Lesson 12

This Number Talk encourages students to rely on properties of operations and what they know about multiplication of a fraction and a whole number to mentally solve problems. The understandings elicited here will be helpful later in the lesson when students complete or create Number Talk activities.

• 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.

• “Hoy vamos a escribir nuestras propias actividades tipo ‘Conversación numérica’. Imaginen que el autor de esta actividad empezó con \(6 \times \frac{1}{4}\). ¿Qué observaron acerca de cómo cambió cada expresión? ¿Qué creen que el autor quería que ustedes notaran?” // “Today we are going to write our own Number Talks. Imagine the writer of this Number Talk started with ​\(6 \times \frac{1}{4}\)​. What did you notice about how each expression changed? What do you think the writer was trying to get you to notice?” (Each expression had a factor that changed by multiplying by 3 for the first two expressions, then by 10 for the last expression. I think they were getting us to notice how you can look for ways to make each expression easier to solve by using products we've already found.)
• As needed, “¿En qué se parece cada expresión a la expresión anterior a ella? ¿Cómo pueden usar lo que cambia para encontrar cada valor nuevo?” // “How is each expression like the expression before it? How can you use that change to find each new value?” (Each expression has one factor that changes. If you think about how many times greater the factor is than the factor before it, you can use that to find the product.)

A Number Talk activity encourages students to look for structure and relationships that can help them reason about expressions. This activity puts students in the mindset of a Number Talk writer. It prompts students to anticipate some ways that others might decompose, rearrange, and regroup numbers, or to otherwise make use of structure to find the value of expressions. The work here will be helpful as students consider the numbers and numerical relationships to use as they write their own expressions later.

• Groups of 2
• “Encuentren al menos dos maneras de hallar mentalmente el valor de cada expresión, sin escribir” // “Find at least two ways to mentally find the value of each expression, without writing.”

• 7–8 minutes: independent work time
• 2–3 minutes: partner discussion

• Invite students to share their mental computation strategies. Collect as many strategies as time permits and record them.
• If no students mentioned making use of one or more preceding expressions in their mental computation, ask them how it could be done. (See student response for examples.)
• Select a couple of students to share their new expressions. Ask the class to find the value of the expressions mentally.

In this activity, students use their understanding of place value and knowledge of operations on numbers to write new multiplication and division expressions. Students study the given expressions and their relationships and consider how they could find the value of each expression mentally. Next, they propose new expressions to complete each Number Talk activity.

• Groups of 2
• “Van a ver dos grupos incompletos de una actividad tipo ‘Conversación numérica’. A cada uno le hace falta una o más expresiones. Encuentren mentalmente el valor de cada expresión y piensen en cómo se relacionan” // “You’ll see two incomplete Number Talk sets, each with one or more missing expressions. Find the value of each expression mentally and think about their relationship.”

• “Escriban una o más nuevas expresiones para completar los grupos. Prepárense para explicar cómo razonaron en cada expresión” // “Write one or more new expressions to complete the sets. Be prepared to explain your reasoning for the expression.”
• 7–8 minutes: group work time
• Monitor for students who:
  • adjust the dividend but keep the divisor
  • adjust the divisor but keep the dividend
  • use place value to multiply or divide
  • use multiplicative relationships such as halving and doubling and reason how it impacts the quotient

• Select students who reasoned differently to share their expressions. Record each set.
• Ask the class to determine the reasoning behind the new expression(s) and ask the writers to respond to the feedback.
• “¿Cómo podría la segunda expresión ayudar a alguien a encontrar el valor de la tercera expresión?” // “How might the second expression help someone find the value of the third expression?”
• “¿Cómo podría la tercera expresión ayudar a alguien a encontrar el valor de la cuarta?” // “How might the third expression help someone find the value of the fourth?”

In this optional activity students are given four expressions—three expressions involve operations of two whole numbers and one involves multiplication of a whole number and a fraction. Students choose one expression and write three new ones to create a Number Talk activity.

Students have considerably more freedom to decide the direction of subsequent expressions, but should be just as prepared to explain the rationale behind their expressions. Expect the increased openness to be a greater cognitive lift for students.

For the activity synthesis, ask each group of 2 to test their set by presenting it to another group and attending to how others reason about their expressions.

• Groups of 2

• “Escojan una expresión con la que van a empezar. Después, con su compañero, escriban tres nuevas expresiones para completar una actividad tipo ‘Conversación numérica’. Prepárense para explicar cómo razonaron en cada expresión” // “Choose a starting expression. Then, work with your group to write three new expressions to make a complete Number Talk activity. Be prepared to explain the reasoning for each expression.”
• “Evaluar cada nueva expresión por sí sola puede ser retador. Puede ser más fácil evaluarla después de trabajar en las expresiones que están antes” // “Each new expression may be more challenging to evaluate on its own, but easier to evaluate after working through the ones before it.”
• 15 minutes: group work time

• “Trabajen con otro grupo. Por turnos presenten su actividad tipo ‘Conversación numérica’” // “Work with another group and take turns presenting your Number Talk activity.”
• “Muestren una expresión a la vez. Continúen solo después de que les hayan dado una respuesta y una explicación. Anoten el razonamiento de sus compañeros” // “Show one expression at a time. Move on only after an answer and an explanation are given. Record their reasoning.”
• After students have a chance to test their Number Talk, discuss questions such as:
  • “¿Sus compañeros razonaron como ustedes esperaban?” // “Did others reason in a way you anticipated?”
  • “¿Hubo expresiones que ajustarían?” // “Were there expressions that you’d revise?”