose of this True or False is for students to demonstrate strategies and understandings they have for determining equivalence of numerical expressions. These understandings help students deepen their understanding of the properties of operations and are helpful as students interpret expressions for volume. In this activity, students have an opportunity to notice and make use of structure (MP7) when they use the properties of operations to determine equivalence without having to calculate.

• Display one statement.
• “Hagan una señal cuando sepan si la afirmación es verdadera o no, y puedan explicar cómo lo saben” // “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

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

• Focus Question: “¿Cómo pueden justificar su respuesta sin evaluar ambos lados?” // “How can you justify your answer without evaluating both sides?” (I could see on the first equation that all of the factors are the same so it is true.)
• Consider asking:
  • “¿Alguien puede expresar el razonamiento de ___ de otra forma?” // “Who can restate ___’s reasoning in a different way?”
  • “¿Alguien quiere agregar algo al razonamiento de _____?” // “Does anyone want to add on to _____’s reasoning?”
  • “¿Podemos hacer alguna generalización a partir de las afirmaciones?” // “Can we make any generalizations based on the statements?”

The purpose of this activity is for students to interpret expressions that represent the volume of a rectangular prism. Students connect the structure in rectangular prisms to the symbols in their related expressions (MP2, MP7). If there is time and you would like to add student movement, have students make a poster to display the sorted cards. Students can walk around and add additional expressions to other posters to represent the volume of the prism.

• Create a set of cards from the blackline master for each group of 2.
• Have connecting cubes available for students who need them.

• Groups of 2
• Distribute one set of pre-cut cards to each group of students.
• “¿Qué observan sobre los prismas en estas tarjetas?” // “What do you notice about the prisms on these cards?” (They don’t have any cubes, It says “units”.)
• “Cuando las medidas están en unidades, los cubos que usamos para llenar el prisma se llaman unidades cúbicas” // “When the measurements are in units, the cubes we use to fill the prism are called cubic units.”

• “En esta actividad, van a clasificar tarjetas en las categorías que ustedes elijan. Cuando clasifiquen las tarjetas, deben escoger las categorías con su pareja” // “In this activity, you will sort some cards into categories of your choosing. When you sort the cards, you should work with your partner to come up with categories.”
• 4 minutes: partner work time
• Select groups to share their categories and how they sorted their cards.
• “Ahora, en parejas, asocien cada prisma a las expresiones que representen su volumen” // “Now work with your partner to match each prism with the expressions that represent the volume.”
• 3 minutes: partner work time

If students do not correctly match expressions to the prisms, ask:

“¿Cómo nos ayuda usar los cubos encajables a asociar las expresiones con los prismas?” // “How can we use the connecting cubes to help you match the expressions to the prisms?”

• Select groups to share their matches.

• Display Prism A:

• “¿Cómo representan estas expresiones el volumen?” // “How do these expressions represent the volume?”
  • \(6\times(5\times3)\)
  • \((6\times5)\times3\)
  • \(15\times6\)
• Display:
  • \((5\times3)\times6\) = \(15\times6\)
• “¿Cómo se relaciona esta ecuación con el prisma A?” // “How does the equation relate to Prism A?” (Both expressions show that the prism has a height of 6. One expression shows the side lengths of the base. The other expression shows the area of the base.)

The purpose of this activity is for students to compare and contrast two different ways to calculate the volume of a rectangular prism: multiplying the area of the base and its corresponding height, and multiplying all three side lengths. Students see that both of these strategies result in the same volume. It is a convention to consider a prism’s base the face it is resting on, however when calculating the volume of a rectangular prism, any face of the prism can be considered a base as long it is multiplied by the corresponding height. Similarly, when calculating the volume of a rectangular prism, any edge can be considered the length, width, or height.

• Groups of 2

• 1 minute: independent work time
• 8 minutes: partner work time

If a student does not write the correct corresponding height for a given base, ask “¿Cómo se relacionan los números de la tabla con el prisma?” // “How do the numbers in the table relate to the prism?” or “¿Cómo decidiste qué números escribir en la tabla?” // “How did you decide which numbers to write in the table?”

• Ask students to share responses to the second problem. Display the expression: \(6\times3\times4\)
• “¿Cómo representa esta expresión el volumen del prisma A?” // “How does this expression represent the volume of prism A?” (The prism's side lengths are 6, 4, and 3 and I multiply them to find the volume.)
• Display expression: \((6\times3)\times4\)
• “¿Cómo representa esta expresión el volumen del prisma A?” // “How does this expression represent the volume of prism A?” (One base has a length of 6 units and a width of 3 units and the height is 4 units.)
• “¿Qué expresión serviría para encontrar el volumen usando la base de 3 unidades por 4 unidades?” // “Which expression could you use to find the volume using the 3 unit by 4 unit base?” (We could use either \((3\times4)\times6\) or \(6\times(3\times4)\). They are equal and they both represent the volume of the prism.
• Display equation: \((6\times3)\times4\) = \((3\times4)\times6\)
• “¿Cómo saben que la ecuación es verdadera?” // “How do you know the equation is true?” (Both expressions represent the volume of the prism and we can see both expressions in the prism. One of them represents a base with the side lengths 6 and 3 and a height of 4. The other expression represents a base with the side lengths 3 and 4 cubes and a height of 6.)

This activity is optional if students need additional practice writing expressions to represent the volume of a rectangular prism. This activity also supports students in identifying the information they need to represent volume. Students are given the opportunity to write and interpret expressions that show that the volume is the same when multiplying the edge lengths or multiplying the area of the base and height.  In the second part of the activity, students reason abstractly and quantitatively when they interpret the meaning of expressions in the context of volume (MP2).

• Groups of 2
• “Con su pareja van a jugar ‘2 verdades y una mentira’ con prismas rectangulares” // “You and your partner are going to play 2 truths and a lie with rectangular prisms.”
• “Cada uno va a escribir expresiones que representen el volumen de dos prismas. 2 deben ser ‘verdaderas’ y una ‘falsa’. Después las van a intercambiar para responder algunas preguntas” // “You will each write expressions, 2 true and one false, to represent the volume of two prisms and then trade to answer some questions.”
• “Un compañero escribe 2 verdades y una mentira para los prismas A y C y el otro las escribe para los prismas B y D” // “One partner writes 2 truths and a lie for Prisms A and C and the other partner writes about Prisms B and D.”

• 5 minutes: independent work time (create expressions)
• “Intercambien la hojas de expresiones y vean si pueden descifrar para cada prisma cuál es la expresión que es una mentira” // “Switch papers with your partner and see if you can figure out the expression that is a lie for each of their prisms.”
• 5 minutes: independent work time on partner’s problems (analyze expressions)

If students do not write any correct expressions that represent the volume of the prism, refer to an expression that does represent the volume of the prism and ask, “¿Puedes explicar cómo esta expresión representa el volumen del prisma?” // “Can you explain how this expression represents the volume of the prism?” 

• Display each of the prisms.
• “¿Cuáles expresiones representan el volumen del prisma, en unidades cúbicas? ¿Cuáles no?” // “Which expressions represent the volume of the prism in cubic units? Which do not?”
• “¿Cómo decidieron cuáles expresiones no representaban el volumen de ninguno de los prismas?” // “How did you decide the expressions that did not represent the volume of a rectangular prism?” (Looking at the different bases and heights and experimenting with expressions. Finding the product and checking that it does not match the volume of any of the prisms.)