Lesson 8

Fracciones y números enteros

Warm-up: Conversación numérica: Dividamos entre 4 (10 minutes)

Narrative

This Number Talk encourages students to rely on their knowledge of multiplication, place value, and properties of operations to mentally solve division problems. The reasoning elicited here helps to develop students' fluency with multiplication and division within 100.

To find the quotients of larger numbers, students need to look for and make use of structure in quotients that are smaller or more familiar, or to rely on the relationship between multiplication and division (MP7).

Launch

• Display one expression.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

Activity

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$12 \div 4$$
• $$24 \div 4$$
• $$60 \div 4$$
• $$72 \div 4$$

Activity Synthesis

• “¿Cómo les ayudaron las primeras expresiones a encontrar el valor de las últimas expresiones?” // “How did the earlier expressions help you find the value of the later expressions?”
• “¿Alguien usó la misma estrategia, pero la explicaría de otra forma?” // “Did anyone have the same strategy but would explain it differently?”
• “¿Alguien pensó en el problema de otra forma?” // “Did anyone approach the problem in a different way?”

Activity 1: Fracciones ubicadas en números enteros (15 minutes)

Narrative

The purpose of this activity is for students to place fractions greater than 1 on the number line and notice how fractions can be written as whole numbers. For example, students will see that for halves, every second half is located at a whole number because it takes 2 halves to make a whole.

Students work in groups. Each member will be assigned a different set of fractions to put on their number line so that the group can look for patterns across halves, thirds, and fourths. Through repeated reasoning, students may notice two types of regularity (MP8):

• It takes 2 halves, 3 thirds, or 4 fourths to make a whole.
• Whole numbers appear regularly (every 2 halves, every 3 thirds).

Launch

• Groups of 3
• Assign one set of fractions to each student in the group.

Activity

• “Tómense unos minutos para ubicar y marcar en la recta numérica las fracciones que les asignaron” // “Take a few minutes to locate and label your assigned fractions on the number line.”
• 2–3 minutes: independent work time
• “Compartan con su grupo su estrategia para ubicar las fracciones y entre todos, busquen patrones en los números” // “Share your strategy for locating the fractions with your group and look for patterns in the numbers together.”
• 4–6 minutes: small-group discussion
• Monitor for students who:
• Notice that every 2 halves, 3 thirds, and 4 fourths ends up at a whole number.
• Notice that the numerator is a multiple of the denominator.

Student Facing

1. Ubica y marca en la recta numérica las fracciones que te asignaron. Prepárate para explicar tu razonamiento.

1.  $$\frac{1}{2}, \frac{2}{2}, \frac{3}{2}, \frac{4}{2}, \frac{5}{2}, \frac{6}{2}, \frac{7}{2}, \frac{8}{2}, \frac{9}{2}, \frac{10}{2}$$
2.  $$\frac{1}{3}, \frac{2}{3}, \frac{3}{3}, \frac{4}{3}, \frac{5}{3}, \frac{6}{3}, \frac{7}{3}, \frac{8}{3}, \frac{9}{3}$$
3.  $$\frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{4}{4}, \frac{5}{4}, \frac{6}{4}, \frac{7}{4}, \frac{8}{4}, \frac{9}{4}, \frac{10}{4}, \frac{11}{4}, \frac{12}{4}$$
2. Hagan una lista de todas las fracciones que estaban ubicadas en un número entero. Incluyan las de las tres rectas numéricas del grupo.
3. ¿Qué patrones ves en las tres rectas numéricas que marcaron?

Activity Synthesis

• Display 3 blank number lines from 0 to 5 to label as students share.
• Select previously identified students to share the patterns they noticed in the fractions that share the same location as the whole numbers.
• “¿Por qué puede tener sentido que las fracciones muestren esos patrones?” // “Why might it make sense that the fractions show those patterns?” (Sample responses:
• because it takes 2 halves, 3 thirds, or 4 fourths to make a whole.
• because there are 2 halves in 1, so there are $$2\times 2$$ or 4 halves in 2, $$3\times 2$$ or 6 halves in 3, and so on.)
• Label the number line as students share, particularly the whole numbers, like: $$1 = \frac{2}{2}$$, $$2= \frac{4}{2}$$, $$3=\frac{6}{2}$$ to highlight the idea that the number of equal parts (2, 3, or 4) in the fractions affects when you end up at a whole number.

