Geometry Goal 3

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GeometryGoal 3

• Grade 8
• NC SCOS objectives
• Sandra Davidson
• NBCT EA Math

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NC SCOS objectives (5 weeks)

• 3.01 Represent problem situations with geometric
  models.
• 3.02 Apply geometric properties and
  relationships, including the Pythagorean
  theorem, to solve problems.
• 3.03 Identify, predict, and describe dilations in
  the coordinate plane.

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1. The Race (NCTM Navigating Geometry)

• Five students in our class want to determine who
  is the fastest runner. The race will take place
  on a large square on the playground. All
  students will begin on X and run to their
  assigned point B, C, D, E, or F. The first
  person to reach their spot will be the winner.
• Is this a fair race?
• Why or why not?
• Is segment XB the same distance as segment XD?

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Looking for Squares (Connected
Mathematics-Looking for Pythagoras)

• Use the 5x5 dot paper to draw 8 squares and find
  the area of these squares. Begin by drawing the
  smallest square you can by connecting the dots to
  make a square with an area of 1 unit. Next draw
  a 2x2 square and find the area. Continue until
  you have found 8 possible squares and their
  areas. Some of these squares will be tilted
  squares.
• How is the area of each square related to the
  side length of each square?

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2. Perfect Squares and Square Roots (p.146 and
p.150)
12 1 22 4 32 9 42 16 52 25 62 36 72
49 82 64 92 81 102 100 112 121 122 144
Can you make a square out of 16 blocks or 49
blocks? Yes, because they are perfect square
numbers. The square root of 16 is 4. The square
root of 49 is 7. Place each of these perfects
squares on the number line. Can you place v32 on
a number line? Yes, it falls between 5 and 6 on
the number line!
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3. Solving Equations with Perfect Squares
(p.148)

• v64
• v121
• 2v16 5
• v(916) 7
• -v25 (v16)
• 2v(100-75)
• v64/4

A chessboard contains 32 black and 32 white
squares. How many squares are along each side of
the game board?
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4. Irrational numbers numbers that cannot be
expressed as a fraction, and decimals that
neither terminate or repeat. p,v3

• Natural numbers the counting numbers 1,2,3
• Whole numbers the set of counting numbers plus
  zero 0,1,2,3
• Integers the set of counting numbers and their
  opposites plus zero -3,-2,-1,0,1,2,3
• Rational numbers numbers that can be expressed
  as a fraction, and decimals that either terminate
  or repeat. 8.325, 0.141414
• Irrational numbers numbers that cannot be
  expressed as a fraction, and decimals that
  neither terminate or repeat. p,v3

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Locating Irrational Numbers thru Estimation
(p.150)

• State whether the number is rational, irrational
  or not a real number, then place it on a number
  line.
• v4
• v72
• -v-2
• -v36
• v22

• Name the two integers that each square root lies
  between.
• v40
• -v72
• v200
• -v340
• Clothes Line Activity

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5. Review the Real Numbers (p.156)

• Rational numbers numbers that can be expressed
  as a fraction.
• ¾
• -5
• 16
• 3.5
• v16
• -6.325

• Irrational numbers numbers that cannot be
  expressed as a fraction.
• p
• v3
• v15
• What about this? v-9
• This is not a Real Number!

Play Radical Match
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6. Pythagoras

• Pythagoras, a Greek mathematician who lived in
  600 B.C., had a devoted group of followers known
  as the Pythagoreans. The Pythagoreans had many
  rituals, and they approached mathematics with an
  almost religious intensity. Their power and
  influence became so strong that some people
  feared that they threatened the local political
  structure, so they were forced to disband.
  However, many Pythagoreans continued to meet in
  secret and to teach Pythagoreans ideas.

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Discovering the Pythagorean Theorem (Adapted from
Connected Mathematics Looking for Pythagoras)
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Discovering Pythagorean Theorem (cont.)

• 1. Using a centimeter ruler, measure the length
  of each hypotenuse on the worksheet, to the
  nearest tenth. Record this measurement in the
  table in Column 3.
• 2. Draw squares on each of the legs of the
  triangles. Record the area of the squares in
  Column 4 and 5.
• 3. Using the pre-cut squares, determine which
  square would match the length of the hypotenuse.
  Trace this square on the hypotenuse. Record this
  squares area in Column 6.
• 4. Using a calculator, find the square root (to
  the nearest tenth) of your answer in Column 6.
  Record this answer in Column 7.

