Lesson 9

The purpose of this True or False is to elicit insights students have about how the commutative property applies to addition and multiplication, but not subtraction. The reasoning students do here helps to deepen their understanding of the properties of operations and how they apply to subtracting within 1,000. It will also be helpful later when students need to recognize the need to decompose hundreds or tens to get more tens or ones.

• Display one equation.
• “Hagan una señal cuando sepan si la ecuación es verdadera o no, y puedan explicar cómo lo saben” // “Give me a signal when you know whether the equation is true and can explain how you know.”
• 1 minute: quiet think time

• Share and record answers and strategy.
• Repeat with each equation.

• “¿En qué se diferencia esta última ecuación?” // “What is different about the last equation?” (If we switch the order in subtraction, then both sides of the equal side aren't the same. If we switch the order when we subtract, we don't get the same number.)
• Consider asking:
  • “¿Alguien puede expresar el razonamiento de _____ de otra forma?” // “Who can restate _____'s reasoning in a different way?”
  • “¿Alguien quiere agregar algo al razonamiento de _____?” // “Does anyone want to add on to _____'s reasoning?”
  • “¿Podemos hacer generalizaciones basándonos en las afirmaciones?” // “Can we make any generalizations based on the statements?”

The purpose of this activity is for students to examine an error in an algorithm in which a larger digit is subtracted from a smaller digit in the same place value position. In such a case, it is common for students to subtract the smaller digit from the larger digit instead, not realizing that subtraction is not commutative. The given algorithm here shows the numbers in expanded form to help students see that it is necessary to first decompose a hundred into tens before the 50 can be subtracted from 20.

When students make sense of and correct Lin’s mistake, they construct viable arguments and critique the reasoning of others (MP3).

• Groups of 2
• Give students access to base-ten blocks.
• Display the image of Lin’s work.
• “Ahora examinemos cómo Lin le restó 156 a 428. Por un minuto, examinen lo que hizo” // “Now let’s look at how Lin subtracted 156 from 428. Take a minute to examine what she did.”
• 1–2 minutes: quiet think time

• “Con su pareja, describan el error y piensen lo que le dirían a Lin para que pudiera ajustar su trabajo” // “Work with your partner to describe the mistake and what you would tell or show Lin so she can revise her work.”
• 5 minutes: partner work time
• Monitor for students who:
  • use base-ten blocks or an algorithm to make sense of Lin’s mistake
  • decompose a hundred into 10 tens before subtracting 50 from 20, and showing this process by exchanging base-ten blocks or rewriting 400 as \(300 + 100\) and combining the 100 with 20
• Identify students who used these strategies and select them to share during synthesis.

If students don’t mention the error in Lin's work, consider asking:

• “¿Qué error cometió Lin cuando restó?” // “What mistake did Lin make when subtracting?”
• “¿Cómo podríamos usar bloques en base diez para ayudarle a Lin a ajustar su trabajo?” // “How could we use base-ten blocks to help Lin revise her work?”

• “¿Cómo describirían el error de Lin?” // “How would you describe Lin’s mistake?” (She tried to subtract 20 from 50 when you’re subtracting 50 from 20. She needed to decompose a hundred to get more tens.)
• Select previously identified students to share what they would tell or show Lin so she can revise her work.
• If no students suggest the following revision to Lin's work, display the algorithm and ask students to explain the revision:

• “Tengan presente el error de Lin mientras practicamos este algoritmo de resta en la próxima actividad” // “Keep Lin’s mistake in mind as we practice using this subtraction algorithm in the next activity.”

The purpose of this activity is for students to practice using the subtraction algorithm introduced in a previous lesson. Provide base-ten blocks for students who choose to use them to support their reasoning about the algorithm.

• Groups of 2
• Display Kiran’s algorithm from the previous lesson.
• “Este es un algoritmo de resta que vieron en una lección anterior. ¿Qué harían primero si fueran a usar este algoritmo para encontrar el valor de la expresión de resta de la actividad?” // “Here’s a subtraction algorithm you saw in an earlier lesson. What might be the first thing you’d do if you are to use this algorithm to find the value of the subtraction expressions in the activity?” (Write the numbers in expanded form and stack them.)
• 1 minute: quiet think time
• Share responses.
• Give students access to base-ten blocks.

• “Tómense un momento en silencio para probar este algoritmo. Si tienen preguntas, discútanlas con su compañero” // “Take some quiet time to try this algorithm. Check in with your partner if you have questions.”
• 5–7 minutes: independent work time
• If students have questions about the notation used to record the decomposition of a hundred or ten into more tens or ones, consider asking:
  • “¿Hay algún lugar en el problema en donde no hay suficientes decenas o unidades?” // “Is there any place in the problem where you don’t have enough tens or ones?”
  • “¿Cómo podrían obtener más decenas (o unidades)?” // “How could you get more tens (or ones)?”
  • “¿Cómo podrían registrar la descomposición de una centena en 10 decenas (o de una decena en 10 unidades)?” // “How could you record a hundred being decomposed into 10 tens (or a ten decomposed into 10 ones)?”

If students do not record the multiple decompositions in the last problem, consider asking:

• “¿Qué unidades en base diez tuviste que descomponer en este problema?” // “What units did you need to decompose in this problem?”
• “¿En dónde tendría más sentido para ti registrar cómo descompusiste una centena en más decenas? ¿Y una decena en más unidades?” // “Where would it make the most sense to you to record how you decomposed a hundred into more tens? A ten into more ones?”

• Select students to share their reasoning for 2–3 problems. Choose problems to focus on based on common questions that came up. Be sure to discuss the last problem, which requires decompositions of both a ten and a hundred.
• For the last problem, ask: “¿Cómo decidieron de qué manera registrar la decena y la centena que se necesitaba descomponer?” //  “How did you decide how to record the ten and hundred that needed to be decomposed?” (I started subtracting the ones and decomposed a ten into 10 ones so I had already crossed off the 30 and written 20. When I decomposed the hundred into tens, I just decided to write the 120 on top of the 20 like I wrote the 20 on top of the 30.)
• Record the completed algorithm, showing the decompositions of tens and hundreds.
• Consider asking:
  • “¿No había suficientes decenas o centenas para restar?” // “Where were there not enough tens or ones to subtract?”
  • “¿Qué se descompuso y cómo se registró?” // “What was decomposed and how was it recorded?”
  • “¿Identificaron lugares en los que podrían haber cometido el error que vimos en el trabajo de Lin?” // “Did you notice any places where you might have made the error we saw in Lin's work?"