Lesson 13

The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. It gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it (MP3). Asking yourself “Does this make sense?” is a component of making sense of problems (MP1), and making an estimate or a range of reasonable answers with incomplete information is a part of modeling with mathematics (MP4).

• 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.
• “De los grupos que estaban en el campo, ¿cuáles incluyeron como parte de la banda de marcha?” // “Which groups on the field did you count as part of the marching band?” (I counted only the performers in the band uniforms. I included everyone on the field.)

• “Observen que muchas de nuestras estimaciones se expresan en términos de decenas y centenas. ¿Por qué será? ¿Por qué no estimamos que hay aproximadamente, por ejemplo, 163 personas o 248 personas en la imagen?” // “Notice that many of our estimates are expressed in terms of tens and hundreds. Why might that be? Why don’t we estimate that there are about, say, 163 people or 248 people in the picture?” (If we are estimating, we aren’t trying to guess the exact number. Tens and hundreds are easy to think about.)
• “Muchas veces ayuda hacerse una idea de una cantidad (cuántos, cuánto, cuánto tiempo, etc.) con grupos de 10 o grupos de 100. Los números que son grupos de 10, como 50 o 90, se llaman ‘múltiplos de 10’. Los números que son grupos de 100, como 200 o 700, se llaman ‘múltiplos de 100’” //  “It is often helpful to get a sense of a quantity (how many, how much, how long, etc.) with groups of 10 or groups of 100. Numbers like 50 and 90 that are groups of 10 can be called ‘multiples of 10.’ Numbers like 200 or 700 that are groups of 100 can be called ‘multiples of 100.’”
• “Cuando hacemos una estimación, usualmente decimos un múltiplo de 10 o de 100 que pensamos que está cerca del número” // “When we estimate, we often name the multiple of 10 or 100 that we think the number is close to.”
• “¿Pero cómo decidimos si un número está cerca de algún múltiplo de 10 o de 100? Pensaremos más en esta pregunta en la siguiente actividad” // “But how do we decide whether a number is close to some multiple of 10 or 100? We’ll think more about this question in the next activity.”

The purpose of this activity is for students to think about what it means for numbers to be close to multiples of 100. There is no definition given about what “close to” means during the activity, so students may interpret the term in different ways.

• Groups of 2
• Display the first problem.
• “Por un minuto, observen cuántas personas hay en varios lugares de una escuela durante un día normal. ¿Qué observan? ¿Qué se preguntan?” // “Take a minute to look at the numbers of people in different parts of a school during a school day. What do you notice? What do you wonder?” (There aren’t many people in the library. There are a lot of people in the school. Are there any other places where people could be in the school?)
• 1 minute: quiet think time
• Share and record responses.

• “Para cada lugar de la escuela, decidan si allí hay aproximadamente 100 personas o no. Escriban los números en la tabla. Prepárense para explicar cómo decidieron si un número es o no es aproximadamente 100” // “For each room, decide if there are about 100 people in the room or not. Record the numbers in the table. Be prepared to explain how you decide a number is or is not about 100.”
• 2 minutes: independent work time
• Share responses.
• “Ahora, decidan con su compañero si el número de personas que hay en cada lugar de la escuela está cerca de 0, cerca de 100 o cerca de 200. Si creen que algún número no pertenece a ninguna columna, no lo pongan en ninguna. Prepárense para explicar cómo razonaron” // “Now, work with your partner to decide if the number of people in each part of the school is close to 0, close to 100, or close to 200. If you don’t think a number belongs in any column, set it aside. Be prepared to explain your reasoning.”
• 2–3 minutes: partner work time
• Monitor for students who provide reasoning for where they would place 36, 52, and 163. Ask them to share during the synthesis.

• “¿Cómo decidieron si un número está cerca de 0, de 100 o de 200?” // “How did you decide whether a number is close to 0, 100, or 200?” (I thought about whether it was almost that number, like 94 is almost 100. I decided between the choices that were there—like for 216, it was closer to 200 than to 0 or 100.)
• “¿Cómo decidieron si un número no pertenece a ninguno de estos grupos?” // “How did you decide if a number doesn’t belong to any of these groups?” (If a number seemed far away from all the choices, then I set it aside. Like 52 wasn’t close to 0 or 100 because it’s almost right in the middle.)
• “¿Qué pasaría si la tabla dijera ‘más cerca de 0’, ‘más cerca de 100’ y ‘más cerca de 200’? ¿Cambiaría el lugar donde va cada número? ¿Podrían poner todos los números en la tabla?” // “What if the table showed ‘closer to 0,’ ‘closer to 100,’ and ‘closer to 200?’ Would it change where each number goes? Would you be able to place all the numbers in the table?” (Yes, because I would just be choosing the number that’s closer than the other numbers, not saying it’s really close to the number.)

