Lesson 8

The purpose of this Number Talk is to elicit strategies students have for subtracting within 1,000. These understandings help students develop fluency and will be helpful later when students choose between using an algorithm or another strategy to subtract.

• 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 las expresiones siguientes?” // “How did the first couple of expressions help you find the value of later expressions?” (When I noticed that the numbers had adjusted a little, it was easy to adjust the difference the same way. Sometimes it looked like a new problem, but both of the numbers had gone up the same amount so the difference was the same.)
• 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 use their knowledge of base-ten diagrams and place value to make sense of a subtraction algorithm. Students notice that in both the base-ten drawing and the algorithm, the subtraction happens by place. We can find the difference of two numbers by subtracting ones from ones, tens from tens, and hundreds from hundreds, and adding these partial differences to find the overall difference.

Students also recall that sometimes a place value unit needs to be decomposed before subtracting. For example, a ten may first need to be decomposed into 10 ones. This decomposition can be seen in both the base-ten drawing and in the algorithm. In the synthesis, students interpret the work and reasoning of others (MP3).

• Groups of 2
• Display Jada’s drawing.
• “Jada y Kiran encontraron el valor de \(391-215\). Tómense un minuto para entender el dibujo de Jada” // “Jada and Kiran found the value of \(391-215\). Take a minute to make sense of Jada’s drawing.”
• 1–2 minutes: quiet think time
• Share responses.

• “Trabajen con su pareja para entender el algoritmo de Kiran y completar las preguntas sobre su trabajo” // “Work with your partner to make sense of Kiran’s algorithm and complete the questions about his work.”
• 5-7 minutes: partner work time

If students don't explain the written algorithm, consider asking:

• “¿Cómo restó cada estudiante?” // “How did each student subtract?“
• “¿Cómo nos puede ayudar el dibujo de Jada a entender el trabajo de Kiran?” // “How could Jada's drawing help us understand Kiran's work?“

• Invite students to share their responses.
• “¿Cómo supo Kiran que necesitaba reescribir 391?” // “How did Kiran know to rewrite 391?” (If he tried to subtract the ones, he would notice he doesn’t have enough, so he needs to decompose a ten into 10 ones.)
• “¿Por qué él puede reescribir \(300 + 90 + 1\) como \(300 + 80 + 11\)?” // “Why is he allowed to rewrite \(300 + 90 + 1\) as \(300 + 80 + 11\)?” (Both of these add up to 391. They’re both showing 391 in different ways.)
• “¿En qué se parece el razonamiento de Kiran al razonamiento de Jada?” // “How is Kiran's reasoning like Jada's reasoning?” (Both of them decomposed the numbers into hundreds, tens, and ones, and subtracted the numbers in each place separately. Both of them decomposed a ten into 10 ones.)
• “¿En qué se diferencia su razonamiento?” // “How is their reasoning different?” (Jada used base-ten drawings to represent the numbers and Kiran wrote them out in expanded form.)
• “¿Cómo terminaron el trabajo de Kiran?” // “How did you finish Kiran’s work?” (I subtracted the ones, the tens, and then the hundreds. I subtracted the hundreds, the tens, and then the ones.)
• Record Kiran’s completed algorithm and keep it posted throughout this lesson and the following lesson.

The purpose of this activity is for students to analyze the connections between algorithms and base-ten diagrams that represent subtraction. In particular, students relate how the two strategies show a hundred decomposed into tens and a ten into ones in order to facilitate subtraction.

As students work, encourage them to refine their descriptions of what is happening in both the diagrams and the algorithms using more precise language and mathematical terms (MP6).

• Create a set of cards from the blackline master for each group of 2.

• Groups of 2
• Distribute one set of pre-cut cards to each group of students.

• “Este grupo de tarjetas incluye diagramas en base 10 y algoritmos que representan la misma resta. Emparejen cada diagrama con un algoritmo. Trabajen con su pareja para explicar sus decisiones” // “This set of cards includes base-ten diagrams and algorithms that represent the same subtraction. Match each diagram to an algorithm. Work with your partner to explain your choices.”
• 8 minutes: partner work time

• Select 2–3 students to share a match they made and how they know the cards go together.
• “¿Por qué a veces las centenas se movieron a la posición de las decenas en los diagramas?” // “In the diagrams, why were the hundreds sometimes moved over into the tens place?” (This happened when there weren’t enough tens to subtract. We used a hundred to get more tens.)
• “¿Cómo se hizo para mostrar esto en el algoritmo?” // “How was this shown in the algorithm?” (A hundred was crossed out and 10 more tens were added to the tens place, like in F, 20 was crossed out and 120 was written above it.)