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Geometry on the Coordinate Plane

Unit Plan

Geometry on the Coordinate Plane

Objectives

In this unit, students will use slope, the distance formula, and the midpoint formula to find properties of geometric figures in the coordinate plane. Students will:

  • learn how the slopes of parallel and perpendicular lines are related.
  • use the distance formula to find lengths of sides of geometric figures.
  • use the midpoint formula to find the middle of a line segment.
  • learn the difference between rigid transformations and dilations.

Essential Questions

  • How can you use coordinates and algebraic techniques to represent, interpret, and verify geometric relationships?

Related Unit and Lesson Plans

Related Materials & Resources

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Formative Assessment

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    Multiple-Choice Items:

    1. The equation of a line is . What is the equation of the line perpendicular to that line?

     

    2. What is the distance between the points (−2, 4) and (6, −2)?

    A

    10

    B

    14

    C

    48

    D

    100

     

     

    3. What is the midpoint of (−5, 12) and (7, −8)?

    A

    (−1, −2)

    B

    (1, 2)

    C

    (−6, −10)

    D

    (6, 10)

     

     

    4. What is the rotated image of the figure below?

         

    5. Which transformation moves an object without changing its orientation?

    A

    Dilation

    B

    Reflection

    C

    Rotation

    D

    Translation

     

     

    6.What is the reflected image of the object below?

     

     

     

    7. Dilations change the ____________ of an object.

    A

    Center

    B

    Orientation

    C

    Shape

    D

    Size

     

     

    8. Which of the following is NOT a dilation of the object below?

     

     

     

     

    9. What are the coordinates of the image point of (4, −6) when it is dilated by a factor of 2?

    A

    (6, −4)

    B

    (8, −12)

    C

    (2, −3)

    D

    (2, −8)

     


    Multiple-Choice Answer Key:

    1. D

    2. A

    3. B

    4. B

    5. D

    6. C

    7. D

    8. A

    9. B

     

     

     

     

     

    Short-Answer Items:

    10. Use the distance formula to find the diameter of the circle below. Then use the midpoint formula to find the center of the circle.

     

     

     

    11. List the transformation(s) that were/was performed to get from Figure X to Figure Y.

     

     

     

     

    12. A triangle in the coordinate plane has vertices at (−4, 8), (−12, −4) and (2, 0). The triangle was dilated by a factor of . What are the vertices of the image?

     

     

    Short-Answer Key and Scoring Rubric:

    10. Use the distance formula to find the diameter of the circle below. Then use the midpoint formula to find the center of the circle.

    diameter = 10 units; center = (−6, −2)

     

    Points

    Description

    2
    • The student used the distance formula correctly and got the right answer.
    • The student used the midpoint formula correctly and got the right answer.

    1
    • The student only used one of the formulas correctly and got one right answer.

    0
    • The student didn’t use either formula correctly and didn’t get a right answer.

     

     

    11. List the transformations that were performed to get from Figure X to Figure Y.

    The figure was:

    • reflected horizontally, or reflected vertically and rotated 180°, or reflected horizontally and translated up or down.
    • rotated 180° clockwise

    Points

    Description

    2
    • The student used the terminology correctly.
    • The student got the right answer.

    1
    • The student used the terminology correctly.
    • The student did not get the right answer.

    0
    • The student did not use the terminology correctly.
    • The student did not get the right answer.

     

     

    12. A triangle in the coordinate plane has vertices at (−4, 8), (−12, −4) and (2, 0). The triangle was dilated by a factor of . What are the vertices of the image?

    (−1, 2), (−3, −1) and (, 0)

    Points

    Description

    2
    • The student correctly multiplied the coordinates by  and got the right answer.

    1
    • The student multiplied the coordinates by 4 instead of .

    0
    • The student didn’t multiply the coordinates by a factor of anything.

     

     

     

    Performance Assessment:

    Give students the Performance Assessment Worksheet (M-G-5_Performance Assessment Worksheet.doc). They will:

    1. Draw a geometric figure on the axes and label the coordinates of the vertices.
    2. Calculate the slope of one of the sides of their figure.
    3. Use the distance formula to calculate the distance from the top of the figure to the bottom.
    4. Use the midpoint formula to calculate the midpoint from the left of the figure to the right.
    5. Perform the following transformations and draw the image after each transformation.
      1. Rotate the figure 90° clockwise.
      2. Translate the figure 4 units down and 3 units to the left.
      3. Reflect the figure over the line x = −2.
      4. Dilate the figure by a factor of 3.


    Performance Assessment Answer Key and Scoring Rubric:

    Specific answers vary; sample response:
    triangle with vertices A (5, 5), B (−2, 4), C (1, 1).

    1. Draw a geometric figure on the axes and label the coordinates of the vertices.



    2. Calculate the slope of one of the sides of their figure.
    slope of

     

    3. Use the distance formula to calculate the distance from the top of the figure to the bottom.

    distance

    4. Use the midpoint formula to calculate the midpoint from the left of the figure to the right.

    midpoint of

     

    5. Perform the following transformations and draw the image after each transformation.

    a. Rotate the figure 90° clockwise.

    A′ (5, −3), B′ (4, 4), C′ (1, 1)

     

    b. Translate the figure 4 units down and 3 units to the left.

    A′ (−5, 0), B′ (2, 1), C′ (−2, −3)

     

     

     

     

    c. Reflect the figure over the line x = −2.

    A′ (−9, 5), B′ (−2, 4), C′ (−5, 1)

     

     

     

    d. Dilate the figure by a factor of 3.

    A′ (15, 15), B′ (−6, 12), C′ (3, 3)

     

     

    Points

    Description

    4

    The student:

    • drew a geometric figure and correctly labeled the vertices.
    • calculated the slope of one of the sides and showed work.
    • calculated the distance from the top of the figure to the bottom and showed work.
    • calculated the midpoint from the left of the figure to the right and showed work.
    • performed all four transformations correctly and drew the image after each one.

    3

    The student:

    • drew a geometric figure and correctly labeled the vertices.
    • calculated the slope of one of the sides.
    • calculated the distance from top to bottom.
    • calculated the midpoint.
    • performed three of the four transformations correctly and drew the image.

    2

    The student:

    • drew a geometric figure.
    • calculated the slope of one of the sides.
    • calculated either the distance or the midpoint but not both.
    • performed two of the four transformations correctly and drew the image.

    1

    The student:

    • drew a geometric figure and didn’t label the vertices.
    • calculated the slope, distance, or midpoint but not all three.
    • performed one of the four transformations correctly.

    0

    The student:

    • drew a geometric figure.
    • did not calculate slope, distance, or midpoint.
    • did not perform any of the four transformations correctly.

     

DRAFT 10/13/2011
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