Lesson 16

The purpose of this Number Talk is to elicit strategies and understandings students have for products of 4 and 6 as they relate to products of 5. These understandings help students develop fluency and will be helpful later when students consider solutions for and solve two-step word problems.

When students use products of 5 to determine products of 4 by thinking of them as one fewer group or one fewer object in each group, or work from products of 5 to determine products of 6 by thinking of them as one more group or one more object in each group, they look for and make use of structure (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.

• “¿Cómo les ayudó saberse el valor de \(5\times7\) a encontrar algunos de los otros productos?” // “How does knowing \(5\times7\) help you find some of the other products?” (I can remove a group of 7 to find \(4\times7\) or add a group of 7 to find \(6\times7\).)
• 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 apply what they learned about rounding in prior lessons to think about all the numbers that would round to a given number. Students should be encouraged to use whatever representations make sense to them. Although the number line is often used to represent rounding, it is also worth sharing other ways that students are representing or thinking about rounding.

• Groups of 2
• “Diego está pensando en un número. Cuando el número de Diego se redondea a la decena más cercana, la respuesta es 40. Digan un número que podría ser el número de Diego. Digan un número que no puede ser el número de Diego” // “Diego is thinking of a number. When you round Diego's number to the nearest ten, the answer is 40. What's a number that could be Diego's number? What's a number that could not be Diego's number?” (38 rounds to 40 so it could be his number. 34 does not round to 40 so it couldn't be his number.)
• 30 seconds: quiet think time
• Share 3–5 responses. Highlight the idea that more than one number can round to 40 and that some numbers are greater than 40 and some are less than 40.

• “Trabajen con su compañero en todos estos problemas. Asegúrense de justificar cómo razonaron” // “Work with your partner on all these problems. Be sure to justify your reasoning.”
• 5–7 minutes: partner work time

If students don’t find all the numbers that round to the given number, consider asking:

• “¿Cómo decidiste redondear estos números a ____?” // “How did you determine that these numbers would round to ____?”
• “¿Cómo podrías usar una recta numérica para encontrar todos los números que se redondean a ___?” // “How could you use a number line to find all the numbers that round to ___?”

• “¿Cómo decidieron qué números se debían redondear a 40?” // “How did you decide what numbers would round to 40?” (We looked at all the numbers that are closer to 40 than 50 or 30.)
• Consider asking:
  • “¿A qué número se redondea 35?” // “What does 35 round to?” (40 because it is halfway between 30 and 40)
  • “¿A qué número se redondea 45?” // “What does 45 round to?” (50 because it is halfway between 40 and 50)
• “Miren sus respuestas a los primeros 2 problemas. ¿Qué patrones ven en los números? ¿Por qué ocurre eso?” // “Look at your responses for the first 2 problems. What patterns do you see in the numbers? Why is that happening?” (I see they each start with a 5 in the ones place below it because it’s halfway to the nearest ten, and the numbers end with a 4 in the ones place because that is closer than the next ten.)
• “¿Cómo usaron lo que aprendieron en los primeros 2 problemas para pensar en el último problema?” // “How did you use what you learned from the first 2 problems to think about the last problem?” (Instead of thinking about fives, we thought about fifties. We looked at all the numbers that are closer to 600 than 500 or 700.)
• Consider asking:
  • “¿A qué número se redondea 550?” // “What does 550 round to?” (600 because it is halfway between 500 and 600.)
  • “¿A qué número se redondea 650?” // “What does 650 round to?” (700 because it is halfway between 600 and 700.)

The purpose of this activity is for students to apply what they’ve learned about rounding to play a game in which each student generates a mystery number with three clues. The three clues describe whether the mystery number is even or odd, what it rounds to, and two numbers that it’s between. It is possible that more than one number can fit the clues provided. In the synthesis, students reflect on which clues were most helpful for determining the mystery number.

• Each student needs an index card.

• Groups of 4
• “Vamos a jugar un juego en el que deben adivinar un número secreto que alguien de su grupo va a escribir” // “We’re going to play a game in which you have to guess a mystery number that someone in your group writes down.”
• Choose a mystery number and give the class three clues. Play a round of the game with the class and discuss the clues. Consider using 275 and these clues:
  • “Mi número secreto es impar” // “My mystery number is odd.”
  • “Mi número secreto se redondea a 300” // “My mystery number rounds to 300.”
  • “Mi número secreto está entre 270 y 278” // “My mystery number is between 270 and 278.”
• “Van a completar tres frases para darle tres pistas a su grupo. La primera pista debe decir si el número es par o impar. Tómense un par de minutos para escoger un número secreto y escribir sus tres pistas” // “You’ll give your group three clues by finishing three sentences. The first clue should tell whether the number is even or odd. Take a couple minutes to choose a mystery number and write down your three clues.”
• 2 minutes: independent work time

• “Ahora van a jugar el juego con su grupo. Todos tendrán una oportunidad de compartir las pistas sobre su número secreto. Si les queda tiempo, todos pueden pensar en un nuevo número secreto con tres pistas nuevas” // “Now, you’re going to play the game with your group. Everyone will get a chance to share the clues for their mystery number. If you have time, you can each create a new mystery number with three new clues.”
• 12–15 minutes: small-group work time

• “Mientras jugaban, ¿cuáles pistas fueron las que más les ayudaron y por qué?” // “As you played the game, what clues were the most helpful and why?” (Knowing how the mystery number would round to the nearest ten was really helpful because that really narrowed it down. Knowing the numbers the mystery number was between was helpful if it was something like 150 and 160, but not if it was between 100 and 200.)