Lesson 3

The purpose of this Choral Count is to invite students to practice counting by a unit fraction and notice patterns in the count. These understandings will be helpful later in this lesson when students recognize every fraction can be written as the product of a whole number and unit fraction.

• “Cuenten de \(\frac{1}{4}\) en \(\frac{1}{4}\), empezando en 0” // “Count by \(\frac{1}{4}\), starting at 0.”

• Record as students count.
• Stop counting and recording at \(\frac{11}{4}\).
• Repeat with \(\frac{1}{8}\).
• Stop counting and recording at \(\frac{15}{8}\).

• “¿Qué patrones observaron?” // “What patterns do you notice?” (In both counts, the numerators go up by 1, and denominators stay the same.)
• “¿Cuántos grupos de \(\frac{1}{4}\) tenemos?” // “How many groups of \(\frac{1}{4}\) do we have?” (11)
• “¿Dónde los ven?” // “Where do you see them?” (Each count represents a new group of \(\frac{1}{4}\).)
• “¿Cómo podemos representar 11 grupos de \(\frac{1}{4}\) con una expresión?” // “How might we represent 11 groups of \(\frac{1}{4}\) with an expression?” (\(11 \times \frac{1}{4}\))
• “¿Cuántos grupos de \(\frac{1}{8}\) tenemos?” // “How many groups of \(\frac{1}{8}\) do we have?” (15)
• “¿Cómo podemos representar 15 grupos de \(\frac{1}{8}\) con una expresión?” // “How might we represent 15 groups of \(\frac{1}{8}\) with an expression?” (\(15 \times \frac{1}{8}\))
• “¿Cómo cambia nuestro conteo si contamos de \(\frac{2}{4}\) en \(\frac{2}{4}\) o si contamos de \(\frac{2}{8}\) en \(\frac{2}{8}\)?” // “How would our count change if we counted by \(\frac{2}{4}\) or \(\frac{2}{8}\)?” (Each numerator would be a multiple of 2 or an even number.)

Students may have previously noticed a connection between the whole number in a given multiplication expression and the numerator of the fraction that is the resulting product. In this activity, they formalize that observation. Students reason repeatedly about the product of a whole number and a unit fraction, observe regularity in the value of the product, and generalize that the numerator in the product is the same as the whole-number factor (MP8).

• Groups of 2
• “Completen las tablas con su pareja. Uno de ustedes debe empezar con el conjunto A y el otro con el conjunto B” // “Work with your partner to complete the tables. One person should start with Set A and the other with Set B.”
• “Después, analicen juntos sus tablas completas y busquen patrones” // “Afterwards, analyze your completed tables together and look for patterns.”

• 5–7 minutes: partner work time on the first two problems
• Monitor for the language students use to explain patterns:
  • The whole number in each expression is only being multiplied by the numerator of each fraction.
  • Language describing patterns in the denominator of the product (The denominator in the product is the same as the unit fraction each time.)
  • “Groups of” language to justify or explain patterns (The number of groups of each unit fraction is going up each time because it is one more group.)
• “Después de que hayan descrito los patrones del segundo problema, hagan una pausa” // “Pause after you've described the patterns in the second problem.”
• Select 1–2 students to share the patterns they observed.
• “Ahora apliquen los patrones que observaron para completar los últimos dos problemas” // “Now let's apply the patterns you noticed to complete the last two problems.” 
• 3 minutes: independent or partner work time

• “¿Cómo usaron los patrones para escribir expresiones para \(\frac{11}{8}, \frac{13}{8}, \frac{2}{12}\) y \(\frac{2}{15}\)?” // “How did you use the patterns to write expressions for \(\frac{11}{8}, \frac{13}{8}, \frac{2}{12}\), and \(\frac{2}{15}\)?” (I knew that each expression had a whole number and a unit fraction. The whole number is the same as the numerator of the product.)
• Select students to share their multiplication expressions for these four fractions.
• “¿Cualquier fracción se puede escribir como una expresión de multiplicación usando su fracción unitaria?” // “Can you write any fraction as a multiplication expression using its unit fraction?” (Yes, because the numerator is the number of groups and the denominator represents the size of each group.)
• “¿Cómo se escribe \(\frac{3}{10}\) como una expresión de multiplicación usando un número entero y una fracción unitaria?” // “What would it look like to write \(\frac{3}{10}\) as a multiplication expression using a whole number and a unit fraction?” (\(\frac{3}{10} = 3\times \frac{1}{10}\))

This activity serves two main purposes. The first is to allow students to apply their understanding that the result of \(a \times \frac{1}{b}\) is \(\frac{a}{b}\). The second is for students to reinforce the idea that any non-unit fraction can be viewed in terms of equal groups of a unit fraction and expressed as a product of a whole number and a unit fraction. 

The activity uses a “carousel” structure in which students complete a rotation of steps. Each student writes a non-unit fraction for their group mates to represent in terms of equal groups, using a diagram, and as a multiplication expression. The author of each fraction then verifies that the representations by others indeed show the written fraction. As students discuss and justify their decisions they create viable arguments and critiqe one another’s reasoning (MP3).

• Groups of 3
• “Ahora usemos los patrones que ya vimos. Usémoslos para escribir ecuaciones verdaderas que muestren un número entero multiplicado por una fracción” // “Let's now use the patterns we saw earlier to write some true equations showing multiplication of a whole number and a fraction.”

• 3 minutes: independent work time on the first set of problems
• 2 minutes: group discussion
• Select students to explain how they reasoned about the missing numbers in the equations.
• If not mentioned in students' explanations, emphasize that: “Podemos interpretar \(\frac{5}{10}\) como 5 grupos de \(\frac{1}{10}\), \(\frac{8}{6}\) como 8 grupos de \(\frac{1}{6}\), etcétera” // “We can interpret \(\frac{5}{10}\) as 5 groups of \(\frac{1}{10}\), \(\frac{8}{6}\) as 8 groups of \(\frac{1}{6}\), and so on.” 
• “En la actividad anterior, aprendimos que podemos escribir cualquier fracción como un número entero multiplicado por una fracción unitaria. Ahora van a mostrar que esto se cumple para las fracciones que escriban los compañeros de su grupo” // “In an earlier activity, we found that we can write any fraction as a multiplication of a whole number and a unit fraction. You'll now show that this is the case using fractions written by your group mates.”
• Demonstrate the 4 steps of the carousel using \(\frac{7}{4}\) for the first step.
• Read each step aloud and complete a practice round as a class.
• “Antes de comenzar, ¿qué preguntas tienen sobre lo que hay que hacer?” // “What questions do you have about the task before you begin?”
• 5–7 minutes: group work time

• See lesson synthesis.