Lesson 2

The purpose of this estimation exploration is for students to use what they know about fractions to estimate how much of the tape is shaded. Students use what they know about division to determine about how much of the bar is shaded.

• 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. 

If students do not have an estimate, encourage them to draw on the diagram or cut it out and fold it. Ask students: “¿Cómo te ayuda dibujar en el diagrama o doblarlo a descifrar cuánto está sombreado?” // “How can drawing on or folding the diagram help you figure out how much is shaded?”  

• “¿Cómo encontraron su estimación razonable?” // “How did you make your about right estimate?” (It looks like there are about 5 of those shaded pieces in the whole rectangle so that’s about \(\frac{1}{5}\).)

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 \(\frac{1}{3}\), as 1 piece where 3 of those equal pieces make the whole. Students also understand division, \(1\div3\), as 1 thing divided into 3 equal shares. 

During the activity synthesis, connect the two expressions, \(1\div3\) and \(\frac{1}{3}\), 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). 

• Groups of 2

• 5–8 minutes: partner work time
• Monitor for students who:
  • notice that the size of each piece is getting smaller as more people share it
  • notice that the denominator in the amount of sandwich each person gets is the number of people sharing sandwich 

If students do not write the correct amount of sandwich each person gets or the correct division expression, encourage them to draw their own diagram of the situation and ask:

• “¿De qué manera tu diagrama representa el número de sándwiches que se comparten?” //“How does your diagram represent the number of sandwiches being shared?”
• “¿De qué manera tu diagrama representa el número de personas que comparten los sándwiches?” // “How does your diagram represent the number of people sharing the sandwiches?”
• “¿De qué manera tu diagrama representa la cantidad de sándwich que cada persona va a recibir? ¿Qué número representa la cantidad de sándwich que va a recibir cada persona?” // “How does your diagram represent the amount of sandwich each person will get? What number represents the amount of sandwich each person will get?”

• Invite selected students to share the patterns they noticed in the table.
• Display student work that shows a diagram of one sandwich shared by 3 people or display the diagram from the student solutions.
• “¿De qué manera el diagrama que dibujaron representa la expresión \(1 \div3\)?” // “How does the diagram you drew represent the expression \(1 \div3\)?” (Each rectangle is divided into 3 equal pieces so that’s \(1 \div 3\).)
• Highlight that the division sign means that the whole is divided into equal pieces.
• “¿De qué manera la fracción \(\frac{1}{3}\) representa la cantidad sombreada?” // “How does the fraction \(\frac{1}{3}\) represent the shaded amount?” (One of the 3 equal-sized pieces in the rectangle is shaded in so that’s \(\frac{1}{3}\).)

This sorting task gives students opportunities to analyze and connect representations, situations, and expressions (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).

• Create a set of cards from the blackline master for each group of 2.

• Groups of 2
• Display image from student workbook.
• “Esta representación muestra la manera como 2 sándwiches se pueden compartir equitativamente entre 5 personas. ¿Qué cantidad de sándwich recibe cada persona? Prepárense para explicar cómo pensaron” // “This representation shows how 2 sandwiches can be shared by 5 people equally. How much sandwich does each person get? Be prepared to share your thinking.” (\(\frac{2}{5}\) since each piece is \(\frac{1}{5}\) of one whole and there are two of them.)
• 1 minute: quiet think time
• Share responses.
• Distribute one set of pre-cut cards to each group of students.

• “Estas tarjetas incluyen diagramas, expresiones y situaciones. Asocien cada diagrama con una situación y una expresión. Es posible que algunas situaciones y expresiones correspondan a más de un diagrama. Con su compañero, justifiquen sus elecciones. Después, respondan las preguntas en su libro de trabajo” // “This set of cards includes diagrams, expressions, and situations. Match each diagram to a situation and an expression. Some situations and expressions will match more than one diagram. Work with your partner to justify your choices. Then, answer the questions in your workbook.”
• 5–8 minutes: partner work time
• Monitor for students who: 
  • notice that the number of large rectangles in the picture and the dividend in the expressions represent the number of sandwiches
  • notice that the number of pieces in each whole and the divisor in the expressions represent the number of people sharing the sandwiches

If students do not match all of the diagrams to a situation or did not match the diagrams correctly, point to one of the diagrams that they did match correctly, and ask: “¿De qué manera este diagrama representa cómo algunas personas comparten unos sándwiches?” // “How does this diagram represent some people sharing some sandwiches?”

• Have students 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 2 personas comparten 3 sándwiches?” // “How does each diagram represent 3 sandwiches being shared 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?” (\(\frac{3}{2}\) or \(1\frac{1}{2}\) because each rectangle is cut into halves and 3 of them are shaded.)

• Display:

  \(3\div2\)

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