Lesson 14

The purpose of this Estimation Exploration is for students to think about what value a point on the number line could represent. The only labeled tick marks are hundreds so students need to reason about what numbers are in between and how far the point is from the labeled numbers. 

• Groups of 2
• Display the image.
• “¿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.

• Consider asking:
  • “¿Alguien tiene una estimación menor que _____? ¿Alguien tiene una estimación mayor que _____?” // “Is anyone’s estimate less than _____? Is anyone’s estimate greater than _____?”
  • “Teniendo en cuenta esta discusión, ¿alguien quiere ajustar su estimación?” // “Based on this discussion does anyone want to revise their estimate?”
• “¿Qué nos ayudaría a estar más seguros de nuestra estimación?” // “What would help us be more sure of our estimate?” (More tick marks that are equally spaced. Marks of multiples of 10.)

Previously, students identified two multiples of 100 that border a given number, reasoned about their relative distance from the number, and then named the nearest multiple of 100. The purpose of this activity is for students to practice naming the nearest multiple of 100 and apply the same reasoning to identify the nearest multiple of 10. They determine two multiples of 10 that are closest to a given number (two intermediate tick marks on the number line) and then identify the multiple of 10 that is closer.

• Groups of 2
• “Ubiquen y marquen cada número de la tabla en la recta numérica. Luego, encuentren el múltiplo de 100 más cercano. Dejen la última columna en blanco por ahora” // “Locate and label each number in the table on the number line. Then, find the nearest multiple of 100. Leave the last column blank for now.”
• 3–5 minutes: partner work time
• Share responses.

• “Ahora completen el segundo problema con su compañero. Prepárense para compartir cómo razonaron” // “Now, complete the second problem with your partner. Be prepared to share your reasoning.”
• 1–2 minutes: partner work time
• “¿Cómo decidieron qué múltiplos de 10 estaban más cerca de 128?” // “How did you decide which multiples of 10 were closest to 128?” (Sample responses: I looked at which multiples of 10 it was in between. I looked for a multiple of ten to the right and a multiple of 10 to the left, these were the closest.)
• Have students label the last column and record the multiple of 10 that was closest to 128.
• “Completen el último problema individualmente” // “Complete the last problem independently.”
• 3–5 minutes: independent work time

• “¿Cómo saben qué múltiplo de 10 es el más cercano a cada número?” // “How do you know which multiple of 10 is the nearest for each number?” (If we label and locate the number on the number line, we can see which tick mark the point is closer to. We can see how many numbers to count up or down to get to a multiple of 10. For example, we only count up once to get from 89 to 90, but we count down 9 times to get down to 80.)

In this activity, students identify the nearest multiples of 10 and 100 for given three-digit numbers. They may do so by using the number lines from earlier, but they may also start to notice a pattern in the relationship between the numbers and the nearest multiples and decide not to use number lines. The work here prepares students to reason numerically in the next lesson.

When students notice and describe patterns in the relationship between the numbers and the nearest multiples of 10 or 100, they look for and express regularity in repeated reasoning (MP8).

• Groups of 2
• “Encontremos más múltiplos de 10 y de 100 que estén cerca de algunos números” // “Let's find more multiples of 10 and 100 that are close to some numbers.”
• “Observen que no nos dieron rectas numéricas. Piensen si aún así pueden encontrar los múltiplos de 10 y de 100 más cercanos. Si lo necesitan, pueden usar una recta numérica de todas maneras” // “Notice that no number lines are given. See if you can still find the nearest multiples of 10 and 100 without them. If you need to, you can still use a number line.”

• “Completen estos problemas con su compañero” // “Work with your partner to complete these problems.”
• 5–7 minutes: partner work time
• Monitor for students who use the following strategies to highlight:
  • Reason about the midpoint between a multiple of 10 or a multiple of 100 (5 or 50) to determine which multiple is closer, such as, “568 está más cerca de 570 porque el punto del medio entre 560 y 570 es 565” // “568 is closer to 570 because 565 would be the middle point between 560 and 570.”
  • Use place value patterns to determine which multiple is closer, such as, “Como el 1 de 712 es menor que 5, esto me dice que el número está más cerca de 700” // “Since the 1 in 712 is less than 5, it tells me that the number is closest to 700.”
• Pause for a brief discussion before students complete the last problem.
• Select previously identified students share the strategies they used to find the nearest multiple of 100 and the nearest multiple of 10.
• “Ahora tómense unos minutos para completar el último problema” // “Now take a few minutes to complete the last problem.”
• 2–3 minutes: independent work time

If students don’t identify the closest multiple of 10 or 100, consider asking:

• “¿Qué has intentado hacer para encontrar el múltiplo de 10 (o de 100) más cercano?” // “What have you tried to find the closest multiple of 10 (or 100)?”
• “¿Cómo podrías usar una recta numérica para encontrar el múltiplo de 10 (o de 100) más cercano?” // “How could you use a number line to find the closest multiple of 10 (or 100)?”

• Invite students to share their responses and reasoning for the last problem.