This Warm-up prompts students to visualize the idea of arranging angles around a point and adding their measurements as more pieces are added. The angles are familiar angles: , , and . Students previously arrived at these benchmarks by decomposing a full turn or . Here, they compose a full turn from angles.
The work here familiarizes students with the context and mathematics that might be involved later in the lesson. In the subsequent activities, students will compose and decompose paper cutouts of angles to determine angle measurements.
Launch
Groups of 2
Display the image.
“¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
1 minute: quiet think time
Activity
“Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
1 minute: partner discussion
Share and record responses.
¿Qué observas? ¿Qué te preguntas?
Student Response
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Advancing Student Thinking
Activity Synthesis
“¿Cuál creen que es la medida de cada ángulo?” // “What do you think is the measurement of each angle?” (They look like , , and angles.)
“¿Cómo lo saben?” // “How do you know?” (If the paper is a rectangle, then the corner pieces are right angles or each. Two of the corner pieces would be . Three pieces would be .)
“¿Qué ángulo obtenemos si agregamos la última pieza de la esquina?” // “What angle would we get if we add the last corner piece?” ()
Activity 1
Standards Alignment
Building On
Addressing
4.MD.C.7
Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
In this activity, students use their knowledge of , , and , and paper cutouts of some acute angles to determine the measurements of those angles. Students then use those measurements to compose and find the measurements of greater angles.
No explicit directions for finding the angles are given, so students have an opportunity make sense of the problem and use what they know about the additivity of angles to find the angle measures (MP7). If requested, give students access to rectangular sheets of paper, the square corners of which could be torn off.
This activity uses MLR5 Co-craft Questions. Advances: reading, writing.
Launch
Groups of 2–4
Give each group the cutouts of the four angles, 4 copies of each angle per group, or if using patty paper, give 1–2 sheets to each student.
If using patty paper, demonstrate that it can be used for tracing the angles.
Activity
MLR5 Co-craft Questions
Display only the image to the first problem, without revealing the question.
“Escriban una lista de preguntas matemáticas que se podrían hacer acerca de esta imagen” // “Write a list of mathematical questions that could be asked about this image.” (What is the size of each angle? Could we put angles together to make a right angle? Would all of the angles make a straight line? Which angle is closest to 90 degrees and how far away from 90 degrees is it?)
2 minutes: independent work time
2–3 minutes: partner discussion
Invite several students to share one question with the class. Record responses.
“¿En qué se parecen estas preguntas? ¿En qué son diferentes?” // “How are these questions alike? How are they different?” (Most questions are related to the sizes of the angles.)
Reveal the task (students open books), and invite additional connections.
Monitor for groups who:
Compose 2 copies of Angle P or 3 copies of Angle Q into a right angle.
Compose 2 copies of Angle R into Q.
Compose Angles P and Q into Angle S.
Pause for a whole-class discussion. Select students or groups, who reasoned as previously outlined, to share their reasoning.
3–4 minutes: independent work time on the second question
2 minutes: group discussion
Monitor for the different ways that smaller angles are used to compose the angles in parts a–d.
Tu profesor te va a dar materiales que te pueden ayudar a encontrar medidas de ángulos.
Usa los materiales y lo que sabes acerca de un ángulo recto para encontrar el tamaño de los ángulos P, Q, R y S. Explica o muestra cómo razonaste.
Después, usa las medidas de los ángulos P, Q, R y S para encontrar las medidas de estos ángulos.
Activity Synthesis
Invite students to share their responses and reasoning for the second question. Display the angles.
For each angle in the second problem, record the different compositions of angles students use to find its measurement. For example:
To compose the angle in the first part of the problem, students may use Angles P, R, and S, or they may use 3 copies of P.
To find the angle in the third part of the problem, students may use 3 copies of Angle S, or they may draw a line to separate 180 of the angle and fit Angle P in the remaining space.
Activity 2
Standards Alignment
Building On
Addressing
4.MD.C.7
Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
In this activity, students find the sizes of the angles created by folding paper several times, and reason about the resulting angles (MP7).
