This activity gives students an opportunity to determine and request the information needed to recreate a design made out of different lines. Thanks to Henri Picciotto for permission to use these designs.

The goal is for students to recognize the information that determines the location of a line in the coordinate plane.  Acknowledge students who use the following strategies, but keep the discussion focused on describing each line using slope and intercepts or coordinates of points on the line.

• Describing one line as a vertical translation up or down of another line

• Describing a line as parallel to another line and containing a particular point

• Giving an equation to describe the line

Students are not expected to communicate by saying the equations of the lines, though there is nothing stopping them from doing so. 

Here are some questions for discussion:

• "What details were important to pay attention to?" (the slope of the line, whether the slope was positive or negative, the vertical and horizontal intercepts)

• "How did you use coordinates to help communicate where the line was?" (Coordinates could tell a location that the line had to go through.)

• "How did you use slope to communicate how to draw the line?" (Slope told my partner how steep to draw the line and if it was going uphill or downhill.)

• "Were there any cases where your partner did not give enough information to know where to draw the line? What more information did you need?" (Answers vary.)

Then display this blank coordinate plane, or one similar, for all to see. 

Plot and label the point on the coordinate axes. Ask students to explain how to draw a line with a slope of -2 that goes through the point on the display. Demonstrate how to use the plotted point as one vertex of a slope triangle with a vertical length of 2 units and a horizontal length of 1 unit, and then draw in the third side of the triangle using a straightedge or ruler. Discuss:

• “Can the point and slope used describe any other line on the coordinate plane?” (No)

• “Can this same line be described in a different way?” (Yes)

• “What are some other ways?” (a line with slope that goes through the point  or any other point on the line, a line with slope with a vertical intercept of 9, the line or equivalent)

This activity provides additional practice for students to calculate the slope of a line given two points. 

The goal of this discussion is for students to understand that when calculating slope, the order of the points does not matter as long as the same order is used for calculating both the vertical and horizontal change. 

Invite previously selected students to share their strategy for calculating the slope of one of the pairs of points. Include one student who subtracted coordinate values of the first point from the second, and another student who subtracted the values of the second point from the first. Here are some questions for discussion: 

• “What do you notice about both approaches?” (Both result in the same value for slope. They subtracted their numbers in the opposite order. They kept the same order when calculating both the vertical and horizontal change.)

• “Are there any advantages to picking one point to use first or second?” (Subtracting the numbers in a certain order may result in “easier” calculations.)

If time allows, invite a student who drew a sketch of the two points to share their reasoning. While students can use any strategy to calculate slope, emphasize the sketch as a good way to check that the sign of the slope matches the direction of the line.