The purpose of this activity is for students to connect their understanding of unit fractions with their understanding of division. Students understand a unit fraction such as , as 1 piece where 3 of those equal pieces make the whole. Students also understand division, , as 1 whole divided into 3 equal shares. 

During the Activity Synthesis, connect the two expressions, and , to a common diagram to show the relationship between the operation of division and the fraction as a quotient. Students relate diagrams, fractions, and division expressions with one another and interpret them within the context of sandwiches (MP2). 

• Invite selected students to share the patterns they noticed in the table.
• Display student work that shows a diagram of one sandwich shared equally by 3 people or display the diagram from the student solutions.
• “¿De qué manera el diagrama que dibujaron representa la expresión  ?” // “How does the diagram you drew represent the expression ?” (Each rectangle is divided into 3 equal pieces so that’s .)
• Highlight that the division sign means that the whole is divided into equal pieces.
• “¿De qué manera la fracción  representa la cantidad coloreada?” // “How does the fraction represent the shaded amount?” (One of the 3 equal-sized pieces in the rectangle is shaded in so that’s .)

This sorting task gives students opportunities to analyze representations, situations, and expressions closely and make connections (MP2, MP7). As students work, encourage them to refine their descriptions of how the diagrams represent the situations and expressions using more precise language and mathematical terms (MP6).

• “¿Qué sabes sobre estas tarjetas?” // “What do you know about these cards?”
• “Todas estas tarjetas representan la misma situación. ¿De qué manera cada una de ellas muestra el número de sándwiches que se comparten, el número de personas que comparten los sándwiches y la cantidad de sándwich que recibe cada persona?” // “These cards all represent the same situation. How does each of them show the number of sandwiches being shared, the number of people sharing the sandwiches and how much sandwich each person gets?”

• Invite previously selected students to share the matches they made and how they know those cards go together.
• Attend to the language that students use to describe their matches, giving them opportunities to describe how the diagrams and expressions represent the situation more precisely.
• Highlight the use of terms like “divide”, “dividend”, “divisor”, “number of pieces”, and “size of each piece”.
• Display Cards B, D, J, and N.
• “¿De qué manera cada diagrama representa que 3 sándwiches se comparten equitativamente entre 2 personas?” // “How does each diagram represent 3 sandwiches being shared equally by 2 people?” (Each of the large rectangles is a sandwich and the shaded part shows how much each person gets.)
• “¿Qué cantidad de sándwich recibe cada persona? ¿Cómo lo saben?” // “How much sandwich does each person get? How do you know?” ( or , because each rectangle is cut into halves and 3 of them are shaded.)

• Display:

• “¿De qué manera esta expresión representa la situación?” // “How does this expression represent the situation?” (3 sandwiches are being shared equally by 2 people.)