This Number Talk encourages students to compose or decompose multiples of 7 and to rely on properties of operations to mentally solve problems. The ability to compose and decompose numbers will be helpful when students divide multi-digit numbers. It also promotes the reasoning that is useful when finding multiples of a number, or when deciding if a number is a multiple of another number.
Launch
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.
Encuentra mentalmente el valor de cada expresión.
Student Response
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Advancing Student Thinking
Activity Synthesis
“¿Qué tienen en común las expresiones?” // “What do the expressions have in common?” (They all involve division by 7. The dividends are all multiples of 7. The results have no remainders.)
“¿Cómo nos ayudaron las primeras tres expresiones a encontrar el valor de la última expresión?” // “How did the first three expressions help us find the value of the last expression?”
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
Standards Alignment
Building On
4.NBT.B.5
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.
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 prompt students to use the relationship between multiplication and division and their understanding of factors and multiples to solve problems about an unknown factor (MP7). Students recognize such problems as division situations. In this activity, the dividends are three-digit numbers and the divisors are one-digit numbers. Students use the context of factors and multiples to interpret division that results in a remainder.
One approach for solving these problems is by decomposing the dividend into familiar multiples of 10 and then dividing the remaining number (which is now smaller) by the divisor. Another is to find increasingly greater multiples of the divisor until reaching the dividend. Both are productive and appropriate. In the Activity Synthesis, help students see the connections between the two paths.
Engagement: Develop Effort and Persistence.If students don’t recognize this as a division situation at first, they may do so when presented with a more accessible value. Consider offering this situation first: “Han comienza a escribir múltiplos de un número. Cuando llega a 12, él ha escrito 3 números” // “Han starts writing multiples of a number. When he reaches 12, he has written 3 numbers.” Invite students to identify what number Han is writing multiples of (4), and how they know. Then ask how they might apply that reasoning to the task as presented here. Supports accessibility for: Conceptual Processing
Launch
“¿Alguien le puede recordar a la clase el significado de ‘múltiplo’?” // “Who can remind the class of the meaning of ‘multiple?’”
Activity
4–5 minutes: independent work time for the first set of questions
Monitor for students who:
Decompose 104 into a multiple of 8 and another number.
Compose 104 from increasingly larger multiples of 8.
Use multiplication or division equations to show their reasoning.
Pause for a discussion before the second set of questions.
Select students who use different decomposition strategies to share responses. Record and display their reasoning.
“Retomemos cada una de las preguntas que acabamos de responder. ¿Qué ecuaciones podemos escribir para representarlas?” // “Let’s revisit each question we just answered. What equations could we write to represent them?”
Display equations and highlight their connection to the questions:
or
or
or
4–5 minutes: independent work time for the second set of questions
Activity Synthesis
Invite students to share responses for the last set of questions.
Highlight that to divide a number by a smaller number—say, divide 150 by 7, we can:
Use familiar multiples or multiplication facts to help us. For example, if we know , we know the result is 21 with a remainder of 3, or .
Think of the dividend in smaller chunks. For example: We can see the 150 as and divide each 140 and 10 by 7 separately, which gives with a remainder of 3.
Activity 2
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 optional activity is for students to continue to use the relationship between multiplication and division to reason about situations that involve division. Students divide three-digit numbers by single-digit divisors and find results with and without a remainder. This activity provides more practice applying the relationship between multiplication and division.
MLR8 Discussion Supports. Before they begin, invite students to make sense of the situation. Monitor and clarify any questions about the context. Advances: Reading, Representing.
Launch
Groups of 2
Activity
5 minutes: independent work time on the first question
2 minutes: partner discussion
2–3 minutes: partner work time on the second question
Monitor for students who clearly show how they use partial products or partial quotients to answer the questions.
Jada escribe múltiplos de un número secreto. Después de escribir algunos números, ella escribe el 126.
Mai dice que el número secreto es 6.
Priya dice que el número secreto es 8.
Andre dice que el número secreto podría ser 9.
¿Con cuál estudiante estás de acuerdo? Usa ecuaciones para mostrar cómo lo sabes.
Jada ofrece otra pista: “Si sigo escribiendo múltiplos, llegaré a 153”.
¿Cuál es el número secreto? Explica o muestra tu razonamiento.
Student Response
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Advancing Student Thinking
Activity Synthesis
“¿Cómo podríamos usar la división como ayuda para encontrar el número secreto?” // “How could we use division to help us find the mystery number?” (If the result has a remainder, then the divisor could not be the mystery number. 153 divided by 6 has 3 as a remainder. 153 divided by 9 has no remainder, so 9 is the mystery number.)
Activity 3
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 the center activity Watch Your Remainder, Stage 1. Students spin a spinner to get the divisor for the round. Each student picks 5 cards and chooses 3–4 of them to create a dividend. Each student finds their quotient. The score for the round is the remainder from each expression. Students pick new cards so they have 6 cards in their hand and then start the next round. The player with the lowest score after 6 rounds wins.
The Activity Synthesis provides suggested cards to display to discuss strategies for the game. Consider having a group of 2 share one of the rounds from their game instead.
Launch
Groups of 2
Give each group a set of Number Cards and 2 copies of the recording sheet.
