Lesson 15

This warm-up prompts students to carefully analyze and compare fractions or expressions containing fractions, relying on what they know about the size of fractions, equivalence, mixed numbers, and addition of fractions. The reasoning also helps students to recall familiar relationships between fractions where one denominator is a factor or a multiple of the other. This awareness will be helpful later when students solve problems that involve combining quantities with different fractional parts. 

• Groups of 2
• Display the numbers and expressions.
• “Escojan una que sea diferente. Prepárense para compartir por qué es diferente” // “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

• “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

• “Encontremos al menos una razón por la que cada una es diferente” // “Let’s find at least one reason why each one doesn’t belong.”
• “Mencionen algunas fracciones que sean equivalentes a los números o las expresiones de A, B y C” // “What are some fractions that are equivalent to the numbers or expressions in A, B, and C?” (\(\frac{3}{2}\), \(\frac{6}{4}\), \(\frac{9}{6}\), \(\frac{15}{10}\), \(\frac{18}{12}\))
• “¿Cuáles son algunas formas de decidir si dos fracciones son equivalentes o no?” // “What are some ways to decide if two fractions are equivalent?” (Sample responses:
  • Think about the relationship of unit fractions—for instance, how many one eighths are in one fourth?
  • Compare the fractions to a benchmark—for instance, if one fraction is greater than 1 and the other less than 1, then they’re not equivalent.
  • See if the numerator and denominator of one fraction could be multiplied by the same factor to get the other fraction.)

In earlier lessons, students found sums and differences of fractions (including mixed numbers) with the same denominator. In this activity, they reason about problems that involve combining or removing fractional amounts with different denominators—2, 4, and 8—in the context of stacking playing bricks. Because the denominators are familiar and are multiples or factors of one another, students can rely on what they know about the relationships of halves, fourths, and eighths to compare amounts (how much more or less one amount than another) or to combine them.

The last question asks students to reason about the height of a tower of bricks created by combining three shorter stacks. Students may arrive at two different answers depending on their familiarity with playing bricks and attention to precision. Some students may notice that each playing brick has studs that disappear into the bottom of another brick when stacked, so the combined height of two stacks will be less than the sum of the heights of individual stacks (MP6). Both answers are acceptable as long as they are supported. 

• Groups of 2
• Display the image of stacked playing bricks. Read the task statement (including the heights of the three stacks) together.
• “La imagen muestra la torre de Priya. Traten de visualizar las torres de Kiran y Lin en la misma imagen. ¿Qué tan altas serían?” // “The picture shows Priya’s tower. Try visualizing Kiran and Lin’s towers in the same picture. How tall would they be?” Invite students to try sketching or describing where the top of each tower would reach.
• Consider asking: “Piensa en la torre más alta de bloques para armar que hayas construido. ¿Qué tan alta era?” // “How tall was the tallest tower of playing bricks you have built?”

• “Trabajen individualmente durante unos minutos. Después, compartan con su compañero cómo pensaron” // “Work independently on the task for a few minutes. Then, share your thinking with your partner.”
• 5–6 minutes: independent work time
• 4–5 minutes: partner discussion
• Monitor for students who:
  • use the fact that there are 2 fourths or 4 eighths in 1 half to reason about sums or differences of the fractions
  • compare the fractional parts of the mixed number to the benchmark of \(\frac{1}{2}\) or 1 (especially in the last problem)
  • found differences and sums by writing equivalent fractions in fourths or eighths

Students may be inclined to add all the mixed numbers but may be unsure about how to proceed given the different fractional parts. Consider asking: “¿Hay otras formas de comprobar si la torre cabría sin sobrepasar el techo?” // “Are there other ways to see if the tower would fit?” and “¿Sería útil pensar en las medidas de los números enteros y en las partes fraccionarias por separado?” // “Would it help to think about the whole number measurements and fractional parts separately?”

• Select students to share their responses and reasoning.
• Focus the discussion on how students found the fractional differences between the measurements. Highlight explanations about the relative sizes of halves, fourths, and eighths, and about equivalence. (See Student Responses.)

Previously, students used their knowledge of equivalence to reason about the sums and differences of fractions with denominators 2, 4, or 8. In this activity, they do the same with fractions with denominators 2, 3, and 6. As before, students are not expected to write addition expressions in which the fractions are written with a common denominator (though some students may choose to do so). Instead, they rely on what they know about the relationship between \(\frac{1}{3}\) and \(\frac{1}{6}\), and between \(\frac{1}{2}\) and \(\frac{1}{6}\), to solve the problems. Students may choose to use visual representations to support their reasoning. When students create and compare their own representations for the context, they develop ways to model the mathematics of a situation and strategies for making sense of and persevering to solve problems (MP1, MP4).

The measurements in the task—\(\frac{1}{2}\), \(\frac{1}{3}\), and \(\frac{1}{6}\)—are given in feet. Because each of them has a whole-number equivalent in inches, some students may choose to reason entirely in inches, which is a valid strategy. Ask these students to think about how they’d approach the problems if the given unit is an unfamiliar one, or one that doesn’t convert handily to whole numbers in another unit.

• Groups of 2
• Display the image of the three foam blocks.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 30 seconds: quiet think time
• 30 seconds: partner discussion

• “En silencio, trabajen unos minutos en la actividad. Después, compartan con su compañero cómo pensaron” // “Take a few quiet minutes to work on the activity. Then, share your thinking with your partner.”
• 7–8 minutes: independent work time
• 3–4 minutes: partner discussion
• Monitor for the different reasoning strategies students use to combine different fractional parts, including use of diagrams, descriptions, and expressions or equations.

Students may not notice a relationship between the fractions in the task. Consider asking “¿Qué sabes sobre la relación entre tercios y sextos?” // “What do you know about the relationship between thirds and sixths?” and “¿Y entre medios y sextos?” // “What about between halves and sixths?” If needed, consider referring to fraction strip diagrams from an earlier unit.

• Invite students to share their strategies for determining whether or how certain combinations of blocks would make a specified height. Record and display their reasoning.
• To highlight different ways to combine different-size fractional parts, consider sketching or displaying diagrams as shown:
  • Making 1 foot with all blocks:

    

    

  • Making \(1\frac{1}{2}\) foot with \(\frac {1}{2}\) and \(\frac{1}{6}\) blocks:

    

    

  • Making \(1\frac{1}{2}\) foot with \(\frac{1}{2}\) and \(\frac{1}{3}\) blocks: