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The Pythagorean Theorem, Distance, Surface Area, and Volume

Unit Plan

The Pythagorean Theorem, Distance, Surface Area, and Volume

Objectives

Students will develop a general understanding of geometric formulas including the Pythagorean theorem and formulas for the surface area and volume of solids. Students will:

  • use the Pythagorean theorem to find the lengths of sides of right triangles.
  • use the Pythagorean theorem to prove that triangles are right triangles given the side lengths.
  • find the distance between points on the coordinate plane.
  • apply problem-solving tools to find surface area and volume.

Essential Questions

  • How can recognizing repetition or regularity assist in solving problems more efficiently?
  • How are spatial relationships, including shapes and dimension, used to draw, construct, model, and represent real situations or solve problems?
  • How can the application of the attributes of geometry shapes support mathematical reasoning and problem solving?
  • How can geometric properties and theorems be used to describe, model, and analyze situations?

Related Unit and Lesson Plans

Related Materials & Resources

Formative Assessment

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    Multiple-Choice Items:

    1. Which of the following must be known to use the converse of the Pythagorean theorem?

    A

    The triangle is a right triangle.

    B

    The measure of all the angles in a triangle.

    C

    The triangle has a longest side.

    D

    The lengths of all three sides of a triangle.

    1. One leg of a right triangle is 15 units long. The length of the hypotenuse of the triangle is
      17 units. What is the length, in units, of the other leg of the right triangle?

    A

    2

    B

    4

    C

    8

    D

    64

    1. The lengths of three sides of a triangle are 20, 21, and 29. Which statement is true?

    A

    The triangle is a right triangle because
    20 + 21 > 29.

    B

    The triangle is a right triangle because
    202 + 212 = 292.

    C

    The triangle is not a right triangle because
    20 + 21 > 29.

    D

    The triangle is not a right triangle because
    202 + 212 = 292.

    1. What is the distance between (0, 0) and (5, 12)?

    A

    5

    B

    7

    C

    12

    D

    13

    1. What is the distance between (3, 4) and (5, 10) to the nearest unit?

    A

    6

    B

    8

    C

    20

    D

    40

    1. The distance between (3, 7) and (x, 12) is between 7 and 8. Which of the following is a possible value for x?

    A

    −5

    B

    −2

    C

    2.5

    D

    10.5

    1. The volume of an object is measured in how many dimensions?

    A

    0

    B

    1

    C

    2

    D

    3

    1. What is the surface area of a sphere with a radius of 4 centimeters?

    A

    64π square centimeters

    B

    16π square centimeters

    C

    64 square centimeters

    D

    16 square centimeters

     

    1. What is the volume of a sphere that has a circumference of 6π feet?

    A

    288π4 cubic feet

    B

    36π cubic feet

    C

    36π4 cubic feet

    D

    288 cubic feet

    Multiple-Choice Answer Key:

    1. D

    2. C

    3. B

    4. D

    5. A

    6. B

    7. D

    8. A

    9. B

     

     

    Short-Answer Items:

    1. A right triangle has one side that is 6 units long and another that is 8 units long. What are the possible lengths of the third side of the triangle?
    2. Point P is located at (−2, 5) and point Q is located at (7, 3). Explain how to use the Pythagorean theorem to find the distance between the two points, and then find the distance.
    3. Find the surface area and volume of a rectangular prism with a length of 3 inches,
      a width of 5 inches, and a height of 20 inches. Include the appropriate labels with your answers. Show all of your work.

    A

    Surface Area =

    B

    Volume =

     

    Short-Answer Key and Scoring Rubrics:

    1. A right triangle has one side that is 6 units long and another that is 8 units long. What are the possible lengths of the third side of the triangle?

    Answer:  and 10  

     

    Points

    Description

    2

    • Student gives both possible lengths of the third side of the triangle.

    1

    • Student gives one of the two possible lengths of the third side of the triangle.

    0

    • Student does not give either possible length of the third side of the triangle.
    • Student demonstrates no understanding of the Pythagorean theorem.

     

    1. Point P is located at (−2, 5) and point Q is located at (7, 3). Explain how to use the Pythagorean theorem to find the distance between the two points, and then find the distance.

