Lesson 20

This warm-up prompts students to analyze an example of subtraction using both the standard algorithm and expanded form. The numbers require decomposing multiple units, which are shown in both strategies. The observations here prepare students to later reason with similar subtraction problems in which more than one decomposition is needed.

• Groups of 2
• Display subtraction calculations.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “¿En qué se parecen los dos ejemplos? ¿En qué son diferentes?” // “How are the two examples alike? How are they different?” (In the second one the numbers are written in expanded form, including the numbers that show regrouping.)
• “Ya hemos visto problemas de resta en donde se descomponen unidades en base diez. ¿En qué son diferentes estos nuevos problemas?” // “We’ve seen subtraction problems with decomposed units before. How are these different?” (In the problems we have seen so far, only one place needed to be decomposed in order to subtract. In these examples, more than one place needs to be decomposed.)

The purpose of this activity is for students to add and subtract multi-digit numbers through the hundred-thousands place. To find the value of some differences, students will decompose more than one unit.

The last question includes problems with a missing addend and a missing subtrahend. Besides making use of the structure of the standard algorithm (MP7), students will need to rely on what they know about the relationship between addition and subtraction to find the missing numbers.

• Groups of 2
• Give students access to grid paper.

• 5–7 minutes: independent work time
• “Revisen sus respuestas del primer problema con su compañero y hagan los ajustes necesarios. Después trabajen juntos en el segundo problema” // “Check your responses to the first problem with your partner and make any adjustments you need to make. Then work on the second problem together.”
• 2–3 minutes: partner discussion
• As students work on the second problem monitor for students who:
  • Find the missing addend by adding up from 67,182 to 129,400, or by subtracting \(129,\!400 - 67,\!182\).
  • Find the missing subtrahend by adding up from 193,710 to 234,650, or by subtracting \(234,\!650 - 193,\!710\)

Students may not recognize the relationship between addition and subtraction, or may not know how to begin the problem involving a missing addend and missing subtrahend. Consider asking: “¿Cómo podríamos escribir este problema como una ecuación que tenga un símbolo que represente el número desconocido?” // “How might we write this problem as an equation with a symbol for the unknown?”

• “¿Cómo pueden saber si sus respuestas son correctas? ¿Cómo las pueden revisar?” // “How can you tell if your answers are correct? How can you check them?” (One way to check the result of subtraction is by adding it back to the number being subtracted. One way to check the result of addition is by subtracting one addend from the sum. Another way of checking is by performing the calculations with the numbers written in expanded form.)

In this activity, students analyze addition and subtraction calculations, identify errors, and explain what makes certain ways of calculating problematic. The work here gives students opportunities to construct logical arguments and critique the reasoning of others (MP3).

• Groups of 2
• Display the first problem.
• 2 minutes: quiet think time
• 2–3 minutes: partner discussion

• 5 minutes: independent work time to complete the second problem.
• 2 minutes: partner discussion
• Monitor for students who describe the errors using place value language and an understanding of how and when to decompose a unit.

If students have trouble making sense of Lin's work, encourage them to find the difference and then compare their work to Lin's.

• Select 2–3 students to share their responses and reasoning.
• Record responses as students share.
• “¿Qué errores se cometen a menudo cuando se restan dos números usando el algoritmo estándar?” // “What are some errors that are commonly done when subtracting two numbers using the standard algorithm?” (Subtracting without lining up the numbers by place value. Using the notation of regrouping without decomposing a unit. Crossing out digits without including them in the regrouping. Subtracting from left to right without decomposing units.)
• “¿Qué ideas tienen para evitar esos errores?” // “What ideas do you have for avoiding those errors?” (Use grid paper to line up the digits by place value. Decompose the unit before regrouping. Check answers using addition.)