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.

• 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 Activity 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.

• 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?” (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 complete drawings of line-symmetric figures given half of each figure, 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.

• 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.)