# Lesson 7

Fracciones no unitarias en la recta numérica

## Warm-up: Conteo grupal: Cuartos (10 minutes)

### Narrative

The purpose of this Choral Count is to invite students to practice counting by $$\frac{1}{4}$$ and notice patterns in the count. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to locate fractions on the number line using their knowledge of unit fractions. Save the recorded count to compare to a count in an upcoming lesson.

### Launch

• “Cuenten de $$\frac{1}{4}$$ en $$\frac{1}{4}$$, empezando en $$\frac{1}{4}$$” // “Count by $$\frac{1}{4}$$, starting at $$\frac{1}{4}$$.”
• Record as students count. Record 4 fractions in each row, then start a new row. There will be 4 rows.
• Stop counting and recording at $$\frac{16}{4}$$.

### Activity

• “¿Qué patrones ven?” // “What patterns do you see?” (The bottom part of the fraction never changes. The top part of the fraction is increasing by 1. The rows end at counts of four in the top like 4, 8, 12, 16.)
• 1–2 minutes: quiet think time
• Record responses.

### Activity Synthesis

• “¿En qué se parece contar de fracción en fracción a contar de número entero en número entero? ¿En qué es diferente?” // “How is counting by fractions the same as counting by whole numbers? How is it different?” (The top part of the fraction is just like counting by whole numbers, going up one. The bottom part is different because it doesn’t change.)
• “¿Alguien puede describir el patrón con otras palabras?” // “Who can restate the pattern in different words?”
• “¿Alguien quiere agregar otra observación sobre por qué ocurre ese patrón aquí?” // “Does anyone want to add an observation on why that pattern is happening here?”
• “¿Están de acuerdo o en desacuerdo? ¿Por qué?” // “Do you agree or disagree? Why?”
• “Es útil hablar de la parte de arriba de la fracción y de la parte de abajo de la fracción. Tenemos nombres para esas partes. La parte de abajo de una fracción se llama el denominador. Nos dice en cuántas partes iguales está partida la unidad. La parte de arriba de una fracción se llama el numerador. Nos dice cuántas de esas partes iguales están descritas. Busquen lugares en la lección de hoy en los que esa terminología les puede ayudar a explicar su razonamiento” // “This is a place where it’s helpful to talk about the top part of the fraction and the bottom part of the fraction. We have words for those parts. The bottom part of a fraction is called the denominator. It tells how many equal parts the whole was partitioned into. The top part of a fraction is called the numerator. It tells how many of the equal parts are being described. Look for places in today's lesson where that terminology might help you explain your reasoning.”
• Display the terms “denominador” // “denominator” and “numerador” // “numerator” and their definitions, and keep displayed throughout the lesson.

## Activity 1: Carrera sobre rectas numéricas (15 minutes)

### Narrative

The purpose of this activity is for students to practice identifying fractional intervals along a number line. This is Stage 2 of the center activity, Number Line Scoot. This activity encourages students to count by the number of intervals (the numerator). Students have to land exactly on the last tick mark, which represents 4, to encourage them to move along different number lines. While this activity does not focus on equivalence, it gives students exposure to this idea before they work more formally with it in the next section. In the synthesis, students relate counting on a number line marked off in whole numbers to their number lines marked off in fractional-sized intervals.

It may be helpful to play a few rounds with the whole class to be sure students are clear on the rules of the game. Keep the number line game boards for center use.

Engagement: Develop Effort and Persistence. Check in and provide each group with feedback that encourages collaboration and community. For example, check in after the first round of Number Line Scoot.
Supports accessibility for: Attention, Social-Emotional Functioning

### Required Materials

Materials to Gather

Materials to Copy

• Number Line Scoot Stage 2 Directions, Spanish
• Number Line Scoot Stage 2 Gameboard

### Required Preparation

• Each group of 2 students needs a number cube.
• Each student needs at least 5 base-ten cubes to use as game pieces.

### Launch

• Groups of 2
• Give each group of 2 a gameboard, two recording sheets, a number cube, and at least 10 base-ten cubes.
• “Ahora van a jugar un juego en el que se van a mover, de fracción en fracción, sobre diferentes rectas numéricas. Para comenzar, cada jugador pone un cubo pequeño sobre el cero de cada recta numérica. El objetivo del juego es llevar tantos cubos como puedan hasta el 4 de cualquiera de las rectas numéricas” // “Now you will play a game where you move, by fractions, along different number lines. To start, each player places a small cube on zero on each number line. The goal of the game is to get as many small cubes as you can to the 4 on any of the number lines.”
• Roll the number cube, demonstrate where to record the rolled number and move that fraction along one of the number lines.