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Discovering Pythagorean Theorem (cont.)

• 5. Look for a pattern in the relationship among
  the area of the square drawn on the legs and the
  area of the hypotenuses square. Make a
  conjecture about the relationship you discover.
• 6. What is the relationship between your answers
  in Column 3 and 7?
• 7. Draw a triangle on your paper with leg lengths
  of 2 and 4. What would be the length of the
  hypotenuse? Use your answers to questions 1 and
  2 to help determine the answer.
• Follow-Up Activity
• On the second worksheet are right triangles with
  the hypotenuse square drawn. Use the
  relationship you discovered above to find the
  area of the square and then find the length of
  the hypotenuse.

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7. The Scarecrows formula The Wizard of OZ

• After receiving his diploma the Scarecrow recites
  the mathematical theorem 
• "The sum of the square roots of any two sides of
  an isosceles triangle is equal to the square root
  of the remaining side." 
• Is the Scarecrow's theorem true?

• In actuality, the Pythagorean Theorem states "The
  square of the Hypotenuse of a right triangle is
  equal to the squares of the two remaining
  sides." 

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Pythagorean Proof (adapted from Navigating
through Geometry NCTM)

• In the center of your graph paper draw a right
  triangle with legs measuring 3 and 4 units.
• Draw a square along each leg. Color the small
  square blue and the other square green.
• Along the bottom of your paper cut out a 3x3,
  4x4, and 5x5 square.
• Color the 3x3 blue, and the 4x4 green.
• Cut the blue and green squares and fill the 5x5
  square. Does the 5x5 fit the hypotenuse?

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Penny Drop Game

• Work in groups of three.
• Place the large coordinate grid on the floor.
• Each student drops a red/yellow disc on the grid.
  Find the distance between the two that land as
  matching colors using the Pythagorean Theorem.
  Record this distance as your score, only the two
  students with matching colors score on that
  round. Highest score wins.

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8. Whats Your Angle, Pythagoras?

• A reading and writing activity.
• Complete the questions using complete sentences
  and proper writing conventions.

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Pythagorean Triples What do you think is meant
by the term Pythagorean triple?

• Cut out graph paper (cm) strips with lengths of
  3, 4, and 5 units.
• Can you form a right triangle with the strips?
• Cut out additional strips with lengths of 6, 7,
  8, 10, 12, 13, 16, 17, and 20.
• Verify that 5-12-13 is a Pythagorean triple by
  forming a triangle with your graph paper strips.
  Continue to complete the table.

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Given one Pythagorean triple, 3-4-5 triangle, how
can you find new ones?

• Find the missing values in each set of
  Pythagorean triples.
• 30, 40, ?
• 15, ?, 25
• ?, 44, 55
• The numbers 6 and 8 are two numbers of another
  Pythagorean triple. Compare the side lengths of
  this triangle with those of the 3-4-5 triangle to
  find the hypotenuse.
• Use what you have discovered above to discover
  some new Pythagorean triples.

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9. Using the Pythagorean Theorem(MathScape
Roads and Ramps)

• Suppose you are going to make a ramp for your
  school. It will start at the sidewalk and end at
  the top of a stairway that leads to the front of
  the school. The ramp is to cover a horizontal
  distance of 40 ft. and a vertical distance of 5
  ft.

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Using the Pythagorean Theorem(Connected Math
Looking for Pythagoras p.41)

• Choose a scale and make an accurate scale drawing
  using a ruler to solve this problem. What is the
  length of the ramp?
• Use the Pythagorean Theorem to solve the same
  problem.
• How do your two results compare? Do you think
  one method is more accurate? Explain.
• How long would a banister for the ramp be if it
  needs to extend past each end of the ramp by 1
  ft.?

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Real World Pythagorean worksheet

• A guy wire is 25m long. It is attached to an
  anchor on the ground 7m from the base of the TV
  tower. How tall is the tower?
• John left his campsite to go on a bike ride. He
  plans to keep in touch with his father by a
  walkietalkie system which has a range of 5
  miles. John walks 4.3 miles north and then 2.4
  miles east. Will he be able to talk to his
  father?

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Review Real World Pythagorean

• 3. A ramp is designed to help individuals in
  wheelchairs move from one level to another. What
  is the height of the ramp?

