Lesson 16

The purpose of this True or False is to elicit insights students have about comparing unit fractions. The reasoning students do helps to deepen their understanding of what the denominator of a fraction means. It will also be helpful later when students compare fractions with the same numerator.

In this activity, students have an opportunity to look for and make use of structure (MP7) because they notice that a larger denominator indicates that a whole is split into more parts. The more parts the whole is split into, the smaller those parts will be.

• Display one statement.
• “Hagan una señal cuando sepan si la afirmación es verdadera o no, y puedan explicar cómo lo saben” //  “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

• Share and record answers and strategy.
• Repeat with each statement.

• Consider asking:
  • “¿Alguien puede expresar el razonamiento de _____ de otra forma?” // “Who can restate _____’s reasoning in a different way?”
  • “¿Alguien quiere agregar algo al razonamiento de _____?” // “Does anyone want to add on to _____’s reasoning?”

The purpose of this activity is for students to represent their thinking visually as they compare pairs of fractions with the same numerator. They also locate fractions with the same numerator on number lines and observe the relative locations of the points. Students see that fractions with larger denominator are smaller in size (or are closer to 0 on the number line). Their reasoning here reinforces the idea that the denominator of a fraction determines how many equal parts are in a whole, and that the more parts there are, the smaller each part is (MP7).

To compare \(\frac{5}{6}\) and \(\frac{5}{8}\), students are likely to draw one diagram or number line for sixths and a separate one for eighths. They may use a single diagram or number line, but find it more difficult to partition and represent both denominators.

• Groups of 2
• “Priya y Tyler comparan dos fracciones. Lean sus conclusiones y decidan con quién están de acuerdo” // “Priya and Tyler are comparing two fractions. Read their conclusions and decide who you agree with.”
• 2 minutes: independent work time

• “Hablen con su compañero sobre con quién están de acuerdo. Usen diagramas o rectas numéricas para mostrar cómo pensaron” // “Talk to your partner about who you agree with. Use diagrams or number lines to show your thinking.”
• 3–5 minutes: partner discussion
• As students work, consider asking:
  • “¿Cómo muestra su representación cuál fracción es mayor?” // “How does your representation show which fraction is greater?”
  • “¿Cómo saben que los octavos son más pequeños que los sextos?” // “How do you know that eighths are smaller than sixths?”
• Monitor for students who use diagrams and those who use number lines.
• Pause for a discussion.
• Select two students, one who uses each representation, to share. Display their work side-by-side for all to see.
• “¿Cómo muestran estas dos representaciones que \(\frac{5}{6}\) es mayor que \(\frac{5}{8}\)?” // “How do these representations both show that \(\frac{5}{6}\) is greater than \(\frac{5}{8}\)?” (Both show that we are looking at 5 parts in each fraction and that sixths are larger than eighths. That means that 5 sixths are larger than 5 eighths.)
• “Ahora hagan los últimos dos problemas” // “Now complete the last two problems.”
• 5–7 minutes: independent or partner work time

If students are not sure how to get started, consider asking: “¿Qué sabemos sobre estas fracciones?” // “What do we know about these fractions?” and “¿Cómo podemos representar estas fracciones?” // “How could we represent these fractions?”

• Display a set of number lines that a student completed.
• Invite the class to share their observations about the locations of the points.
• “¿Cómo puede ayudar la ubicación de los puntos a decidir cuál es mayor, \(\frac{5}{3}\) o \(\frac{5}{4}\)?” // “How can the locations of the points help you decide which is greater, \(\frac{5}{3}\) or \(\frac{5}{4}\)?” (I can see that \(\frac{5}{3}\) is located to the right of \(\frac{5}{4}\), so \(\frac{5}{3}\) is greater.)
• Consider asking: “¿Por qué la fracción con el denominador más pequeño, 2, es la fracción más grande del grupo?” // “Why is the fraction with the smallest denominator, 2, the greatest fraction in the set?” (There are only 2 parts in 1 whole. With every other fraction the denominator is larger than 2 so there are more parts in the whole which makes the parts smaller.)

The purpose of this activity is for students to compare two fractions with the same numerator. Students use any representation to reason about the number and size of the parts of each fraction. In the last problem, students may notice that more than one denominator can sometimes work, but do not need to generalize about all the denominators that make each statement true. There is an opportunity for that kind of generalization in a future lesson.

• Groups of 2
• “Acabamos de ver varias formas en las que podemos representar comparaciones de fracciones que tienen el mismo numerador. Cuando comparen fracciones que tienen el mismo numerador en esta actividad, pueden usar diagramas o rectas numéricas o pueden escribir para explicar su razonamiento” // “We just saw several ways we could represent comparisons of fractions with the same numerator. As you compare fractions with the same numerator in this activity, you can use diagrams or number lines or write to explain your reasoning.”

• 8–10 minutes: independent work time
• “Compartan con su compañero cuál es su forma favorita de representar su razonamiento” // “Share your favorite way to represent your reasoning with your partner.”
• 3–5 minutes: partner discussion
• Monitor for students who choose different denominators in the last problem.

If students don’t determine which fraction is greater, consider asking: “¿Qué sabemos sobre estas fracciones?” // “What do we know about these fractions?” and “¿Cómo podemos representar estas fracciones?” // “How could we represent these fractions?”

• Select 3–4 students to share the denominators they chose for one part in the last problem. Ask them to explain why their chosen denominator makes the statement true.