“Vamos a aprender a jugar un juego que se llama ‘Atento a tu residuo’” // “We are going to learn a game called Watch Your Remainder.”
Give students several minutes to read over the directions.
Answer any clarifying questions from students.
Consider playing a sample round with the class.
Activity
10 minutes: partner work time
Monitor for students who play a round where there is no remainder and one where there is a remainder.
Instrucciones:
Gira la ruleta para obtener tu divisor. Este es un número de un dígito.
Cada compañero:
Toma tarjetas y usa 3 o 4 de ellas para crear un dividendo.
Escribe una ecuación de multiplicación para representar el cociente. (Por ejemplo, para , escribirías y tu puntaje sería 1).
Revisa el trabajo de tu compañero para asegurarte de que estás de acuerdo.
Tu puntaje en cada ronda es el residuo.
Toma otras tarjetas de modo que tengas 4 tarjetas para empezar la siguiente ronda.
Gana la persona que tenga menos puntos cuando se haya llenado la hoja de registro.
Student Response
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Advancing Student Thinking
Activity Synthesis
Invite selected students to show the number cards for the round with no remainder or display the number cards 0, 1, 2, 3, 7 with a divisor of 5.
“¿Cómo usarían estas tarjetas para obtener el menor residuo?” // “How would you use these cards to get the smallest remainder?”
Invite selected students to show the number cards for the round with a remainder or display the number cards 0, 1, 2, 3, 7 with a divisor of 9.
“¿Cómo usarían estas tarjetas para obtener el menor residuo?” // “How would you use these cards to get the smallest remainder?”
Lesson Synthesis
“Hoy tratamos de averiguar si un número es un múltiplo o un factor de otro número. Por ejemplo, ¿267 es un múltiplo de 8?” // “Today we tried to find out if a number is a multiple or a factor of another number. For example: Is 267 a multiple of 8?”
“¿Esta pregunta corresponde a un problema de división?” // “Is this question a division problem?” (Yes)
“¿Por qué?” // “Why?” (We’re looking for how many 8s are in 267.)
“¿Qué estamos dividiendo?” // “What are we dividing?” (267 by 8)
“La pregunta se puede responder usando hechos de multiplicación conocidos o encontrando productos parciales. ¿Cómo empezarían?” // “One way to answer the question is by using familiar multiplication facts or by finding partial products. How would you start?” (One way is to start with 10 x 8 or its multiples, build the products up to 267 or close to it, and then try smaller multiples of 8.)
“¿Por qué nos puede ayudar empezar con los múltiplos de 10?” // “Why might it be helpful to start with multiples of 10?” (They’re easy to find and easy to add.)
“¿Podemos usar la división para responder la pregunta?” // “Can we use division to answer the question?” (We can start with a number close to 267 that is a multiple of 8, divide it by 8, see what is left, and find a multiple of 8 that is close to that number. For example: . After taking 160 away, there’s 107 left. . After taking 80 away, there’s 27 left. . After taking 24 away, there’s 3 left, which is not enough to make 8.)
“¿En qué se parecen las dos estrategias?” // “How are the two approaches alike?” (They involve using smaller multiples of a number to see if a larger number is a multiple of that number.)
Student Section Summary
Resolvimos distintos problemas en los que tuvimos que dividir números enteros.
Recordamos dos formas de pensar en la división.
Por ejemplo, supongamos que representa una situación en la que se ponen 274 marcadores en grupos iguales. El valor de nos puede decir:
Cuántos marcadores hay en cada grupo si hay 8 grupos.
Cuántos grupos se pueden formar si hay 8 marcadores en cada grupo.
Aprendimos que en , el se llama el dividendo y el se llama el divisor. Después, identificamos varias formas de encontrar el valor de un cociente (es decir, el resultado de la división). Para calcular , podemos:
Pensar si un número es un múltiplo o un factor de otro número. Por ejemplo, “¿274 es un múltiplo de 8?” u “¿8 es un factor de 274?”.
Dividir de acuerdo al valor posicional y pensar en poner 2 centenas, 7 decenas y 4 unidades en 8 grupos iguales.
Dividir por partes y encontrar cocientes parciales. Por ejemplo, podemos encontrar primero (que es 20), después (que es 10) y después (que es 4).
Pensar en términos de la multiplicación. Por ejemplo, podemos pensar en , y así sucesivamente.
Esta es una forma de escribir la división usando cocientes parciales:
Divide. 2 hundred seventy 4 divided by 8, 11 rows. First row: 34. Second row: 4. Third row: 10. Fourth row: 20. Fifth row: 8, long division symbol with 2 hundred seventy 4 inside. Sixth row: minus 1 hundred 60. Side note, 8 times 20. Horizontal line. Seventh row: one hundred 14. Eighth row: minus 80, side notes 8 times 10. Horizontal line. Ninth row: 34. Tenth row: minus 32, side note 8 times 4. Horizontal line. Eleventh row: 2.
A veces, al dividir, sobra algo que no podemos poner en grupos iguales ni alcanza para formar un grupo nuevo. A lo que sobra lo llamamos un residuo. El resultado de dividir 274 entre 8 es 34, con un residuo de 2.
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.
