Lesson 8

The purpose of this warm-up is to activate what students know about the use of number lines to represent fractional values, preparing them to use number lines to reason about addition of fractions in a later activity. While students may notice and wonder many things about the diagram, be sure to highlight the meaning of each interval and where the numbers 1 and 2 are located on the number line.

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

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

• “¿Qué representa el espacio que hay entre dos marcas consecutivas?” // “What does the space between any two tick marks represent?” (A third) “¿Cómo lo saben?” // “How do you know?” (If five spaces represent 5 thirds and the spaces are the same size, then each space is 1 third.)
• “¿En qué lugar de la recta numérica está el 1?” // “Where is 1 on the number line?” (The third tick mark from 0) “¿En qué lugar está el 2?” // “Where is 2?” (The sixth tick mark from 0, or 1 tick mark to the right of \(\frac{5}{3}\))
• “Hoy vamos a usar rectas numéricas para ayudarnos a razonar sobre sumas de fracciones” // “Today we’ll use number lines to help us reason about sums of fractions.”

This activity prompts students to use number lines to illustrate the decomposition of a fraction into sums of other fractions, reinforcing their work from an earlier lesson. Along the way, students recognize that one way to decompose a fraction greater than 1 is to write it as a sum of a whole number and a fraction less than 1. This insight prepares students to interpret and write mixed numbers in later activities.

• Display or draw this number line:

  

  

• “¿Qué número describe el punto?” // “What number does the point describe?” (8)
• “¿Qué piensan que representan los ‘saltos’?” // “What do you think the ‘jumps’ represent?” (The numbers that are put together to go from 0 to 8)
• “¿Qué ecuaciones podemos escribir para representar la combinación de saltos?” // “What equations can we write to represent the combination of jumps?” (\(6 + 2 =8\) or \(2 + 6 = 8\))
• “Examinemos los saltos en otras rectas numéricas y tratemos de entender lo que pueden representar” // “Let’s look at the jumps on some other number lines and see what they may represent.”

• Groups of 2
• “Durante unos minutos, trabajen individualmente en la actividad. Después, compartan sus respuestas con su pareja” // “Work independently on the activity for a few minutes. Afterwards, share your responses with your partner.”
• 5–7 minutes: independent work time
• 2 minutes: partner discussion

Students may not make a connection between the fractions in the problems and the number lines because the latter show no fractional labels. Consider asking students to label the number line to show fractions or asking, “¿En qué lugar de esta recta numérica piensas que estaría marcado \(\frac{1}{6}\)?” // “Where do you think \(\frac{1}{6}\) would be labeled on this number line?”

• Invite students to share their equations. Record them for all to see.
• Focus the discussion on part b: writing \(\frac{7}{3}\) as a sum of a whole number and a fraction. Students are likely to write \(1 + \frac{4}{3}\) and \(2 + \frac{1}{3}\).
• Explain that \(2 + \frac{1}{3}\) can be written as \(2\frac{1}{3}\), which we call a mixed number. This number is equivalent to \(\frac{7}{3}\).
• “¿Por qué creen que este se llama un número mixto?” // “Why might it be called a mixed number?” (It is a mix of a whole number and a fraction.)

In this activity, students use number lines to represent addition of two fractions and to find the value of the sum. The addends include fractions greater than 1, which can be expressed as a sum of a whole number and a fraction. Students practice constructing a logical argument and critiquing the reasoning of others when they explain which of the strategies they agree with and why (MP3).

• Groups of 2
• Draw students’ attention to the four addition expressions in the first problem.
• “¿Qué observan sobre los números? Hagan algunas observaciones” // “What do you notice about the numbers? Make some observations.” (Sample responses:
  • They all have 8 for the denominator but different numbers for the numerator.
  • Some fractions are less than 1 and others are greater than 1.
  • There is one mixed number.)
• “¿Qué observan sobre las rectas numéricas?” // “What do you notice about the number lines?” (They are identical. They are all partitioned into eighths.)

• “Trabajen individualmente de 5 a 7 minutos. Luego, discutan sus respuestas con su pareja” // “Take 5–7 minutes to work on the task independently. Then, discuss your responses with your partner.”
• 5–7 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for students who can explain why Priya, Kiran, and Tyler might each be correct.

Students may not make a connection between the fractions in the problems and the number lines because the latter show no fractional labels. Consider asking: “¿Qué crees que representan los espacios que hay entre dos marcas consecutivas?” // “What do you think the spaces between the tick marks represent?” and “¿En qué lugar de la recta numérica estaría \(\frac{1}{2}\)?” // “Where would \(\frac{1}{2}\) be on the number line?” Encourage students to label every tick mark or every other one as eighths, including those that represent whole numbers or benchmarks such as \(\frac{1}{2}\) and \(1\frac{1}{2}\).

• Invite students to share their responses to the first problem.
• “Miren las sumas que encontraron. ¿Qué observan acerca de los números de cada suma? ¿Cómo se relacionan con los números de las fracciones que se están sumando?” // “Look at the sums you found. What do you notice about the numbers in each sum? How do they relate to the numbers in the fractions being added?” (The denominator of the sum is 8. The numerator of the sum is the result of adding the numerators of the addends.)
• Select previously identified students to share their responses to the second problem.
• If no students use number lines to make their case, consider sketching or displaying these diagrams.

  

  

  

  

  

  

• Point out that although it is true that \(\frac{11}{5} = 1 + \frac{6}{5}\), we don’t usually write \(1\frac{6}{5}\) as the mixed number equivalent to \(\frac{11}{5}\). Because we can make another 1 whole with \(\frac{6}{5}\), or \(1+1+\frac{1}{5}\) , we'd instead write \(2\frac{1}{5}\).

This optional activity gives students an additional opportunity to practice using number lines to decompose fractions into sums of other fractions and to record the decompositions as equations.

The fractions on the cards (shown here) contain no whole numbers or mixed numbers, but some students may use them to help them find the second addend (and to avoid counting tick marks on the number line). Some may also choose to label each number line with whole numbers beyond 1 to facilitate their reasoning and equation writing. 

\(\frac{1}{3} \qquad \frac{2}{3} \qquad \frac{3}{3} \qquad \frac{4}{3} \qquad \frac{5}{3} \qquad \frac{6}{3} \qquad \frac{7}{3} \qquad \frac{8}{3}\)

• Create a set of cards from the blackline master for each group of 2.

• Groups of 2
• Give each group a set of fraction cards from the blackline master.
• Display the four number lines.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 30 seconds: quiet think time
• 30 seconds: partner discussion

• “Marquen cada punto de la recta numérica con una fracción que esté representada por el punto. Este punto va a ser su objetivo” // “Label each point on the number line with a fraction it represents.”
• 1–2 minutes: independent work time
• “Van a dar dos saltos en la recta numérica para ir desde 0 hasta el objetivo. Van a escribir una ecuación que represente sus saltos” // “You will make two jumps on the number line to go from 0 to the point and write an equation to represent your moves.”
• Explain how to use the cards and how to complete the task. Consider demonstrating with an example and allowing students to ask clarifying questions before they begin.
• Monitor for students who:
  • use whole numbers and mixed numbers in their equations and those who don’t
  • label their number lines with whole numbers beyond 1

• Select previously identified students to share their responses to the first couple of diagrams (or more if time permits). Start with students who used no whole numbers or mixed numbers. Ask them to explain why they chose to write the numbers the way they did.
• Consider discussing the merits and challenges (if any) of expressing the fractions as whole or mixed numbers.