Lesson 3

The purpose of this warm-up is for students to think about points on the axes. In previous lessons they have plotted points with non-zero coordinates. Thinking about the points with zero prepares them for plotting points on the horizontal and vertical axes which they will do in this lesson. 

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses. 

• “¿Cómo podemos describir la ubicación del punto?” // “How can we describe the location of the point?” (It is at the bottom left of the grid.)
• “Las coordenadas de este punto son \((0, 0)\). ¿Cuáles serían las coordenadas del punto si lo moviéramos 2 unidades hacia arriba?” // “The coordinates of this point are \((0, 0)\). What would be the coordinates of the point if we moved it up 2 units?” (0, 2)

The purpose of this activity is for students to plot several points with the same vertical or horizontal coordinate and observe that they lie on a horizontal or vertical line respectively (MP7). Students also plot points on the axes for the first time. Before plotting the points on a grid with gridlines, students first estimate the location of the points. This encourages them to think about the coordinates as distances (from the vertical axis for the first coordinate and from the horizontal axis for the second coordinate).

• Groups of 2
• “Ustedes y su compañero van a completar cada uno un grupo distinto de 4 problemas. Cuando terminen, discutan su trabajo con su compañero” // “You and your partner will each complete a different set of 4 problems independently. After you’re done, discuss your work with your partner.”

• 5–7 minutes: independent work time
• 5 minutes: partner discussion
• Monitor for students who:
  • use the halfway point on each axis as a benchmark for the coordinate grid without gridlines
  • start at zero and count spaces along each axis for the marked coordinate grid with gridlines
  • recognize the points should be aligned because they share a common horizontal or vertical coordinate

If students do not reasonably estimate a location for a point on the blank grid, point to the location half way between 0 and 5 on the horizontal axis and ask, “¿Qué número podría ir aquí?” // “What number might go here?”

• Ask previously identified students to share their thinking.
• “¿Qué podemos decir sobre un conjunto de puntos cuando ellos tienen la misma primera coordenada?” // “What can we say about a set of points when they share the same first coordinate?” (They will be on the same vertical line.)
• Display image from student solution showing points with first coordinate 5.
• “¿Cómo supieron dónde ubicar el punto que tenía coordenadas \((5,0)\)?” // “How did you know where to put the point with coordinates \((5,0)\)?” (I put it on the horizontal axis. I went over 5 but did not go up at all.)
• “¿Qué ocurre cuando los puntos de un conjunto tienen la misma segunda coordenada?” // “What happens when a set of points share the same second coordinate?” (They will be on the same horizontal line.)
• Display image from student solution showing points with second coordinate 3.
• “¿Qué nos dice el cero que aparece en (0,3)?” // “What does the zero in (0,3) tell us?” (It means the point will be on line zero of the horizontal axis, which is the vertical axis.)
• “\((0, 0)\) es un punto importante porque es donde empezamos cuando ubicamos un punto en la cuadrícula de coordenadas. Encuentren \((0, 0)\) en la cuadrícula con la que han estado trabajando” // “\((0, 0)\) is an important point because it's where we start when we plot a point on the coordinate grid. Find \((0, 0)\) on the grid you have been working with.”

In the previous activity, students noticed that points with the same vertical coordinate lie on a horizontal line and points with the same horizontal coordinate lie on a vertical line. The purpose of this activity is to reinforce this idea by asking students to plot points on a blank coordinate grid given a single point. Using this point, they can plot other points with the same horizontal or vertical coordinate. Some students may keep one coordinate the same and double or halve the other as is shown in the student solution. Other students may label points on the axes. The given point, for example, allows students to identify \((4,0)\) and \((0,2)\). An advantage to working with points on the axes is that these are essentially number lines with which students have been working for several years. The synthesis highlights the special nature of the points \((1,0)\) and \((0,1)\). All of the points where the gridlines meet can be measured off exactly after \((1,0)\) and \((0,1)\) are plotted. 

•  Groups of 2

• 3–5 minutes: independent work time
• 5 minutes: partner discussion

If students need support getting started with the task, ask, “El punto que está marcado es \((4,2)\). ¿Qué sabes sobre él?” // “What do you know about the point \((4,2)\) that is plotted?”

• Invite students to share the points that they plotted and their reasoning.
• Plot the points as they discuss their reasoning.
• “¿Cómo ubicaron \((1,0)\)?” // “How did you plot \((1,0)\)?” (I know where \((2,0)\) is on the vertical axis because it has the same horizontal coordinate as \((2,4)\). Then I just halved the distance to the vertical axis and that's \((1,0)\).)
• “Si ya saben dónde está \((1,0)\), ¿qué otros puntos pueden ubicar en el eje vertical?” // “Once you know where \((1,0)\) is, what other points can you locate on the vertical axis?” ((2,0), (3,0), (4,0),... I can just keep marking off that distance like I do when I am on a number line.)