Lesson 6

This warm-up prompts students to compare four images. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another. During the synthesis, ask students to explain the meaning of any terminology they use, such as parts, partitions, mark, label, thirds, or fourths.

• Groups of 2
• Display the image.
• “Escojan una que sea diferente. Prepárense para compartir por qué es diferente” // “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

• “Discutan con su pareja cómo pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

• “Para que su explicación sea clara cuando ubican y marcan fracciones en una recta numérica, ¿qué cosas importantes deben incluir?” // “To make your reasoning clear while locating and labeling fractions on a number line, what are some important things to include?” (Partitions of the equal parts, a dot and label at the fraction.)
• “En una lección anterior, aprendimos que podemos marcar las fracciones en la recta numérica de la misma manera en la que marcábamos los números enteros” // “We learned in a previous lesson that we label fractions on the number line at the tick marks just like we have labeled whole numbers on the number line.”
• “En qué se diferencia esto de la forma como marcábamos nuestras tiras de fracciones, como la de la figura A?” // “How is this different from how we labeled our fraction strips like in A?” (In diagrams of fraction strips we labeled the part that has size or length \(\frac{1}{3}\). On the number line we are labeling the number \(\frac{1}{3}\).)

The purpose of this activity is for students to make sense of partitioning number lines that extend beyond one. Clare and Diego's work surfaces 2 common misconceptions that students often make while partitioning number lines into fractions. Clare partitions the entire number line into fourths and Diego places 4 tick marks to show fourths. Students analyze these misconceptions (MP3) before they locate and label unit fractions on number lines of various lengths in the next activity.

• Groups of 2
• “Hoy vamos a partir rectas numéricas para ubicar fracciones unitarias. Tómense un minuto para examinar la forma como Clare, Andre y Diego partieron sus rectas numéricas en cuartos” // “Today we are going to partition number lines to locate unit fractions. Take a minute to look at how Clare, Andre, and Diego have partitioned their number lines into fourths.”
• 1–2 minutes: quiet think time

• “Con su compañero, decidan cuál partición tiene más sentido para ustedes y por qué” // “Work with your partner to decide whose partitioning makes the most sense to you and why.”
• 3–5 minutes: partner work time
• Monitor for students who can explain why Andre’s partitioning makes sense and why the others do not show fourths.

• Ask students to share why Andre’s partitioning makes sense to them.
• Consider asking:
  • “¿Alguien pensó en el razonamiento de Andre de otra forma?” // “Did anyone think of Andre’s reasoning in a different way?”
  • “¿Cómo sabemos que la recta numérica de Andre está partida en cuartos?” // “How do we know that Andre’s number line is partitioned into fourths?”
• Ask students to explain why Clare and Diego’s partitioning does not show fourths.
• “Aprendimos que cuando se parte la recta numérica, se debe poner atención a dónde están el 0 y el 1, y hay que asegurarse de partir ese pedazo en el número correcto de partes iguales” // “We learned that when you partition the number line, you have to pay attention to where 0 and 1 are and make sure to partition that into the right number of equal-length parts.”

The purpose of this activity is for students to partition the interval from 0 to 1 into equal parts to locate and label unit fractions. Students see number lines that vary in length, from 1 unit to 4 units, which provides an opportunity for them to practice accurately partitioning the unit on the number line, rather than the entire number line (MP6). Some number lines show numbers greater than one which gives students the opportunity to think about fractions greater than one even though they are not explicitly addressed in this lesson.

• Groups of 2
• “Ahora que hemos pensado en algunos errores comunes que se pueden cometer al partir rectas numéricas, van a tener la oportunidad de partir rectas numéricas para ubicar y marcar fracciones unitarias” // “Now that we’ve thought about some common mistakes about partitioning number lines, you are going to have a chance to partition number lines to locate and label unit fractions.”

• “Individualmente, partan cada recta numérica. Luego, ubiquen y marquen cada fracción” // “Work independently to partition each number line and locate and label each fraction.”
• 3–5 minutes: independent work time
• “Ahora compartan con su compañero la forma en la que partieron cada recta numérica y el lugar en el que ubicaron y marcaron cada fracción” // “Now, share how you partitioned each number line and where you located and labeled each fraction with your partner.”
• “Asegúrense de hacerle sugerencias a su compañero sobre la forma en la que partió las rectas numéricas o de pedirle sugerencias para las particiones que fueron retadoras” // “Be sure to share tips on how you partitioned or ask for tips for any of the partitions that were challenging.”
• 3–5 minutes: partner discussion
• Monitor for students who disagree on how to partition one of the number lines.

If students create the same number of tick marks as the denominator or partition the entire number line instead of the interval between 0 and 1, consider asking:

• “Dime cómo partiste tu recta numérica” // “Tell me about how you partitioned your number line.”
• “¿Qué aprendimos en la última actividad sobre cómo partir rectas numéricas?” // “What did we learn in the last activity about how to partition number lines?”

• “¿Hubo rectas numéricas que su compañero y ustedes no estaban seguros de cómo partir o no estaban de acuerdo en cómo hacerlo? ¿Cómo resolvieron su confusión o desacuerdo?” // “Were there any number lines that you and your partner were not sure how to partition or disagreed about? How did you resolve your confusion or disagreement?” (We weren’t sure how to partition the number line that goes up to 4 when we were locating \(\frac{1}{2}\). We talked together about partitioning just the space from 0 to 1 into half.)
• Consider asking: “Hemos visto cómo partir, ubicar y marcar fracciones en rectas numéricas que tienen números mayores que 1. También lo hemos hecho en rectas numéricas que solo llegan hasta 1. ¿En qué son diferentes?” // “What was different about partitioning, locating, and labeling fractions on number lines with numbers greater than 1 than on number lines that just go up to 1?” (You have to be careful to just partition the one whole, not the whole number line.)
• Display the same unit fraction on a number line with length 1 and length 2, such as:

• “¿Qué observan?” // “What do you notice?” (The top number line just has 0 to 1. The bottom number line has the 0 to 2. The number 1 is located in the same place on both number lines. The number \(\frac{1}{6}\) is located in the same place on both number lines.)
• “La ubicación de los números en la recta numérica no cambia si extendemos la recta numérica. El número \(\frac{1}{6}\) está ubicado entre 0 y 1, así la recta numérica llegue hasta 1 o llegue hasta otro número” // “The location of any number on the number line doesn’t change just because we extend the number line. The number \(\frac{1}{6}\) is located between 0 and 1 whether the number line goes up to 1 or it goes up to another number.”