Lesson 3

This Number Talk encourages students to rely on what they know about place value, multiples of 3 and 4, and properties of operations to find the value of products mentally. The reasoning elicited here will be helpful as students use multiplication to find the area and perimeter of rectangles and look for patterns in these measurements. 

• 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.

• “¿Cómo nos puede ayudar alguna de las expresiones anteriores a encontrar el valor de \(42\times3\)?” // “How can one of the previous expressions help us find the value of \(42\times3\)?” (We can take \(21\times3\) and double it to get \(42\times3\). We can take \(40\times3\) and add 2 threes.)

In this activity, students examine a pattern of rectangles and consider different numerical patterns that could represent the rectangles. Students begin by analyzing claims about how the rectangles are growing and work to make the claims clearer and more precise (MP6). In doing so, they notice that the numerical patterns that represent side lengths, perimeter, or area of the rectangles each display a different rule. Students use the rules they observe to predict the value in later terms in the sequence.

• Groups of 2
• Display the image of the rectangles.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 30 seconds: quiet think time
• 30 seconds: partner discussion
• Provide access to graph paper, in case requested.

• Read aloud Priya, Noah, and Lin’s claims about the rectangles and the first question.
• “Revisemos juntos la afirmación de Priya” // “Let’s look at Priya’s statement together.”

MLR3 Clarify, Critique, Correct

• Display and read aloud Priya’s claim: “En cada paso, aumenta 1” // “Each step increases by 1.”
• “¿Qué creen que Priya está tratando de decir? ¿Hay algo que no sea claro?” // “What do you think Priya is trying to say? Is anything unclear?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• “Con su compañero, ajusten la afirmación de Priya para que lo que ella intenta decir sea más claro” // “With your partner, work together to write a revised statement that Priya could say so that her intention is clearer.”
• Display and review the following criteria:
  • Write in a complete sentence.
  • Include mathematical vocabulary when possible.
  • Include an example, if possible.
• 2–3 minutes: partner work time
• Select 1–2 groups to share their revised explanation with the class. Record responses as students share.
• “¿En qué se parecen y en qué son diferentes los ajustes que le hicieron a la afirmación de Priya?” // “What is the same and different about the revisions to Priya’s claim?”
• “Tómense unos minutos para analizar en silencio las afirmaciones de Noah y Lin, y ajústenlas para que lo que intentan decir sea más claro. Luego, completen la actividad con su grupo” // “Take a few quiet minutes to analyze Noah and Lin’s claims and revise them so that their intentions are clearer. Then work with your group to complete the activity.”
• 3–4 minutes: independent work time
• 5–6 minutes: group work time
• Monitor for students who attend to precision and clarity as they revise Noah’s and Lin’s claims. Select them to share during the synthesis.

Students may count individual squares within rectangles to find the area. Consider asking:

• “¿Cómo podrías encontrar, sin contar, el número de cuadrados que hay en el rectángulo que acabas de dibujar?” // “How might you find out the number of squares in the rectangle you just drew without counting?”
• “¿Cómo podrías encontrar el tamaño del siguiente rectángulo sin dibujarlo?” // “How might you find out the size of the next rectangle without drawing it?”

• Invite previously selected students to share their explanations and revisions of Noah and Lin’s claims. Record the revised claims for all to see.
• Select other students to share the numerical patterns they wrote to represent the rectangles, and how they made predictions for the 20th number in each pattern. Record their responses for all to see.

This optional activity gives students an additional opportunity to reason about patterns in the side lengths, area, and perimeter of rectangles that follow a rule.

• Groups of 2
• Display the image of the rectangles.
• “¿Cuántas columnas verticales ven en los rectángulos?” // “How many vertical columns do you see in the rectangles?” (One in step 1, two in step 2, and three in step 3.)
• “¿Cómo está cambiando el número de columnas?” // “How is the number of columns changing?” (It grows by 1 each time.)
• “Además de las columnas verticales, ¿qué otras características de los rectángulos podríamos contar o medir?” // “Besides vertical columns, what other features of the rectangles could we count or measure?” (Sample responses: Number of rows, area or number of square units, side lengths, perimeter)
• “Veamos qué otros patrones pueden encontrar en esta colección de rectángulos” // “Let’s see what other patterns you can find in this set of rectangles.”

• “Trabajen individualmente en la actividad durante unos minutos” // “Work on the activity independently for a few minutes.”
• 5 minutes: independent work time

MLR1 Stronger and Clearer Each Time

• “Busquen un compañero que haya escrito otro patrón numérico” // “Find a partner who wrote a different numerical pattern.”
• “Por turnos, uno habla y el otro escucha. Si es su turno de hablar, compartan su patrón numérico y su explicación para el último problema. Si es su turno de escuchar, hagan preguntas y comentarios que ayuden a su compañero a mejorar su explicación para el último problema” // “Take turns being the speaker and the listener. If you are the speaker, share your numerical pattern and your explanation for the last problem. If you are the listener, ask questions and give feedback to help your partner improve their explanation for the last problem.”
• 3–5 minutes: structured partner discussion.
• Repeat with 1–2 other partners who chose a different feature than you did.
• “Ajusten su respuesta inicial a la última pregunta basándose en los comentarios que les hicieron sus compañeros” // “Revise your initial response to the last question based on the feedback you got from your partners.”
• 2–3 minutes: independent work time

Students continue to analyze and describe patterns related to rectangles. In this activity, the rectangles show no grid. To reason about possible patterns in the features of the rectangles, students rely on what they know about the relationship between side lengths of a rectangle and its perimeter and area.

• Groups of 2
• Read the opening paragraph and the first question as a class.
• “En silencio, tómense un minuto para dibujar los rectángulos que hacen falta. Marquen los lados” // “Take a quiet minute to sketch the missing rectangles. Label the sides.”
• 1 minute: independent work time
• Share responses and display the completed sequence of four rectangles.
• Display this sequence of numbers: 3, 3, 3, 3
• Ask students: “¿Cómo están representados los rectángulos en este patrón numérico?” // “How does this numerical pattern represent the rectangles?” (The length of the shorter side of the rectangle, which doesn’t change.)
• “Pensemos en otros patrones numéricos con los que podemos representar los rectángulos” // “Let’s think about other numerical patterns that can represent the rectangles.”

• 8–10 minutes: independent work time
• 3 minutes: partner discussion
• Monitor for the different ways students reason about the last set of problems.

• See lesson synthesis.