Lesson 7

This Number Talk encourages students to use multiplicative reasoning and to rely on properties of operations to mentally find the value of products of a whole number and a fraction. The reasoning elicited here will be helpful later in the lesson when students find the perimeter of a figure with fractional side lengths. 

• 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

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

• “¿Qué tienen en común estas expresiones?” // “What do these expressions have in common?” (The first number in each sequence is a multiple of 3 and a multiple of 6. The second number is a fraction with 3 in the denominator.)
• “Para encontrar cada producto, ¿cómo les ayudaron estas observaciones sobre los números?” // “How did these observations about the numbers help you find each product?”
• 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 la expresión de otra forma?” // “Did anyone approach the expression in a different way?”
  • “¿Alguien quiere agregar algo a la estrategia de _____?” // “Does anyone want to add on to _____’s strategy?”

In this activity, students find the perimeter of several shapes and write expressions that show their reasoning. Each side of the shape is labeled with its length, prompting students to notice repetition in some of the numbers. The perimeter of all shapes can be found by addition, but students may notice that it is efficient to reason multiplicatively rather than additively (MP8). For example, they may write \(4 \times 6\frac{1}{3}\) instead of \(6\frac{1}{3} + 6\frac{1}{3} + 6\frac{1}{3} + 6\frac{1}{3}\).

• Groups of 2–4

• 3–4 minutes: independent work time
• 2 minutes: group discussion

• Invite students to share the expressions they wrote for the perimeter of each shape. Record the expressions.
• For each shape, ask: “¿En qué se parecen estas expresiones? ¿En qué son diferentes?” // “What is the same about these expressions? What is different?” (They each use the side lengths in the shape. They each show the total length around the shape. Some use addition, some use multiplication, some use both multiplication and addition.)
• Highlight that when two or more sides have the same length, the perimeter can be found by adding, multiplying, or a combination of both. 

Earlier, students found the perimeter of polygons when all the side lengths were given. In this activity, only some of the sides are labeled with their length, but students are given some information about the attributes of the shapes (presence or absence of parallel sides and symmetry). Students use that information to determine the length of the unlabeled sides and the perimeter. They may also conclude that the length of the perimeter cannot be determined.

When students interpret and analyze Mai's and Andre's reasoning about the quadrilateral perimeters they critique the reasoning of others (MP3).

This activity uses MLR3 Clarify, Critique, and Correct. Advances: reading, writing, representing

• Groups of 2
• Provide access to patty paper.
• 2 minutes: quiet think time for the first question

• 2 minutes: partner discussion about Mai and Andre's reasoning
• Monitor for students who use symmetry and what they know about parallel sides in their response.
• 3–4 minutes: independent work time

• Read Mai’s reasoning aloud.
• “¿Qué piensan que Mai quiere decir? ¿Qué no está claro? ¿Hay errores?” // “What do you think Mai means? What is unclear? Are there any mistakes?”
• 1 minute: quiet think time
• 2 minute: partner discussion
• “Escriban con su compañero una explicación ajustada para Mai” // “With your partner, work together to write a revised explanation for Mai.”
• Display and review the following criteria:
  • “No podemos encontrar el perímetro de un cuadrilátero si faltan una o más longitudes de lado y _____” // “We can’t find the perimeter of a quadrilateral if one or more side lengths are missing and _____.”
• 3–5 minute: partner work time
• Select 1–2 groups to share their revised explanation with the class.
• Record responses as students share.
• “¿Cómo supieron que el perímetro de C se podía encontrar al sumar dos veces 4 cm y dos veces \(7\frac{1}{2}\) cm o \((2 \times 4) + (2 \times 7\frac{1}{2})\)?” // “How did you know that the perimeter of C can be found by combining twice 4 cm and twice \(7\frac{1}{2}\) cm or \((2 \times 4) + (2 \times 7\frac{1}{2})\)?” (We saw earlier that when quadrilaterals have two pairs of parallel sides, opposite sides have the same length. If the sides are parallel, they are the same distance apart.)
• “¿Cómo supieron que el lado sin marcar en la figura D también medía \(7\frac{1}{2}\) cm?” // “How did you know the unlabeled side in Shape D is also \(7\frac{1}{2}\) cm long?” (The figure has a line of symmetry through the midpoint of the top and bottom segment.)
• “¿Qué expresión podemos escribir para el perímetro de D?” // “What expression can we write for the perimeter of D?” (\(4 + 5\frac{3}{4} + 7\frac{1}{2} + 7\frac{1}{2}\) or \(4 + 5\frac{3}{4} + (2 \times 7\frac{1}{2})\) or equivalent)

This optional activity gives students an additional opportunity to use the attributes and their emerging understanding of the properties of some categories of shapes to find their perimeter, or to conclude that the perimeter cannot be determined.

• Groups of 2
• Provide access to patty paper.

• 3–4 minutes: quiet think time
• 2 minutes: partner discussion
• Monitor for students who use symmetry and the properties of parallelograms in their response.

If students begin to label all opposite sides with the same length, consider asking:

• “¿Cómo sabes que estos dos lados tienen la misma longitud?” // “How do you know these two sides have the same length?”
• “¿Qué necesitas saber sobre una figura para asegurarte de que cualquier lado sin marcar tiene la misma longitud que los lados marcados?” // “What do you need to know about a shape to be sure that any unlabeled sides have the same length as the labeled sides?”

• Invite students to share how they knew for which shapes it is possible to find the perimeter and for which it is not.
• Select other students to share how they matched the expressions and the shapes.