This Number Talk encourages students to look for and make use of the structure of numbers in base-ten to mentally solve division problems (MP7). The reasoning elicited here will be helpful later in the lesson when students divide large numbers using increasingly more abstract strategies. In explaining their reasoning, students practice being precise in their use of language (MP6).
Launch
Groups of 2
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
Activity
Record answers and strategy.
Keep expressions and work displayed.
Repeat with each expression.
Student Task Statement
Encuentra mentalmente el valor de cada expresión.
Student Response
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Advancing Student Thinking
Activity Synthesis
“¿Cómo les ayudó cada expresión a encontrar la siguiente?” // “How did each expression help you find the next one?”
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 la expresión de otra forma?” // “Did anyone approach the expression in a different way?”
“¿Alguien quiere agregar algo a la estrategia de ____?” // “Does anyone want to add on to____’s strategy?”
Activity 1
20 mins
Descompongamos dividendos
Standards Alignment
Building On
Addressing
4.NBT.B.6
Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
The purpose of this activity is for students to encounter a way to divide a multi-digit number by using partial quotients and writing equations for them. They analyze and interpret the equations and consider how it is like and unlike finding quotients using base-ten representations. In the next activity, students will be introduced to a way to record partial quotients vertically.
MLR8 Discussion Supports. Provide students with the opportunity to rehearse describing their strategy for finding with a partner before they share with the whole class. Advances: Speaking
Launch
Groups of 4.
Give students access to base-ten blocks.
Activity
3–5 minutes: independent work time on the first question
2–5 minutes: Pause after the first question and discuss students’ responses. Record and display responses for all to see.
5–7 minutes: independent work time on remaining problems
Student Task Statement
Encuentra el valor de . Explica o muestra cómo razonaste. Puedes usar bloques en base diez si crees que te pueden ayudar.
Priya encontró el valor de así:
Describe los pasos del método de Priya.
¿En qué se parecen el método de Priya y el tuyo?
Usa el método de Priya para encontrar el valor de .
Activity Synthesis
Invite students to share their interpretations of Priya’s work and compare it to their reasoning in the first question.
“¿Cómo descompuso Priya el número 465?” // “How did Priya decompose the number 465?” (By place value. )
“¿Qué hizo Priya después de escribir las tres primeras ecuaciones?” // “What does Priya do after writing the first three equations?” (She adds up the quotients.)
“Podemos encontrar un cociente por partes, dividiendo una porción del dividendo a la vez, hasta que no haya más partes para dividir (o hasta que no haya suficiente cantidad para dividir)” // “We can find a quotient in parts—dividing a portion of the dividend at a time—until there is no more (or until there is not enough) of the dividend to divide.”
“Cada cociente se llama un cociente parcial” // “Each quotient is called a partial quotient.”
Activity 2
15 mins
El método de Tyler
Standards Alignment
Building On
Addressing
4.NBT.B.6
Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
The purpose of this activity is to introduce students to an algorithm that uses partial quotients, a vertical method of recording partial quotients. Students compare and contrast this approach with other ways of dividing numbers using partial quotients. They also use partial quotients to divide multi-digit numbers.
When students analyze Priya and Tyler's work and explain their reasoning, they critique the reasoning of others (MP3).
Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy for problem 2 before they begin. Students can speak quietly to themselves, or share with a partner. Supports accessibility for: Conceptual Processing, Organization
Launch
Groups of 4
Display Priya and Tyler’s methods.
Activity
“Tyler usó otro método para registrar . Analicen lo que está pasando en su método. Piensen en las semejanzas y las diferencias entre los dos métodos” // “Tyler used a different method to record . Analyze what is happening in his method. Think about how the two methods are alike and different.”
3 minutes: independent work time on the first two questions
3 minutes: small-group discussion
Invite students to share their analyses on the two methods.
If not mentioned by students, highlight that both Priya and Tyler divided in parts, but reasoned and recorded differently. Both methods are simply two ways to record partial quotients, but they are not fundamentally different.
Priya recorded the partial quotients with division equations.
The partial quotients in Tyler’s work are recorded as factors being multiplied by 5, and also listed above the dividend.
Tyler kept dividing in parts and subtracting until there’s nothing left of the dividend to divide.
Clarify the meaning of the numbers in Tyler’s method before students work on the last question.
3 minutes: independent work time to find the value of .
Student Task Statement
Tyler usa otro método para encontrar el valor de . Comparemos el trabajo de Priya y el de Tyler.
El método de Priya
El método de Tyler
¿En qué se parecen estos métodos? ¿En qué son diferentes? Haz una lista de todas las semejanzas y otra de todas las diferencias que puedas encontrar.
¿Por qué crees que Tyler hace restas en su método?
Muestra cómo podría Tyler registrar el proceso para encontrar el valor de .
Student Response
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Advancing Student Thinking
Activity Synthesis
Invite students to share their responses. Highlight the different ways to decompose 428 or the different partial quotients that could be used to find .
Make sure students see that Priya’s equations and Tyler’s method are simply two ways to record partial quotients, but they are not fundamentally different.
“El método de Tyler, en el que registra de forma vertical, es otro tipo de algoritmo” // “Tyler’s vertical recording method is another type of algorithm.”
Lesson Synthesis
“Hoy aprendimos a usar un algoritmo en el que se usan cocientes parciales para dividir números” // “Today we learned to use an algorithm that uses partial quotients to divide numbers.”
“¿Cómo le explicarían los ‘cocientes parciales’ a un compañero que no haya venido hoy?” // “How would you explain ‘partial quotients’ to a classmate who might be absent today?” (We can find a quotient in parts—dividing a portion of the dividend at a time—until there is no more or until there is not enough of the dividend to divide. Each quotient is called a partial quotient.)
“Supongamos que queremos encontrar el valor de y sabemos que podemos descomponer el 738 en partes. ¿Cómo sabríamos cuáles números escoger?” // “Suppose we’d like to find the value of and know we could decompose the 738 into parts. How would we know what numbers to choose?” (Look for multiples of 9. Try to start with the largest multiple of 9 and 10 within 738.)
“¿Qué formas hay de descomponer 738 en múltiplos de 9?” // “What are some ways to decompose 738 into multiples of 9?” (, or , among others.)
Display:
“Vimos dos maneras de anotar los cocientes parciales: escribiendo varias ecuaciones y escribiendo los pasos de la división de forma vertical. ¿En qué lugar de cada una vemos los cocientes parciales?” // “We saw two ways of recording partial quotients—by writing a series of equations and by recording the steps of division vertically. Where can we see the partial quotients in each one?”
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Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
