Lesson 21
Los ceros en el algoritmo estándar
Warmup: Cuál es diferente: Números con 0, 2 y 5 (10 minutes)
Narrative
This warmup prompts students to carefully analyze and compare features of multidigit numbers. In making comparisons, students have a reason to use language precisely (MP6), especially place value names. The activity also enables the teacher to hear how students talk about the meanings of nonzero digits in different places of a multidigit number.
Students observations will support their reasoning in the next activity when they subtract a number with nonzero digits from the four numbers listed.
Launch
 Groups of 2
 Display numbers.
 “Escojan uno que sea diferente. Prepárense para explicar por qué es diferente” // “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
 1 minute: quiet think time
Activity
 “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
 2–3 minutes: partner discussion
 Share and record responses.
Student Facing
¿Cuál es diferente?
 2,050
 2,055
 205.2
 20,005
Student Response
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Activity Synthesis
 “¿Qué tal si le restáramos 44 a estos números enteros?” // “What if we subtracted 44 from each whole number?” Record “– 44” under each whole number.
 “¿A cuál número sería más fácil restarle 44?” // “Which number would it be easiest to subtract 44?” \((2,\!055  44\) because you don’t have to decompose units.)
 “¿Cómo podríamos restarle 44 a los otros números enteros?” // “How could we subtract 44 from the other whole numbers?” (We would have to decompose other place values.)
Activity 1: ¿Qué hacemos si no hay nada para descomponer? (20 minutes)
Narrative
The purpose of this activity is to examine subtraction cases in which nonzero digits are subtracted from zero digits. In some cases, students could simply look at the digit to the left of a 0 and decompose 1 unit of that number. But in other cases, the digit to the left is another 0 (or more than one 0), which means looking further to the left until reaching a nonzero digit. Students learn to decompose that unit first, and then move to the right, decomposing units of smaller place values until reaching the original digits being subtracted. The problems are sequenced from fewer zeros to more zeros to allow students to see how to successively decompose units.
Recording all of the decompositions can be challenging. For the last problem, two sample responses are given to show two different ways of recording the decompositions. The important point to understand is that because there are no tens, hundreds, or thousands to decompose, a tenthousand must be decomposed to make 10 thousands. Then, one of the thousands is decomposed to make 10 hundreds, and so on, until reaching the ones place. Those successive decompositions can be lined up horizontally, but this can make it hard to see what happened first. A second way shows more clearly the order in which the decompositions happen, but it may be challenging to see which place the successive decomposed units are in.
To add movement to this activity, the second problem could be done as a gallery walk where each group completes one problem and then walk around the room to look for similarities and differences in others’ posters.
Advances: Conversing, Representing
Supports accessibility for: Conceptual Processing, Language, Organization
Launch
 Groups of 2–4
 5 minute: independent think time
 3 minute: partner share
Activity
 “Tómense un momento para pensar en el primer problema y luego discútanlo con su compañero” // “Take some time to think about the first problem and then discuss it with your partner.”
 2 minutes: quiet think time
 3 minutes: partner discussion
 Pause for clarifying questions, as needed.
 “Ahora, con su compañero, completen los siguientes 2 problemas” // “Now, work with your partner to complete the next 2 problems.”
 5 minutes: partner work time
Student Facing
Estos son algunos números que viste antes. Cada número tiene al menos un 0. A todos los números se les resta 1,436.

Dale sentido a los problemas y explícaselos a un compañero.
 Usa la estrategia que se muestra en el primer problema para encontrar estas dos diferencias:

Encuentra el valor de cada diferencia. Prepárate para explicar cómo razonaste. Si tienes dificultades, trata de restar usando la forma desarrollada.
Student Response
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Advancing Student Thinking
Students may lose track of the units when multiple rounds of decompositions are required. Offer baseten blocks and prompt students to show multiple decompositions when finding the value of \(100  1\) using a large square baseten block that represents 100. Consider asking:
 “¿Cómo puedes mostrar la resta de 1?” // “How can you show subtraction of 1?” (Exchange it for 100 ones.)
 “¿Qué haríamos si solo pudiéramos intercambiar 1 unidad en base diez por 10 unidades en base diez a la vez?” // “What if we could only exchange 1 unit for 10 units at a time?” (Exchange the hundred for 10 tens, then exchange 1 ten for 10 ones, so that we’d have 9 tens and 10 ones. Then, we can remove 1.)
 “Usando el algoritmo estándar, ¿cómo mostrarías la forma de restarle 1 a 100?” // “How would you show the subtraction of 1 from 100 using the standard algorithm?”
Activity Synthesis
 Ask students to share responses and to demonstrate consecutive decomposing when multiple zeros are involved.
 If needed, use expanded form to represent the decompositions students are explaining.
Activity 2: ¿Cuál es tu edad? (15 minutes)
Narrative
In this activity, students solve contextual problems that involve subtracting numbers with nonzero digits from numbers with one or more zero digits. Students may choose other ways to find the difference (for example, adding up and using a number line to keep track), but are asked to use the standard algorithm at least once.
In the launch, students subtract their age from the current year. This provides an opportunity for students to notice the relationship between this difference and their current age (MP7).
Required Materials
Materials to Gather
Launch
 Groups of 2
 Give students access to grid paper.
 “Escriban el año en el que nacieron y réstenle ese año al año en el que estamos” // “Write the year that you were born down and subtract that year from the current year.”
 “Compartan con un compañero lo que descubrieron” // “Share with a neighbor what you found out.” (Students will notice that the number that results is their current age or the age they will be on their upcoming birthday.)
 “Jada usó este método para encontrar la edad de algunos de sus familiares” // “Jada used this method to find the age of some of her relatives.”
Activity
 10 minutes: partner work time
Student Facing
miembro de la familia  año de nacimiento 

abuela  1952 
abuelo  1948 
bisabuela  1930 
bisabuelo  1926 
Student Response
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Activity Synthesis
 Ask students to share responses.
 “¿Cuáles años les parecieron más fáciles de restar? ¿Cuáles les parecieron más difíciles? ¿Por qué?” // “Which years did you find easiest to subtract? Which were more difficult? Why?” (Subtracting a year without decomposing any units was easier than subtracting a year that required decomposing.)
 Prompt students to check and compare ages found using the standard algorithm and those found using other ways of reasoning.
Lesson Synthesis
Lesson Synthesis
Display these expressions. “Estas son tres expresiones” // “Here are three expressions.”
“¿En qué se parecen las expresiones?” // “How are the expressions alike?” (They all involve 2,222 that is subtracted from a sixdigit number with three 5s and three 0s. Finding each difference requires multiple regroupings.)
“¿En qué son diferentes?” // “How are they different?” (The fives and zeros are in different positions in each number. In the first expression, only one unit needs to be decomposed before 2 ones could be subtracted from it. In the second expression, two units need to be decomposed. In the third expression, three units need to be decomposed before 2 ones could be subtracted.)
“Un amigo no está seguro de cómo resolver la segunda expresión, \(505,\!500  2,\!222\). Explícale a un compañero cómo usarías el algoritmo estándar para encontrar el valor de la diferencia” // “A friend is unsure how to solve the second expression, \(505,\!500  2,\!222\). Explain to a partner how you would use the standard algorithm to find the value of the difference.”
2 minutes: partner discussion
2 minutes: wholegroup discussion
Cooldown: Encuentra algunas diferencias (5 minutes)
CoolDown
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