Lesson 3

The purpose of this How Many Do You See is to allow students to use subitizing or grouping strategies to describe the image they see. Listen for the language students use to describe how they count and define the line segments in the image.

• Groups of 2
• “¿Cuántos ven? ¿Cómo lo saben?, ¿qué ven?” // “How many do you see? How do you see them?”
• Display the image.
• 1 minute: quiet think time

• Display the image.
• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “¿Cuántos segmentos de recta tiene esta imagen?” // “How many line segments does this image have?” (9)
• “¿Cómo decidimos dónde comienzan y dónde terminan los segmentos de recta?” // “How do we determine where line segments start and end?” (They are between pairs of points or vertices in the figure.)
• Consider asking:
  • “¿Alguien puede expresar con otras palabras la manera en la que _____ vio la figura?” //  “Who can restate the way _____ saw the figure in different words?”
  • “¿Alguien vio la figura de la misma manera, pero lo explicaría de otra forma?” // “Did anyone see the figure the same way but would explain it differently?”
  • “¿Alguien quiere compartir otra observación acerca de la manera en la que _____ vio la figura?” // “Does anyone want to add an observation to the way _____ saw the figure?”

The purpose of this activity is to draw students’ attention to intersecting lines and parallel lines. Students may not have had a reason to consider how to describe the characteristics of lines that cross. To motivate them to do so, they are given three intersecting lines and asked to add a fourth one—first to make any quadrilateral, and then to make a rectangle.

Students notice that creating a quadrilateral is not a problem, but creating a rectangle is. They analyze the given lines and consider the attributes of a rectangle that make the second task difficult. Along the way, students are likely to recognize that, to form a rectangle, they would need two pairs of lines where each pair is always pointing in the same direction and never converging, and where the first pair make square corners when they cross the second pair (MP7).

• Groups of 2–4
• Give access to rulers or straightedges.
• “¿Qué observan acerca de las rectas del primer problema?” // “What do you notice about the lines in the first problem?” (There are three of them. They cross each other. Some are straight and some are slanted. The segments make a triangle.)
• “Las tres rectas forman un triángulo. Describan los segmentos de recta que conforman el triángulo. ¿En dónde comienzan? ¿En dónde terminan?” // “The three lines form a triangle. Describe the line segments that make up the triangle. Where do they start? Where do they end?” (The line segment starts and ends where it crosses the other lines.)
• 1–2 minutes: partner discussion
• Share responses.
• Record student observations and say “las rectas que se cruzan son rectas que se intersecan” // “lines that cross are intersecting lines.”
• “¿En dónde más han escuchado la palabra intersecan?” //“Where else have you heard the word intersect?” (Streets have intersections or places where they cross each other.)

• 4–5 minutes: independent work time
• 3 minutes: group discussion
• Monitor for students who notice that:
  • the slanted line can’t be a side of a rectangle because it intersects the horizontal and vertical lines, adding another line won’t help
  • to make a rectangle, we need two pairs of lines, where the lines in each pair never cross, but the first pair make square corners with the second pair when they cross

Students may add a line to the drawing, but not yet identify how it creates (or does not create) a quadrilateral. Consider asking:

• “¿En dónde agregaste tu nueva recta? ¿En dónde se cruza tu nueva recta con las rectas que estaban ahí?” // “Where did you add your new line? Where does you new line cross the lines that were there?”
• “¿Qué figuras nuevas se ven debido a esa recta adicional?” // “What new shapes do you see because of that extra line?“
• “¿En dónde más podrías dibujar tu recta para que se forme una nueva figura?” // “Where else could you draw your line so that it would make a new shape?”

• Display the image on the task statement. Invite students to share their responses and reasoning.
• “¿Qué tiene que ser cierto acerca de las cuatro rectas para que estas formen un rectángulo?” // “What must be true about the four lines for them to form a rectangle?” (Sample responses:
  • The lines need to make square corners whenever they cross.
  • We need 2 pairs of lines, each pair going in the same direction and never coming together, and where they cross, they need to make square corners.)
• Record examples to match student explanations or invite students to draw examples of their thinking for all to see.
• “Las rectas que nunca se intersecan, como las que están en lados opuestos de un rectángulo, se llaman rectas paralelas” // “Lines that never intersect—like those on opposite sides of a rectangle—are called parallel lines.”
• Display or add to student examples to emphasize lines and line segments:

  

  

• If time permits, consider asking, “¿Pueden encontrar un par de rectas paralelas aquí en nuestro salón de clase?” // “Can you find a pair of parallel lines here in our classroom?” (Lines on either side of a row of floor tiles, the top and bottom sides of the board, or the left and right sides of the door.)

In this activity, students are prompted to draw intersecting and parallel lines, and to explain how they know a pair of parallel lines would never intersect.

Students are not expected to formally justify that two lines are parallel. They are expected to make a case that goes beyond appearance (such as “it looks like they would never cross”) and notice that the parallel lines maintain the same distance apart (MP3). Students are also introduced to the convention of naming lines with letters to support precision when describing and comparing lines. They are not expected to formally name lines or line segments with letters.

• Groups of 2
• Give each student access to a ruler or a straightedge.
• Display a field of dots.
• Select a student to draw a line in the field.
• “Algunas veces marcamos las rectas como ayuda para indicar las diferentes partes de una figura” // “Sometimes we label lines to help communicate about different parts of a figure.”
• Demonstrate labeling the line with a letter.
• “Podemos llamarla ‘recta a’ porque la marcamos con una ‘a’” // “We can call this ‘line a’ because we labeled it with an ‘a’.”

• 5 minutes: independent work time
• 5 minutes: Discuss differences in drawings and make revisions.

Students may say that lines are intersecting only if they cross in the field of dots or that they are parallel if they do not cross in the field of dots. Consider asking:

• “¿Me puedes mostrar cómo se verían tus rectas si las extendieras por fuera del campo de puntos?” // “Can you show me what your lines would look like if you extended them outside the field of dots?”
• “¿Qué observas? Si sigues extendiendo las rectas, ¿en algún momento se van a intersecar?” // “What do you notice? If you kept extending the lines, would they ever intersect?”

• Discuss the first three questions. Select a few students to show or explain how they know their lines would never cross.
• If students suggested that they extended the lines and the lines didn’t cross, ask if there is a chance that the lines would cross if they continue to be extended.
• “¿Cómo supieron que sus rectas nunca se cruzarían?” // “How did you know that your lines would never cross?” (Sample reasoning:
  • I lined up one side of a ruler with line \(h\) or line \(g\) and traced the other side of the ruler to make a new line. The two sides of a ruler never cross.
  • The distance between the dots is always the same, so if one line is always, for instance, 2 dots away horizontally from the other line, then they would never cross.)
• If students discovered after extending lines that the lines would cross, ask: “¿Qué observan acerca de la distancia entre las dos rectas que en algún momento se van a cruzar?” // “What do you notice about the distance between the two lines that would eventually cross?” (They are getting closer.)
• “¿Qué observan acerca de la distancia entre dos rectas paralelas?” // “What do you notice about the distance between two parallel lines?” (The distance doesn’t change.)
• Discuss students’ responses to the last question. Ask students to support their predictions by extending the sides of the trapezoid (or display the image in the student response).