Lesson 7

Relacionemos multiplicación y división

Warm-up: Cuántos ves: Decenas (10 minutes)

Narrative

The purpose of this How Many Do You See is for students to use grouping strategies to describe the images they see.

When students use grouping to find the total in a multiple of tens, they look for and make use of structure (MP7).

Launch

• Groups of 2
• “¿Cuántos ven? ¿Cómo lo saben?, ¿qué ven?” // “How many do you see? How do you see them?”
• Flash the image.
• 30 seconds: quiet think time

Activity

• Display the image.
• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

Student Facing

¿Cuántos ves? ¿Cómo lo sabes?, ¿qué ves?

Activity Synthesis

• “¿Qué expresiones podríamos escribir para representar las diferentes formas en las que los estudiantes vieron las decenas?” // “What expressions could we record for the different ways that students saw the tens?” ($$6 \times 10$$, because some students saw 6 groups of 10. $$3 \times (10 \times 2)$$, because some students saw 2 rows of 10, then multiplied by 3. $$(3 \times 10) \times 2$$, because some students multiplied 3 times 10 for each column, then multiplied by 2.)
• “¿Alguien puede expresar con otras palabras la forma en la que _____ vio las decenas?” // “Who can restate in different words the way _____ saw the tens?”
• “¿Alguien vio las decenas de la misma forma, pero lo explicaría de otra manera?” // “Did anyone see the tens the same way but would explain it differently?”
• “¿Alguien quiere compartir otra observación sobre la forma en la que _____ vio las decenas?” // “Does anyone want to add an observation to the way _____ saw the tens?”

Activity 1: Mesa redonda de división (20 minutes)

Narrative

The purpose of this activity is for students to solidify what they have learned about the relationship between multiplication and division. Students start by creating a drawing of equal groups. They then get a drawing created by another student in their group and write a division situation to match it. Then, they pass their paper and use the drawing of equal groups and the situation to write a multiplication equation. In the final round of this “carousel” structure, students write a division equation to match the other representations.

When students relate drawings, situations, and equations they reason abstractly and quantitatively (MP2). As students look through each other’s work, they add to the representations and can defend different points of view. Students are able to critique the work of others and construct viable arguments (MP3).

Students work on the same box on a graphic organizer as the other students in their group, so if they struggle, encourage them to talk to their group. Remind students that what they are creating should match what has already been filled in.

Engagement: Develop Effort and Persistence. Check in and provide each group with feedback that encourages collaboration and community. For example, supporting students in participating, passing the paper to the right, and writing the symbol.
Supports accessibility for: Social-Emotional Functioning, Language

Required Materials

Materials to Copy

• Division Round Table, Spanish

Launch

• Groups of 4
• Give each student a recording sheet.
• “En el primer recuadro de su hoja, hagan un dibujo que muestre grupos iguales de objetos. Otros estudiantes de su grupo van a usar este dibujo para completar los otros recuadros” // “In the first box on your sheet, create a drawing that shows equal groups of objects. This drawing will be used by other students in your group to fill in the other boxes.”
• 3 minutes: independent work time

Activity

• “Pásenle su hoja al compañero que está a su derecha. En el recuadro 2, escriban una descripción de una situación de división que corresponda al dibujo que les acabaron de pasar” // “Pass your paper to your right. In Box 2, write a description of a division situation that matches the drawing you were just passed.”
• 3 minutes: independent work time
• “Pasen su hoja a la derecha. En el recuadro 3, escriban una ecuación de multiplicación que corresponda al dibujo y a la situación de división que acabaron de recibir. Usen un símbolo para representar la cantidad desconocida” // “Pass your paper to your right. In Box 3, write a multiplication equation that matches the drawing and division situation you just received. Use a symbol for the unknown quantity.”
• 2 minutes: independent work time
• “Pasen su hoja a la derecha. En el recuadro 4, escriban una ecuación de división que corresponda al dibujo, a la situación de división y a la ecuación de multiplicación que acabaron de recibir. Usen un símbolo para representar la cantidad desconocida” // “Pass your paper to your right. In Box 4, write a division equation that matches the drawing, division situation, and multiplication equation you just received. Use a symbol for the unknown quantity.”
• 2 minutes: independent work time
• “Pasen su hoja una vez más. Deberían tener su dibujo original de vuelta” // “Pass your paper one more time. You should have your original drawing back.”
• “Hablen con su grupo sobre cuál recuadro les pareció más difícil de completar. Compartan ideas acerca de qué fue lo que más les ayudó durante esta actividad” // “Talk to your group about which box was the most difficult for you to fill in. Share ideas about what helped you most during this activity.”
• 5 minutes: small-group discussion

Student Facing

Tu profesor te dará una hoja de papel con 4 recuadros y te pedirá que dibujes o escribas algo en cada recuadro.

Después de trabajar en cada recuadro, haz una pausa y espera que el profesor te dé las instrucciones para el siguiente recuadro.

