In this Warm-up, students practice identifying obtuse angles in an image. They may, for instance, rely on the symmetry of the figure or on a grouping strategy, or otherwise scan the figure in a methodical way.
Launch
Groups of 2
“¿Cuántos ángulos ven? ¿Cómo lo saben?, ¿qué ven?” // “How many angles do you see? How do you see them?”
Display the image.
1 minute: quiet think time
Activity
Display the image.
“Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
1 minute: partner discussion
Record responses.
¿Cuántos ángulos ves en el corazón de papel doblado?
Student Response
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Advancing Student Thinking
Activity Synthesis
“¿Cómo se aseguraron de haber tenido en cuenta todos los ángulos?” // “How did you make sure all the angles are accounted for?”(I put a mark through them or numbered them.)
“¿Cuántos ángulos obtusos hay en esta imagen?” // “How many obtuse angles are in this image?” (10)
Label each obtuse angle with reasoning from students.
Consider asking:
“¿Alguien puede expresar con otras palabras la manera en la que _____ vio los ángulos?” // “Who can restate in different words the way _____ saw the angles?”
“¿Alguien vio los ángulos de la misma manera, pero lo explicaría de otra forma?” // “Did anyone see the angles the same way but would explain it differently?”
“¿Alguien quiere compartir otra observación sobre la manera en la que _____ vio los ángulos?” // “Does anyone want to add an observation to the way _____ saw the angles?”
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.
Previously, students found numerous angle sizes by reasoning, without using a protractor. They have done so with problems with and without context. In this activity, students consolidate various skills and understandings gained in the unit and apply them to solve problems that are more abstract and complex. They rely, in particular, on their knowledge of right angles and straight angles to reason about unknown measurements. (Students may need a reminder that an angle marked with a small square is a right angle.)
The angles with unknown measurements are shaded but not labeled, motivating students to consider representing them (or their values) with symbols or letters for easier reference. Students also may choose to write equations to show how they are thinking about the problems.
When students use the fact that angles making a line add up to and that angles making a right angle add up to , they make use of structure to find the unknown angle measures (MP7).
MLR8 Discussion Supports. Display sentence frames to support small-group discussion: “Primero, yo _____ porque . . .” // “First, I _____ because . . .,” “Observé _____ entonces yo . . .” // “I noticed _____ so I . . .,” and “¿Por qué tú . . .?” // “Why did you . . . ?” Advances: Conversing, Representing
Representation: Internalize Comprehension. Synthesis: Invite students to identify what they had to look for in the pictures to solve each problem. Display the sentence frame: “La próxima vez que encuentre la medida de un ángulo sin un transportador, buscaré . . .” // “The next time I find the measurement of an angle without a protractor, I will look for . . . .“ Record responses, and invite students to refer to them in the next activity. Supports accessibility for: Conceptual Processing, Memory, Attention
Launch
Groups of 2
Activity
5 minutes: independent work time
2 minutes: partner discussion
Monitor for students who:
Use symbols or letters to represent unknown angles.
Write equations to help in reasoning about the angle measurements.
Encuentra la medida de los ángulos que están sombreados. Muestra cómo lo sabes.
Student Response
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Activity Synthesis
Display the angles. Select students to share their responses. Record and display their reasoning.
Highlight equations that illuminate the relationship between the known angle, the unknown angle, and the reference angle (, , or ). For instance: , , , and so on.
Label the diagrams with letters or symbols, as needed, to facilitate equation writing.
When discussing the last question, highlight that finding unknown values sometimes involves multiple steps, and some steps may need to happen before others.
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.
This activity features an Information Gap (Info Gap) routine in which students solve abstract multi-step problems involving an arrangement of angles with several unknown measurements. By now students have the knowledge and skills to find each unknown value, but the complexity of the diagram and the Info Gap structure demand that they carefully make sense of the visual information and look for entry points for solving the problems. Students need to determine what information is necessary, ask for it, and persevere if their initial requests do not yield the information they need (MP1). The process also prompts them to refine the language they use to ask increasingly more precise questions until they get useful input (MP6).
Launch
Groups of 2
MLR4 Information Gap
Display the Task Statement, which shows a diagram of the Info Gap structure.