The purpose of this activity is for students to locate two- and three-digit numbers on a series of number lines. The endpoints of each number line are multiples of 100, and the space between them is partitioned into ten equal intervals. As they locate the numbers, students recognize each tick mark as a multiple of 10. Later in the activity, students use a number line to name the closest multiple of 100 to a given number. When students choose the correct number line and accurately place each number on the number line they attend to precision and show an understanding of place value (MP6, MP7).

• Groups of 4
• “¿Qué saben sobre la recta numérica?” // “What do you know about the number line?” (Each point on the number line can represent a number. You can add or subtract by moving right or left on the number line. It can show distance between numbers, like the number 10 is 10 away from 0.)
• 1 minute: quiet think time
• Share and record responses.
• “Observen las rectas numéricas del primer problema. ¿Qué observan acerca de ellas? ¿Qué se preguntan?” // “Take a look at the number lines in the first problem. What do you notice about them? What do you wonder?” (Students may notice: Each number line has two multiples of 100. There are tick marks between the numbers. Students may wonder: Why don’t the number lines go higher or lower? What numbers do the tick marks represent?)
• 30 seconds: quiet think time
• Share responses.
• Display the number line:

  

  

• “¿Ven múltiplos de 100 en esta recta numérica?” // “Do you see multiples of 100 in this number line?” (Yes, 100 and 200)
• “¿Qué números creen que están representados por las marcas sin números?” // “What numbers do you think the unlabeled tick marks represent?” (Tens, groups of 10, numbers that we get if we count up by 10 starting from 100, multiples of 10) “¡Nombrémoslos!” // “Let’s name them!” (100, 110, . . . , 200)
• Label the first few tick marks.
• “¿Pueden estimar en qué lugar de la recta numérica está 113?” // “Can you estimate where 113 goes on the number line?” (Between the second and third tick marks, but closer to the second tick mark. Or between 110 and 120, but closer to 110.)
• Assign one set of numbers (A, B, C, D, or E) to each group of 4.

• “Decidan con su grupo en cuál recta numérica debe ir cada número. Ubiquen cada número en la recta numérica usando un punto y márquenlo. Luego, completen el segundo problema” // “Work with your group to decide on which number line each number should go. Locate each number on the number line with a dot, and label it. Then, complete the second problem.”
• 5–7 minutes: small group work time
• After students place 364 on a number line, pause for a discussion. Ask each group to share one of their numbers and how they knew on which number line to place it.
• “¿En cuál recta numérica ubicaron 364?” // “On which number line did you place 364?” (The number line with 300 and 400.)
• “¿Cómo decidieron qué múltiplo de 100 era el más cercano?” // “How did you decide which multiple of 100 was the closest?” (Once I located 364 on my number line I could tell it was closer to 400. I counted the tick marks back to 300 and up to 400 and it was less tick marks to get to 400, so it was closer.)
• “Completen el último problema individualmente” // “Complete the last problem on your own.”
• 2–3 minutes: independent work time

If students place a number on a number line on which it doesn’t belong, such as placing 216 on the number line that goes from 100 to 200, consider asking:

• “Dime cómo decidiste ubicar 216 en esa recta numérica” // “Tell me how you decided to place 216 on that number line?”
• “¿Me puedes mostrar dónde iría 216 en esa recta numérica?” // “Can you show me where 216 would be on that number line?”

• “¿Cómo pueden saber qué múltiplo de 100 está mas cerca de un número dado?” // “How can you tell which multiple of 100 a number is closest to?” (The endpoints of the number lines are multiples of 100. From the number line, we can tell whether a point is closer to one end or the other. We can tell if a point is in the lower half or upper half of the number line. We can count the tick marks to each multiple of 100 to decide which one is closer.)