Lesson 10

The purpose of this Choral Count is to invite students to practice counting by \(\frac{1}{2}\) and notice patterns in the count. These understandings help students develop fluency and will be helpful later in this lesson when students recognize and generate equivalent fractions. In the synthesis, students have the opportunity to notice that \(\frac{2}{2}\) and \(\frac{4}{4}\) are both equal to 1 whole.

• Have recording of choral count by one-fourth available, from a previous lesson.

• “Cuenten de \(\frac{1}{2}\) en \(\frac{1}{2}\), empezando en \(\frac{1}{2}\)” // “Count by \(\frac{1}{2}\), starting at \(\frac{1}{2}\).”
• Record as students count. Record 2 fractions in each row, then start a new row. There will be 4 rows.
• Stop counting and recording at \(\frac{8}{2}\).

• “¿Qué patrones ven?” // “What patterns do you see?”
• 1–2 minutes: quiet think time
• Record responses.

• Display count by \(\frac{1}{4}\) from the previous lesson. There should be 4 rows and 4 fractions in each row with the count ending at \(\frac{16}{4}\).
• “¿En qué se parecen estos dos conteos? ¿En qué son diferentes?” // “How are these two counts the same? How are they different?” (The denominator stays the same in both counts—4 for the last count, and 2 for today’s count. The numerators change in the same way because they both count by one. They start a new line at \(\frac{2}{2}\) and \(\frac{4}{4}\), which are both whole numbers.)
• Consider asking:
  • “¿Quién puede describir el patrón con otras palabras?” // “Who can restate the pattern in different words?”
  • “¿Alguien quiere compartir otra observación sobre por qué ocurre ese patrón aquí?” // “Does anyone want to add an observation as to why that pattern is happening here?”
  • “¿Están de acuerdo o en desacuerdo? ¿Por qué?” // “Do you agree or disagree? Why?”

The purpose of this activity is for students to consider equivalent fractions using diagrams. One half has been chosen to introduce equivalent fractions because there are many ways to see and represent fractions that are equivalent to \(\frac{1}{2}\). Many students may be familiar with the concept of halves and justify equivalence by saying 2 is half of 4. This reasoning is helpful with 1 half and 2 fourths but may not be generalizable to other cases of equivalence. For this reason, the activity synthesis focuses on justifications about whether or not the shaded parts are the same size. The idea that \(\frac{1}{2}\) and \(\frac{2}{4}\) are the same size is used to define equivalent fractions as fractions that are the same size.

Students need to use language carefully as they explain why the shaded parts of a shape show \(\frac{1}{2}\) (MP6). For example, they may say that 2 of 4 equal parts in shape D are shaded, but if they combine those parts, the total shaded amount is the same as in the shape where 1 of 2 equal parts is shaded.

• Groups of 2
• “¿Qué saben sobre \(\frac{1}{2}\)?” // “What do you know about \(\frac{1}{2}\)?” (There are 2 equal parts. The parts have to be the same size. One of the parts would be shaded.)
• 1 minute: quiet think time
• Share and record responses.

• “Ahora, con su compañero, seleccionen todas las figuras en las que la sección sombreada representa \(\frac{1}{2}\) de la figura. Luego, expliquen cómo puede haber más de una figura en la que esto sucede” //  “Now work with your partner to select all the shapes where the shaded portion represents \(\frac{1}{2}\) of the shape and explain how there are more than one shape where this is the case.”
• 5–7 minutes: partner work time
• Monitor for students who explain that the shading in A and D both represents \(\frac{1}{2}\) of the shape.

• Invite students to share their responses.
• Display C and D.
• “¿Cómo es posible que la sección sombreada de cada uno muestre \(\frac{1}{2}\) si los cuadrados se partieron en un número diferente de partes iguales?” // “How can the shaded portion in each show \(\frac{1}{2}\) when the squares have been partitioned into a different number of equal parts?” (The shaded part is the same size even though they look different. The same amount of the square is shaded.)
• “Aunque C está partido en medios y D está partido en cuartos, podemos decir que está sombreado \(\frac{1}{2}\) de cada cuadrado porque está sombreada la misma cantidad en los cuadrados C y D, lo que significa que las dos fracciones tienen el mismo tamaño” // “Even though C is partitioned into halves and D is partitioned into fourths, we can say that \(\frac{1}{2}\) of each square is shaded because the same amount is shaded as in squares C and D, which means the two fractions are the same size.”
• “Dos números que tienen el mismo tamaño son equivalentes, así que las fracciones \(\frac{2}{4}\) y \(\frac{1}{2}\) son fracciones equivalentes” // “Two numbers that are the same size are equivalent, so the fractions \(\frac{2}{4}\) and \(\frac{1}{2}\) are equivalent fractions.”

The purpose of this activity is for students to use fraction strips to identify equivalent fractions and explain why they are equivalent. Highlight explanations that make clear that the parts that represent the fractions are the same size and the parts of the fractions refer to the same whole.

• Students need the fraction strips they made in a previous lesson.

• Groups of 2
• Ask students to refer to the fraction strips they made in an earlier lesson.

• “Encuentren tantas fracciones como puedan que sean equivalentes a las fracciones de la lista. Para lograrlo, usen sus tiras de fracciones” // “Use your fraction strips to find as many fractions as you can that are equivalent to the listed fractions.”
• 5–7 minutes: independent work time
• If students have extra time, encourage them to use their fraction strips to find other pairs of fractions that are equivalent.
• “Ahora compartan con su compañero las fracciones equivalentes que encontraron. Asegúrense de compartir cómo razonaron” //  “Now, share the equivalent fractions you found with your partner. Be sure to share your reasoning.”
• 3–5 minutes: partner discussion
• Monitor for students who explain equivalence by saying that the fractions are the same size.

If students don’t generate an equivalent fraction for one of the given fractions, consider asking:

• “¿Cómo representaste la fracción con las tiras de fracciones?” // “How did you represent the fraction with the fraction strips?”
• “¿Cómo puedes usar las tiras de fracciones para hacer una fracción equivalente?” // “How could you use the fraction strips to make an equivalent fraction?”

• Invite students to share pairs of equivalent fractions and why they are equivalent. Highlight that the fractions are equivalent because the part of the strips that represent the fractions are the same size.
• Display a set of fraction strip diagram for all to see.
• As students share, mark up the fraction strip diagram to illustrate the equal size of the parts (for example, by drawing lines or circling the parts). Then, record pairs of equivalent fractions using the equal sign like: \(\frac{1}{2} = \frac{3}{6}\).