Lesson 5

The purpose of this Estimation Exploration is for students to practice the skill of making a reasonable estimate for the number represented by a point on a number line. They give a range of reasonable answers when given incomplete information and have the opportunity to revise their thinking as additional information is provided.

After students have made estimates based on the first image, draw a tick mark at the halfway point and label with 20. Students can revise their thinking based on this additional information. Revealing the actual number represented by the point is not necessary because leaving it open-ended encourages students to focus on reasonableness and not just one right answer.

• Group of 2
• Display image.

• “El punto representa un número en la recta numérica. ¿Qué número podría ser?” // “The point represents a number on the number line. What number could this be?”
• “¿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 compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.
• Draw a tick mark at the halfway mark and label with 20.
• “Teniendo en cuenta esta nueva información, ¿quieren ajustar o cambiar sus estimaciones?” // “Based on this new information, do you want to revise, or change, your estimates?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Record responses.

• “¿Cómo les ayudó la segunda imagen a ajustar su estimación?” // “How did the second image help you revise your estimate?” (When I saw it was less than 20, but close to 20, I changed my about right estimate from 14 to 18.)

The purpose of this activity is for students to estimate the number represented by a point on the number line and justify their reasoning. Encourage students to explain why their estimates are reasonable. Monitor for students who are using the location of other numbers to determine reasonable estimates. Students may choose to estimate and label other numbers (for example, multiples of 5 or 10). For each successive number line, the given tick marks are farther apart so students need to rely more on their understanding of properties of the number line and the accuracy with which they can locate the given numbers depends on how much extra work they do thinking about other numbers which they can locate accurately (MP1).

• Groups of 2

• “Miren cada recta numérica y escriban una estimación del número representado por el punto” // “Look at each number line and record an estimate of the number that the point represents.”
• 5 minutes: independent work time
• “Comparen cada estimación con su compañero y expliquen por qué creen que su respuesta es razonable” // “Compare each estimate with your partner and explain why you believe your answer is reasonable.”
• 7 minutes: partner work time
• Monitor for students who add tick marks or labels, including multiples of 10 or 5, to help identify the number.

If a student's estimates are outside the range of reasonable estimates, consider asking:

• “¿Cómo decidiste qué número está representado por este punto?” // “How did you decide what number this point represents?”
• “¿Cómo usaste los números que están marcados como ayuda para pensar en el número representado por el punto?” // “How did you use the numbers that are labeled to help you think about the number represented by the point?”
• “Si sabes que esta marca representa ___ y esta marca representa ___, ¿cómo puedes ubicar y marcar otros números?” // “If you know this tick mark represents ___ and this tick mark represents __, how could you locate and label other numbers?”

• “¿De cuál estimación están más seguros? ¿Por qué?” // “Which estimate are you most confident in? Why?”
• Invite previously identified students to share their strategies for an estimate they are confident in.
• “¿De cuál estimación están menos seguros? ¿Por qué?” // “Which estimates are you least confident in? Why?”
• “¿Qué los ayudaría a ser más precisos con sus estimaciones?” // “What would help you to be more precise with your estimates?” (If I knew more numbers that were closer to the point. If the 10s and 5s were labeled.)

The purpose of this activity is for students to locate numbers on a number line without tick marks to represent each number. Students use what they know about multiples of 10, the relative position of numbers on the number line, and comparing length to locate and label a set of numbers on the number line. They start by organizing number cards on a number line and make adjustments to their positions after each card is placed. After they place all of their cards, they locate and label the numbers on the number line. In the synthesis, students compare the number lines that are created and discuss, using the structure of the number line, why some numbers were placed more precisely than others (MP7). This also gives them a chance to construct viable arguments for how they placed the numbers and to critique the reasoning of others (MP3).

This activity uses MLR7 Compare and Connect. Advances: representing, conversing.

• Create a number line on chart paper for each group of students. 
• On each number line, draw tick marks at the beginning (label 0) and the end (label 40)
• On each number line, draw tick marks and label: 10, 20, 30.
• Create a set of number line cards from the blackline master for each group of 3 (each set should include 10 cards).

• Groups of 3
• Give each group chart paper, markers, and a set of number cards.

• “Van a trabajar en grupo para organizar las tarjetas de números en la recta numérica” // “You will be working with your group to arrange the number cards on the number line.”
• “Por turnos, escojan una tarjeta y ubíquenla cerca de su lugar en la recta numérica” // “Take turns picking a card and placing it near its spot on the number line.”
• “Expliquen cómo decidieron dónde ubicar su tarjeta” // “Explain how you decided where to place your card.”
• “Si creen que necesitan cambiar de lugar otras tarjetas, expliquen por qué” // “If you think you need to rearrange other cards, explain why.”
• “Cuando se hayan puesto de acuerdo en que ubicaron todos los números en los lugares correctos, marquen cada uno de los números en sus tarjetas con un punto en la recta numérica. Escriban debajo de cada punto el número que le corresponde” // “When you agree that you have placed all the numbers in the right spots, mark each of the numbers on your cards with a point on the number line. Label each point with the number it represents.”
• 10 minutes: small-group work time
• Consider asking:
  • “¿Por qué ubicaron su tarjeta ahí?” // “Why did you place your card there?”
  • “¿Dónde dibujarían un punto para representar este número?” // “Where would you draw a point to represent this number?”
  • “¿Cuáles tarjetas decidieron ubicar primero? ¿Por qué?” // “Which cards did you choose to place first? Why?”

• “Comprueben que todos sus números estén representados en los lugares que quieren que estén en la recta numérica” // “Check to make sure all of your numbers are represented in the spots you want them on the number line.”
• 1–2 minutes: group work time
• 5–7 minutes: gallery walk
• “¿En qué se parecen y en qué se diferencian las distintas rectas numéricas?” // “What is the same and what is different between the different number lines?”
• Consider asking:
  • “¿Cuáles números fueron ubicados por muchos grupos en el mismo lugar en sus rectas numéricas?” // “Which numbers did most groups have in the same spot on their number lines?”
  • “¿Cuáles números parecen estar en lugares diferentes?” // “Which numbers look like they are in different spots?”