### Activity

• 10 minutes: partner work time
• As students work, monitor for students who count by the numerator once they have chosen a number line.

### Activity Synthesis

• Display a gameboard with a marker on $$\frac{3}{4}$$.
• “Si yo sacara 4 y escogiera mover $$\frac{4}{4}$$, ¿cómo contarían la movida?” // “If I rolled a 4, and chose to move $$\frac{4}{4}$$, how would you count the move?” (I would count 1, 2, 3, 4.)
• “¿Cómo supieron que se habían movido $$\frac{4}{4}$$?” // “How did you know you have moved $$\frac{4}{4}$$?” (Because each space is $$\frac{1}{4}$$, so I need to move 4 times.)
• Display a number line marked with only 0, 1, 2, 3, 4.
• “¿En qué se parecen contar sobre esta recta numérica y contar sobre sus rectas numéricas? ¿En qué son diferentes?” // “How is counting along this number line the same and different than counting along your number lines?” (On the whole number one each space is 1 so we just count 1, 2, 3, 4. On our number lines we still count the jumps, but now each space is smaller than 1 so we need the denominator to tell us the size of each space.)

## Activity 2: Fracciones en la recta numérica (10 minutes)

### Narrative

The purpose of this activity is for students to locate a variety of fractions on the number line. Students are given a fraction less than 1 and greater than 1 with the same denominator to locate on each number line. The activity synthesis focuses on counting the number of unit fractions in a fraction to locate it on a number line and how to determine whether fractions are less than 1 or greater than 1. As they locate the fractions on the number lines, students strengthen their understanding of the meaning of the numerator and denominator of a fraction (MP6).

MLR8 Discussion Supports. Synthesis: Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.

### Launch

• Groups of 2
• Display the number line for fourths from Number Line Scoot.
• “¿Qué saben sobre $$\frac{3}{4}$$ y $$\frac{6}{4}$$?” // “What do you know about $$\frac{3}{4}$$ and $$\frac{6}{4}$$?” (They both have 4 on the bottom. Three is less than 4 and 6 is greater than 4. They are both numbers of fourths.)
• Share and record responses.

### Activity

• “Con su compañero, ubiquen las fracciones en cada recta numérica y contesten las preguntas sobre su trabajo” // “Work with your partner to locate the fractions on each number line and answer the questions about your work.”
• 4–6 minutes: partner work time
• Monitor for students who locate non-unit fractions on the number line by partitioning into equal parts of size $$\frac{1}{b}$$ and count the number of those parts.

### Student Facing

1. Ubica y marca $$\frac{3}{4}$$ y $$\frac{6}{4}$$.

2. Ubica y marca $$\frac{7}{8}$$ y $$\frac{12}{8}$$.

3. Ubica y marca $$\frac{2}{3}$$ y $$\frac{4}{3}$$.

4. Ubica y marca $$\frac{2}{6}$$ y $$\frac{7}{6}$$.

5. ¿Cómo partiste la recta numérica cuando ibas a ubicar los números $$\frac{7}{8}$$ y $$\frac{12}{8}$$? Explica tu razonamiento.
6. ¿Qué patrones observaste en las fracciones que ubicaste?

### Student Response

If students partition the interval from 0 to 2 into fourths instead of the interval from 0 to 1 (or a similar error with another fraction), consider asking:

• “Dime cómo encontraste $$\frac{3}{4}$$ en la recta numérica” // “Tell me about how you found $$\frac{3}{4}$$ on the number line?”
• “¿Cómo nos ayuda el denominador a saber cómo partir la recta numérica?” // “How does the denominator help us know how to partition the number line?”

### Activity Synthesis

• “¿Qué patrones observaron en las fracciones que ubicaron?” // “What patterns did you notice in the fractions you located?”
• “¿Cómo saben cuándo una fracción es menor que 1?” // “How do you know when a fraction is less than 1?”
• “¿Cómo saben cuándo una fracción es mayor que 1?” // “How do you know when a fraction is greater than 1?”
• “¿Cómo les ayudó contar de fracción unitaria en fracción unitaria a ubicar las otras fracciones en la recta numérica?” // “How did counting by unit fractions help you locate the other fractions on the number line?” (I counted by $$\frac{1}{4}$$ three times to find $$\frac{3}{4}$$. I counted by $$\frac{1}{6}$$ as I moved along the number line, like $$\frac{1}{6}$$, $$\frac{2}{6}$$, $$\frac{3}{6}$$, $$\frac{4}{6}$$, $$\frac{5}{6}$$, $$\frac{6}{6}$$, $$\frac{7}{6}$$ to find $$\frac{7}{6}$$.)

