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. 

If students find values other than those listed in the student responses, consider asking:

• “¿Cómo encontraste el valor que hace que la ecuación sea verdadera?” // “How did you find the value that makes the equation true?”
• “¿Cómo podrías usar el diagrama de Priya para encontrar el valor que hace que la ecuación sea verdadera?” // “How could you use Priya's diagram to find the value that makes the equation true?”

• 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: .
• “¿Cómo se muestra en el diagrama de Priya?” // “How does Priya’s diagram show ?” (It is the shaded part. We know it is 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 Noah's work and explain their reasoning, they critique the reasoning of others (MP3).

This activity uses MLR3 Critique, Correct, Clarify. Advances: reading, writing, representing

MLR3 Critique, Correct, Clarify

• Display Noah's response for students to consider:

  

  because I divided into 2 equal parts and of is shaded in.
• Read the explanation aloud.
• “¿Qué partes de esta respuesta no son claras, son incorrectas o están incompletas?” // “What parts of this response are unclear, incorrect, or incomplete?” (If you divide into 2 pieces, the answer will be smaller than and is larger than . The equation should be .)
• 2 minutes: partner discussion
• Invite 2–3 groups share what they discussed. Record for all to see.
• “Con su compañero, escriban una explicación ajustada” // “With your partner, work together to write a revised explanation.”
• Display and review the following criteria:
  • Correct equation
  • Labeled diagram
• 2–3 minutes: partner work time
• Select 1–2 groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each group shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.
• “¿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 Noah’s work or use the one from student responses.
• “¿Dónde vemos ?” // “Where do we see ?” (The shaded section shows one of the pieces if you divide into 2 equal pieces.)
• “¿Dónde vemos ?” // “Where do we see ?” (The shaded section also shows of .)
• “¿Qué fracción del diagrama completo está coloreada?” // “What fraction of the whole diagram is shaded in?” ()
• Display:
• “¿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 .)

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 . This relationship is not explicitly brought up in the Activity Synthesis, but if students describe this relationship, connect it to the student work that is discussed, as appropriate. When students notice a pattern, they look for and express regularity in repeated reasoning (MP8).

• Display: 
  
  
  
• “¿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?” (We are dividing into more pieces each time, so the size of each piece will be smaller.)