Lesson 12

This Number Talk encourages students to rely on what they know about fractions to mentally find the value of differences with mixed numbers. 

• 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 les ayudaron las primeras expresiones a encontrar el valor de la última expresión?” // “How did the first few expressions help you find the value of the last expression?”
• “Al restar \(1\frac{7}{8}\), ¿por qué puede ser útil pensar primero en restar 2?” // “When subtracting \(1\frac{7}{8}\), why might it be helpful to first think about subtracting 2?” (\(2\frac{3}{8}\) has a whole number and a fraction, so we can easily subtract 2 from the whole number and then put back the extra \(\frac{1}{8}\) that we took out.)
• Consider asking:
  • “¿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?”

In this activity, students find the number that makes addition and subtraction equations with mixed numbers true without a context. The equations are designed to encourage students to decompose or write equivalent fractions for one or more numbers to find the unknown value, but students may choose to reason without doing either. When students share their strategies with their group they construct viable arguments (MP3).

• Groups of 2–4

• “Individualmente, encuentren el número que debe ir en cada espacio en blanco para que cada ecuación sea verdadera” // “Work independently to find the number that should go in the blank to make each equation true.”
• 5 minutes: independent work time on the first problem
• “Ahora piensen en cómo empezaron a encontrar cada número desconocido. Escriban una frase que describa su primer paso en el proceso de completar la ecuación” // “Now think about how you started finding each missing number. Write a sentence to describe your first step in completing each equation.”
• 5 minutes: independent work time on the second problem
• “Compartan sus primeros pasos con su grupo. En cada ecuación, ¿empezaron a encontrar el número desconocido de la misma forma? Discutan por qué sí o por qué no” // “Share your first steps with your group. For each equation, did you start to find the missing number the same way? Discuss why or why not.”
• 5 minutes: group discussion

• Invite students to share how they went about finding the missing numbers. Display or record their reasoning for all to see.
• To invite others into a discussion, consider asking:
  • “¿Alguien encontró el número desconocido de esta ecuación de la misma manera?” // “Did anyone find the missing number in this equation the same way?”
  • “¿Alguien usó la misma estrategia o una similar, pero la explicaría de otra forma?” // “Who used the same or a similar strategy but would explain it differently?”

In this activity, students analyze a set of addition and subtraction expressions and consider whether it is helpful or necessary to decompose a number in order to find the value of the expressions. In doing so, they practice looking for structure in expressions, which they can in turn use to find sums or differences more effectively (MP7). Note that there is more than one way to sort the expressions, as students may have different ways of reasoning about these expressions.

• Groups of 3–4
• “En actividades anteriores, entendimos que a veces es útil descomponer un número entero o un número mixto antes de restarle una fracción” // “In earlier activities, we saw it was sometimes useful to decompose a whole number or a mixed number before subtracting a fraction from it.”
• “Al mirar una expresión de suma o de resta, ¿pueden saber si sería útil (o necesario) descomponer un número o reescribirlo antes de sumar o restar? ¡Averigüémoslo!” // “Can you tell—by looking at an addition or subtraction expression—whether it would be helpful (or necessary) to decompose a number or rewrite it before we could add or subtract? Let’s find out!”

• “Con su grupo, clasifiquen las expresiones en dos categorías” // “Work with your group to sort the expressions into two categories.”
• 5 minutes: group work time
• “Cuando terminen, escojan por lo menos una expresión de cada categoría y encuentren su valor individualmente” // “When you are done, choose at least one expression from each category and work independently to find its value.”
• “Cada miembro del grupo debe escoger una expresión diferente de cada categoría” // “Every group member should choose a different expression from each category.”
• 5 minutes: independent work time

• Select groups to share the expressions they put in each category and invite others to agree or disagree.
• Ask other students to choose 1–2 expressions that they evaluated from each category and show that it is or it is not helpful or necessary to decompose a number. (For example, if they put expression A in the 'not necessary to decompose' category, their work should show that the difference can be found without decomposing.)