Lesson 13

The purpose of this warm-up is to elicit the idea that number lines and tape diagrams can be used to represent the same relationships between numbers, which will be useful when students use tape diagrams and number lines in a later activity to interpret and solve story problems. While students may notice and wonder many things about these representations, the important discussion points are the way both representations show quantities and how they could be used to represent addition.

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

• “¿Cuál ecuación piensan que le corresponde mejor a estos diagramas?” // “What equation do you think best matches these diagrams?” (\(5+4=9\))
• “¿Dónde vemos el 5 en ambos diagramas?” // “Where do we see the 5 in both diagrams?” (the 5 on the number line is the distance from 0 to the point at 5)
• “¿Dónde ven el 4 en ambos diagramas?” // “Where do you see the 4 in both diagrams?” (the 4 on the number line is the 4 spaces that were moved to the right)

The purpose of this activity is for students to make connections between different representations of story problems. Students match stories, equations, number lines, and tape diagrams (MP2, MP7). The synthesis focuses on how the representations are the same and different. Students recognize some representations help to make sense of the story, while others help to show their thinking when finding solutions.

• Create a set of cards from the blackline master for each group of 3.

• Groups of 3
• Give each group a set of cards (A–P).

• “Hemos representado la suma y la resta en la recta numérica” // “We have been representing addition and subtraction on the number line.”
• “Hoy vamos a pensar en cómo podemos usar diagramas para dar sentido a historias y usar rectas numéricas para mostrar cómo pensamos” // “Today we are going to think about how we can use diagrams to make sense of stories and use number lines to show our thinking.”
• “Lin y sus amigos estaban haciendo trenes de cubos encajables. Estas historias son sobre todas las formas como ellos construyeron sus trenes” // “Lin and her friends were making connecting cube trains. These stories are about all the ways they built their trains.”
• “Agrupen cada historia con un diagrama, una recta numérica y una ecuación” // “Match each story to a diagram, a number line, and an equation.”
• 10 minutes: small-group work time

• Display cards A, E, G, O.
• “¿En qué se parecen estas representaciones del mismo problema? ¿En qué son diferentes?” // “How are these representations of the same problem the same? How are they different?” (I can see the whole length of the train in the diagram, but I only see a point at 37 on the number line and I don’t see anything but the numbers in the equation. They all show 37 as one part and they all show addition.)

The purpose of this activity is for students to solve story problems using any strategies and representations that make sense to them. In previous lessons, students have used number lines, tape diagrams, and equations to represent situations and solve problems. They will have the opportunity to apply their understanding and make choices about how to make sense of problems and show their thinking. In the synthesis, students reflect on which representations are most helpful to them and make connections across representations (MP2). This question will likely elicit a variety of responses from students which can emphasize the point that students should use representations that make sense to them.

• Groups of 2
• Give students access to the blackline master.

• “Mientras clasificaban las tarjetas, tuvieron la oportunidad de leer historias y de encontrar las representaciones que les correspondían” // “While doing the card sort, you had a chance to read stories and find matching representations.”
• “Ahora van a resolver problemas-historia y a representar cómo piensan. Háganlo de una manera que tenga sentido para ustedes” // “Now you are going to solve story problems and represent your thinking in a way that makes sense to you.” 
• “Si les ayuda, hay rectas numéricas sin saltos y diagramas que pueden usar” // “There are blank number lines and diagrams that you can use if it helps.”
• “Van a leer los problemas con su compañero, pero luego los van a resolver solos” // “You and your partner will read each problem together and then solve on your own.”
• “Prepárense para explicar cómo pensaron” // “Be prepared to explain your thinking.”
• 8 minutes: partner work time
• As students work, consider asking: “¿En qué se parecen o en qué son diferentes la forma en que ustedes mostraron cómo pensaron y la forma cómo su compañero lo hizo?” // "What is the same or different about the way you showed your thinking and the way your partner showed theirs?"
• Monitor for students who use a number line and tape diagram for the same problem. 

• Invite previously selected students display their work side by side for all to see.
• “¿En qué se parecen o en qué son diferentes las representaciones de _____ y _____?” // “What was the same or different between _____ and _____ representations?”
• Help students make connections across representations by highlighting the places where they see the solution in each.
• “¿Cuáles representaciones les parecen más útiles?” // “Which representations do you find most helpful?”