Lesson 3

The purpose of this warm-up is to elicit ideas about what a number line can be used to represent. The sequence of diagrams emphasizes the position of multiples of 5 and 10 on the number line, which will be useful when students represent numbers on number lines that only include labeled tick marks at these positions. While students may notice and wonder many things about these number line diagrams, ideas about how the diagrams may represent counting are the important discussion points.

• Groups of 2
• Display the number lines.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “Estas rectas numéricas nos ayudan a ver cómo se ve cuando contamos a saltos usando diferentes números” // “These number lines help us see what it looks like when we count by different numbers.”
• “Contemos de 5 en 5 hasta 30, empezando en 0” // “Count to 30 by 5 starting with 0.”
• “¿Cuál recta numérica representa nuestro conteo? Expliquen” // “Which number line represents our count? Explain.” (B because the arrows show moving from 0 to 5 to 10 to 15...)
• “Contemos de 10 en 10 hasta 30, empezando en 0” // “Count to 30 by 10 starting with 0.”
• “¿Cuál recta numérica representa nuestro conteo? Expliquen” // “Which number line represents our count? Explain.” (C because the arrows show moving from 0 to 10 to 20 to 30).

The purpose of this activity is for students to work with number lines that only have multiples of 5 or 10 labeled and do not start with 0. Students reason about how a number line can be accurate without all the numbers labeled. Students use the numbers that are labeled to locate specific numbers on number lines. Students rely on the regular structure of the number line and the counting sequence in order to accurately place numbers (MP7).

• Groups of 2
• Display the first number line.

• “¿Qué observan acerca de esta recta numérica?” // “What do you notice about this number line?” (Not all the tick marks are labeled. Only the fives are labeled.)
• 30 seconds: quiet think time
• Share responses.
• “¿Las marcas con números están en los lugares correctos en la recta numérica? Expliquen” // “Are the labeled tick marks in the right spots on the number line? Explain.” (Yes because 5 is 5 length units from 0. 10 is 10 units from 0. There are 5 length units between each labeled mark.)
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Share responses.
• “Hoy vamos a darles sentido y a usar rectas numéricas que no tienen números en cada marca” // “Today we are going to make sense of and use number lines that do not label every tick mark.”

• “Ubiquen cada número en la recta numérica y márquenlo con un punto. Prepárense para mostrarle a su compañero cómo ubicaron los números y cómo saben que están en los lugares correctos” // “Locate each number on the number line and mark it with a point. Be ready to show your partner how you know you located the numbers and how you know they are in the right spots.”
• 4 minutes: independent work time
• 4 minutes: partner work time
• Consider asking:
  • “¿Cómo ubicaron este número?” // “How did you locate this number?”
  • “¿Cómo usaron las marcas con números?” // “How did you use the labeled tick marks?”
  • “¿Cómo saben que su número está en la longitud correcta medida desde 0?” //“How do you know your number is at the right length from 0?”
• Monitor for students who show their number is at the right position by:
  • counting on from 0 or referencing the length from 0
  • counting on or back from a labeled tick mark
  • describing the length between each labeled tick mark

If students count every tick mark to locate numbers, consider asking:

• “¿Cómo puedes usar las marcas con números como ayuda para encontrar un número?” // “How could you use the labeled tick marks to help you find a number?”

• Display the number line labeled 20, 30, 40, 50, 60.
• Invite 1–2 previously identified students to share how they located 48.
• “¿Cómo saben que su punto está a la distancia correcta desde 0?” // “How do you know your point is the right distance from 0?” (It’s like when we measured from different spots on a ruler. The tick marks are spaced the same space apart and count by 1 length unit. We could draw the line back and find where 0 is.)
• “La representación de una recta numérica no puede representar todos los números con marcas, puntos o etiquetas. Podemos usar lo que sabemos basándonos en los números que están marcados para ubicar otros números” // “A number line representation cannot represent all numbers with tick marks, points, or labels. We can use what we know based on the numbers that are labeled to locate other numbers.”

The purpose of this activity is for students to locate numbers up to 100 on a number line. They complete number lines that are labeled with multiples of 5 or 10 by using what they know about length on the number line and counting by 5 and 10. They use the labeled tick marks to locate and represent numbers within 100. When students explain to one another how the located different numbers on the number lines they construct viable arguments  and may critique each other's reasoning (MP3).

• Groups of 2

• “Individualmente, completen cada recta numérica: llenen los espacios con el número que la marca representa. Después, ubiquen cada número, márquenlo con un punto y escriban debajo del punto el número que le corresponde” // “On your own, complete each number line by filling in the missing labels with the number the tick mark represents. Then, locate each number, mark it with a point, and label the point with the number it represents.”
• “Cuando terminen, piensen cómo pueden explicarle a su compañero cómo saben que sus números y puntos están en los lugares correctos en las rectas numéricas” // “When you finish, think of how you can explain to your partner how you know your labels and points are at the right spots on the number lines.”
• 5 minutes: independent work time
• “Compartan su trabajo con un compañero. Asegúrense de que están de acuerdo sobre sus respuestas” // “Share your work with a partner. Make sure you agree on your answers.”
• 5 minutes: partner discussion
• Monitor for students who:
  • explain their labeled tick marks based on counting by 5 or 10
  • explain their labeled tick marks based on the equal lengths between each labeled tick mark
  • use labeled tick marks to explain how they locate numbers

If students' labels are numbers other than the specified numbers, consider asking:

• “Veo que escribiste ___ debajo de este punto. ¿Cómo puedes usar la recta numérica para demostrar que esa es la ubicación de ese número?” // “I see you labeled this point as ___. How can you use the number line to prove that is the location of that number?”

• Display the image of an incomplete number line labeled with 10 and 50.
• “¿Cómo podemos encontrar el 35 sin completar los números que faltan?” // “How could you find 35 without filling in the missing labels?” (We could count back from 50. We could count on from 10.)
• Invite previously identified students to explain how they found the labels and used them to locate 35.
• Complete the number line as students explain.