Solve+problems+involving+the+distance+between+points

Algebra I SPI3102.4.3 Solve problems involving the distance between points or midpoint of a segment. James Tant

This lesson involves using a circle on the coordinate plane to help students understand distance and midpoint. The students should be familiar with the distance and midpoint formula as well as diameter and center of a circle before this lesson is used. The graphs of the circles are created in Geometry Sketchpad and are pasted into this Wiki as a printable network graphic.

Algebra I Standards 2009 to 2010 SPI 3102.4.3 Solve problems involving the distance between points or midpoint of a segment.

Using the graph shown below calculate the diameter and center of the circle using the distance formula and the midpoint formula..

Created with GSP and posted as a printable network graphic.

The students should be able to readily see that the diameter of this circle goes from (-5,0) to (5,0) on the x axis and the diameter is 10 units. The students should also see that the center of the circle is located at (0,0). The same is true for the points (0,-5) and (0,5) on the y axis. Divide the students in half and ask half to use the x-axis points and half to use the y-axis points with the distance formula to verify that the distance is indeed 10 units. Ask the two groups of students to use the midpoint formula to verify that the center of the circle is at (0,0). Break the students into four groups and assign each group a quadrant to work in. Explain that there are two easily discernible points in each quadrant other than the x-axis and y-axis points. Together the group is to find the two points and match them to two points in the opposite quadrant that create a diameter. Then the group is to divide into two smaller equal sized groups. One group is to use one set of points, that form the diameter, and the other group is to use the other set of points, that form the diameter, along with the distance formula and the mid-point formula and then calculate the diameter and the center of the circle. Have each group go to the whiteboards and show their work on with the pair of points they are using. Each group will be responsible for explaining how their set of points works in the distance and midpoint formula. Return the students to a whole group setting and discuss their learning.

The next graph you encounter will be exactly like the first except the center of the graph has moved from the origin to the point (2,1).

This graph was created with GSP and posted as a printable network graphic.

The students should be able to readily see that the diameter of this circle goes from (-3,1) to (7,1) parallel to the x axis and the diameter is 10 units. The students should also see that the center of the circle is located at (2,1). The same is true for the points (2,-4) and (2,6) parallel to the y axis. Divide the students in half and ask half to use the horizontal diameter points and half to use the vertical diameter points with the distance formula to verify that the distance is indeed 10 units. Ask the students to use the midpoint formula to verify that the center of the circle is at (2,1). Break the students into four groups and assign each group a quadrant to work in. Explain that there are two easily discernible points in each quadrant other than the horizontal diameter and vertical diameter points. Together the group is to find the two points and their matching points in the opposite quadrant. Then the group is to divide into two smaller equal sized groups. One group is to use one set of points and the other group is to use the other set of points along with the distance formula and the mid-point formula and then find the diameter and the center of the circle. Have each group go to the whiteboards and show their work with the pair of points they are using. Each group will be responsible for explaining how their set of points works in the distance and midpoint formula. Return the students to a whole group setting and discuss their learning.

This lesson will be more dynamic if Geometry Sketchpad is used live during the class. Different pages could be created and students challenged to create their own circles in on the coordinate plane.