This Warm-up prompts students to carefully analyze and compare attributes of two-dimensional figures with attention to the number of sides, symmetry, and presence of parallel and perpendicular lines. In making comparisons, students have a reason to use language precisely (MP6). The activity enables the teacher to observe the attributes that students notice intuitively and hear the terminologies they feel comfortable using.
Launch
Groups of 2
Display the image.
“Escojan 3 que vayan juntas. Prepárense para compartir por qué van juntas” // “Pick 3 that go together. Be ready to share why they go together.”
“Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
2–3 minutes: partner discussion
Share and record responses.
¿Cuáles 3 van juntas?
Student Response
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Advancing Student Thinking
Activity Synthesis
“¿En qué características de las figuras se fijaron cuando intentaban encontrar las 3 que iban juntas?” // “What attributes of the figures did you pay attention to as you tried to find which 3 go together?"
As students share, highlight the attributes on the images that they are comparing.
If no students mention parallel sides as an attribute to consider, ask them about it.
Activity 1
Standards Alignment
Building On
Addressing
4.G.A.3
Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
In this activity, students are given the result of folding a figure along one or more lines of symmetry and they are asked to reason about the original figure. No lines of symmetry are specified, so students must consider all sides of a folded figure as a possible line of symmetry and visualize the missing half accordingly.
The first question offers opportunities to practice choosing tools strategically (MP5). Some students may wish to trace the half-figures on patty paper, to make cutouts of them, or to use other tools or techniques to reason about the original figure. Provide access to the materials and tools they might need.
During the Activity Synthesis, discuss the different ways students approach the second question. Consider preparing cutouts of Figures A–F to facilitate the discussion. (The figures are provided in the blackline master.)
Action and Expression: Internalize Executive Functions. Invite students to plan a strategy, including the tools they will use, for the task. If time allows, invite students to share their plan with a partner before they begin. Supports accessibility for: Conceptual Processing, Organization, Attention
Launch
Groups of 2
Give a ruler or a straightedge to each student.
Provide access to protractors, patty paper, scrap paper, and scissors.
Activity
5 minutes: independent work time
2–3 minutes: partner discussion
Monitor for the different strategies students use to identify the original shape of the half-shapes (as noted in the Activity Narrative).
Mai tiene una hoja de papel. Ella puede obtener cada una de estas 2 figuras al doblar la hoja una vez a lo largo de una línea de simetría. ¿Qué forma tiene la hoja de papel sin doblar?
Diego dobló una hoja de papel una vez a lo largo de una línea de simetría y obtuvo este triángulo rectángulo.
¿Qué formas pudo tener la hoja de papel antes de ser doblada? Explica o muestra cómo razonaste.
Activity Synthesis
Invite students to share their responses and strategies.
When discussing the second question, ask students why B, C, and E are not possible figures of the original piece of paper even though there’s a line that breaks each figure into two right triangles that match the given triangle. (B and E have no line symmetry. C has lines of symmetry but not diagonally from corner to corner. If you fold from corner to corner, the triangles would not be on top of one another.)
Activity 2
Standards Alignment
Building On
Addressing
4.G.A.3
Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.
Previously, students reason about line-symmetric figures that have been folded once along a line of symmetry. In this activity, students encounter figures that have been folded more than once, each time along a line of symmetry, and reason about the perimeter of the original figure. They think about how a given set of expressions could represent the original perimeter of a twice-folded figure, looking for and making use of structure (MP7).
MLR8 Discussion Supports. Students who are working toward verbal output may benefit from access to mini-whiteboards, sticky notes, or spare paper to write down and show their responses to their partner. Advances: Writing, Representing
Launch
Groups of 2–4
Give a ruler or a straightedge to each student.
Provide access to patty paper, scrap paper, and scissors.
Activity
3–4 minutes: independent work time for the first set of questions
Pause for a brief class discussion on possible shapes of the original piece of paper and possible expressions for its perimeter.
5 minutes: independent work time for the second question
As needed, invite students to fold a piece of paper twice in different ways. Consider asking:
“¿Qué observan acerca de los rectángulos que obtuvieron?” // “What do you notice about the rectangles you created?”
“¿Qué representan los números de cada expresión? Si es necesario, usen la hoja doblada como ayuda” // “What do the numbers in each expression represent? Use your folded paper to help you if needed.”
2–3 minutes: group discussion
Monitor for students who use the drawings or expressions from the first question to help them reason about the second question. Select them to share during the Activity Synthesis.
Jada dobló una hoja de papel a lo largo de una línea de simetría y obtuvo este rectángulo.
¿Cómo pudo verse la hoja antes de ser doblada? Haz uno o más dibujos.
Escribe una expresión que represente el perímetro de la hoja antes de ser doblada.
Kiran dobló otra hoja de papel dos veces, cada vez a lo largo de una línea de simetría. Al doblar la hoja, Kiran obtuvo el mismo rectángulo que Jada.
Muestra que cada expresión podría representar el perímetro de la hoja de Kiran antes de ser doblada.
Activity Synthesis
Select students to share their responses and reasoning to the second question.
Consider asking: “Vimos tres hojas distintas que corresponden a los distintos perímetros, ¿se les ocurre otra figura que pueda tener la hoja de papel original?” // “Aside from the three figures whose perimeters are represented here, are there other possible figures that the original piece of paper could have?” (No)
“¿Cómo lo saben?” // “How do you know?” (The folded rectangle has two pairs of sides of the same length. There are only three possible pairs of lines of symmetry: both along the 182 mm side, both along the 105 mm side, and once along each 182 and 105 mm side. All three are already represented by the given expressions.)
“Si las distintas figuras originales se pueden doblar y formar la misma figura, ¿quiere decir esto que las figuras originales tienen el mismo perímetro?” // “If different original figures can be folded into the same figure, does that mean the original figures have the same perimeter?” (No)
Lesson Synthesis
“Hoy practicamos cómo visualizar figuras que han sido dobladas a lo largo de una línea de simetría y razonamos sobre el perímetro de las figuras originales” // “Today we practiced visualizing shapes that have been folded along a line of symmetry and reasoning about the perimeter of the original shapes.“
Display:
“Supongamos que este triángulo rectángulo fue el resultado de doblar una figura una vez a lo largo de una línea de simetría. ¿Qué estrategias podemos usar para decidir qué figuras posibles se tenían antes de doblar?” // “Suppose this right triangle is a result of folding once along a line of symmetry. What strategies could we use to determine the possible shapes before they were folded?” (Reflect the triangle along each of the sides—mentally, using tracing paper, or cutting out two copies of the triangle and arranging them so they mirror each other.)
“Para encontrar el perímetro de la figura original, ¿podemos simplemente duplicar el perímetro de la figura doblada? ¿Por qué sí o por qué no?” // “To find the perimeter of the original shape, could we just double the perimeter of the folded shape? Why or why not?” (No, because there is one side—along the folding line—that is not part of the perimeter of the original shape.)
“¿Cuáles podrían ser los perímetros de las figuras originales que se doblaron para obtener este triángulo?” //“What could be the perimeters of the original shapes that fold into this triangle?” (If folded along the longest side: it will be 28, or . If folded along the side that is 8 units long, it will be 32, or . If folded along the shortest side, it will be 36, or .)
Standards Alignment
Building On
Addressing
4.G.A.1
Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.
Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
