Lesson 12

The purpose of this Estimation Exploration is for students to think about dividing a unit fraction into smaller pieces. In the lesson, students will be given extra information so they can determine the exact size of shaded regions like the one presented here. 

• 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.

• Display the image:

  

  

• “¿En qué se parecen el diagrama de cinta y este diagrama de área? ¿En qué son diferentes?” // “How is the tape diagram the same as this area diagram? How is it different?” (Both diagrams show \(\frac{1}{4}\) of the whole and then a piece of that \(\frac{1}{4}\) is shaded. The tape diagram is long and narrow and the shaded piece is an entire vertical slice. The shaded piece in the area diagram is cut horizontally.)

In this activity, students interpret division of a unit fraction by a whole number using tape diagrams. In future lessons, students use tape diagrams to understand division of a whole number by a unit fraction. The first two activities are structured so students attend to the structure of the tape diagram and recognize how it can be used to show both a fractional part of a whole being divided into a whole number of pieces and also the size of each resulting piece in relation to the whole. The third activity provides an opportunity for students to begin to notice structure in equations when dividing a fraction by a whole number. 

• Groups of 2

• Monitor for students who:
  • can explain how Mai’s diagram shows \(\frac {1}{3}\) divided into 4 equal pieces.
  • can explain how Priya’s diagram shows that the size of each piece is \(\frac {1}{12}\).

If students do not explain how Priya’s diagrams represent the expression \(\frac{1}{3} \div 4\), suggest they draw their own diagram to represent the expression and ask, “¿En qué se parecen y en qué son diferentes tu diagrama y el diagrama de Priya?” // “How is your diagram the same and different from Priya’s diagram?”

• Ask previously selected students to share how Priya and Mai’s diagrams are the same and how they are different.
• Display the diagrams that Priya and Mai drew and this equation: \(\frac {1}{3} \div 4 = \frac {1}{12}\)
• “¿Cómo se muestra \(\frac {1}{12}\) en el diagrama de Priya?” // “How does Priya’s diagram show \(\frac {1}{12}\)?” (It is the shaded part. We know it is \(\frac {1}{12}\) of the whole because Priya divided all the thirds into 4 pieces.)

In the previous activity, students explained how tape diagrams represent equations and they used diagrams to find the value of division expressions. In this activity, students examine a mistake in order to recognize the relationship between the number of pieces the fraction is being divided into and the size of the resulting pieces. When students decide whether or not they agree with Priya’s work and explain their reasoning, they critique the reasoning of others (MP3).

This activity uses MLR3 Collect and Display. Advances: Reading, Writing, Representing.

• Groups of 2

• 5 minutes: independent work time
• 3 minutes: partner discussion

If students do not find the correct value of \(\frac{1}{3} \div 2\), prompt them to draw a diagram to represent the expression.

• Display the following partially correct answer and explanation:

  

  

  \(\frac {1}{3} \div 2 = \frac {1}{2}\) because that is how much is shaded in.
• Read the explanation aloud.
• “¿Qué creen que Priya quiere decir?” // “What do you think Priya means?” (She shaded in \(\frac {1}{2}\) of \(\frac {1}{3}\).)
• “¿Hay algo que no esté claro?” // “Is anything unclear?” (If you divide \(\frac {1}{3}\) into 2 pieces, the answer will be smaller than \(\frac {1}{3}\) and \(\frac {1}{2}\) is larger than \(\frac {1}{3}\).)
• “¿Hay algún error?” // “Are there any mistakes?” (The equation should be \(\frac {1}{3} \div 2 = \frac {1}{6}\).)
• 1 minute: quiet think time
• 2 minutes: partner discussion
• “Con su compañero, escriban una explicación ajustada” // “With your partner, work together to write a revised explanation.”
• Display and review the following criteria:
  • explanation for each step
  • correct solution
  • labeled diagram
• 3–5 minutes: partner work time
• Select 1–2 groups to share their revised explanation with the class. Record responses as students share.
• “¿En qué se parecen y en qué son diferentes las explicaciones?” // “What is the same and different about the explanations?”
• Display a revised diagram for Priya’s work or use the one from student responses.
• “¿Dónde vemos \(\frac {1}{3} \div 2\)?” // “Where do we see \(\frac {1}{3} \div 2\)?” (The shaded section shows one of the pieces if you divide \(\frac {1}{3}\) into 2 equal pieces.)
• “¿Dónde vemos \(\frac {1}{2} \times \frac {1}{3}\)?” // “Where do we see \(\frac {1}{2} \times \frac {1}{3}\)?” (The shaded section also shows \(\frac {1}{2}\) of \(\frac {1}{3}\).)
• “¿Qué fracción del diagrama completo está sombreada?” // “What fraction of the whole diagram is shaded in?” (\(\frac {1}{6}\))
• Display: \(\frac {1}{3} \div 2 = \frac {1}{3} \times \frac {1}{2}\)
• “¿Cómo sabemos que esto es verdadero?” // “How do we know this is true?” (We can see both expressions in the diagram and they are both equal to \(\frac {1}{6}\).)

In this activity, students notice as the divisor increases for a given dividend, the quotient gets smaller. Students may recognize and explain the relationship between multiplication and division. For example, they may notice that dividing one quarter into two equal pieces is the same as finding the product of \(\frac{1}{2}\times\frac{1}{4}\) . This relationship is not explicitly brought up in the synthesis, but if students describe this relationship, connect it to the student work that is discussed in the synthesis, as appropriate. When students notice a pattern, they look for and express regularity in repeated reasoning (MP8).

• Groups of 2

• 1–2 minutes: independent think time
• 3–5 minutes: partner work time

If students do not explain how they would find the value of \(\frac{1}{4}\) divided by any whole number, prompt them to find the value of \(\frac{1}{4} \div 5\) and \(\frac{1}{4} \div 6\) and ask, “¿Qué relación hay entre el divisor y el cociente?” // “What is the relationship between the divisor and the quotient?”

• Display: 
  \(\frac {1}{4} \div 2 = \frac {1}{8}\)
  \(\frac {1}{4} \div 3 = \frac {1}{12}\)
  \(\frac {1}{4} \div 4 = \frac {1}{16}\)
• “¿Qué patrones observan?” // “What patterns do you notice?” (The quotient is getting smaller. The denominator of the quotient is getting bigger. The denominator in the quotient increases by 4. The denominator in the quotient is equal to 4 times the number you are dividing by.)
• “¿Por qué el cociente se hace más pequeño?” // “Why is the quotient getting smaller?” (Because we are dividing \(\frac {1}{4}\) into more pieces each time, so the size of each piece will be smaller.)