Lesson 11

This warm-up prompts students to carefully analyze subtraction expressions containing two fractions or a whole number and a fraction. To compare the values of the expressions, students need to perform subtraction and apply their knowledge of equivalence. The reasoning here will also be helpful later as students reason about differences of two mixed numbers or a mixed number and a fraction.

• Groups of 2
• Display the 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 pareja cómo 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.”

In this activity, students solve contextual problems that involve subtracting fractions in which at least one value is a mixed number and it is necessary to decompose one or both numbers. Students find differences in any way that makes sense to them. They may use number line diagrams, reason in terms of addition, or perform repeated partial subtractions (without necessarily writing expressions or equations). For instance, to find the difference between \(9\frac{4}{8}\) and \(\frac{7}{8}\), they may:

• draw a number line, partition it into eighths, locate \(9\frac{4}{8}\), and count back 7 spaces to the left to represent subtraction of \(\frac{7}{8}\)
• reason: “What number plus \(\frac{7}{8}\) gives \(9\frac{4}{8}\) or \(\underline{\hspace{.5in}} + \frac{7}{8} = 9\frac{4}{8}\)?”
• first subtract \(\frac{4}{8}\) from \(9\frac{4}{8}\), and the subtract another \(\frac{3}{8}\)
• first subtract 1 from \(9\frac{4}{8}\), and then add \(\frac{1}{8}\) back

Students may also use the insights they gained previously about rewriting and decomposing whole numbers to facilitate subtraction and consider whether they could be applied to subtraction of a fraction from a mixed number. When students recognize the mathematical features of things in the real world and ask questions that arise from a presented situation, they reason abstractly and quantitatively (MP2).

• Groups of 2
• Consider asking: “¿Qué cosas han hecho, dado o recibido para celebrar sus amistades?” // “What are some things you have made, given, or received that celebrate your friendships?”
• Explain that macramé is a way of making textile by knotting and is at least a few thousand years old. The name came from the Arabic word “miqramah,” with one meaning being “fleco ornamental o decorativo” // “ornamental or decorative fringe.”
• “Las pulseras de macramé son una forma popular de celebrar amistades. ¿Han visto o hecho alguna?” // “Macramé bracelets are a popular way to celebrate friendships. Have you seen or made one?” (If possible, prepare a couple of bracelets—a finished one and an unfinished one—for students to see, or show images or a video of a bracelet being made.)
• “Usemos lo que sabemos sobre restar fracciones para resolver algunos problemas sobre pulseras de la amistad” // “Let’s use what we know about subtracting fractions to solve some problems about friendship bracelets.”

• “En silencio, trabajen durante 5 minutos. Luego discutan sus ideas y completen el resto con su compañero” // “Take 5 quiet minutes to work on the task, and then discuss your thinking and complete the rest with your partner.”
• 5 minutes: independent work time
• 5 minutes: partner discussion and work time
• Monitor for the different ways students go about finding differences. Identify those who write addition or subtraction expressions or equations.

Students may be unsure what quantity is unknown in each problem or how it is related to what’s given. Consider asking:

• “¿Qué sabemos sobre la pulsera?” // “What do we know about the bracelet?”
• “¿Qué no sabemos, pero queremos saber?” // “What do we not know but want to know?”
• “¿Qué necesitamos hacer para averiguarlo?” // “What do we need to do to find out?”

• Select 2–3 students who use different methods to briefly share their responses. Record subtraction and addition expressions or equations that students wrote or mentioned.
• Students will take a closer look at the same numbers in the next activity.

Previously, students found differences of two fractions, including mixed numbers, using any way that made sense to them. This activity formalizes and makes explicit how such differences can be found by writing equivalent fractions and decomposing a whole number or a mixed number. When students share their responses with a partner and revise them based on the feedback they receive, they construct viable arguments and critiqe the reasoning of others (MP3).

This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing

• Groups of 2
• Display the four expressions in the task statement.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 30 seconds: quiet think time
• Share responses. If not mentioned by students, point out that finding the values of these expressions is a way to answer the questions in the first activity.
• “Estudiemos más a fondo cómo podemos restar fracciones como estas sin usar diagramas ni un contexto que nos ayuden” // “Let’s take a closer look at how we can subtract fractions like these without using diagrams or a context to help us.”

• “En silencio, analicen el primer grupo de cálculos durante 2 minutos. Luego, hablen con su compañero sobre lo que piensan que ocurre en los cálculos” // “Take 2 quiet minutes to study the first set of calculations. Then, talk to your partner about what you think is happening in the calculations.”
• 2 minutes: quiet think time
• 1–2 minutes: partner discussion
• Share responses.
• “¿Por qué piensan que \(9\frac48\) aparece descompuesto en distintas sumas?” // “Why do you think \(9\frac48\) is decomposed into different sums?” (See Student Responses.)
• “¿Qué muestran las últimas dos expresiones?” // “What do the last two expressions show?” (The first one shows the \(\frac{7}{8}\) being subtracted from the \(\frac{12}{8}\), which is a part of \(9\frac{4}{8}\). The second shows the result of that subtraction being rejoined with the 8 from the \(9\frac{4}{8}\).)
• “Ahora es su turno de restar números mixtos. Si es necesario, descompónganlos” // “Now it’s your turn to subtract some mixed numbers, by decomposing them, if needed.”
• “Con su pareja, completen los tres cálculos incompletos del segundo problema” // “Work with your partner to complete the three incomplete calculations in the second problem.”
• 5–7 minutes: partner work time

Students may lose track of the subtraction in each problem because the minuend and subtrahend are expressed as sums. Consider asking students, “¿Qué podrían representar el primer número y el segundo número en un problema-historia?” // “What might the first and second numbers represent in a story problem?” Allow students to describe the subtraction in terms of the context to support reasoning.

• Ask each student to find a new partner.
• “Explíquenle a su nuevo compañero su respuesta a la parte c del último problema” // “Explain your response to part c of the last problem to your new partner.”
• “Por turnos, uno habla y el otro escucha. Si es su turno de hablar, compartan sus ideas y lo que han escrito hasta el momento. Si es su turno de escuchar, hagan preguntas y comentarios que ayuden a su compañero a mejorar su trabajo” // “Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 3–4 minutes: structured partner discussion.
• Repeat with at least one other partner.
• “Ajusten su respuesta inicial basándose en los comentarios que les hicieron sus compañeros” // “Revise your initial response based on the feedback you got from your partners.”
• 2 minutes: independent work time