The purpose of this Estimation Exploration is for students to apply their understanding of dividing a whole number by a fraction from previous lessons. The dividend in this expression is much larger than those that students have previously worked with to encourage students to use multiplication to estimate.
Launch
Groups of 2
Display the expression.
“¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high? Too low? About right?”
1 minute: quiet think time
Activity
“Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
1 minute: partner discussion
Record responses.
Escribe una estimación que sea:
muy baja
razonable
muy alta
Student Response
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Advancing Student Thinking
Activity Synthesis
“¿Cómo saben que el valor de es menor que 500?” // “How do you know the value of is less than 500?" (It's less than and that's 500)
Activity 1
Standards Alignment
Building On
Addressing
5.NF.B.7
Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.
The purpose of this activity is for students to reason about the size of quotients, involving a unit fraction and a whole number, by carefully analyzing the relative sizes of the dividend and divisor rather than finding the value of the expressions. As students work, listen for the language they use to explain why they think the value of an expression is greater than or less than 1. Highlight the language during the Activity Synthesis. When students explain to each other how they decided whether a quotient is greater than 1 or less than 1 they construct viable arguments (MP3).
This activity uses MLR1 Stronger and Clearer Each Time. Advances: Reading, Writing.
Engagement: Provide Access by Recruiting Interest. Synthesis: Optimize meaning and value. Invite students to share if a previously selected expression is less than or greater than one and how they determined the value of the expression (display of their work on a problem, strategy they used, verbal explanation with a classmate who missed the lesson). Supports accessibility for: Attention, Conceptual Processing, Memory
Launch
Groups of 2
Activity
1–2 minutes: quiet think time
5–8 minutes: partner work time
Sin calcular el valor de las expresiones, escribe cada expresión en la categoría correcta.
El valor de la expresión es menor que 1
El valor de la expresión es mayor que 1
Explica tu estrategia para decidir si un cociente es menor que 1 o mayor que 1.
Student Response
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Advancing Student Thinking
Activity Synthesis
MLR1 Stronger and Clearer Each Time
“Compartan con su compañero su explicación sobre cómo decidir si un cociente es menor que 1 o mayor que 1. Por turnos, uno habla y el otro escucha. Si es su turno de hablar, compartan sus ideas y lo que han escrito hasta el momento. Si es su turno de escuchar, hagan preguntas y comentarios que ayuden a su compañero a mejorar su trabajo” // “Share your explanation for determining whether a quotient is less than or greater than 1 with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
3–5 minutes: structured partner discussion.
Repeat with 2–3 different partners.
If needed, display question starters and prompts for feedback.
“¿Puedes darme un ejemplo que ayude a mostrar . . . ?” // “Can you give an example to help show . . . ?”
“¿Cómo puedes usar las palabras divisor y dividendo en tu explicación?” // “How can you use the words divisor and dividend in your explanation?”
“Ajusten su borrador inicial basándose en los comentarios que les hicieron sus compañeros” // “Revise your initial draft based on the feedback you got from your partners.”
2–3 minutes: independent work time
Activity 2
Standards Alignment
Building On
Addressing
5.NF.B.7
Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.
The purpose of this activity is for students to order the quotients from the previous activity from least to greatest, without calculating. The quotients of a whole number by a unit fraction have the same dividend so students reason that the expression with the smallest unit fraction divisor represents the largest quotient. In the same way, the quotients of a unit fraction by a whole number all have the same divisor so the expression with the largest unit fraction dividend is the largest.
Launch
Groups of 2
Activity
5 minutes: independent work time
5 minutes: partner discussion
Monitor for students who:
Explain that the greatest quotient is because it represents the largest number of pieces.
Explain that the smallest quotient is because it represents the smallest sized pieces.
Change their response after the partner discussion.
Sin calcular el valor de las expresiones, ordénalas de menor a mayor.
Escoge 2 expresiones y encuentra el valor de esas expresiones.
Student Response
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Advancing Student Thinking
If students confuse the strategies for dividing a whole number by a unit fraction with dividing a unit fraction by a whole number, consider asking:
“¿Puedes explicar cómo ordenaste las expresiones?” // “Can you explain how you put the expressions in the order?”
Refer to one of the expressions. “¿El cociente va a ser mayor o menor que el dividendo?” // “Will the quotient be greater than or less than the dividend?”
Activity Synthesis
Ask previously selected students to explain their reasoning.
“¿Qué fue un reto en esta actividad?” // “What was challenging about this activity?” (It was hard to explain without finding any values.)
“¿Cómo les ayudó razonar sobre el orden de las expresiones a encontrar el valor de 2 de las expresiones?” // “How did reasoning about the order of the expressions help you find the value of 2 of the expressions?” (I knew the value was going to be a unit fraction [or whole number].)
Lesson Synthesis
Display:
“Si el número del cuadro fuera un número entero, ¿qué sabríamos sobre el valor de esta expresión?” // “What do we know about the value of this expression if the number in the box is a whole number?” (It is going to be greater than 25. It is going to be a multiple of 25.)
Display:
“Si el número del cuadro fuera un número entero, ¿qué sabríamos sobre el valor de esta expresión?” // “What do we know about the value of this expression if the number in the box is a whole number?” (It is going to be a unit fraction. The denominator is going to be a multiple of 25.)
Student Section Summary
Aprendimos a dividir con números enteros y fracciones unitarias. Primero, usamos diagramas para resolver problemas en los que se dividía una fracción unitaria entre un número entero.
Ejemplo:
El diagrama A muestra que es igual a . Encontramos el tamaño de una parte si se divide en 4 partes iguales.
Diagrama A
Después, observamos la relación que hay entre la división y la multiplicación.
A partir del diagrama A, sabemos que porque .
Luego, usamos diagramas para resolver problemas en los que se dividían números enteros entre fracciones unitarias. También escribimos ecuaciones para representar estos problemas.
Ejemplo:
El diagrama B muestra que si una tira de papel de 2 pies de largo se corta en pedazos de de pie, habrá 12 pedazos. Por eso, porque encontramos cuántos pedazos de de pie tienen juntos una longitud de 2 pies.
Diagrama B
Por último, observamos patrones al dividir números enteros y fracciones unitarias.
Observamos que cuando un número entero se divide entre una fracción unitaria, el cociente es mayor que 1.
Ejemplo:
El valor de es mayor que 1. El número de pedazos de en los que se divide una longitud de 12 (o cualquier número entero) es mayor que 1.
También observamos que cuando una fracción unitaria se divide entre un número entero, el cociente es menor que 1.
Ejemplo:
El valor de es menor que 1. Cuando un número entero menor que 1 se divide en muchas partes, cada parte es menor que 1.
Standards Alignment
Building On
Addressing
5.NF.B.7.b
Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for , and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that because .
