Lesson 14

The purpose of this warm-up is to elicit the idea that tape diagrams are similar to bar graphs and can be used to represent the same data, which will be useful as students make sense of the tape diagram throughout this section. While students may notice and wonder many things about these images, recognizing that tape diagrams can show comparisons like a bar graph is the important discussion point.

When students make connections between the different ways the representations represent the same categories and quantities, they reason abstractly and quantitatively and look for and make use of structure (MP2, MP7).

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

• “Las 2 imágenes de abajo se llaman diagramas” // “The 2 images at the bottom are called diagrams.”
• “¿Qué creen que representan el 15 y el 8 de los diagramas de abajo?” // “What do you think the 15 and 8 represent in the diagrams at the bottom?” (Its how many students that picked apple juice and how many students picked grape juice.)
• “¿Qué representa el signo de interrogación del diagrama?” // “What does the question mark in the diagram represent?” (The difference between the number of students that picked apple juice and grape juice, how many more students picked apple juice than grape juice, how many fewer students picked grape juice than apple juice.)

The purpose of this activity is for students to use their understanding of bar graphs to make sense of tape diagrams. Students physically create a tape diagram from a bar graph and compare two of the categories. Students recognize that even without a given scale, they can use this visual structure to compare amounts and reason about the difference between the two (MP2).

Students revisit the meaning of a diagram—a drawing or picture that represents quantities.

• Groups of 3–4
• Display the Party Time bar graph from the blackline master.

  

  

• “¿Qué podemos saber a partir de esta gráfica?” // “What can we learn from this graph?”
• Share responses.
• “Un equipo de fútbol va a celebrar una fiesta. El entrenador preguntó a los jugadores si les gustaría comer perros calientes, hamburguesas o pizza. Sus respuestas se muestran en esta gráfica de barras” // “A soccer team is having a party. The coach asked the players if they would like hot dogs, burgers, or pizza. Their responses are shown in this bar graph.“
• Give each student a copy of the blackline master, scissors, and tape or glue.

• “Recorten cada una de las barras de la gráfica y pongan las tiras una al lado de la otra. ¿Qué afirmaciones de comparación pueden hacer acerca de las barras?” // “Cut out each of the bars in the graph and put the strips next to each other. What comparison statements can you make about the bars?”
• 5 minutes: small-group work time
• “¿Qué afirmaciones de comparación hicieron acerca de los datos? ¿Alguien tuvo alguna confusión sobre lo que cada barra representaba? ¿O alguna confusión sobre cuántos hay en cada barra?” // “What comparison statements did you make about the data? Did anyone get confused about what each bar represented? Did anyone get confused about how many are in each bar?”
• 3–4 minutes: class discussion
• “Podemos usar las barras para hacer un diagrama como el que vimos en el calentamiento. Un diagrama nos ayuda a representar y comparar cantidades. Al igual que las gráficas, los diagramas deben tener etiquetas para que podamos entender lo que representan. Usen sus barras para hacer un diagrama que compare el número de estudiantes que escogieron perros calientes con el de aquellos que escogieron hamburguesas. Cuando terminen, escriban dos afirmaciones distintas que comparen los perros calientes con las hamburguesas” // “We can use the bars to make a diagram, like the one we saw in the warm-up. A diagram can be used to help us represent and compare quantities. Just like graphs, diagrams need labels so we can understand what they represent. Use your bars to create a diagram to compare the number of students who wanted hot dogs to those who wanted burgers. When you’re done, write two different statements that compare hot dogs to burgers.”
• 5 minutes: independent work time
• Monitor for students who label the categories and quantities.

If students label the diagram incorrectly, consider asking:

• “¿Cómo te muestra este diagrama el valor más grande?” // "How does this diagram show the greater value?"
• “¿En qué parte del diagrama ves la diferencia entre los valores?” // "Where do you see the difference between the values in the diagram?"

• Display a student’s completed tape diagram.
• “¿Cómo les ayuda su diagrama a ver qué comida recibió el mayor número de votos?” // “How does your diagram make it easy to see which food received the most votes?” (We wrote labels to show which bar matches which food.)
• “¿Qué deben agregar a las barras para responder preguntas como '¿Cuántos estudiantes más escogieron perros calientes que hamburguesas?'” // “What do you need to add to the bars to answer questions like ‘How many more students chose hot dogs than burgers?’” (Label what number the bar represents.)
• 1 minute: quiet think time
• 1 minute: partner discussion
• Share responses.
• As needed, draw a diagram that resembles the tape diagram in the warm-up.
• “Los diagramas pueden ayudar a representar problemas de comparación. Pueden usar rectángulos como las barras de una gráfica de barras para representar la cantidad más grande, la cantidad más pequeña y la diferencia” // “Diagrams can help represent Compare problems. You can use rectangles like the bars in a bar graph to represent the bigger amount, the smaller amount, and the difference.”

The purpose of this activity is for students to complete a tape diagram and describe the features of tape diagrams. Throughout the activity, students use what they know about bar graphs to make sense of what each number in a tape diagram represents and the role of the question mark. They also connect the structure of the tape diagram to equations that represent a comparison of two quantities (MP2). As students connect the labels on the tape diagram to the question, the features on the graph, and the context, they attend to precision and reason abstractly and quantitatively (MP6).

• Groups of 2

• “Ahora van a usar los datos de la gráfica ‘Hora de festejar‘ para completar más diagramas" // “Now you are going to use the data from the Party Time graph to complete some more diagrams.”
• 3 minutes: quiet work time
• 6 minutes: partner discussion

If the student assigns the larger value to the smaller part or places the question mark in a place other than the difference, consider asking:

• “¿Puedes explicarme tu representación?” // "Can you explain your representation to me?"
• “¿Cómo puedes usar la gráfica de barras como ayuda para marcar el diagrama?” // "How can you use the bar graph to help you label the diagram?

• “¿Cómo averiguaron cuál comida se estaba comparando con la pizza?” // “How did you figure out what food was being compared to pizza?” (We looked for the food that was picked by 14 students.)
• “¿Qué representa el signo de interrogación en el diagrama?” // “What does the question mark represent in the diagram?” (It shows the difference between hot dogs and pizza. It shows how many more students would need to pick hot dogs to be the same as the number of students who picked pizza.)
• Point out the dashes representing the difference, but not an actual quantity.
• “¿Qué ecuación podemos escribir para representar esta diferencia?” // “What equation could we write to represent this difference?” (\(19-{?} =14\), \(14 + {?} = 19\)) 
• Record responses.
• “¿Cómo ven representado a cada número en el diagrama?” // “How do you see each number represented in the diagram?” (19 shows how many picked pizza, 14 shows how many picked hot dogs. The question mark shows the difference.)