Lesson 20

The purpose of this Number Talk is to elicit strategies and understandings students have for using multiplication to help them divide. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to find the value of quotients.

• 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 ayuda pensar en la multiplicación cuando dividimos?” // “How does thinking about multiplication help you divide?” (I can think about what number multiplied by 5 will be 65. I can break that into smaller products I know.)
• 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?”
  • “¿Alguien quiere agregar algo a la estrategia de _____?” // “Does anyone want to add on to _____’s strategy?”

The purpose of this activity is for students to transition from reasoning about division concretely or visually (using base-ten diagrams) to doing so more abstractly (by writing equations). It also reinforces the connections between multiplication and division.

Students make sense of three different strategies of dividing 78 by 3 and attend to the connections between the visual and numerical representations of the same quotient. As they do so, they practice reasoning quantitatively and abstractly (MP2).

During the synthesis, discuss how place value units play a role in all three strategies.

• Groups of 2
• “Tómense un par de minutos para darle sentido al trabajo de Lin, de Priya y de Tyler” // “Take a couple minutes to make sense of Lin, Priya, and Tyler’s work.”
• 2 minutes: quiet think time

• “Con su compañero, denle sentido al trabajo de Lin, de Priya y de Tyler, y completen la actividad” // “Work with your partner to make sense of Lin, Priya, and Tyler’s work and to complete the activity.”
• 7–8 minutes: partner work time
• Monitor for students who:
  • make connections between the equations and the base-ten diagram
  • recognize how Tyler’s equations and reasoning are different from Priya’s equations
• Identify students who can explain these connections or distinctions to share during the synthesis.

• Invite students to share how Lin, Priya, and Tyler’s work are alike and how they are different.
• Select previously identified students to share additional connections that they noticed.
• “¿Por qué tiene sentido que Priya y Tyler escribieran ecuaciones de multiplicación para encontrar el valor de un cociente?” // “Why does it make sense that Priya and Tyler wrote multiplication equations to find the value of a quotient?” (Multiplication and division can represent the same equal-groups situation. To divide 78 by 3 is to find how many are in each of 3 groups or how many groups of 3 there are. We can multiply up to 78 to find the answer.)
• Consider asking: “¿Qué ideas nuevas sobre la división de números aprendieron y les gustaría intentar? Discutan con un compañero por qué les gustaría intentarlas” // “What new ideas about dividing numbers did you learn and would like to try? Talk to a new partner about why you’d like to try them.”

The purpose of this activity is for students to practice finding the value of division expressions using any strategy that makes sense to them. They may divide the dividend into equal groups or use the divisor to multiply up to the given dividend. They may choose to represent the division or multiplication with base-ten blocks or by drawing diagrams.

During the synthesis, highlight strategies that rely on place value, properties of operations, and the relationship between multiplication and division (MP7).

• Groups of 2
• Give students access to base-ten blocks and grid paper.

• “Encuentren el valor de cada cociente. Pueden usar la estrategia o la representación que prefieran” // “Find the value of each quotient. You can use whatever strategy or representation you prefer.”
• 8–10 minutes: independent work time
• Monitor for a variety of strategies and representations. Identify students to share in the synthesis.
• “Compartan con su compañero cuál es su forma favorita de dividir” // “Share your favorite way to divide with your partner.”
• 2–3 minutes: partner discussion

• Have previously selected students share their responses. Display or record the strategies and representations students use.
• For each problem, consider polling the class on the strategy they used.

The purpose of this optional activity is for students to practice evaluating division expressions in order to make comparisons. Compare is a center that focuses on the procedural skills needed to solve single- and multi-step word problems. In this stage, students will use division to evaluate and compare quotients within 100.

This stage of the Compare center is used in grades 3, 4, and 5. When used in grade 3, remove the cards with two-digit divisors.

• Create a set of cards from the blackline master for each group of 2. Remove the cards with two-digit divisors.

• Groups of 2
• Give each group of students a set of cards from the blackline master.
• “Ahora juguemos ‘Compara’ para practicar lo que han aprendido sobre dividir números” // “Now let’s play Compare to practice what you learned about dividing numbers.” 
• “Tómense un minuto para leer las instrucciones con su compañero” // “Take a minute to read the directions with your partner.”
• 1 minute: quiet think time
• Answer any questions about the directions. If needed, play a turn with the class.

• 5–7 minutes: partner work time
• If time permits, provide extra time to play Compare.

• Display: \(92 \div 4\) and \(72 \div 3\).
• “Supongamos que estas son las dos tarjetas que tomaron. ¿Cómo decidirían cuál expresión tiene el mayor valor?” // “Suppose these are the two cards you draw. How would you decide which expression has the greatest value?”