Lesson 17

This warm-up prompts students to carefully analyze and compare the features of four fractions. They may consider size (of the fraction, the numerator, or the denominator), equivalence, relationship to benchmark numbers, and more. The reasoning here will be helpful later in the lesson, as students classify sums of fractions by their size and relationship to 1.

• Groups of 2
• Display the four 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.

• Consider asking: “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.”
• “¿Algunas de estas son iguales a 1?” // “Are any of these equal to 1?” (No)
• “¿Cuáles de estas fracciones son mayores que 1? ¿Cómo lo saben?” // “Which of these fractions are greater than 1? How do you know?” (\(\frac{120}{100}\), because it is greater than \(\frac{100}{100}\).)

The purpose of this activity is for students to practice adding tenths and hundredths, by sorting a set of expressions based on whether their values are less than, equal to, or greater than 1. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP7). They decide whether it is necessary to write equivalent fractions, and if so, whether to use tenths or hundredths.

Through repeated reasoning, students build their ability to compare the size of hundredths to tenths and to 1 (MP8). They also have an opportunity to look for and make use of structure (MP7). For instance, students may conclude that certain expressions are greater than 1 by noticing that one of the addends is greater than 1.

Here is a list of the expressions on the blackline master, for reference:

• Create a set of cards from the blackline master for each group of 2-4 students.

• Groups of 2–4
• Give each group one set of cards from the blackline master and a couple of sticky notes.

• “En grupo, clasifiquen las tarjetas en tres categorías, dependiendo de si las expresiones son menores que 1, iguales a 1 o mayores que 1” // “Work with your group to sort the cards into three groups, based on whether the expressions are less than 1, equal to 1, or greater than 1.”
• “Prepárense para explicar o mostrar cómo saben a qué categoría pertenece cada suma” // “Be prepared to explain or show how you know where each sum belongs.”
• 8–10 minutes: group work time
• Monitor for:
  • the ways students decide whether to write equivalent fractions in tenths or hundredths
  • students who estimate the value of the expressions by looking at the relative size of the addends, without finding the sum
• “Cuando terminen, vayan a ver la forma en la que otro grupo clasificó la colección. Examínenla y dejen una nota con cualquier pregunta que tengan” // “When you finish, visit another group’s sorted collection. Examine it and leave a note about any questions you have.”
• 3–4 minutes: Visit another group.
• “Vuelvan a su colección. Respondan las preguntas que les dejaron o reconsideren lo que habían pensado. Después, escriban las expresiones que van en cada categoría” //   “Return to your collection. Address any questions left for you or revise your thinking. Then, record what’s in each group.”

Some students may have a better intuition for how tenths or hundredths relate to \(\frac{1}{2}\) than how they relate to one another, or how they relate to 1. Consider asking: “¿Cuál fracción de la pareja está cerca de \(\frac{1}{2}\)?” // “Which fraction in the pair is close to \(\frac{1}{2}\)?” and “¿Esto te ayuda a decidir cómo se relaciona esta expresión con 1?” // “Can this help you determine how this expression relates to 1?”

• Discuss questions such as:
  • “¿Cómo decidieron dónde debía ir cada expresión? ¿Escribieron siempre una fracción equivalente?” // “How did you decide where each expression should go? Did you always write an equivalent fraction?”
  • “¿Cuándo fue necesario escribir una fracción equivalente? ¿Cuándo no fue necesario?” // “When was it necessary to write an equivalent fraction? When was it not?”
  • “¿Hubo expresiones que pudieron clasificar sin reescribir ninguna fracción ni sumar? ¿Qué características de esas expresiones lo hicieron posible?” // “Were there expressions you were able to sort without rewriting any fractions or adding anything? What was it about those expressions that made that possible?”

In previous activities, students learned to combine tenths and hundredths. In this activity, students complete addition equations to make them true. To do so, they rely on a range of understandings and skills: how to write equivalent fractions, how to add fractions, and how to decompose a fraction into a sum. Though many of the equations involve an unknown addend, students are not expected to find them by subtraction.

• Groups of 2
• Display these equations:
  • \(\frac{1}{2} + \frac{4}{2} = 2\)
  • \(\frac{9}{10} + \underline{\hspace{0.5in}} = 1\)
• “¿Estas ecuaciones son verdaderas? Tómense un minuto para pensar en esto” // “Are these equations true? Take a minute to think about it.”
• 1 minute: quiet think time
• Discuss responses.
• “¿Por qué la primera ecuación no es verdadera?” // “Why is the first equation not true?” (The sum of the fractions on the left is \(\frac{5}{2}\), which does not equal 2.)
• “¿Por qué tanto \(\frac{1}{10}\) como \(\frac{10}{100}\) hacen que la última ecuación sea verdadera?” // “Why are \(\frac{1}{10}\) and \(\frac{10}{100}\) both true for the last equation?” (They are equivalent, so when added to \(\frac{9}{10}\) both result in 1.)
• “Encontremos otras fracciones que harían que las ecuaciones fueran verdaderas” // “Let’s find some other fractions that would make equations true.”

• “Trabajen individualmente. Completen al menos tres ecuaciones del primer problema y tres del segundo problema antes de discutir con su compañero” // “Work independently to complete at least three equations from the first problem and three from the second before discussing with your partner.”
• 6–7 minutes: independent work time
• 3–4 minutes: partner work time
• Monitor for the equations that seem to be challenging to many students or to be prone to errors. Discuss them during synthesis.

• “¿Qué ecuaciones fueron difíciles de completar? ¿Qué características de las fracciones dadas hicieron que fuera difícil encontrar los números que faltaban?” // “Which equations were difficult to complete? What about the given fractions made it hard to find the missing numbers?”
• “¿Qué fue más retador: encontrar las décimas desconocidas o las centésimas desconocidas? ¿Por qué?” // “Which did you find more challenging: finding missing tenths or missing hundredths? Why might that be?”

This optional activity allows students to practice adding tenths and hundredths (and to reinforce their ability to compare fractions) through a game. Students use fraction cards to play a game in groups of 2, 3, or 4. To win the game is to draw pairs of cards with the greater (or greatest) sum, as many times as possible.

Consider arranging students in groups of 2 for the first game or two (so that students would need to compare only 2 sums at a time), and arranging groups of 3 or 4 for subsequent games. Before students begin playing, ask them to keep track of and record pairs of fractions that they find challenging to add.

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

• Groups of 2 for the first game or two, then groups of 3–4 for subsequent games, if time permits
• Give each group one set of fraction cards from the blackline master.
• Tell students that they will play one or more games of Fraction Action.
• Demonstrate how to play the game. Invite a student to be your opponent in the demonstration game.
• Read the rules as a class and clarify any questions students might have.

• “Jueguen una partida con su compañero” // “Play one game with your partner.”
• “Mientras juegan, puede que encuentren parejas de fracciones que sean difíciles de sumar. Anoten todas estas parejas. Prepárense para explicar cómo descifraron finalmente cuál suma era mayor” //  “As you play, you may come across one or more pairs of fractions whose sums are hard to find. Record those fractions. Be prepared to explain how you eventually figured out which sum is greater.”
• “Si terminan antes de tiempo, jueguen otra partida con el mismo compañero o con los jugadores de otra pareja” // “If you finish before time is up, play another game with the same partner, or play a game with the players from another group.”
• 15 minutes: group play time

• Invite groups to share some of the challenging expressions they recorded and how they eventually determined the sums.
• As one group shares, ask others if they have other ideas about how the fractions could be added.