Lesson 13

The purpose of this warm-up is to draw students’ attention to inequality statements. It reminds them of the meaning of inequality symbols and how to read the statements, which will be useful when students compare fractions later in the lesson. The warm-up also elicits observations that an equation or inequality can be true or false. While students may notice and wonder many things, highlight observations about comparison and about the meaning of the symbols and statements.

• Groups of 2
• Display the four statements.
• “¿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 dice cada afirmación?” // “What does each statement say?”
• “¿Cuáles de estas afirmaciones son verdaderas? ¿Cuáles no?” // “Which of these statements are true? Which ones are not?” (The first and last are true. The second and third are false.)
• “¿Por qué son falsas?” // “Why are they false?” (\(\frac{9}{2}\) is equal to, not greater than, 4 wholes and \(\frac{1}{2}\). Four is greater than \(\frac{3}{2}\).)

Previously, students classified fractions based on their relationship to \(\frac{1}{2}\) and 1 (whether they are less than or more than these benchmarks). They used these classifications to compare fractions. In this activity, students are presented with fractions that are in the same group (for example, both less than \(\frac{1}{2}\), or both greater than \(\frac{1}{2}\) but less than 1), so they need to reason in other ways to make comparisons.

Students can reason in a number of ways—by thinking about size and number of parts, drawing a diagram or number line, or reasoning numerically, but in most cases, they need to also rely on the idea of equivalence.

• Groups of 2
• “Estas son unas fracciones que ya habíamos organizado en una lección anterior. Las comparamos con \( \frac{1}{2}\) y con 1.” // "Here are some fractions you’ve sorted in an earlier lesson. We compared them to \( \frac{1}{2}\) and 1.”
• “¿Qué tienen en común las fracciones del grupo 3? ¿Por qué creen que están en el mismo grupo?” // “What do the fractions in group 3 have in common? Why might they be in the same group?” (They are all greater than 1.)
• “¿En qué son diferentes las fracciones del grupo 1 a las fracciones del grupo 2?” // “How are the fractions in group 1 different than those in group 2?” (Those in group 1 are less than \( \frac{1}{2}\), and those in group 2 greater than \( \frac{1}{2}\) but less than 1.)
• “Podemos darnos cuenta de que las fracciones del grupo 2 son mayores que las del grupo 1 y que las fracciones del grupo 3 son mayores que las de los otros dos grupos” // “We can tell that the fractions in group 2 are greater than those in group 1, and the fractions in group 3 are greater than those in the other groups.”
• “Ahora comparen las fracciones dentro de cada grupo” // “Now compare the fractions in each group.”

• “Tómense unos minutos en silencio para trabajar en los problemas. Después, compartan sus respuestas con su pareja” // “Take a few quiet minutes to work on the problems. Afterward, share your responses with your partner.”
• 7–8 minutes: independent work time
• 5 minutes: partner discussion
• Monitor for students who:
  • reason by drawing number lines or tape diagrams
  • reason about the distance of each fraction from 0, \(\frac{1}{2}\), or 1
  • reason about equivalent fractions (even if they don’t write out the multiplication numerically)
  • reason about equivalent fractions numerically by writing out the multiplication

• Select students who used different strategies to share their responses. 
• “¿Cómo comparamos dos fracciones que están en el mismo grupo (por ejemplo, dos menores que \( \frac{1}{2}\) o dos mayores que 1)?” // “How do we compare two fractions that are in the same group—say, both less than \( \frac{1}{2}\) or both greater than 1?” (We can think about how close or far away from \( \frac{1}{2}\) each fraction is.)
• Highlight how equivalent fractions came into play in each strategy. For example, ask, “Cuando comparamos \(\frac{2}{3}\) y \(\frac{7}{12}\), ¿por qué les ayudó pensar en \(\frac{2}{3}\) como \(\frac{8}{12}\)?” // “When comparing \(\frac{2}{3}\) and \(\frac{7}{12}\), why was it helpful to think of the \(\frac{2}{3}\) as \(\frac{8}{12}\)?” Or, “Cuando comparamos \(\frac{7}{10}\) y \(\frac{4}{5}\), ¿por qué pensaron en \(\frac{4}{5}\) como \(\frac{8}{10}\)?” // “When comparing \(\frac{7}{10}\) and \(\frac{4}{5}\), why did you think of \(\frac{4}{5}\) as \(\frac{8}{10}\)?”
• If no students mention that it is often easier to compare two fractions when they have the same denominator, ask them about it.

The purpose of this activity is for students to compare pairs of fractions by writing one or more equivalent fractions. In all pairs of fractions given here, one denominator is a factor or a multiple of the other, which encourages students to convert one into an equivalent fraction with the same denominator as the other fraction. On repeated reasoning, students see that writing an equivalent fraction can facilitate the comparison (though in some cases, students may still find it efficient to reason in other ways).

This is the first time in grade 4 that students use the symbols \(<\) and \(>\) to express comparison, so some supports for reading aloud inequality statements are suggested in the launch.

• Groups of 2
• Read together the four statements in the first question. 
• Consider writing out in words the meaning of the symbols \(<\) and \(>\) (“es mayor que” // “is greater than” and “es menor que” // “is less than”) and display them for students’ reference.

• 7–8 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for students who make comparisons by:
  • using the relationship and distance to benchmark numbers
  • writing an equivalent fraction either by dividing or multiplying the numerator and denominator by a number

• Select students to share their responses and how they reasoned about them.