Lesson 15

The purpose of this warm-up is to elicit the idea that the size and the number of unit fractions can help us compare fractions. Students can see that the two diagrams have same-size parts but not how much of one diagram is shaded, prompting them to think about the number of shaded parts. While students may notice and wonder many things about these images, what fractions could be represented by the partially hidden strip is the important discussion point.

• 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.

• “¿Cuántas partes podrían estar sombreadas en la tira de arriba? ¿Podría estar sombreado menos de \(\frac{3}{4}\)? ¿Podría estar sombreado más de \(\frac{3}{4}\)?” // “How many parts could be shaded on the top strip? Could less than \(\frac{3}{4}\) be shaded? Could more than \(\frac{3}{4}\) be shaded?” (If 2 parts are shaded, that's \(\frac{2}{4}\), which is less than \(\frac{3}{4}\). If 3 parts are shaded, that's \(\frac{3}{4}\). If the whole strip is shaded, that's \(\frac{4}{4}\), which is more than \(\frac{3}{4}\).)

The purpose of this activity is for students to compare two fractions with the same denominator. Students may use any representation to reason about how the size or length of the parts in the two fractions are the same because the denominator is the same, but that there are different numbers of those parts because the numerator is different (MP2). Students are also reminded about the meaning of the symbols > and <.

• Groups of 2
• Display the first problem.
• “Tómense unos minutos para decidir cuál fracción es mayor en cada una de estas parejas” // “Take a few minutes to decide which fraction is greater for each of these pairs.”
• 2–3 minutes: independent work time
• “Compartan sus ideas con su compañero” // “Share your ideas with your partner.”
• 1–2 minutes: partner discussion
• Share and record responses.
• Display: < and >
• “Refresquemos nuestra memoria sobre los símbolos ‘menor que’ y ‘mayor que’. ¿Cómo se leen estos símbolos?” // “Let’s refresh our memories about ‘less than’ and ‘greater than’ symbols. How do we read these symbols?” (“is less than” and “is greater than”)
• Display: \(\frac{1}{2} < 1\) and \(\frac{5}{4} > 1\)
• “¿Cómo se leen estas afirmaciones?” // “How do we read these statements?” (One-half is less than 1. Five-fourths is greater than 1.)
• “Usando estos símbolos, ¿qué expresiones podemos escribir acerca de \(\frac{1}{2}\) y \(\frac{3}{2}\) y acerca de \(\frac{2}{8}\) y \(\frac{3}{8}\)?” // “What expressions could you write about \(\frac{1}{2}\) and \(\frac{3}{2}\) and \(\frac{2}{8}\) and \(\frac{3}{8}\) using these symbols?” (\(\frac{1}{2} < \frac{3}{2}\), \(\frac{3}{2} > \frac{1}{2}\), \(\frac{2}{8} < \frac{3}{8}\), \(\frac{3}{8} > \frac{2}{8}\))

• Share and display responses. Ask students to read aloud each statement that is shared.

• “Con su compañero, comparen las fracciones del siguiente problema y usen estos símbolos. Asegúrense de explicar o mostrar cómo razonaron” // “Work with your partner to compare the fractions in the next problem and use these symbols. Be sure to explain or show your reasoning.”
• 5–7 minutes: partner work time
• Monitor for students who explain their reasoning with:
  • an area diagram
  • a fraction strip diagram
  • a number line
  • written description about the size or length of parts

• Select previously identified students to share different representations for comparing fractions with the same denominator.
• Consider asking: “¿En qué se parecen estas representaciones? ¿En qué son diferentes?” // “How are these representations alike? How are they different?”

The purpose of this activity is for students to practice comparing fractions with the same denominator while playing a game. Students spin a spinner for the numerator of their fractions and then locate and label the fractions on a number line to determine which fraction is greater.

• Each group of 2 needs a paper clip for their spinner.

• Groups of 2
• Give each group a paper clip, colored pencils, a spinner, and a sheet of number lines.
• “Ahora van a jugar un juego en el que comparan fracciones que tienen el mismo denominador. Para empezar, un jugador escogerá un denominador para la primera ronda” // “Now you will play a game in which you compare fractions with the same denominator. To start, one player will choose a denominator for the first round.”
• “Cada uno va a girar la ruleta para obtener el numerador de su propia fracción (que tiene el denominador que ya escogieron). Después, van a ubicar y marcar sus fracciones en la misma recta numérica y van a decidir cuál fracción es mayor” //  “You will each spin for the numerator of your own fraction with that denominator. Then you’ll locate and label your fractions on the same number line and determine whose fraction is greater.”
• “Si su fracción es mayor, pueden escoger el denominador de la fracción de la siguiente ronda. El objetivo del juego es ganar todas las rondas que puedan” // “If your fraction is greater, you get to choose the denominator for the next round. The goal of the game is to win as many rounds as you can.”
• Ask the class to choose a denominator (2, 3, 4, 6, or 8). Then spin the spinner and discuss how to represent the fraction on a number line on the recording sheet.

• 10 minutes: partner work time
• As students work, monitor for students who notice patterns as they play.

• “¿Qué tipo de número querían sacar en su turno? ¿Por qué?” // “What kind of number did you want to spin on your turn? Why?” (I wanted to spin a large number because a large numerator means a greater fraction. If you spin a small number, then the small numerator makes a smaller fraction.)