Lesson 11

The purpose of this Number Talk is to elicit strategies and understandings students have for relating the operations of addition and subtraction. The equations \(27 + 13 = 40\), \(40 - 13 = 27\), and \(40 - 27 = 13\) each show the same relationship between numbers. Students will use this idea during the lesson when they are given a tape diagram and discuss why an addition or subtraction equation might match.

• Display one expression.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

• Record answers and strategies.
• Keep expressions and work displayed.
• Repeat with each expression.

• “¿Cómo pueden usar su respuesta a \(27 + 13\) para encontrar el valor de \(40 - 13\)?” // “How can you use your answer for \(27 + 13\) to find the value of \(40 - 13\)?” (\(27 + 13\) is 40 so \(40 - 13\) is 27.)
• “¿Cómo pueden usar su respuesta a \(27 + 13\) para encontrar el valor de \(40 - 27\)?” // “How can you use your answer for \(27 + 13\) to find the value of \(40 - 27\)?” (\(27 + 13\) is 40 so \(40 - 27\) is 13.)

The purpose of this activity is to match story problems with equations and tape diagrams. The numbers were selected so that there is a pair of stories, tapes, and equations using the same set of numbers. In order to determine the matching sets, students will need to think about and describe how the expression and diagrams represent the context of the story (MP2).

The goal of the activity synthesis is to discuss matching representations, highlighting how the tape diagram can help students understand the story and visually represent the equation.

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

• Groups of 2
• Give each group a set of cards.

• “Hay 4 problemas-historia, 4 diagramas y 4 ecuaciones. Su objetivo es formar 4 grupos, cada uno con tarjetas que estén asociadas” // “You have a set of 4 story problems, 4 diagrams, and 4 equations. Your goal is to find 4 matching sets.”
• 5 minutes: independent work time
• 5 minutes: partner discussion
• Monitor for how students distinguish the pairs of stories using the same numbers, especially those who use the tape diagrams to figure out the operation and equation.

• “¿En qué se parecen las tarjetas de este grupo? ¿En qué son diferentes?” // “What is the same about the cards in this group? What is different?”
• “¿Cómo le corresponde la ecuación a la historia? ¿Cómo le corresponde al diagrama de cinta?” // “How does the equation match the story? How does it match the tape diagram?”
• “¿Cómo le corresponde el diagrama a la historia? ¿Cómo le corresponde a la ecuación?” // “How does the diagram match the story? How does it match the equation?”

• Display tape diagrams for stories A and B.

• “¿En qué se parecen los dos diagramas? ¿En qué son diferentes?” // “How are the two diagrams the same? How are they different?” (Both diagrams have a 53 and a 25. One diagram shows two separate rectangles comparing quantities. 53 is the total for one of the rectangles. In the other diagram 53 is one part and the total is unknown.)
• “¿Cuál problema le corresponde al primer diagrama? ¿Cómo lo saben?” // “Which problem does the first diagram match? How do you know?” (The baseball and tennis ball problem. The first bar shows 53 and the second bar shows 25 less than 53.)
• “¿En qué son diferentes este problema y el problema de las piñas?” // “How is this different from the pine cone problem?” (53 is how many pine cones are left after 25 were taken away.)
• “¿Por qué el segundo diagrama le corresponde al problema de las piñas?” // “Why does the second diagram match the pine cone problem?” (It shows that 53 are left if you take away 25.)

The purpose of this activity is to write story problems that match given tape diagrams (MP2). One diagram represents a one-step Compare problem while the other two show two-step problems. For the tape diagram with three addends, students might:

• write Put Together/Take Apart or Add To stories with an unknown addend
• write Take Away stories with the change or difference unknown
• write a Compare problem that compares the full tape to the sum of the first two sections

For the tape diagrams that represent Compare problems, students may phrase the question using “How many more . . . ?” or using “How many fewer . . . ?” Although these diagrams are typically used to represent Compare problems, students may also choose to write Put Together/Take Apart or Take Away problems.

The lesson synthesis highlights different types of stories students write that match the same tape diagram.

• Groups of 3
• “Ahora vamos a escribir historias matemáticas. Mencionen algunas cosas de nuestra escuela que pueden contar y sobre las que pueden escribir historias” // “We are now going to write math stories. What are some things in our school that you could count and write stories about?” (kids, teachers, chairs, tables, classrooms, sports equipment, posters, windows)
• Make a list of student ideas on a chart and save the chart for the next day’s lesson.

• “Ahora van a mirar unos diagramas de cinta y a escoger un diagrama para escribir un problema-historia que le corresponda. Si quieren, usen el póster para obtener ideas” // “Now you are going to look at a few tape diagrams and choose one diagram to write a story problem that matches it. Use the chart for ideas if you’d like.”
• “En su grupo de 3, asegúrense de que cada uno trabaje con un diagrama distinto. Trabajen individualmente durante 5 minutos. Luego, van a tener tiempo para compartir su problema-historia con su grupo y para que sus compañeros les hagan comentarios sobre él” // “In your group of 3, make sure everyone is working on a different diagram. You will have 5 minutes to work on your own, then you will have time to share your story problem with your group and get feedback on it.”
• “Antes de reunirse con toda la clase, tómense un momento para ajustar sus historias después de que hayan recibido los comentarios” // “Before we come back together as a whole group, take time to make any revisions to your stories after you receive feedback.”
• 5 minutes: independent work time
• 5 minutes: group work time
• Monitor for students who write different types of stories for the same tape diagram.

• Invite previously identified students to share their stories for the same tape diagram or share one story and have the class generate a different story after discussing the student work.
• “¿Cómo supieron que el segundo diagrama representaba una comparación?” // “How did you know that the second diagram represents a comparison?” (It has two different rectangles. We know the larger amount, but it shows a question mark for the smaller amount.)
• “¿Cómo resolvieron el problema?” // “How did you solve the problem?” (I subtracted 20 from 63 and then subtracted 6. Or, I added 4 to 26 to make 30, then added 3, and then added 30 more.)