The purpose of this activity is for students to analyze pairs of fractions to determine if they are equivalent. Students may use any representation that makes sense to them. 

Monitor for and select to share in the Activity Synthesis students who used the following approaches to determine if  and are equivalent:

• Show that the fractions are not the same size, by drawing a diagram or by using fraction strips.
• Show that the fractions are not at the same location on the number line.
• Reason that since both fractions represent a different number of sixths, they cannot be equivalent.

The approaches are sequenced from more concrete to more abstract reasoning to help students make sense of the different ways they may compare fractions with the same numerator, or the same denominator, throughout the section. Monitor for the same approaches as students determine if  and  are equivalent, and invite students to connect and compare their representations if there is time. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven't shared recently.

• Invite previously selected students to share in the given order. Record or display their work for all to see.
• Connect students’ approaches to the learning goal by asking:
  • “Estas son maneras distintas de decidir si  y  son equivalentes, ¿pero en qué se parecen?” // “These are different ways of thinking about whether  and  are equivalent, but how are they the same?” (All show they are not equivalent. The diagram and the number line show both fractions.)
  • “¿Cómo se mostró en cada representación que  y  no son equivalentes?” // “How did each representation show that and are not equivalent?” (In the diagram, you can see that has more space shaded. On the number line, they are not at the same location.)
• If there is additional time, invite previously selected students to share their approaches for  and .

The purpose of this activity is for students to recognize that fraction comparisons are valid only when they refer to the same-size whole. Previously, students analyzed pairs of fractions to determine if they were equivalent. In doing so, they likely used comparison language, such as “larger than” or “smaller than” or “greater than” or “less than.” In this activity, students encounter a pair of fractions they saw earlier ( and ) and compare them more explicitly. The student work in this activity uses the number line, but this might also come up with student-drawn diagrams.

In order to interpret Lin’s argument that is greater than , students need to articulate the meaning of fractions and highlight the fact that the two wholes Lin is comparing are not equal (MP6).

• Select previously identified students to share how Han and Lin could make different comparison statements for the same fractions.
• “Cuando comparamos fracciones, es importante recordar que esas fracciones deben hacer referencia a una unidad del mismo tamaño” // “It is important to remember when we are comparing fractions that those fractions need to refer to the same-size whole.”
• “Cuando la unidad de 0 a 1 es del mismo tamaño, podemos ver que  es menor que ” // “When the whole from 0 to 1 is the same size, we can see that is less than .”
• “Esto es cierto independientemente de que dibujemos una recta numérica, usemos tiras de fracciones o dibujemos un diagrama” // “This is true whether we are drawing a number line, using fraction strips, or drawing a diagram.”
• Consider asking, “¿Qué entendieron Lin y Han sobre representar fracciones en una recta numérica?” // "What did both Lin and Han understand about representing fractions on a number line?” (The number lines need to be partitioned into equal parts. The denominator tells us the number of parts. The numerator tells us how many parts to count to locate a point. Points farther to the right are greater than those to the left.)