Lesson 14

The purpose of this Number Talk is to elicit strategies and understandings students have for adding and subtracting multi-digit numbers. These understandings help students develop fluency and will be helpful in later units as students will need to be able to add and subtract multi-digit numbers fluently using the standard algorithm.

When students make adjustments and create multiples of ten for mental addition they are looking for and making use of the base ten structure of numbers (MP7). 

• 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.

How did you use multiples of ten, for example 20, 40, and 60 to help add these numbers mentally? (I changed the addends by adding one more to each addend and the subtracting the extra two from the final sum.)

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 are given several sets of fractions and some clues about the size of a particular fraction in each set. To identify a fraction that meets certain size requirements or falls within a specified range, students need to use multiple comparison strategies they have learned. For example, they can use comparisons to benchmarks such as \(\frac{1}{2}\) and 1 to eliminate some fractions, and then use equivalent fractions to compare the remaining ones.

• Groups of 3–4
• Read the opening paragraph of the task as a class.
• “Hay seis conjuntos de fracciones en esta actividad. Cada conjunto viene con unas pistas. Su tarea en cada conjunto es encontrar una fracción que cumpla con las tres pistas” // “There are six sets of fractions in the activity. Each set comes with some clues. Your task is to find one fraction that meets all three clues in each set.”

• “Trabajen en grupo para encontrar las fracciones desconocidas” // “Work with your group to find the mystery fractions.”
• “Cada miembro del grupo debe empezar con un conjunto de fracciones distinto y debe encontrar las fracción desconocida de al menos dos conjuntos antes de discutir sus respuestas” // “Each group member should start with a different set, and should find at least two mystery fractions before discussing their responses with the group.“
• 5–6 minutes: independent work time
• 7–8 minutes: group work time

• “¿Cuáles pistas les ayudaron a eliminar fracciones más rápido?” // “Which clues helped you eliminate fractions the fastest?” (clues about size relative to 1)
• “¿Qué estrategias usaron para comparar fracciones?” // “What strategies did you use to compare fractions?” (compare fractions to \(\frac{1}{2}\), 1, or another benchmark, write equivalent fractions to compare two fractions, compare fractions with the same numerator or denominator)
• “¿En algún caso tuvieron que usar más de una estrategia para comparar fracciones?” // “Did you ever have to use more than one strategy to compare fractions?” (Yes, two or three were often needed to find the mystery fraction.)

This activity has two purposes: to give students an opportunity to solve fraction comparison problems in context, and to reinforce the idea that two fractions can be compared only if they refer to the same whole. To serve the former, students compare fractional distance measurements. To serve the latter, they investigate fractional measurements in two different units of distance: Chinese “li” and kilometer. 

When comparing the distances in the first question, students can rely on a number of familiar strategies. Two of the fractional values are close to 2. Some students are likely to use that benchmark for efficient comparison. For example, they may note that the school and the market are both a little over 1 li from home, the library is more than 2 li, and the badminton club is a little under 2 li.

Focus the synthesis on the last two questions about interpreting fractional measurements in two different units.

• Groups of 3–4
• “¿Cuáles son algunas unidades que usamos para medir distancia? Digamos todas las que se nos ocurran” // “What are some units that we use for measuring distance? Let’s name as many as we can think of.” (Sample responses: inches, feet, miles, meters, kilometers)
• Share and record responses.
• “Hoy vamos a pensar en distancias medidas en ‘li’, una unidad que se usa con frecuencia en China” // “Today we’ll look at distances measured in ‘li,’ a unit commonly used in China.”

• “Tómense unos minutos para trabajar en silencio en las preguntas. Prepárense para explicar su razonamiento” // “Take a few quiet minutes to work on the questions. Be prepared to explain your reasoning.”
• “Después, discutan sus respuestas con su grupo y trabajen juntos para terminar la actividad” // “Afterward, discuss your responses with your group and work together to complete the activity.”
• 4 minutes: independent work time
• 4 minutes: group work time
• Monitor for students who:
  • attend to the location of 1 whole when representing \(\frac{4}{5}\) and \(\frac{7}{5}\) on the two number lines (rather than partitioning both number lines into the same 5 parts)
  • recognize that \(\frac{4}{5}\) and \(\frac{7}{5}\) in the two cases refer to different wholes and can articulate their reasoning

• Ask previously selected students to share their responses to the last two questions. Display their number lines, or display the number lines from the activity for them to annotate while they explain.
• Emphasize that just as 1 km is not the same distance as 1 li, \(\frac{4}{5}\) km is not the same distance as \(\frac{4}{5}\) li. We can’t compare two fractions that refer to different wholes.