The purpose of this activity is for students to relate multiplying a non-unit fraction by a whole number to multiplying a unit fraction by the same whole number. After finding the value of in a way that makes sense to them, they then consider the value of the products and . In the Activity Synthesis students address how they can use the value of to find the value of other expressions.

• Ask previously selected students to share their solutions.
• Display: , ,
• “¿En qué se parecen las expresiones?” // “How are the expressions the same?” (They all have a 3. They all have some fifths and there is a product.)
• “¿En qué son diferentes las expresiones?” // “How are the expressions different?” (The number of fifths is different. There is 1 and then 2 and then 3.)
• “¿Cómo pueden usar el valor de como ayuda para encontrar el valor de ?” // “How can you use the value of to help find the value of ?” (I can just double the result because it’s instead of .)
• “¿Y para ?” // “What about ?” (That’s just another .)
• Display diagram from student solution or a student generated diagram like it.
• “¿De qué manera el diagrama muestra ?” // “How does the diagram show ?” (There is 3 total and of it is shaded.)
• Display: .
• “¿Cómo podrían adaptar el diagrama para mostrar ?” // “How could you adapt the diagram to show ?” (I could fill in 2 of the fifths in each whole instead of 1.)
• “En la siguiente actividad, vamos a estudiar un diagrama para ” // “In the next activity we will study a diagram for more.”

The purpose of this activity is to interpret diagrams in multiple ways, focusing on different multiplication and division expressions. The repeating structure in the diagrams allows for many different ways to find the value and interpret the meaning of the expressions. Encourage students to use words, diagrams, or expressions to explain how the diagram represents each of the expressions.

Monitor for students who:

• Explain that the diagram represents the multiplication expression because it shows 3 groups of .
• Explain that the diagram represents because there are 3 wholes divided into 5 equal pieces and 2 of the pieces in each whole are shaded.
• Explain how the diagram represents the relationship between and .

This activity gives students an opportunity to generalize their learning about fractions, division and multiplication. Students see shaded diagrams in different ways, representing different operations, and begin to see the operations as a convenient way to represent complex calculations (MP8).

This activity uses MLR2 Collect and Display. Advances: Conversing, Reading, Writing.

• Display the expression:
• “¿De qué manera el diagrama representa la expresión?” // “How does the diagram represent the expression?” (It shows 3 groups of .)
• Display the expression:
• “¿De qué manera el diagrama representa la expresión?” // “How does the diagram represent the expression?”
• Display:
• “¿Cómo sabemos que esto es cierto?” // “How do we know this is true?” (We can see both of them in the diagram. is the same as and ).
• “¿Qué otras palabras, frases o diagramas importantes deberíamos incluir en nuestra presentación?” // “Are there any other words, phrases, or diagrams that are important to include on our display?”
• As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.