Lesson 12

The purpose of this Number Talk is to elicit strategies students have for finding products of single-digit factors. These reasoning strategies help students develop fluency and will be helpful later in this unit when students solve two-step word problems.

When students use strategies based on the properties of multiplication to find unknown products, they look for and make use of structure (MP7). Students may reverse the order of the factors to create a multiplication fact they know. Students may think about “one more group” as they move from the first expression to the second expression (or the third to the fourth). Also, students may say that they “just know” the product. All of these responses are acceptable because students will be in different stages as they progress toward fluency.

• 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 pensar en productos de 2 a encontrar productos de 3?” // “How did thinking about products of 2 help you find products of 3?” (I could think about 2 groups, then add one more group. I could think about 2 in each group, then one more in each group.)
• 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 choose a strategy or algorithm to subtract within 1,000. Students should attend to the details of numbers in the problems that could indicate whether a particular strategy or algorithm is most useful. The important thing is that students choose an algorithm or another strategy that they can use efficiently and accurately for the given problem. As students choose strategies to find the values of each expression, they look for common structure and observe regularity in repeated reasoning (MP7, MP8).

• “Hemos estado aprendiendo sobre algoritmos de resta. Recuerden que los algoritmos son solo una de las maneras en las que podemos resolver problemas. También podemos usar otras estrategias o representaciones” // “We’ve been learning about subtraction algorithms. Remember that algorithms are just one way we can solve problems. We can also use other strategies or representations.”
• “¿Cómo describirían en qué se diferencia un algoritmo de otras estrategias?” // “How would you describe the difference between an algorithm and other strategies?” (A strategy like adding up might work for the one problem you are solving, but an algorithm has steps that work for any problem.)
• 1 minute: partner discussion
• Share responses.
• “Van a tener la oportunidad de encontrar el valor de cada una de estas diferencias usando la estrategia o el algoritmo que prefieran” // “You’re going to have an opportunity to find the value of each of these differences using a strategy or algorithm of your choice.”

• “Individualmente, encuentren el valor de cada diferencia. Después, tendrán la oportunidad de compartir su trabajo” // “Work independently to find the value of each difference, then you’ll have a chance to share your work.”
• 7–10 minutes: independent work time
• Identify students who use the same strategy to subtract and those who use different ones.
• Choose a few problems for students to discuss. Consider selecting \(382-190\) (the second expression) and \(600-478\) (the fourth expression), which lend themselves to be evaluated with an algorithm and another strategy, respectively.
• “Encuentren un compañero que haya restado de la misma forma que ustedes. Discutan cómo razonaron” // “Find a partner who subtracted the same way you did. Discuss your reasoning.”
• 1–2 minutes: partner discussion
• “Ahora encuentren un compañero que haya restado de una forma diferente a la de ustedes. Discutan cómo razonaron” // “Now find a partner who subtracted the problem in a different way from you. Discuss your reasoning.”
• 2–3 minutes: partner discussion
• Repeat the discussion with 1-2 expressions or as many as time permits.

• Invite 4–5 students share a strategy or algorithm they saw.
• “¿Qué estrategias o algoritmos quieren practicar más?” // “What strategies or algorithms do you want to practice more?”

The purpose of this activity is for students to play a game that enables them to practice using strategies and algorithms to subtract within 1,000. Students decide whether they will try to make the smallest or greatest difference, then spin a paper clip on a spinner to generate two three-digit numbers. Students use their choice of strategy or algorithm to subtract the numbers.

When students use place value to create a pair of numbers with a specific type of difference, they are looking for and making use of structure (MP7).

• Each group of 2 will need a paper clip.

• Groups of 2
• Give each group 1 copy of Greatest Difference, Smallest Difference.
• “Tómense un momento para leer las instrucciones del juego con su pareja” // “Take a minute and read the directions to the game with your partner.”
• 1 minute: partner work time
• Play one round of the game against the class to illustrate how the game should be played.
• “¿Tienen alguna pregunta sobre el juego?” // “Are there any questions about the game?”
• Answer any questions students have about the game.

• “Ahora jueguen el juego con su pareja por un rato” // “Now, take some time and play the game with your partner.”
• 7–10 minutes: partner work time

• Display: 601 and 398
• “Si estos fueran los números que ustedes formaron, ¿cómo encontrarían la diferencia y por qué?” // “If these were the numbers you made, how would you find the difference and why?” (Sample responses: I would count up because 398 is so close to 400. I would use an algorithm because I know that if I follow the steps it would work every time.)