Lesson 5

This Number Talk encourages students to think about subtraction with expressions that may require decomposing and to rely on using what they know about place value and counting up or back to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students subtract one-digit numbers from two-digit numbers.

When students notice that they can use the value of previously found expressions to find new values, they look for and make use of structure (MP7). This Number Talk provides opportunities for students to notice they can subtract to get to a ten, then subtract the rest (\(17 - 8 = 17 - 7 - 1 = 10 - 1 = 9\)).

• 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 pueden usar la tercera expresión como ayuda para encontrar la diferencia de la última expresión?” // “How could you use the third expression to help you find the difference of the last expression?” (I know that taking away 6 gets me to 20, then I just took 2 more away.)

The purpose of this activity is for students to subtract in a way that makes sense to them. Students use a method of their choice and share their methods with one another. This can serve as a formative assessment of how students approach finding the value of a difference when a ten must be decomposed when subtracting by place. Although students may use many methods to subtract, including those based on counting or compensation, the synthesis focuses on connecting these methods to those based on place value where a ten is decomposed.

Monitor and select students with the following methods to share in the synthesis:  

• Uses connecting cubes to make 82 and removes 9 blocks. Counts back or counts all to find the difference.
• Subtracts 2 from 82 to get to a ten, 80, and then subtracts 7 from 80 by counting back (with or without blocks).
• Uses base-ten blocks to show 82 and decomposes a ten to get 12 ones. Subtracts 9 ones from 12 ones and counts the remaining blocks.

• Groups of 2
• Give students access to connecting cubes and base-ten blocks.

• “Encuentren la diferencia: \(82 - 9\)” // “Find the difference for \(82 - 9\).”
• “Muestren cómo pensaron. Usen dibujos, números o palabras” // “Show your thinking using drawings, numbers, or words.”
• “Prepárense para compartir lo que pensaron” // “Be prepared to share your thinking.”
• As student work, consider asking:
  • “¿Qué hicieron primero? ¿Por qué?” // “What did you do first? Why?”
  • “¿Qué quitaron primero? ¿Por qué?” // “What did you take away first? Why?”
  • “¿Cómo representaron el 82 al comienzo? ¿Tuvieron que hacer algún cambio para quitar 9? Expliquen” // “How did you show 82 at first? Did you have to make any changes to take away 9? Explain.”
  • “¿Qué herramienta usaron? ¿Por qué?” // “What tool did you use? Why?”

• Ask selected students to share in the given order.
• “¿Por qué  _____ tuvo que cambiar 82 a 7 decenas y 12 unidades para restarle unidades a unidades?” // “Why did _____ need to change 82 to 7 tens and 12 ones to subtract ones from ones?” (There are too many ones to take away, so she traded a ten for 10 ones to show decomposing a ten.)

The purpose of this activity is for students to subtract a one-digit number from a two-digit number. In the previous activity, students shared many ways to subtract including using connecting cubes or base-ten blocks to show decomposing a ten. They build on this understanding as they use base-ten blocks to represent the starting number and subtract an amount that requires them to decompose a ten.

• Groups of 2
• Give each group access to base-ten blocks.

• “Diego estaba representando números usando bloques en base diez. Con un compañero, sigan lo que Diego hizo para ver lo que él descubrió” // “Diego was representing numbers using base-ten blocks. Work with a partner to follow along and see what Diego discovered.”
• “Primero, usen sus bloques para mostrar lo que hace Diego. Después, contesten las preguntas” // “Use your blocks first to show what Diego does. Then answer any questions.”
• 8 minutes: partner work time
• Monitor for students who talk about “exchanging” or “trading” a ten for ten ones.

Some students may get the ones they need, but also keep the ten. Consider asking:

• “¿Por qué agregaste 10 unidades a los bloques de Diego?” // “Why did you add 10 ones to Diego’s blocks?”
• “¿Cuál es el valor de los bloques ahora?” // “What is the value of the blocks now?”
• “Cuando descompones una torre de diez, ¿qué le pasa a la torre? ¿Cómo puedes mostrar esto con los bloques en base diez?” // “When you decompose a tower of ten, what happens to the tower? How could you show this with the base-ten blocks?”

• “¿Qué tuvieron que hacer con los bloques cuando Diego quitó 8 unidades?” // “What did you need to do with the blocks when Diego took away 8 ones?”
• Select previously identified students to share.
• “¿En qué es diferente usar bloques en base diez comparado con usar torres de diez?” // “What is different about using base-ten blocks compared to the towers of ten?” (With the towers we could break it apart, but with the base-ten blocks we had to give a ten and take out 10 ones.)
• “Cuando restan usando el valor posicional, a veces se debe descomponer una decena para restar unidades. Al usar bloques encajables podemos ver esto cuando separamos o descomponemos una torre de diez” // “When you subtract by place, sometimes you need to decompose a ten to subtract ones. When we use connecting cubes, we can show those by breaking apart or decomposing a tower of ten.”
• “Cuando usamos bloques en base diez, no podemos separar la decena para formar unidades, pero de todos modos podemos mostrar cómo descomponer reemplazando 1 decena por 10 unidades” // “When we use base-ten blocks, we can't break apart the ten into ones, but we can still show decomposing it by replacing 1 ten with 10 ones.”