1. Dibuja grupos iguales en el recuadro 1 de tu hoja de registro.
2. En el recuadro 2 de la hoja que acabaste de recibir, escribe una descripción de una situación de división que corresponda al dibujo.
3. En el recuadro 3 de la hoja que acabas de recibir, escribe una ecuación de multiplicación que corresponda al dibujo y a la situación de división. Usa un símbolo para representar la cantidad desconocida.
4. En el recuadro 4 de la hoja que acabas de recibir, escribe una ecuación de división que corresponda al dibujo, a la situación de división y a la ecuación de multiplicación. Usa un símbolo para representar la cantidad desconocida.

Activity Synthesis

• “¿Qué estrategias compartieron en su grupo?” // “What strategies were shared in your group?” (When I wasn’t sure about writing a situation, I looked back at the drawing and tried to imagine something I could be dividing that looks like the drawing. When I was writing an equation it helped me to imagine the situation happening.)
• “Al examinar su hoja, ¿qué conexiones observan entre la multiplicación y la división?” // “As you look at your paper, what are some connections you notice between multiplication and division?” (I can use both multiplication and division to represent the same drawing or situation. The multiplication equations are all missing a factor, but the division equations are all missing the quotient.)

Activity 2: Grupos de útiles escolares (15 minutes)

Narrative

The purpose of this activity is for students to represent and solve problems involving equal groups. Students can solve the problem first or write the equation first, depending on the order that makes the most sense to them. Students write equations with a symbol standing for the unknown quantity to represent each problem, but can write either a multiplication equation or a division equation. A multiplication equation and a division equation that represent the same problem are highlighted in the synthesis.

MLR8 Discussion Supports: Prior to writing the equations, invite students to make sense of the situations and take turns sharing their understanding with their partner. Listen for and clarify any questions about the context.

Launch

• Groups of 2
• “Todas estas situaciones son acerca de cosas que pueden encontrar en un pupitre o alrededor del pupitre. ¿Qué cosas podrían encontrar en un pupitre o a su alrededor?” // “These situations are all about things that you could find on a desk or around a desk. What are some things that you could find on a desk or around a desk?”
• 30 seconds: quiet think time
• Share responses.

Activity

• “Lean cada situación y escriban una ecuación para cada una. Usen un símbolo para representar la cantidad desconocida. Después, resuelvan la ecuación y encuentren el número desconocido. Pueden resolver el problema primero o escribir una ecuación primero dependiendo de qué orden tiene más sentido para ustedes. Prepárense para explicar su razonamiento” // “Read through each situation and write an equation with a symbol that represents the unknown quantity for each situation. Then, solve and determine the unknown number in each equation. You can solve the problem first or write an equation first depending on what order makes the most sense to you. Be prepared to explain your reasoning.”
• 7–10 minutes: independent work time
• Monitor for students who write a division equation and a multiplication equation for the same situation to share during the synthesis.
• “Ahora, compartan sus ecuaciones y soluciones con su compañero. Por turnos, compartan sus ecuaciones y soluciones” // “Now, share your equations and your solutions with your partner. Take turns sharing your equations and solutions.”
• 3–5 minutes: partner discussion

Student Facing

a.  Escribe una ecuación que represente la situación. Usa un símbolo para representar la cantidad desconocida.

b.  Resuelve el problema y encuentra el número desconocido de la ecuación. Prepárate para explicar tu razonamiento.

1. Kiran tenía 32 clips. Le dio 4 clips a cada estudiante. ¿Cuántos estudiantes recibieron clips?

1. Ecuación: _______________________

2.

2. Hay 28 libros en 4 pilas. Si cada pila tiene la misma cantidad de libros, ¿cuántos libros hay en cada pila?

1. Ecuación: _______________________

2.

1. Ecuación: _______________________

2.

1. Ecuación: _______________________

2.

Student Response

If students don’t find a solution to the problems, consider asking: “¿De qué se trata este problema?” // “What is this problem about?” and “¿Cómo puedes representar el problema?” // “How could you represent the problem?”

Activity Synthesis

• Have students share a division equation and a multiplication equation that were written to represent the same division situation and display for all to see.
• Discuss differences in equations students wrote.
• “¿Cómo está representada la situación por cada número de las ecuaciones?” // “How does each number in the equations represent the situation?”
• “_____ escribió _____ y _____ escribió _____ para representar el mismo problema. ¿En qué se parecen y en qué se diferencian esas ecuaciones?” // “_____ wrote _____ and _____ wrote _____ to represent the same problem. How are those equations the same and different?” (One of the equations is a division equation, but the other equation is a multiplication equation with an unknown factor. They used different symbols for the unknown amount. Both symbols in the equation represent the missing _____ in the situation.)
• Have students share strategies they used to solve the problem.

Lesson Synthesis

Lesson Synthesis

Display: $$24 \div 4 = {?}$$

“¿Cuál sería la ecuación de multiplicación relacionada?” // “What would be the related multiplication equation?” ($$4 \times {?} = 24$$ or $${?} \times 4 = 24$$)

“¿Cómo se relacionan?” // “How are they related?” (The missing number in the division equation is the number of groups or the number in each group and that’s what the missing number in the multiplication equation represents.)

Display: $$4 \times {?} = 28$$

“¿Cuál sería la ecuación de división relacionada?” // “What would be the related division equation?” ($$28 \div 4 = {?}$$)

“¿Cómo se relacionan?” // “How are they related?” (The multiplication equation is missing the number in each group and that is what the quotient represents in the division equation.)