School Improvement in Maryland

Lesson Plan: Lesson plans were written by Maryland mathematics educators and could be used when teaching the concepts.

Goal 2 Geometry, Measurement, And Reasoning

Expectation 2.1 The student will represent and analyze two- and three-dimensional figures using tools and technology when appropriate.

Indicator 2.1.2 The student will identify and/or verify properties of geometric figures using the coordinate plane and concepts from algebra.

Lesson Content

Squares and Parallelograms in the Coordinate Plane

Objective

Students will be able to find the coordinates of the vertices of special quadrilaterals placed in the coordinate plane.

Approximate Time

One 45-minute lesson (Other quadrilaterals may be done as you study them.)

Prerequisite Concepts Needed

Students should be able to plot ordered pairs. All quadrilaterals in this lesson will be drawn in the first quadrant and will be labeled in alphabetical sequence, counter-clockwise. (HSA labels figures in alphabetical sequence, clockwise).

Materials Needed

Lesson Structure

    Warm-Up/Opening Activity

    Plot segment AB. Given A (2, 0) and B (6, 0), draw a square in the first quadrant that has AB as one side.
     
    The teacher should draw AB on the overhead sheet. After students have drawn their square, have them label the vertices so that they have square ABCD. They should also write the pair of coordinates near the appropriate vertex. Students can draw and label as the teacher graphs on the overhead sheet.
     
    Answers: C (6, 4) and D (2, 4)

    Development of Ideas

    Key to the lesson: Today we are going to find the coordinates of missing vertices of squares and parallelograms. We will apply properties of quadrilaterals and what we know about graphing to find these vertices.
     
    Let's try another square that is not as easy. Plot A (3, 5) and B (9, 5).
     
    Now try to find the coordinates for C and D so that a square is formed.
    Answers: C (9, 11) and D (3, 11)
     
    After students have finished, discuss the relationships between the sides of the square. The length of AB is 6 so they know that the length of each side will be 6.
     
    What is special about the coordinates of C? or what pattern do you see that can help you find C?
     
    Assist students to see that the x value will be the same as the x value for B. The y value of C is computed by adding the length 6 to the y value of B.
     
    What patterns can we see for point D?
     
    Show:
    D (3, 5 + 6) or (3, 11)
    A (3, 5)
    C (9, 5 + 6) or (9, 11)
    B (9, 5)


    How do we know we have a square? Use mathematics to justify your answer.
     
    Students should be able to tell you that you need to show 4 congruent sides as well as at least one right angle. You can then divide up the labor of using the distance formula and the slope formula. Side length = 6 and slopes of 2 consecutive sides: 0 and undefined. Then have students write a justification, i.e., Because ABCD has 4 congruent sides and a right angle, we know from the theorems of parallelograms and by definition of a square that ABCD is a square.
     
    Let's find the coordinates of the missing vertices when we are given variable coordinates. Given the coordinates A (m, n) and B (g, n), we can roughly plot these points. It is helpful to students to draw and label the axes with the variables as shown below:
     
    Chart 1
     
    What is the length of AB? (g - m)
     
    What are the coordinates of C and D? Give students time to work on this. Working in pairs might also be helpful.
    Solution: C (g, n + (g - m)) and D (m, n + (g - m)).
     
    Worksheet: Finding Missing Vertices of Squares and Parallelograms

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