Lesson 20

The purpose of this Estimation Exploration is to estimate a whole given the value of a fraction of the whole. This prepares students for the type of problem they will solve in this lesson.

• Groups of 2
• Display the image.
• “¿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

• “Discutan con su pareja cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “¿Qué estrategias usaron para decidir qué número va en el espacio en blanco?” // “What strategies did you use to determine the number that goes in the blank?” (I tried to see how many 15's fit in the whole rectangle.)

The purpose of this activity is to solve problems about how many students are in the whole fifth grade. Students should use whatever strategy makes sense to them. As students consider which equations represent the problem, they may use the context, the relationship between multiplication and division, or computations to make sense of the equations. When students interpret the meaning of their answer in a context, they are reasoning abstractly and quantitatively (MP2).

• Groups of 2

• 1–2 minutes: independent think time
• 6–8 minutes: partner work time
• Monitor for students who:
  • describe the total number of students as 27 because there are 3 groups of 9 students
  • describe the total number of students as 27 because when 27 is divided into 3 equal groups, there will be 9 in each group
  • describe the total number of students in the class as 27 because \(\frac {1}{3} \times 27\) is 9

• Ask previously identified students to explain how each equation can represent the situation.
• Display:
  \(27 \div 3 = 9\)
  \(3 \times 9 = 27\)
• “¿Cuál es la relación que hay entre estas dos ecuaciones? Discutan sus ideas con un compañero” // “What is the relationship between these two equations? Discuss with a partner.” (They are like opposites. 27 divided into 3 equal groups is 9 and 3 groups of 9 is 27.)
• Display:
  \(\frac {1}{3} \times 27 = 9\)
• “¿De qué manera esta ecuación representa la situación?” // “How does this equation represent the situation?” (We know that \(\frac {1}{3}\) of the class is 9, so the class must have 27 kids in it, because \(\frac {1}{3}\) of 27 is 9.)

The purpose of this activity is for students to reason about which equations represent a situation. They use their understanding of the relationship between multiplication and division to make their selections.

• Groups of 2

• 5–8 minutes: partner work time
• Monitor for students who use the relationship between multiplication and division to make their selections.

If students do not select appropriate equations, show these equations and ask them to explain how they are the same and different.

• \(24 = \frac {1}{4} \times 96\)
• \(\underline{\hspace{1 cm}} \div 4 = 24\)

• Ask previously selected students to share their reasoning and solutions. 
• Display:
  \(\frac {1}{4} \div 24 = \underline{\hspace{1 cm}}\)
• “¿Cómo sabemos que esta ecuación no representa la situación?” // “How do we know this equation does not represent the situation?” (\(\frac {1}{4}\) divided into 24 groups is going to be really small. There will be \(\frac{1}{96}\) in each group.)
• Display:
  \(24 = \frac {1}{4} \times \underline{\hspace{1 cm}}\)
• “¿De qué manera esta ecuación representa la situación?” // “How does this equation represent the situation?” (We know there are 24 students in Jada’s class and we know that is \(\frac {1}{4}\) of the whole grade, but we don’t know how many students are in the whole grade.)
• Display:
  \(4 \times 24 = \underline{\hspace{1 cm}}\)
• “¿Cómo nos ayuda esta ecuación a saber cuántos estudiantes hay en todo el grado?” // “How does this equation help us figure out how many students are in the whole grade?” (If 24 is \(\frac {1}{4}\) of the grade, we can multiply 24 by 4 to figure out how many are in the whole grade.)

The purpose of this activity is for students to solve more “how many in one group” division problems in which the dividend is a whole number and the divisor is a unit fraction. The numbers are larger but still well-suited for a tape diagram representation. No method of solution is suggested or requested so students may draw a picture or a tape diagram or write an equation. For the second problem, the distance context may encourage students to use a number line representation to solve the problem.

• Groups of 2
• 2 minutes: quiet think time

• 8 minutes: partner work time
• Monitor for students who use a tape diagram or number line to represent and solve each problem.

Students may not be familiar with the contexts in this task. Consider incorporating a launch into the activity that supports students’ understanding of the context, for example, an image of daily recommended nutritional values, or a video of a rocket launching into space.

• Invite previously selected students to share their solutions.
• Display:
  \(60 = \frac {1}{5} \times \underline{\hspace{1 cm}}\)
  \(60 \div \frac {1}{5}\) = _______
• “Estas ecuaciones representan el problema del cohete. Podemos resolver ambas ecuaciones al multiplicar 60 por 5. ¿Por qué multiplicamos por 5?” // “These equations represent the rocket problem. We can solve both of these equations by multiplying 60 by 5. Why do we multiply by 5?” (60 days is \(\frac{1}{5}\) of the trip, so \(5 \times 60\) is the whole trip.)