Lesson 20

The purpose of this warm-up is to elicit strategies and understandings students have for identifying multiples of small numbers, primarily by looking for and making use of structure (MP7). These understandings will be helpful later when students solve division problems that involve distinguishing quotients with and without a remainder.

• “Cuenten de 2 en 2, empezando en 90” // “Count by 2 starting at 90.”
• Record as students count.
• Stop counting and recording at 112.

• “¿Qué patrones observan en cada uno de los conteos que anotamos?” // “What patterns do you notice in each of the recorded counts?”
• Repeat with 3 and 5.
• “Cuenten de 3 en 3, empezando en 90” // “Count by 3 starting at 90.” Stop counting and recording at 114.
• “Cuenten de 5 en 5, empezando en 90” // “Count by 5 starting at 90.” Stop counting and recording at 115.

• “¿105 es un múltiplo de 2?, ¿de 3?, ¿de 5? ¿Cómo lo saben?” // “Is 105 a multiple of 2, 3, or 5? How do you know?”
• “¿105 es un múltiplo de 15?” // “Is 105 a multiple of 15?”

This activity encourages students to interpret the quantities in situations, represent them mathematically, use their representations to find solutions, and then interpret their solutions in context (MP2). The dividends here are limited to three-digit numbers.

• Groups of 2

• 5–6 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for students who appropriately interpret remainders based on the situation.

• Invite students to share their responses and reasoning.
• For problem 1, highlight that both 94 and 95 boxes are plausible if they could be defended and the assumptions are made clear. (For example, students might say that the bakers could have two leftover muffins rather than trying to sell them in boxes, so 94 boxes are enough.)
• “¿Qué ecuación podemos escribir para describir la relación que hay entre el número de muffins y el número de cajas llenas?” // “What equation could we write to describe the relationship between the number of muffins and the number of full boxes?” (One possible equation: \( (94 \times 4) + 2 = 378\).)
• For the last problem, ask: “¿Qué ecuación podemos escribir para describir la relación que hay entre el número de filas y el número de asientos?” // “What equation could we write to describe the relationship between the number of rows and the number of seats?” (One possible equation: \((28 \times9) + 6 = 258\).)

In this activity, students continue to solve contextual problems that involve division (MP2). Here, the dividends extend to four-digit numbers and the problems demand a greater lift.

In the second half of the activity, students are asked to reason in the opposite direction: given a division expression, they are to invent a situation that it can represent and interpret the value of the expression in context.

If time permits, consider asking students to create a visual display of the situation they invent for problem 2, so they can present the situation and their reasoning to the class.

• Groups of 2

• 3–4 minutes: independent work time on the first problem
• Invite students to share responses and reasoning.
• “¿Qué ecuación o ecuaciones podemos escribir para representar la relación que hay entre la cantidad ahorrada cada mes, los ahorros, el número de meses de ahorro y la cantidad de dinero necesaria para el jardín?” // “What equation(s) can we write to represent the relationship between the amount saved each month, the savings, the number of months of saving, and the amount needed for the garden?” (One possible equation: \((8 \times 158) + 6 = 1,\!270)\)
• 5–7 minutes: independent work time on problem 2

• Students find a partner who chose a different expression and take turns presenting their work. The person listening should consider whether the response makes sense and check if the quotient is correct.
• If time permits, students share with a different student or partnership.