## Activity 3: ¿Cuál es la fracción? (10 minutes)

### Narrative

The purpose of this activity is for students to determine how a number line is partitioned and what fraction is marked on it with only 0, 1, and 2 labeled. Students partition and locate and mark, but don’t label, a fraction on a number line and then trade with a partner to determine the fraction their partner has marked. Remind students to mark, but not to label their partitions and their fraction, so that their partner only has the 0, 1, and 2 to use to determine what fraction is on their number line.

### Launch

• Groups of 2
• “Completen la primera parte de la actividad individualmente. Partan la recta numérica y pongan una marca que represente una fracción, pero no escriban el número. No le digan a su compañero cómo partieron la recta numérica ni qué número representa la marca” // “Complete the first part of the activity on your own. Partition the number line and mark, but don’t label, a fraction on the number line. Don’t tell your partner how you are partitioning or what number you are marking.”
• 2 minutes: independent work time

### Activity

• “Ahora, intercambien rectas numéricas con su pareja y respondan las preguntas sobre la recta numérica de su compañero. Cuando ambos terminen, compartan cómo razonaron” // “Now, trade number lines with your partner and answer the questions about their number line. When you are both finished, share your reasoning”
• 1–2 minutes: independent work time
• 1–2 minutes: partner work time

### Student Facing

1. Haz una partición de la recta numérica en cualquier número de partes de igual tamaño. Escoge una fracción y ubícala. Pon una marca que la represente, pero no escribas el número.

2. Intercambia tu recta numérica con un compañero.

1. ¿Cómo partió su recta numérica tu compañero?
2. ¿Qué número representa la marca de tu compañero en su recta numérica? Explica tu razonamiento.

Si te queda tiempo: juega otra vez.

### Activity Synthesis

• Display a number line partitioned by a student.
• “Hablen con su compañero sobre qué fracción está representada” // “Talk to your partner about what fraction is represented.”
• Share and record responses.
• “¿Cómo decidieron de qué manera partir su recta numérica y qué fracción poner en su recta numérica?” // “How did you decide how to partition your number line and what fraction you’d put on your number line?”
• “¿Cómo decidieron de qué manera estaba partida la recta numérica de su compañero y qué fracción estaba marcada?” // “How did you decide how your partner’s number line was partitioned and decide what fraction was marked?”

## Lesson Synthesis

### Lesson Synthesis

“Hoy ubicamos más fracciones en la recta numérica. En una lección anterior, aprendimos cómo se construyen las fracciones a partir de fracciones unitarias. ¿Cómo vemos esto en la recta numérica?” // “Today we located more fractions on the number line. In an earlier lesson, we learned how fractions are built from unit fractions. How do we see this on the number line?” (I counted the unit fractions, like 3 one-fourths, to get to $$\frac{3}{4}$$. I partition the number line into unit fractions and then I can count parts up to the fraction I am locating.)

Draw or have students draw a number line with $$\frac{3}{4}$$ marked, such as: Trace or have them trace through and count the 3 one-fourths to get to $$\frac{3}{4}$$, such as: “Recuerden que cuando ubicamos $$\frac{3}{4}$$ en la recta numérica, es útil mostrar o pensar en las tres partes (3 cuartas partes). Luego, ponemos la marca y escribimos el número $$\frac{3}{4}$$ al final de esas partes. Cuando ubicamos y marcamos fracciones, no hay que marcar la longitud, se puede simplemente contar las fracciones unitarias y, luego, poner el punto al final y marcarlo” // “Remember, when we are locating a fraction on the number line, it might be helpful to think about or show the 3 one-fourth parts, and then we mark and label the number $$\frac{3}{4}$$ at the end of those parts. When we locate and label fractions, you don’t have to mark the length, you can just count the unit fractions and then mark and label the point at the end.”

Point to the location of each fraction on the number line and count: “$$\frac{1}{4}$$, $$\frac{2}{4}$$, $$\frac{3}{4}$$.”