Lesson 2

This Number Talk encourages students to use their understanding of mixed numbers and properties of operations to mentally solve problems. The strategies elicited here will be helpful as students develop their fluency in performing operations on fractions.

• 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 se relaciona cada expresión con la que está antes?” // “How is each expression related to the one before it?”
• “¿Cómo nos pueden ayudar las primeras tres expresiones a encontrar el valor de la última expresión?” // “How can the first three expressions help us find the value of the last expression?”
• 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?”

The purpose of this activity is for students to describe and sort triangles based on the length of their sides and the size of their angles. Students are asked to find all the triangles (from the same set of cards used in the previous lesson) that have specific attributes.

Along the way, students notice attributes that all triangles seem to share (for instance, having three angles and at least two acute angles), attributes that some triangles share (for instance, having two equal angles or two equal sides), and those that no triangles seem to have (for example, parallel sides or multiple obtuse angles). With these repeated observations, students deepen their understanding of the properties of triangles and their sub-groups (MP8).

In the synthesis, students are introduced to right triangles.

• Each group needs a set of shape cards from the previous lesson. If time permits, separate the triangle cards from each set in advance.
• Gather the Collect and Display chart from the previous lesson for display in the activity synthesis.

• Groups of 2
• Give each group a set of cards from the first lesson.
• Provide access to rulers and protractors.
• Make available the chart with vocabulary from the previous lesson for reference during this lesson.
• “Para esta actividad, usen solamente las tarjetas que tienen triángulos” // “Use only the cards with triangles for this activity.”
• “Completen la búsqueda de triángulos con un compañero. Cuando terminen, comparen sus respuestas con otro grupo” // “Complete the triangle hunt with a partner. When your group is done, compare responses with another group.”

• 5 minutes: group work time
• 3 minutes: discussion with another group
• 2 minutes: individual work time on the last question

If students place each triangle in only one category, consider asking:

• “¿Cómo te aseguraste de haber encontrado todos los triángulos que tienen cada característica?” // “How did you make sure you found all the triangles that have each attribute?”
• “¿Cómo revisarías si un triángulo solo tiene una de las características de la lista o si tiene más de una?” // “How would you check if a triangle has only one of the listed attributes or more than one?”

• Add new vocabulary and sketches to the chart from the previous lesson.
• “¿Qué observan acerca de los triángulos H, L y V?” // “What do you notice about triangles H, L and V?” (They all have right angles.) 
• “Los triángulos que tienen un ángulo recto son triángulos rectángulos” // “Triangles that have a right angle are right triangles.”
• “¿Cómo supieron si un ángulo de un triángulo es un ángulo recto o no?” // “How did you know whether an angle in a triangle is a right angle?” (I measured it with a protractor or an object with a square corner.)
• “¿Pueden encontrar otros triángulos rectángulos en el grupo?” // “Can you find other right triangles in the set?” (Triangle Y)
• Invite students to share their completed sentences for the last question. Record their responses and invite the class to agree or disagree.
  • Some triangles . . .
  • All triangles . . .
  • No triangles . . .
• If time permits, consider asking: “En nuestro grupo de triángulos, no vimos ninguno que tuviera más de un ángulo recto ni más de un ángulo obtuso. Creen que es posible que un triángulo tenga: // “In our set of triangles, we didn’t see any with more than one right angle or more than one obtuse angle. Do you think it’s possible for a triangle to have:
  • ¿Dos ángulos rectos?” // Two right angles?” (No, two of the sides won’t meet to make a triangle.)
  • ¿Dos ángulos obtusos?” // Two obtuse angles?” (No, two of the sides won’t meet to make a triangle.)

In this activity, students identify right triangles. They also explain why certain given shapes are not right triangles. As they determine whether a triangle is a right triangle, students may use the corner of an index card or a piece of paper, use a protractor, or trace an angle in a triangle to compare with an angle elsewhere that they know to be a right angle. In doing so, students practice choosing tools strategically (MP5).

• Groups of 2
• Provide access to index cards, patty paper, and protractors.

• 5 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for students who use tools to check angles and those who estimate them visually. Ask the latter: “¿Cómo podemos revisar que los ángulos que parecen ser rectos sí midan \(90^\circ\)?” // “How might we check that the angles that look like right angles are \(90^\circ\)?”

Students may not identify triangles C or D because of the orientation. Consider asking:

• “¿Cómo podemos mostrar que un triángulo tiene un ángulo recto?” // “How can we show that a triangle has a right angle?”
• “¿Dónde más ves triángulos que pueden tener lados perpendiculares?” // “Where else do you see triangles that may have perpendicular sides?”
• “¿Qué herramientas podrías usar para ver si los triángulos tienen algún ángulo recto?” // “What tools could you use to see if each triangle has any right angles?”

• Invite students to share their responses and how they found the right triangles in the set.
• “¿Hubo triángulos que parecían ser triángulos rectángulos, pero que no lo eran, o que parecían no ser triángulos rectángulos, pero que sí lo eran?” // “Were there any triangles that appeared to be right triangles but are not, or don’t appear to be right triangles but are?” (Triangle H looks like one but isn’t.)