Lesson 5

This Number Talk encourages students to think about place value and to rely on the structure of multi-digit numbers and properties of operations to mentally add multiple addends. The strategies elicited here help students develop fluency in adding multi-digit numbers. They will also be helpful when students reason about the perimeter and angles in line-symmetric figures.

• 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 strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

• “¿Cómo usaron la primera expresión como ayuda para resolver las siguientes expresiones?” // “How did you use the first expression to help solve the expressions that follow?” (\(43 + 57 = 100\), so anywhere we see numbers with that combination being added, we can add 100 instead.)
• Consider asking:
  • “¿Alguien puede expresar el razonamiento de _____ de otra forma?” // “Who can restate _____’s reasoning in a different way?”
  • “¿Alguien usó la misma estrategia, pero la explicaría de otra forma?” // “Did anyone have the same strategy but would explain it differently?”
  • “¿Alguien pensó en la expresión de otra forma?” // “Did anyone approach the expression in a different way?”
  • “¿Alguien quiere agregar algo a la estrategia de _____?” // “Does anyone want to add on to _____’s strategy?”

In an earlier lesson, students observed that a line of symmetry decomposes a figure into two halves that match up exactly if the figure is folded along the line. This activity highlights that having two identical halves on each side of a line doesn’t necessarily make a figure symmetrical. It encourages students to use their understanding of symmetry and the line of symmetry to articulate why this is so as they critique supplied reasoning (MP3).

The given grid enables students to reason about the size and position of the attributes of each figure and provides structure for drawing the vertices and segments of each missing half.

• Groups of 2
• Give patty paper and a ruler or straightedge to each student.

• 2 minutes: quiet think time
• 3 minutes: partner discussion
• As students work, take note of students’ language when describing why Clare’s figures are incorrect.
• 3 minutes: independent work time

MLR 1 Stronger and Clearer

• “¿Cómo saben que sus dibujos tienen una línea de simetría?” // “How do you know that your drawings have line symmetry?”
• “Compartan su explicación con 2 o 3 compañeros diferentes. Si es necesario, ajusten sus ideas para que su explicación sea cada vez más sólida” // “Share your explanation with 2–3 different partners. Revise your thinking, if needed, to make your explanation stronger each time.”
• 3–5 minutes: structured partner discussion

If students agree with Clare’s drawings or create another figure that has no line symmetry, consider asking:

• “¿Cómo encontraste las líneas de simetría en la lección de ayer? ¿Cómo podrías usar esa estrategia para comprobar si la figura de Clare tiene líneas de simetría?” // “How did you find lines of symmetry in yesterday’s lesson? How could you use that strategy to check if Clare’s figure has line symmetry?”
• “¿Cómo podrías usar esa estrategia para crear una figura que sea simétrica con respecto a una línea?” // “How could you use that strategy to create a figure with line symmetry?”

• Invite 1–2 students to share their completed drawings. Compare and contrast them to Clare’s drawings.
• “¿Cuál podría ser una razón para pensar que los dibujos de Clare muestran las figuras completas correctas?” // “What might be a possible reason for thinking that Clare’s drawings show the correct whole figures?” (The new halves are exactly the same figure and size as the given halves.)
• “¿Cómo podemos saber que las figuras completas de Clare no son simétricas con respecto a una línea?” // “How can we tell that Clare’s completed figures do not have line symmetry?” (The given half and the new half don’t match up when the figure is folded along the line.)

In this activity, students continue to reason about the missing half of a line-symmetric figure given half of the figure and a line of symmetry. A grid is given in some cases, but in others, students may choose an appropriate tool (MP5)—patty paper, paper cutouts, rulers, or protractors—to help them draw whole figures with some precision.

In the first problem, students use patty paper to help them draw line-symmetric figures. For some figures, students may rotate and slide—instead of reflect or flip—a traced figure to show the missing half. During the synthesis, highlight that sliding and rotating are not reliable for completing all line-symmetric figures. (For example, it is not possible to slide a copy of the half-star figure to show the full star.) Students learn that flipping the given half across the line of symmetry will consistently complete the symmetrical figure.

• Groups of 2
• Give a ruler or straightedge to each student.
• Provide access to patty paper, paper for cutting, protractors, and scissors.

• 6–8 minutes: independent work time
• Monitor for students who use tools to complete their drawings more precisely.
• 5–7 minutes: group discussion

If students draw freehand to complete the figures on the grid, consider asking:

• “¿Cómo te aseguras de que la figura completa que haces sea simétrica con respecto a una línea?” // “How are you making sure the whole figure you create will have line symmetry?”
• “¿Cómo puedes dibujar segmentos de línea usando la cuadrícula para hacer que tu figura sea simétrica con respecto a una línea?” // “How can you use the grid to draw line segments so that your figure will have line symmetry?”

• Invite previously identified students to share how they used the patty paper and other tools to complete the figures with the triangle, trapezoid, and rhombus halves.
• “¿Cómo los ayudaron las herramientas al dibujar las figuras completas?” // “How did the tools support you in drawing the whole figures?” (Sample responses:
  • I traced or cut out one half of the figure and flipped it across the line of symmetry to make the whole figure.
  • I measured the lengths and distances of different segments and copied the measurements to the other side to be sure the parts were equal.)
• “¿Usaron herramientas al dibujar las figuras completas que estaban en la cuadrícula? ¿Por qué sí o por qué no?” // “Did you use tools when drawing the whole figures that were on the grid? Why or why not?” (No, because the figures are on a grid. I could count the number of units to draw each missing half.)

In the previous activities, students completed drawings of line-symmetric figures given half of each figure and a line of symmetry. In this optional activity, students do so again, but no lines of symmetry are given. Students are likely to find it intuitive to choose a side of the given shape—a triangle—and use it as a line of symmetry. The task prompts them to consider something less intuitive—that there may be multiple possible whole figures they could draw given half a figure.

A cutout of the triangle is given to encourage students to physically flip the shape along its different sides and trace the reflection, though students could also use other tools or methods to complete the task.

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

• Groups of 2–4
• Give a set of triangle cutouts from the blackline master to each group.
• Give a ruler or straightedge to each student.
• Provide access to patty paper, rulers, protractors, and scissors, in case requested.

• 4 minutes: independent work time
• 2 minutes: group discussion

• Invite students to share their completed drawings and their strategies.
• For each triangle, ask: “¿Es posible que haya otras figuras completas que aún no se hayan mostrado?” // “Are there other possible whole figures that are not yet shown?” Encourage students to identify any missing possibilities.
• “¿Alguien usó una estrategia que no fuera darle la vuelta al recorte a lo largo de un lado y dibujarlo? ¿Cuál fue esa estrategia? ¿Les ayudó?” // “Did anyone use a strategy other than flipping the cutout along one side and tracing it? What was the strategy? How did it help you?”
• “¿Dónde está la línea de simetría de cada dibujo que completaron?” // “Where is the line of symmetry in each completed drawing?” (It is the side used to reflect the given triangle, or the side shared by the two triangles.)