Lesson 3

Patrones en la multiplicación

Warm-up: Conteo grupal: $\frac{1}{4}$ y $\frac{1}{8}$ (10 minutes)

Narrative

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.

Launch

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

Activity

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

Student Response

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Activity Synthesis

  • “¿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.)

Activity 1: Describamos el patrón (15 minutes)

Narrative

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).

Representation: Develop Language and Symbols. Provide students with access to a chart that shows definitions and examples of the terms that will help them articulate the patterns they see, including whole number, fraction, numerator, denominator, unit fraction, and product.
Supports accessibility for: Language, Memory

Launch

  • 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.”

Activity

  • 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

Student Facing

  1. Estas son dos tablas que tienen algunas expresiones. Encuentra el valor de cada expresión. Si te ayuda, usa un diagrama.

    Por ahora, deja las dos últimas filas de cada tabla en blanco.

    Conjunto A

    expresión valor
    \(1 \times \frac{1}{8}\)
    \(2 \times \frac{1}{8}\)
    \(3 \times \frac{1}{8}\)
    \(4 \times \frac{1}{8}\)
    \(5 \times \frac{1}{8}\)
    \(6 \times \frac{1}{8}\)

    Conjunto B

    expresión valor
    \(2 \times \frac{1}{3}\)
    \(2 \times \frac{1}{4}\)
    \(2 \times \frac{1}{5}\)
    \(2 \times \frac{1}{6}\)
    \(2 \times \frac{1}{7}\)
    \(2 \times \frac{1}{8}\)

  2. Estudia las tablas que ya completaste. ¿Qué patrones ves en la forma como se relacionan las expresiones y sus valores?
  3. Escribe \(\frac{11}{8}\) y \(\frac{13}{8}\) en las dos últimas filas de la tabla del conjunto A, en la columna de “valor”. Escribe las expresiones que tienen esos valores.
  4. Escribe \(\frac{2}{12}\) y \(\frac{2}{15}\) en las dos últimas filas de la tabla del conjunto B, en la columna de “valor”. Escribe las expresiones que tienen esos valores.

Student Response

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Activity Synthesis

  • “¿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?” (\(\frac{3}{10} = 3\times \frac{1}{10}\))

Activity 2: ¿Qué hace falta? (20 minutes)

Narrative

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).

MLR8 Discussion Supports. Display sentence frames to support small-group discussion after checking their fraction diagram and equation: “Estoy de acuerdo porque . . .” // “I agree because . . .”, “Estoy en desacuerdo porque . . .” // “I disagree because . . . .”
Advances: Conversing

Required Materials

Materials to Gather

Launch

  • 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.”

Activity

  • 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

Student Facing

  1. Completa cada ecuación para que sea verdadera, usando los patrones que observaste antes.

    1. \(5 \times \frac{1}{10} =  \underline{\hspace{0.5in}}\)

    2. \(8 \times \frac{1}{6} = \underline{\hspace{0.5in}}\)

    3. \(4 \times  \underline{\hspace{0.5in}} = \frac{4}{5}\)

    4. \(6 \times  \underline{\hspace{0.5in}} = \frac{6}{10}\)

    5. \( \underline{\hspace{0.5in}} \times \frac{1}{4}= \frac{3}{4}\)

    6. \( \underline{\hspace{0.5in}} \times \frac{1}{12}= \frac{7}{12}\)

  2. Vas a trabajar en un grupo de 3. Tu profesor le va a dar una hoja a cada uno. Completa el paso 1 y pásale tu hoja al compañero que está a tu derecha. Completa el paso 2 en la hoja que recibas y pásale tu hoja al compañero que está a tu derecha. Y así, hasta completar todos los pasos.

    • Paso 1: Escribe una fracción que tenga un numerador distinto de 1 y un denominador que no sea mayor que 12. 
    • Paso 2: Escribe la fracción que recibiste como el producto de un número entero por una fracción unitaria. 
    • Paso 3: Dibuja un diagrama que represente la expresión que recibiste.
    • Paso 4: Recoge tu hoja original. Si crees que el trabajo es correcto, explica por qué la expresión y el diagrama representan la fracción que tú escribiste. Si no la representan, discutan qué ajustes deben hacer.

Student Response

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Activity Synthesis

  • See lesson synthesis.

Lesson Synthesis

Lesson Synthesis

“Hoy vimos dos conjuntos de expresiones de multiplicación. En la primera, el número de grupos cambiaba, mientras que la fracción unitaria era la misma. Encontramos un patrón en los valores de las expresiones” // “Today we looked at two sets of multiplication expressions. In the first set, the number of groups changed while the unit fraction stayed the same. We found a pattern in their values.”

“Después, vimos expresiones en las que la fracción unitaria cambiaba y el número de grupos era el mismo. También allí encontramos un patrón” // “Then we looked at expressions in which the unit fraction changed and the number of groups stayed the same. We found a pattern there as well.”

Display the two tables that students completed in the first activity.

“En la primera tabla, ¿por qué tiene sentido que el numerador del producto sea el mismo número que el factor entero?” // “In the first table, why does it make sense that the numerator in the product is the same number as the whole-number factor?” (Because there are as many groups of \(\frac{1}{8}\) as the whole-number factor)

“En la segunda tabla, ¿por qué tiene sentido que el numerador del producto siempre sea 2?” // “In the second table, why does it make sense that the numerator in the product is always 2?” (Because all the expressions represent 2 groups of a unit fraction.)

“También discutimos cómo podemos escribir cualquier fracción como el producto de un número entero por una fracción unitaria. Díganle a un compañero cómo se puede escribir \(\frac{8}{3}\) como el producto de un número entero por una fracción” // “We also discussed how we could write any fraction as a product of a whole number and unit fraction. Tell a partner about how we could write \(\frac{8}{3}\) as a product of a whole number and a fraction.” (​​\(\frac{8}{3} =8 \times \frac{1}{3}\))

Cool-down: Multiplica por una fracción (5 minutes)

Cool-Down

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