1–2 minutes: quiet think time
Read the steps of the routine aloud.
“Les voy a dar una tarjeta de problema o una tarjeta de datos. Lean su tarjeta en silencio. No se la lean ni se la muestren a su compañero” // “I will give you either a problem card or a data card. Silently read your card. Do not read or show your card to your partner.”
Distribute the cards.
“El diagrama no está dibujado con precisión, así que no les recomiendo usar un transportador para medir” // “The diagram is not drawn accurately, so using a protractor to measure is not recommended.”
1–2 minutes: quiet think time
Remind students that after the person with the problem card asks for a piece of information, the person with the data card should respond with: “¿Por qué necesitas saber _____?” // “Why do you need to know _____ [that piece of information]?”
Activity
5 minutes: partner work time
After students solve the first problem, distribute the next set of cards. Students switch roles and repeat the process with Problem Card 2 and Data Card 2.
Tu profesor te dará una tarjeta de problema o una tarjeta de datos. No se la muestres ni se la leas a tu compañero.
Information Gap routine. Step 1, both students read the problem card. Cycle. Problem card student says, can you tell me, fill in the blank? Data card student says, why do you need to know, fill in the blank? Problem card student says, I need to know, fill in the blank, because dot, dot, dot. Data card student listens to partner's reason and answers with information from the data cards. Repeat cycle until the problem card student can state, I have enough information to solve this problem. Both solve the problem independently and continue to ask questions if more information is needed. Finally, share data card, then compare strategies and solutions.
Haz una pausa aquí para que tu profesor pueda revisar tu trabajo. Pídele al profesor un nuevo grupo de tarjetas. Intercambia roles con tu compañero y repite la actividad.
Student Response
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Advancing Student Thinking
Activity Synthesis
Select students to share how they found each angle measure. Record their reasoning, and highlight equations that clearly show the relationships between angles.
“¿Para cuáles ángulos fue fácil encontrar sus medidas? ¿Qué hizo que fuera fácil?” // “Which angle measurements were easy to find? What made them easy?” (P and D, because it was fairly easy to see that each of them and a neighboring angle make a straight angle.)
“¿Cuáles fueron un poco más complicados? ¿Por qué?” // “Which ones were a bit more involved? Why?” (E, because there are 5 angles that meet at that point. We needed to find A or D before finding E.)
Lesson Synthesis
“Hoy resolvimos problemas sobre ángulos en los que necesitamos varios pasos, todos sin medir con un transportador” // “Today we solved angle problems involving multiple steps, all without measuring with a protractor.”
Display the two diagrams on the Info Gap cards. Label the angles with measurements given on the data cards. ( for U, for C, for S, and for A.)
Focus the discussion on how equations could be used to represent students’ reasoning process and to help find the unknown angle measurements.
“¿Qué ecuaciones podemos escribir que nos ayuden a encontrar el valor de P? ¿Y de D?” // “What equations can we write to help us find the value of P? What about D?” (See sample equations in Student Responses.)
Student Section Summary
Aprendimos formas de nombrar ángulos basándonos en sus medidas.
Los ángulos agudos son ángulos que miden menos de 90º.
Los ángulos rectos son ángulos que miden 90º.
Los ángulos obtusos son ángulos que miden más de 90º.
Los ángulos llanos son ángulos que miden 180º.
También resolvimos problemas sobre ángulos.
Ejemplos:
Si dos ángulos forman un ángulo recto o un ángulo llano, podemos usar el tamaño de un ángulo para encontrar el otro.
El ángulo sombreado debe medir porque forma un ángulo recto cuando se junta con el ángulo de .
Como sabemos que el giro completo de un reloj mide , concluimos que la manecilla larga gira:
Un ángulo de cada hora.
Un ángulo de cada media hora.
Un ángulo de cada 15 minutos.
Un ángulo de cada 10 minutos.
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.
If students solve for the angles in the last diagram by first trying to find the measurement of the acute angle, consider asking:
“¿Qué sabes sobre los ángulos llanos?” // “What do you know about straight angles?”
“¿Cómo puedes usar lo que sabes sobre ángulos llanos para decidir cuál de los ángulos desconocidos encontrar primero?” // “How can you use what you know about straight angles to pick which unknown angle to find first?”
