Lesson 5

The purpose of this True or False is for students to demonstrate strategies and understandings they have for interpreting a fraction as division of the numerator by the denominator and vice versa. These strategies help students deepen their understanding of the relationship between division and fractions where the unknown is the numerator, denominator, or the value of the quotient.

• Display one statement.
• “Hagan una señal cuando sepan si la afirmación es verdadera o no, y puedan explicar cómo lo saben” // “Give me a signal when you know whether the statement is true and can explain how you know.”

• 1 minute: quiet think time

• Share and record answers and strategy.
• Repeat with each statement.

• Display:
  • \(5\frac{1}{2}\)
  • \(\frac{5}{2}\)
• “¿En qué se parecen estas expresiones? ¿En qué son diferentes?” // “How are these expressions the same? How are they different?” (They both have 5 and 2 in them. They both show halves. \(5\frac{1}{2}\) means 5 wholes plus one half and \(\frac{5}{2}\) means 5 groups of one half.)

The purpose of this activity is for students to analyze several different situations about the same context of sharing pounds of blueberries and generalize what they have learned about the relationship between fractions and quotients.  Students rely on their understanding of the relationship between division and fractions to choose numbers that make sense based on the constraints listed in the table. During the synthesis, students generalize their understanding of the relationship between division situations and fractions greater than 1, less than 1, and equal to one whole. 

• Groups of 2
• Display table from student workbook. Refer to the corresponding parts of the table and read, “cada persona recibe exactamente 1 libra de arándanos” // “each person gets exactly 1 pound of blueberries” and “_____ personas comparten ______ libras de arándanos” // “ _____ people share ______ pounds of blueberries.”
• “¿Qué números podemos escribir en los espacios en blanco para que cada persona reciba exactamente 1 libra de arándanos?” // “What numbers can we write in the blanks so that each person will get exactly 1 pound of blueberries?” (Sample responses: 3 and 3, 4 and 4, etc.)
• Record responses for all to see.
• “¿Qué es cierto sobre todas las parejas de números que usamos?” // “What is true about all of the pairs of numbers we used?” (Each blank has the same number in it.)
• “¿Por qué los números que van en los espacios en blanco tienen que ser los mismos?” // “Why do the numbers in the blanks have to be the same?” (In order for each person to get exactly one pound of blueberries, the number of people sharing has to be the same as the number of pounds of fruit.)
• “Vamos a resolver más problemas como este. En cada fila de la tabla, escriban números en los espacios en blanco para que se cumpla la regla que está marcada” // “We are going to solve more problems like this one. For each row in the table, write numbers in the blanks to fit the rule that is checked.”

• 5 minutes: partner work time
• “En su recorrido, observen en qué se parecen los números de las tablas y en qué son diferentes” // “As you walk, notice how the numbers in the tables are the same and different.”
• 5 minutes: gallery walk
• Match each group of 2 with another group of 2 so there are groups of 4.
• “En grupo, hagan un póster sobre los números que se usaron en las tablas” // “Work with your group to make a poster about the numbers that were used in the tables.”
• 5 minute: group work time
• Monitor for groups who:
  • show or explain why the number of people is always less than the number of pounds of blueberries when each person gets more than one pound of blueberries.
  • show or explain why the number of people is always more than the number of pounds of blueberries when each person gets less than one pound of blueberries.
  • show or explain why the number of people is always \(\frac{1}{2}\) the number of pounds of blueberries when each person gets \(\frac{1}{2}\) pound of blueberries.

• Ask previously selected groups to share their posters.
• “¿En qué se parecen las parejas de números que representan situaciones en las que cada persona recibe más de una libra de arándanos?” // “What is the same about the pairs of numbers that represent each person getting more than one pound of blueberries?” (There are always more pounds of blueberries than there are people sharing the blueberries.)
• “¿En qué se parecen las parejas de números que representan situaciones en las que cada persona recibe menos de una libra de arándanos?” // “What is the same about the pairs of numbers that represent each person getting less than one pound of blueberries?” (There are always less pounds of blueberries than there are people.)
• “¿En qué se parecen las parejas de números que representan situaciones en las que cada persona recibe exactamente \(\frac{1}{2}\) libra de arándanos?” // “What is the same about the pairs of numbers that represent each person getting exactly \(\frac{1}{2}\) pound of blueberries?” (The number of people sharing the blueberries is always double the number of pounds of blueberries.)
• “En la siguiente actividad vamos a pensar más sobre estas ideas” // “We are going to think more about these ideas in the next activity.”

The purpose of this activity is for students to explain why \(a \div b = \frac{a}{b}\) for any whole numbers \(a\) and \(b\) when \(b\) is not 0. Students may use words, equations, or diagrams to explain why this is true. In order to see a wide variety of interpretations, students take a gallery walk to observe their classmates’ work. Then they discuss how words and diagrams help show the equation \(a \div b = \frac{a}{b}\) for different values of \(a\) and \(b\).

Constructing an argument that works for any pair of numbers requires thinking carefully about the meaning of the dividend, \(a\), and the divisor, \(b\). Students may use diagrams or situations to help communicate their thinking but will need to explain why these make sense for any numbers \(a\) and \(b\) (MP3).

This activity uses MLR7 Compare and Connect. Advances: Representing, Conversing.

• Groups of 2

• “Completen el primer problema individualmente” // “Complete the first problem on your own.”
• 1-2 minutes: independent work time
• 2-3 minutes: partner discussion

MLR7 Compare and Connect

• “¿Por qué toda expresión de división se puede interpretar como una fracción? Creen una presentación visual que muestre cómo pensaron sobre esto. Incluyan detalles, como palabras, diagramas, expresiones, etc., para ayudar a los demás a entender sus ideas” // “Create a visual display that shows your thinking about why every division expression can be interpreted as a fraction. You may want to include details such as words, diagrams, expressions, etc. to help others understand your thinking.”
• 3-5 minutes: partner work time
• 5 minutes: gallery walk
• “¿En qué se parecen y en qué son diferentes las explicaciones?” // “What is the same and what is different between the different explanations?”
• 30 seconds quiet think time
• 1 minute: partner discussion

If students need more opportunities to explain the relationship between division and fractions, refer to work that was displayed during the gallery walk and ask students to explain the representations in their own words.

• Display the image from the first problem in Student Responses or use student work.
• “¿Cómo ayudan los diagramas a entender que \(? \div 2 = \frac{?}{2}\)?” // “How do diagrams help see that \(? \div 2 = \frac{?}{2}\)?” (They show a set of objects divided in half and also show the same number of halves as there are objects.)
• “¿Cómo ayudan los diagramas a entender que \({2} \div {?} = \frac{2}{?}\)?” // “How do the diagrams help see that \({2} \div {?} = \frac{2}{?}\)?” (They show 2 things divided into equal parts and that’s the same as 2 of those equal parts.)
• Invite students to share contexts that they used to help understand the relationship between division and fractions.