    Answer: To find the distance, I would draw a horizontal line and vertical line so they form the legs of a right triangle through P and Q. The horizontal leg has a length of 9 and the vertical leg has a length of 2, so the Pythagorean theorem says 92 + 22 = c2 where c is the distance between P and Q. The distance is .

     

    Points

    Description

    2

    • Student explains how to use the Pythagorean theorem to find the distance between two points.
    • Student finds the distance between P and Q.

    1

    • Student makes minor mathematical or computation errors.
    • Student explains how to use the Pythagorean theorem to find the distance between two points but does not find the distance correctly. OR Student finds the distance correctly but does not adequately explain how to use the Pythagorean theorem to do so.

    0

    • Student does not explain how to use the Pythagorean theorem to find the distance between two points and does not find the distance correctly.
    • Student demonstrates no understanding of how to use the Pythagorean theorem to find the distance between two points.

     

     

    1. Find the surface area and volume of a rectangular prism with a length of 3 inches, a width of 5 inches, and a height of 20 inches. Include the appropriate labels with your answers. Show all of your work.

    A

    Surface Area =

    B

    Volume =

    Answer: A. SA = 350 inches2; V = 300 inches3

    Points

    Description

    2

    • Answers are correct with no mathematical or calculation errors.
    • Units are named correctly.
    • Student demonstrates thorough understanding of surface area and volume of rectangular prisms.

    1

    • Minor mathematical or calculation errors,  but units are correctly reported.
    • Answers are correct, but units are incorrect.
    • Student demonstrates partial understanding of surface area and volume of rectangular prisms.

    0

    • Answers are incorrect with major mathematical or calculation errors.
    • Units are reported incorrectly.
    • Student demonstrates no understanding of surface area and volume of rectangular prisms.

     

    Performance Assessment:

    The following table contains some information about a few of the planets in our solar system. Determine the missing information by using the formulas for surface area and volume of spheres.

     

    Planet

    Diameter

    Circumference

    Surface Area

    Volume

    Mercury

    7,523 miles

     

     

     

    Earth

     

    24,900 miles

     

     

    Jupiter

    44,423 miles

     

     

     

     

    What measurement do you prefer to use to calculate answers to this problem and why?

     

    Performance Assessment Answer Key and Scoring Rubric:

    The following table contains some information about a few of the planets in our solar system. Determine the missing information by using the formulas for surface area and volume of spheres.

    Planet

    Diameter

    Circumference

    Surface Area

    Volume

    Mercury

    7,523 miles

    23,634.20 miles

    177,800,098.13 miles2

    222,931,689,708.38 miles3

    Earth

    7925.92  miles

    24,900 miles

    197,355,312.53 miles2

    260,703,610,340.86 miles3

    Jupiter

    44,423 miles

    139,558.97 miles

    6,199,628,144.32 miles2

    45,901,013,509,180.34 miles3

    What measurement do you prefer to use to calculate answers to this problem and why?

    Answers may vary.

    Diameter or circumference because it’s easier to multiply than to find the square root or cube root.

    Points

    Description

    4

    • No math/calculation errors are made.
    • Explanations are complete and correct.
    • Student demonstrates thorough understanding of the surface area and volume of spheres, scientific notation, and irrational numbers.

    3

    • No major math errors or conceptual/procedural errors are made.
    • One of the explanations or values is incorrect or incomplete.
    • Student demonstrates substantial understanding of the surface area and volume of spheres, scientific notation, and irrational numbers.

    2

    • Response has some parts missing or contains several minor errors or one or more serious math errors or conceptual/procedural errors.
    • Explanations are incomplete or more than one answer is incorrect.
    • Student demonstrates partial understanding of the surface area and volume of spheres, scientific notation, and irrational numbers.

    1

    • Major mathematical or computational errors are made.
    • Student only shows understanding of one of the concepts.
    • Student demonstrates minimal understanding of the surface area and volume of spheres, scientific notation, and irrational numbers.

    0

    • All sections are unfinished or incorrect, or they contain major mathematical errors or serious conceptual/procedural errors.
    • Explanations are incorrect or missing.
    • Student demonstrates no understanding of the surface area and volume of spheres, scientific notation, and irrational numbers.
Final 04/26/13
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