In this activity, students extend the vertical and horizontal axes to create the 4 regions, called quadrants of the coordinate plane. The coordinate plane is a system that can be used to communicate the locations of points. Students plot points and identify in which quadrant they are located.

The key idea for students to understand is that the coordinate plane is formed by two axes, which are vertical and horizontal number lines. Just as number lines were extended to include negative numbers, these axes have been extended to create the 4 quadrants. Points in these quadrants can be described by using negative and positive numbers as the - and -coordinates. Invite students to share their reasoning about how to identify the quadrants for the points , , and . As time allows, consider asking the following questions:

• “If a point has a negative -coordinate, what quadrants could it be in?” (Quadrants II and III)
• “If a point has a negative -coordinate, what quadrants could it be in?” (Quadrants III and IV)
• “If a point has a positive -coordinate, what quadrants could it be in?” (Quadrants I and IV)

To involve more students in the conversation, consider asking:

• “Do you agree or disagree? Why?”
• “Who can restate ___’s reasoning in a different way?”
• “Does anyone want to add on to _____’s reasoning?”

In this activity students draw their own axes for different sets of coordinates. They must decide which of the four quadrants they need to use and how to scale the axes (MP6).

Monitor for groups who scale the axes in these different ways for the first set of coordinates, ordered from more common to less common.

• Divide the grid into 4 equal-size quadrants. Scale both the - and -axes using the same scale.
• Divide the grid into 4 equal-size quadrants. Scale the -axis and the -axis using different scales.
• Divide the grid into 4 unequal-size quadrants.

The key takeaway from this discussion is that defining axes and scale is a process of reasoning, and that details such as the amount of empty space and the size of the numbers being plotted need to be considered.

Invite previously selected students to share their scaled axes for the first set of coordinates. Sequence the discussion of the approaches in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions, such as:

• “Does this set of axes allow for all the coordinates to be plotted?” (Yes.)
• “What is different from previous approaches?” (The - or -axis counts by a different amount. The - or -axis is in a different location [higher, lower, farther left or right].)
• “How does this change show the coordinates differently?” (The locations of points are more accurate because each point can be plotted at the intersection of grid lines. There is less empty space. Points are located more in the center of the space instead of at the edges.)
• “What strategies can we come up with to help when drawing and labeling axes in the future?” (Look at the minimum and maximum values that need to be displayed, and count how many grid lines there are available.)

In this activity, students locate and express coordinates in all four quadrants as they navigate through a maze. Students plan their route through the maze and strategically choose coordinates to correctly execute their plans (MP1). Consider using this activity if students would benefit from additional practice naming coordinates in all four quadrants.

The goal of this discussion is for students to share how they determined the coordinates for each point. Tell students to compare their coordinates for each point in the maze with their partner. Consider discussing the following questions:

• “Were there any disagreements between you and your partner? How did you resolve them?” (Answers vary.)
• “Did you notice anything about the coordinates of points at the ends of the same line segment?” (They always had either the - or -coordinate the same.)
• “Why do the two points at the end of a line segment always share a common - or -coordinate?” (The line segment is part of a horizontal or vertical line.)
• Can you think of any other situations where coordinates are used to communicate location?” (In modern navigation systems, directions are precisely given in terms of coordinates and are then translated into a visual display for the driver. Precise coordinates are also used to navigate through virtual space in computer simulations.)