In an earlier lesson, students saw that whole numbers could be written as fractions. The purpose of this activity is for students to recognize fractions that are equivalent to whole numbers, using patterns in number lines to support their reasoning. To identify fractions that are equivalent to whole numbers on number lines, students may:

• Use what they know about 2 halves, 3 thirds, and 4 fourths to identify 1, and then circle fractions at the same intervals down each number line.
• Use what they know about 2 halves, 4 halves, and 6 halves to identify 1, 2, and 3 on the first number line, and then circle fractions in the same locations on the other number lines.
• Use the relationship between parts and wholes (for instance, 3 thirds make 1, 6 thirds make 2, and 9 thirds make 3).

Students then record equations that show fractions that are equivalent to whole numbers. Finally, given a list of fractions, students determine which are equivalent to whole numbers.

When students use patterns to identify fractions that are equivalent to whole numbers, they look for and express regularity in repeated reasoning (MP8).

• Invite students to share their responses and their reasoning for the set of fractions in the last problem.
• As students share, record fractions that are equivalent to whole numbers as equations, and highlight that the equations can be written with the fraction first or the whole number first (for example, or ).
• “¿Cómo supieron si cada fracción era equivalente a un número entero?” // “How did you know if each fraction was equivalent to a whole number?”
• “¿Qué patrones ven que pueden ayudar a encontrar más fracciones que son números enteros?” // “What patterns do you see that could be helpful for finding more fractions that are whole numbers?”
• Highlight strategies based on an understanding of the number of equal parts that are in 1 whole and on observed patterns. (For instance, the thirds that are equivalent to whole numbers have a numerator that is a number we get when counting by 3.)

The purpose of this activity is for students to write whole numbers as fractions. Students may reason in any way that makes sense to them, which includes the use of patterns they noticed previously. When students observe patterns as they write whole numbers as fractions, they look for and make use of structure (MP7).

This activity uses a “carousel” structure in which students complete a rotation of tasks. Consider demonstrating the steps before students begin.

MLR8 Discussion Supports. Synthesis: During group presentations, invite the students who are not speaking to follow along and point to the corresponding parts of the display.
Advances: Speaking, Representing

• “Dime cómo escribirías 1 como una fracción” // “Tell me how you would write 1 as a fraction.”
• “¿Cómo puedes usar la fracción que es equivalente a 1 para escribir este número como una fracción?” // “How could you use the fraction that's equivalent to 1 to write this number as a fraction?”

• Invite a group of students to display their completed table.
• Select previously identified students to share how they knew what fractions to write for each whole number.
• Invite the class to share other strategies for generating equivalent fractions.
• “¿Qué patrones nuevos observan en la tabla que ya completaron?” // “What new patterns do you notice in the completed table?” (In fractions that are equivalent to 4, the numerator can be found by skip-counting by the denominator 4 times. In the row with thirds, the numerator increases by 3. In the row for fourths, it increases by 4.)