The first fold decomposes the paper into two congruent shapes, the edges of which line up exactly, and students can see how the fold splits two of the angles into equal halves. The subsequent folds decompose an angle into two equal angles, but this may not be obvious to students because the shapes of the two resulting parts are different. (The edges or creases that form the angles are of different lengths.) Students need to remember that the measure of an angle is not determined by the lengths of the segments that form it, and reason accordingly.
Some students may need support in folding paper precisely. Consider providing a larger sheet of paper or a straightedge to facilitate the folding.
Representation: Access for Perception. Walk students through the steps to fold the paper into a kite, demonstrating with your own paper. Before beginning, and then after each step, invite students to share what they notice about the angles on the paper. Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Launch
Groups of 2–4
Give each student a sheet of origami paper or square paper.
Display the paper folding diagrams.
Activity
2–3 minutes: independent work time
2–3 minutes: group discussion
Monitor for students who recognize that two angles are equal if the edges or creases that form them line up exactly (even if one crease or edge is longer than the other).
Tu profesor te va a dar una hoja de papel cuadrada. Sigue los pasos para doblar tu papel y formar una cometa. Trata ser lo más preciso posible al hacer los dobleces.
¿Puedes encontrar la medida de cada uno de los ángulos que están marcados en la cometa? Si es así, muestra cómo razonaste. Si no, explica por qué no.
Student Response
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Activity Synthesis
Invite students to share their responses and reasoning.
“¿Cómo podemos saber si se formaron parejas de ángulos del mismo tamaño con el primer doblez?” // “How can we tell if the first fold resulted in pairs of equal-size angles?” (The two triangles are identical and match up exactly, which means the angles in the two halves are the same.)
“Las figuras que se forman con el segundo y con el tercer doblez no son la misma. ¿Esos dobleces también forman ángulos del mismo tamaño?” // “The shapes that result from the second and third folds are not the same. Do those folds produce equal-size angles as well?” (Yes. The edges of the resulting angles match up exactly and meet at the same starting point, so the angles are the same size.)
“¿Cuál ángulo es mayor: B o E?” // “Which angle is greater, B or E?” (They are the same, both are . Angle B is half of 90. Angle E is twice .)
Lesson Synthesis
“Hoy usamos diferentes operaciones para encontrar las medidas de varios ángulos” // “Today we used different operations to find the measurements of different angles.”
Display:
“Estos son algunos ángulos a los que intentamos encontrarles sus medidas: el ángulo P, el ángulo S y algunos ángulos compuestos por ángulos más pequeños. Usamos diferentes operaciones para encontrar las medidas desconocidas” // “Here are some angles whose measurements we tried to find: Angle P, Angle S, and some angles composed of smaller angles. We used different operations to find the unknown measurements.”
“¿Cuál de estos ángulos podemos encontrar si usamos la división?” // “Which of these angles can we find by using division?” (Angle P: If we know that 2 copies of P make a right angle, which is , then dividing by 2 gives us the measure of P.)
“¿Cuál ángulo desconocido podemos encontrar si multiplicamos?” // “Which unknown angle can we find by multiplication?” (The angle made up of four angles has a measurement of .)
“¿Cuál ángulo desconocido podemos encontrar si le restamos un ángulo a otro?” // “Which unknown angle can we find by subtracting one angle from another?” (Angle S: We can subtract from and divide by 2 to find the measure of S, which is .)
“¿Cuál ángulo desconocido podemos encontrar si sumamos los ángulos que conocemos?” // “Which unknown angle can we find by adding known angles?” (Once we know the measure of Angle S, we can find the last angle: , which is .)
Standards Alignment
Building On
Addressing
4.MD.C.7
Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
If students find the sizes of Angles P–S by estimating, consider asking:
“¿Cómo encontraste el tamaño de cada ángulo?” // “How did you find the size of each angle?”
“¿Cuáles ángulos puedes usar varias veces para formar un ángulo recto? ¿Cómo puedes usar eso para encontrar el tamaño del ángulo?” // “Which angles can you use multiple times to make a right angle? How can you use that to determine the size of the angle?”
