Lesson 13

This Estimation Exploration prompts students to practice making a reasonable estimate based on experience and known information. In this case, it is not practical to count the paletas, but students could reason about groups of paletas by color, or estimate the complete rows and columns of paletas and extend their estimate to the whole set. Some students might also make an estimate based on their familiarity with how paletas are usually arranged in cases.

• Groups of 2
• Display the image. 
• “Estas son paletas. Se originaron en México y típicamente se hacen con muchas frutas diferentes” // “These are ice pops called paletas. They originated in Mexico and are typically made with many different fruits.”
• Ask students to estimate without counting.
• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

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

• “¿Alguien hizo una estimación menor que 30?, ¿mayor que 80?” // “Is anyone’s estimate less than 30? Greater than 80?”
• “¿Cómo supieron que 30 era una estimación muy baja y que 80 era una estimación muy alta?” // “How did you know that 30 (or another number) would be too low and 80 (or another number) too high?”
• “Teniendo en cuenta esta discusión, ¿alguien quiere ajustar su estimación?” // “Based on this discussion does anyone want to revise their estimate?”

In this activity, students recall what they know about division from grade 3. The context allows students to connect lived experiences to the math of the activity. By inviting students to consider treats that they enjoy in their homes or neighborhoods, they share experiences and foster connections that build community.

The first question gives students an opportunity to co-craft mathematical questions based on a situation before answering a question based on a division equation. Students divide a two-digit number by a one-digit divisor, as they did in grade 3, in a way that makes sense to them. The activity synthesis highlights different representations students made and relates them to the situation. The term dividend is re-introduced in this lesson to describe a number being divided into equal groups.

This activity uses MLR5 Co-craft Questions. Advances: writing, reading, representing

• Groups of 2
• “¿Cuáles son algunas de sus golosinas o meriendas favoritas que comen en casa?” // “What are some of your favorite treats or snacks from home?”
• 30 seconds: quiet think time
• 1 minute: partner discussion

MLR 5: Co-craft Questions

• Display the opening paragraph and the first question.
• “Escriban una lista de preguntas matemáticas que se puedan hacer sobre esta situación” // “Write a list of mathematical questions that could be asked about this situation.”
• 2 minutes: independent work time
• 2–3 minutes: partner discussion
• Invite several students to share one question with the class. Record responses.
• “¿En qué se parecen estas preguntas?” // “How are these questions alike?” (The questions involve multiplying or dividing.) “¿En qué son diferentes?” // “How are they different?” (The questions can be answered using different operations.)
• “Veamos la siguiente pregunta de la actividad” // “Let’s look at the next question in the activity.”

• 3–4 minutes: quiet work time

If students are unsure how to write a division question for \(84 \div 7\), consider asking:

• “Si fueras a actuar el significado de esta expresión, ¿qué harías?” // “If you were to act out the meaning of this expression, what would you do?”
• “Si fueras a explicar su significado, ¿qué dirías?” // “If you were to explain its meaning, what would you say?”
• “¿En qué tipo de situaciones ocurrirían estos tipos de acciones?” // “In what type of situations would these types of actions take place?”

• Display two questions that students wrote for the equation \(84 \div 7 = {?}\) .
• “¿Qué representa el 84 en ambos problemas?” // “What does the 84 represent in both problems?” (The amount being divided into equal groups.) 
• “En matemáticas, el número que se está dividiendo se conoce como el dividendo” // “In mathematics, the number being divided is known as the dividend.”
• Invite students to share their strategies for the last question. Highlight strategies that show equal-size groups and reasoning that relates multiplication and division.

In this activity, students continue to use any strategy to solve division problems in context and to recall the two interpretations of division. One problem involves finding how many in each group and the second involves finding the number of groups. Students work with two- and three-digit dividends and encounter division that results in a number with a remainder. They consider what the leftover means in the given context.

During the synthesis, highlight that both multiplication and division can be used to reason about the solutions, and elicit equations can be written to represent students' reasoning about equal-size groups. Introduce remainders as “leftovers” or quantities that remain after dividing a number into equal groups. 

• Groups of 2
• Display images of snacks in the activity.
• “Miren las imágenes. ¿Qué observan? Qué se preguntan?” // “Take a look at the images. What do you notice? What do you wonder?”
• Share responses.
• “Los gulab jamuns son golosinas populares en India, Pakistán y sus países vecinos de Asia del Sur” // “Gulab jamuns are sweet treats that are popular in India, Pakistan, and their neighboring countries in South Asia.”
• “Los palitos de pan cubiertos con chocolate, crema de fresa u otros sabores son meriendas populares en Japón, Taiwán y otros países de Asia del Este” // “Breadsticks that are covered with chocolate, strawberry cream, or other flavors are popular snacks in Japan, Taiwan, and other East Asian countries.” 
• Ask a student to read the first problem aloud.
• “Los problemas de esta actividad son sobre golosinas que disfrutan los estudiantes de distintos lugares de todo el mundo” // “The problems in this activity involve treats that students enjoy from different places around the world.”

• 6–8 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for students who reason in terms of partial products or partial quotients and record their thinking as equations.
• Monitor for the different ways students describe and decide what to do with the 2 leftover breadsticks.

• Invite students to share their response for the first problem.
• If no students wrote equations to show their thinking, ask: “¿Qué ecuaciones pueden representar la forma como encontraron la solución al primer problema?” // “What equations can represent how you found the solution to the first problem?“
• Record both multiplication and division equations for all to see. For example:
  • \(5 \times 10 = 50\) and \(5 \times 7 = 35\), so \(5 \times 17 = 85\).
  • \(50 \div 5 = 10\) and \(35 \div 5 = 7\), so \(85 \div 5 = 17\).
• Invite other students to discuss the second problem.
• “Cuando dividimos, a veces nos sobran elementos que no alcanzan para formar un grupo nuevo ni para poner un elemento adicional en cada grupo. A lo que sobra lo llamamos residuo” // “When we divide, sometimes we have leftovers that are not enough to make a new group or not enough to put an additional item to each group. We call the leftovers remainders.”
• “¿Cuál fue el residuo cuando se dividieron 110 palitos de pan entre 6?” // “What was the remainder when 110 breadsticks were divided by 6?” (2 breadsticks were leftover)
• “¿En qué se parecen las preguntas de las dos situaciones (la de los gulab jamuns y la de los palitos de pan)?” // “How are the questions in the two situations—gulab jamuns and breadsticks—alike?” (They involved division into equal groups.)
• “¿En qué son diferentes?” // “How are they different?” (The first looks for the number of groups. The second looks for the size of a group. The second involves some leftovers.)