Activity 2: Ubiquemos el 1 en la recta numérica (20 minutes)

Narrative

The purpose of this activity is for students to use the location of a unit fraction to locate 1 and 2 on a number line. It is likely students will reason about repeating the size of the unit fraction to locate 1. To locate 2 on the number lines, they may continue to count unit fraction size parts or use the location of 1 to locate 2.

MLR1 Stronger and Clearer Each Time. Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to “¿Cómo ubicaron 1 cuando les daban la ubicación de una fracción unitaria?” // “How did you locate 1 when given the location of a unit fraction?” Invite listeners to ask questions, to press for details and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation or representation based on the feedback they receive.
Advances: Writing, Representing, Speaking, Listening
Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were needed or most useful to solve the problem. Display the sentence frame, “La próxima vez que ubique el 1 en una recta numérica, buscaré / prestaré atención a . . .” // “The next time I locate 1 on a number line, I will look for/pay attention to . . . ”
Supports accessibility for: Conceptual Processing

Launch

• Groups of 2
• Display the number line with $$\frac{1}{2}$$ marked.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (Students may notice: The number line only has 0 on one end and no whole numbers on the other end. One-half is labeled. Students may wonder: Is the number line partitioned into halves? Where is 1? What other numbers are on the number line? Why is nothing marked after $$\frac{1}{2}$$?)
• 1 minute: quiet think time
• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

Activity

• “Tómense unos minutos para ubicar el 1 en estas rectas numéricas” // “Take a few minutes to locate 1 on these number lines.”
• 3–5 minutes: independent work time
• “Compartan sus estrategias con su pareja y hablen sobre cómo pueden ubicar el 2 en estas rectas numéricas” // “Share your strategies with your partner and talk about how you might locate 2 on these number lines.”
• 3–5 minutes: partner work time
• Monitor for students who:
• iterate the size of the unit fraction using tick marks
• make unit fraction size jumps to count up to 1
• realize there will be 4 one-fourths in 1, for example, and place the 1 before placing $$\frac{2}{4}$$ and $$\frac{3}{4}$$

Student Facing

1. Ubica y marca el 1 en cada recta numérica. Prepárate para explicar tu razonamiento.

2. ¿Cómo puedes ubicar el 2 en las rectas numéricas del problema anterior?

Student Response

If students don’t locate 1, consider asking:

• “Dime qué has intentado hacer para ubicar el 1” // “Tell me about what you’ve tried to locate 1.”
• “¿Cuántos medios (tercios, cuartos u octavos) hay en 1? ¿Cómo podemos usar eso para ubicar el 1?” // “How many halves (or thirds, fourths, or eighths) are in 1? How could we use that to locate 1?”

Activity Synthesis

• Invite students to share a variety of strategies or representations of the number line for locating 1 when given the location of a unit fraction.
• “¿Alguien lo pensó de una forma parecida?” // “Did anyone think about it in a similar way?”
• “¿Alguien quiere agregar algo al razonamiento de ____?” // “Does anyone want to add on to ____ 's reasoning?”
• “¿Qué observaron acerca de cómo ubicaron el 1 las distintas personas?” // “What did you notice about how different people located 1?” (Sample responses: They marked off the lengths of the unit fraction until reaching 1 whole. They used a multiple of a unit fraction and marked off that length as many times as needed to get 1 whole.)
• “¿Qué estrategias usaron para ubicar el 2 después haber ubicado el 1?” // “What strategies did you have for locating 2 once you had located 1?”

Lesson Synthesis

Lesson Synthesis

“Hoy vimos que algunas fracciones se ubicaban en el mismo lugar que los números enteros. ¿Cuáles son algunos ejemplos de fracciones que se ubicaban en el mismo lugar que los números enteros?” // “Today we saw that some fractions were located at the same location as whole numbers. What were some examples of fractions that were located at the same location as whole numbers?” ($$\frac{2}{2}$$, $$\frac{6}{3}$$, $$\frac{8}{4}$$)

“¿Cómo podemos explicar por qué había fracciones y números enteros en la misma ubicación en la recta numérica?” // “How could we explain how fractions and whole numbers were in the same location on the number line?” (Every 2 halves (or 3 thirds or 4 fourths) you are at a whole number, so if you go 2 halves you are at 1. If you moved another 2 halves (or 3 thirds or 4 fourths) you would be at $$\frac{4}{2}$$ which is at the next whole number, which is 2.)