Lesson 13

The purpose of this warm-up is for students to describe the rectangles in the representation of a quilt, which will be useful when students divide strips of paper into unit fraction sized pieces in a later activity. While students may notice and wonder many things about this image, the variety of lengths and colors of fabric strips is the important discussion point. 

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “Estas imágenes muestran mujeres de Gee's Bend, Alabama, que han hecho edredones por generaciones. ¿Cómo describirían el edredón en el que ellas están trabajando?” // “These pictures show women from Gee's Bend, Alabama, who have been making quilts for generations. How would you describe the quilt they are working on?” (It is colorful. There are rectangles. There are different colored pieces of fabric.)
• Consider showing students examples of abstract or improvised quilts by Gee’s Bend Quiltmakers from the website of Souls Grown Deep.

The purpose of this activity is for students to solve problems about dividing a whole number by a unit fraction in a way that makes sense to them. The context of quilt making is used so students can visualize a strip of paper that is a whole number length being cut into fractional sized pieces. As students describe how the problems are similar and different, listen for the authentic language they use to describe division. The paper strip, or tape, is a helpful diagram to use when dividing a whole number by a unit fraction because students recognize important relationships between the divisor, dividend, and quotient (MP7). For example, if the length of the strip stays the same, but the size of the piece gets smaller, then the number of pieces will get bigger. 

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

• Groups of 2
• Refer to the picture from the warm up.
• “Si la tira de tela azul que está bajo el mentón de la mujer mide 1 metro de largo, ¿aproximadamente qué tan larga es la tira gris corta que está al lado?” // “If the blue strip of fabric under the woman’s chin is 1 meter long, about how long is the short gray strip next to it?” (\(\frac{1}{6}\) meter)

• 1–2 minutes: independent think time
• 8–10 minutes: partner work time

• Circulate, listen for, and collect the language students use to describe what was the same and different about the strategies they used to determine the number of pieces of paper for each color. Listen for:
  • The size of the piece changed.
  • The pieces were shorter.
  • There were more pieces.
  • I made more cuts.
• Record students’ words and phrases on a visual display and update it throughout the lesson.

Students may not immediately make a connection with the situation and division, since the number of pieces that results (the quotient) is greater than the number of feet being divided (the dividend). Consider asking, “¿Cómo se representa la división en esta situación?” // “How does this situation represent division?” Allow students to recognize that the cuts are similar to partitions in a diagram, or sharing. They may also recognize that the quotient represents the number of groups, while the fraction being divided is the size of each group. Have them write a division equation for each situation, if it helps. They will write division equations in the next activity.

• “¿Qué otras palabras o frases importantes deberíamos incluir en nuestra presentación?” // “Are there any other words or phrases 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.
• Display:
  \(2 \div \frac {1}{2} = 4\)
  \(2 \div \frac {1}{3} = 6\)
  \(2 \div \frac {1}{4} = 8\)
• “¿Cómo representan estas ecuaciones los problemas sobre las tiras de papel?” // “How do these equations represent the problems about the paper strips?” (The 2 is for 2 feet of paper, and the fractions show the size of the pieces that the paper is being cut into. The 4, 6, and 8 are the number of pieces of each color of paper.)

The purpose of this activity is for students to represent division of a whole number by a unit fraction with diagrams and equations. The context is the same as the previous activity so students can use a tape diagram to solve the problem, if they choose. In the previous activity, students recognized that when the length of paper stays the same and the size of the piece gets smaller, there are more pieces of paper. In this activity, students will consider what happens when the length of the paper changes, but the size of the pieces stays the same.

• Groups of 2

• 1–2 minutes: independent think time
• 6–8 minutes: partner work time
• Monitor for students who:
  • determine that Kiran will have 12 pieces of paper
  • can explain how the equation \(2 \div \frac{1}{6} = 12\) represents the yellow strip of paper being cut into \(\frac16\)-foot strips
  • describe the equation \(3 \div \frac {1}{6} = 18\) as representing a 3 foot strip of paper being cut into 18 pieces that are each \(\frac {1}{6}\) of a foot long

If students do not write an equation that represents the situation, show them \(2 \div \frac{1}{6}\) and ask, “¿Cómo se representa la situación en esta expresión?” // “How does this expression represent the situation?”

• Ask previously identified students to share their solutions.
• Display: \(2 \div \frac {1}{6} = 12\)
• “¿Cómo se representa la tira de papel amarillo en esta ecuación?” // “How does this equation represent the yellow strip of paper?” (The strip of paper is 2 feet long and it is cut into pieces that are \(\frac {1}{6}\) of a foot long so there will be 12 pieces.)
• Display: \(3 \div \frac {1}{6} = 18\)
• “¿Cómo esta ecuación representa una tira diferente que se corta en pedazos de igual tamaño?” // “How does this equation represent a different strip of paper being cut into equal sized pieces?” (A 3 foot piece of paper is cut into pieces that are \(\frac {1}{6}\) of a foot long so there are 18 pieces.)
• “¿Por qué en ambas ecuaciones el cociente es más grande que el dividendo?” // “Why is the quotient larger than the dividend in both of these equations?” (Because you are cutting a whole number length into fractional sized pieces, so there will be more pieces than when you started.)