The purpose of this Choral Count is to invite students to practice counting by a unit fraction and to notice patterns in the count. These understandings will be helpful later in this lesson when students recognize that every fraction can be written as the product of a whole number and a unit fraction.
Launch
“Cuenten de en , empezando en 0” // “Count by , starting at 0.”
Activity
Record as students count.
Stop counting and recording at .
Repeat with .
Stop counting and recording at .
None
Student Response
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Advancing Student Thinking
Activity Synthesis
“¿Qué patrones observaron?” // “What patterns do you notice?” (In both counts, the numerators go up by 1, and the denominators stay the same.)
“¿Cuántos grupos de tenemos?” // “How many groups of do we have?” (11)
“¿Dónde los ven?” // “Where do you see them?” (Each count represents a new group of .)
“¿Cómo podemos representar 11 grupos de con una expresión?” // “How might we represent 11 groups of , with an expression?” ()
“¿Cuántos grupos de tenemos?” // “How many groups of do we have?” (15)
“¿Cómo podemos representar 15 grupos de con una expresión?” // “How might we represent 15 groups of , with an expression?” ()
“¿Cómo cambia nuestro conteo si contamos de en o si contamos de en ?” // “How would our count change if we counted by or ?” (Each numerator would be a multiple of 2, or an even number.)
Activity 1
Standards Alignment
Building On
Addressing
4.NF.B.4
Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
Understand a fraction as a multiple of . For example, use a visual fraction model to represent as the product , recording the conclusion by the equation
Students previously may have 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 “número entero”, “fracción”, “numerador”, “denominador”, “fracción unitaria” y “producto” // “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” // “Afterward, 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, such as:
The whole number in each expression is only multiplied by the numerator of each fraction.
The denominator in the product is the same as in the unit fraction each time.
The number of groups of each unit fraction is going up each time because it is 1 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
Estas dos tablas tienen 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
Conjunto B
expresión
valor
Mira las tablas que ya completaste. ¿Qué patrones ves en la forma como se relacionan las expresiones y los valores?
En las dos últimas filas de la tabla del conjunto A, escribe en una fila y en la otra, en la columna de “valor”. Escribe las expresiones que tienen esos valores.
En las dos últimas filas de la tabla del conjunto B, escribe en una fila y en la otra, en la columna de “valor”. Escribe las expresiones que tienen esos valores.
Student Response
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Advancing Student Thinking
Activity Synthesis
“¿Cómo usaron los patrones para escribir expresiones para y ?” // “How did you use the patterns to write expressions for , and ?” (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 que incluya 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 como una expresión de multiplicación de un número entero por una fracción unitaria?” // “What would it look like to write as a multiplication expression, using a whole number and a unit fraction?” ()
Activity 2
Standards Alignment
Building On
Addressing
4.NF.B.4.a
Understand a fraction as a multiple of . For example, use a visual fraction model to represent as the product , recording the conclusion by the equation
This activity serves two main purposes. The first is to allow students to apply their understanding that the result of is . 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 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 critique one another’s reasoning (MP3).
MLR8 Discussion Supports. Display sentence frames to support small-group discussion after checking students’ fraction diagrams and equations. Examples: “Estoy de acuerdo porque . . .” // “I agree because . . . ,” “Estoy en desacuerdo porque . . .” // “I disagree because . . . .”
Advances: Conversing
Launch
Groups of 3
“Ahora usemos los patrones que ya vimos para escribir ecuaciones que muestren un número entero multiplicado por una fracción” // “Let's now use the patterns we saw earlier to write equations that show 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 como 5 grupos de , como 8 grupos de , etcétera” // “We can interpret as 5 groups of , as 8 groups of , 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 escriba 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.”
Demonstrate the 4 steps of the carousel, using 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
Activity Synthesis
See the 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 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 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 a unit fraction. Tell a partner how we could write as a product of a whole number and a fraction.” ()
Standards Alignment
Building On
Addressing
4.NF.B.4
Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
Completa cada ecuación para que sea verdadera. Usa los patrones que observaste antes.
Vas a trabajar en grupo. 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 ecuación que recibiste.
Paso 4: Recoge tu hoja original. Si crees que todo es correcto, explica por qué la expresión y el diagrama representan la fracción que escribiste. Si no, discute con tus compañeros qué ajustes deben hacer.
