Lesson 6

The purpose of this Notice and Wonder is for students to look at different number lines that all start at 0 but show different decimals. The number lines are nested, that is each successive one is contained in the previous one. The key points for students to notice are that the number lines all have decimals on them and that the size of those decimals is getting smaller. In the lesson, they will plot decimals on number lines like these.

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

• “¿Qué representan las marcas de la recta numérica de arriba?” // “What are the tick marks on the top number line?” (one tenth, two tenths, three tenths, and so on)
• “Hoy vamos a usar rectas numéricas como estas para ubicar varios números decimales” // “Today we are going to use number lines like these to locate different decimals.”

The purpose of this activity is for students to plot the same number on different number lines and recognize that the location of the number on the number line can only be accurately determined when it lies on a tick mark. As they work on locating the number, students reinforce their understanding of place value as the tick marks on the number lines are tenths, hundredths, and thousandths (MP7). Students may struggle to locate 0.001 on the first two number lines. The important take-away is that when a decimal does not lie on a tick mark estimation is needed to locate the number and it can be difficult or impossible to locate it precisely (MP6).

• Groups of 2

• 5 minutes: independent work time
• 5 minutes: partner discussion
• Monitor for students who reason about place value to:
  • label each tick mark
  • locate 0.001 on each number line

• Ask previously identified students to share.
• “¿Cómo decidieron qué escribir debajo de las marcas de la primera recta numérica?” // “How did you decide what to label the tick marks on the first number line?” (There are ten tick marks and there are ten tenths in one whole so I counted by tenths.)
• “¿En qué lugar de la primera recta numérica está ubicado diez décimas?” // “Where is ten tenths on the first number line?” (The number 1 is ten tenths.)
• “¿Cómo decidieron qué escribir debajo de las marcas de la segunda y de la tercera recta numérica?” // “How did you decide what to label the tick marks on the second and third number lines?” (There are 10 hundredths in 1 tenth. There are ten thousandths in 1 hundredth.)
• “¿En qué fue diferente ubicar 0.001 en cada una de las rectas numéricas?” // “How was locating 0.001 different for each of the number lines?” (It was so close to 0 on the first number line that I could not plot it. I could estimate its location on the second number line and then it was the first tick mark on the last number line.)

The purpose of this activity is for students to label number lines where the end tick marks are tenths or hundredths written as decimals. They will use their understanding of place value when they label the tick marks (MP7). Students also choose one of the number lines to compare two numbers, preparing them for the comparison work in the next activity and in future lessons.

• Groups of 2

• 4–5 minutes: independent work
• 2–3 minutes minutes: partner discussion
• Monitor for students who accurately label the number lines with hundredths and thousandths.

• Ask previously identified students to share their solutions and reasoning.
• “¿Qué recta numérica escogerían para comparar 0.534 y 0.537?” // “Which number line would you choose to compare 0.534 and 0.537?” (I liked the middle one because those numbers were labeled tick marks and I could see which one was further to the right.)
• “¿Cuál número es mayor: 0.534 o 0.537? ¿Por qué?” // “Which number is greater, 0.534 or 0.537? Why?” (0.537 because it is further to the right on the number line.)
• Display inequality: \(0.534 < 0.537\)
• “También podemos usar símbolos para decir que 0.534 es menor que 0.537” // “We can also say that 0.534 is less than 0.537 with symbols.”

The purpose of this activity is for students to compare decimal numbers using the number line for support. All of the numbers lie on tick marks and students will use their understanding of place value to accurately place the decimals. They will also use their understanding that one number is greater than another when it lies farther to the right on the number line.

• Groups of 2

• 5 minutes: independent work
• 2 minutes: partner discussion

• “¿Cómo nos ayudan las rectas numéricas a comparar números decimales?” // “How do number lines help compare decimals?” (We can put the decimals exactly on tick marks and then see which number is farther to the right.)