Lesson 7

The purpose of this Choral Count is to invite students to practice counting by \(\frac{3}{4}\) and notice patterns in the count. The patterns they recognize here will be helpful when students decompose fractions into sums of \(\frac{3}{4}\)s in a problem later in the lesson. The exercise also draws students’ attention to multiples of \(\frac{3}{4}\) that are equivalent to whole numbers, which will be helpful as students compose and decompose mixed numbers.

• “Cuenten de \(\frac{3}{4}\) en \(\frac{3}{4}\), empezando en \(\frac{3}{4}\)” // “Count by \(\frac{3}{4}\), starting at \(\frac{3}{4}\).”
• Record as students count.
• Stop counting and recording at \(\frac{48}{4}\).

• “¿Qué patrones ven?” // “What patterns do you see?” (Sample responses:
  • The numerator is increasing by 3 each time.
  • The numerators are multiples of 3.
  • In every fourth fraction in the list, the numerator is a multiple of 4.)
• 1–2 minutes: quiet think time
• Record responses.

• “¿Cuáles de estas fracciones son equivalentes a números enteros?” // “Which of these fractions are equivalent to whole numbers?” (\(\frac{12}{4}, \frac{24}{4}, \frac{36}{4}, \frac{48}{4}\))
• “¿A qué números enteros son equivalentes?” // “To what whole numbers are they equivalent?” (3, 6, 9, 12)
• “¿Por qué piensan que ocurren estos patrones?” // “Why do you think these patterns are happening?” (The number of parts and the size of each part being added stay the same each time, so the numerator is always increasing by 3 and the denominator always staying the same.)

Previously, students considered non-unit fractions in terms of equal groups of unit fractions or as a product of a unit fraction and a whole number. This activity prompts students to think about non-unit fractions as being sums of other fractions. The given context—about measuring fractional amounts using measuring cups of certain sizes—allows students to continue thinking in terms of equal groups, but also invites them to consider a fractional quantity as a sum of two or more fractions with the same denominator. For instance, students may see \(\frac{5}{4}\) as 5 groups of \(\frac{1}{4}\) or as \(5 \times \frac{1}{4}\), but they may also see that \(\frac{5}{4}\) is equal to \(\frac{3}{4} + \frac{1}{4} + \frac{1}{4}\). Students record such a decomposition as an equation. When students connect the quantities in the story problem to an equation, they reason abstractly and quantitatively (MP2).

Students may not be familiar with the use of measuring cups. Consider demonstrating how to use a 1-cup measuring cup to obtain different whole numbers of cups. 

• Gather \(\frac{1}{4}\)-cup and \(\frac{3}{4}\)-cup measuring cups, if available.

• Groups of 2
• “Hoy vamos a ver la receta de una sopa” // “Today we’ll look at a soup recipe.”
• Ask students to share with a partner:
  • “¿Cuál es su sopa favorita?” // “What is your favorite soup?”
  • “¿Qué ingredientes tiene su sopa favorita?” // “What is in your favorite soup?”
  • “Si estuvieran escribiendo la receta de esta sopa, ¿qué diría la receta?” // “If you were writing a recipe for this soup, what would it say?”
• 2 minutes: partner discussion
• 1 minute: share responses
• “Veamos una receta de sopa de cebada. Un familiar de Lin escribió las cantidades de la receta usando cuartos para que fuera más fácil medirlas” // “Let’s look at a recipe for barley soup. Someone in Lin's family wrote the amounts in the recipe in fourths to make measuring easier.”
• If possible, show examples of uncooked barley and make \(\frac{1}{4}\)-cup and \(\frac{3}{4}\)-cup measuring cups available.

• “Trabajen individualmente durante unos minutos. Luego, compartan sus respuestas con su compañero” // “Work independently on the task for a few minutes. Then, share your responses with your partner.”
• 6–7 minutes: independent work time
• Monitor for students who express the amounts in terms of:
  • number of times a measuring cup is filled
  • products of a unit fraction and a whole number
  • sums of unit fractions
  • sums of unit and non-unit fractions
• 3–4 minutes: partner discussion

If students create drawings to show how they would obtain the correct quantities, ask: “¿Cómo muestran tus dibujos lo que Lin podría hacer para obtener las cantidades correctas?” // “How do your drawings show what Lin could do to get the right amounts?” and “¿Cómo podrías usar expresiones para mostrar la misma información?” // “How could you use expressions to show the same information?”

• Select students to share their responses in the order shown in the activity notes (from informal or concrete to formal or symbolic, from multiplication to addition).
• Record the different ways of expressing the quantities in the recipe.
• Consider drawing diagrams and annotating them to help students relate the expressions and the quantities.
• Highlight that each quantity can be written as a product of a whole number and a unit fraction, but it can also be written as a sum of smaller fractions. For example:
  • Barley: \(3 \times \frac{1}{4}\), or \(\frac{1}{4} + \frac{1}{4} + \frac{1}{4}\)

    

    

  • Broth: \(9 \times \frac{1}{4}\) or \(3 \times \frac{3}{4}\), or \(\frac{3}{4} + \frac{3}{4} + \frac{3}{4}\)

    

    

In the previous activity, students saw that a fraction can be decomposed into a sum of fractions with the same denominator and that it can be done in more than one way. In this activity, they record such decompositions as equations. The last question prompts students to consider whether any fraction can be written as a sum of smaller fractions with the same denominator. Students see that only non-unit fractions (with a numerator greater than 1) can be decomposed that way. Students observe regularity in repeated reasoning as they decompose the numerator, 9, into different parts while the denominator in all cases is 5 (MP8).

• “Antes, vimos varias formas de descomponer fracciones en cuartos y escribirlas como sumas de fracciones más pequeñas” // “Earlier, we saw different ways to decompose fractions in fourths and write them as sums of smaller fractions.”
• “¿Cómo podemos escribir la fracción \(\frac{9}{5}\) como una suma de fracciones unitarias?” // “How can we write the fraction \(\frac{9}{5}\) as a sum of unit fractions?” (\(\frac{9}{5} = \frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5}\))
• “Descompongamos \(\frac{9}{5}\) en sumas de otros quintos y \(\frac{4}{3}\) en sumas de tercios” // “Let’s decompose \(\frac{9}{5}\) into sums of other fifths and \(\frac{4}{3}\) into sums of thirds.“

• “Tómense unos minutos para completar la actividad en silencio. Luego, compartan sus respuestas con su compañero” // “Take a few quiet minutes to complete the activity. Then, share your responses with your partner.”
• 5–6 minutes: independent work time
• 3–4 minutes: partner discussion
• Monitor for different explanations students offer for the last question.

• Invite students to share their equations. Display or record them for all to see.
• Next, discuss students' responses to the last question. Select students with different explanations to share their reasoning.
• If not mentioned by students, highlight that fractions with a numerator of 1 (unit fractions) cannot be further decomposed into smaller fractions with the same denominator because it is already the smallest fractional part. Other fractions with a numerator other than 1 (non-unit fractions) can be decomposed into fractions with the same denominator.