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Review Real World Pythagorean

• 4. Mark is building a pyramid with a square base.
  Each side of the base is 7 feet long. The
  isosceles triangles that make the sides of the
  pyramid have an altitude of 6 feet. If Mark is
  53 tall, can he stand up in the pyramid?

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Review Real World Pythagorean

• 5. A conveyor belt moves boxes up this ramp. If
  the ramp has the dimensions shown, how far do the
  boxes move along the belt?

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Review Real World Pythagorean

• 6. Pat believes that Interstate 36 makes a right
  angle with Interstate 18 in Stewartsville. She
  is traveling along I-36 and plans to get on I-18
  in Stewartsville and then travel down I-18 to
  Morton. She is now 38 miles from Stewartsville.
  Morton is another 15 miles from Stewartsville.
  Pat has found a country road going from her
  present location straight to Morton. If she gets
  off the interstate, how many miles will she save.

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Review Real World Pythagorean

• 7. How high up on the house does the ladder
  reach?

• 8. Stop Sneaky Sally as she tires to steal from
  first base to second base. If it is 90 ft.
  between bases, how far must the catcher throw the
  ball to get her out at second?

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10. Real World Algebra - Pythagorean Theorem

• Choose a level and complete the worksheet for
  that level.
• (level I) p. 109-110, do problems 1-10
• (level II) p. 111-112, do problems 1-10
• (level III) p. 113-114, do problems 1-10
  (complete 6)

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11,12. Review and Test

• Internet Practice A
• http//regentsprep.org/Regents/math/fpyth/PracPyth
  .htm
• Textbook Practice (choose one from each section)
• Section A
• (level I) p. 292 (1-7)
• (level II) p. 292 (8-15)
• Section B
• (level I) p. 293 problems 31 and 33
• (level II) p. 293 problem 32
• (level III) p. 293 problems 30 and 36
• Analyze the Wheel of Theodorus (p.54-55) C.M.
  Looking for Pythagoras Labsheet 5.1

Test Tomorrow
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13. Dilating Figures (using a center
point) (Navigating through Geometry - NCTM)

• Sheri found a good way to change the size of a
  figure. She is dilating pentagon ABCDE 200 to
  get a second pentagon ABCDE.
• Describe how you think Sheri is making her
  drawing.
• Complete the dilation, using T as the center.
• What do you notice about the two figures?
• What is the scale factor?

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Using Scale Factors(NCTM Navigating Geometry)

• Draw rays from the projection point through the
  vertices. Multiply the length of this line by
  the scale factor given to locate the vertices of
  the new scaled figure.

Using point P as the projection point and a scale
factor of 2, locate the scaled image ABC of
triangle ABC.
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Using Scale Factors (cont.)

• Draw rays from the projection point through the
  vertices. Multiply the length of this line by
  the scale factor given to locate the vertices of
  the new scaled figure.

Use point P as the projection point and a scale
factor of 3/4, locate the scaled image ABCD
of rectangle ABCD.
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14. Dilating Figures (using Coordinate Points)

• The dilated image of figure QRST is figure
  QRST.
• What is the ordered pair for each coordinate
  point?
• What is the scale factor for the dilation
  QRST?

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Let Your Little Light ShineNC DPI Strategies

• List the ordered pairs for each point labeled on
  the graph. Now create a dilation of the
  lighthouse.
• Using a scale factor of ½, ¾, 2, or 3, make a
  list the new ordered pairs.
• Draw a second lighthouse on a new sheet of graph
  paper.
• What happened to the area of the small rectangle
  near the top?

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15. Coordinate Plane Dilations NCDPI Classroom
Strategies

• Working in groups of four students, you will be
  assigned the ordered pairs of a specific shape.
• Each student in the group will be given a
  different scale factor, and will draw their
  scaled shape on graph paper.
• Use patty paper to compare angles for congruency
  and measure side lengths for proportionality.
• Each group will then create one large graph of
  the original shape and all four dilations.

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16. Dilation Match Game

• Your group has been given several sets of
  different dilations on graph paper index cards.
  Each card has an original figure and a dilation
  of the original.
• Your group will get a set of index cards. One
  person will be the checker for the group. Each
  set of cards will have an answer key that only
  the checker is allowed to see.
• The different scale factors are on cardstock.
  Lay the scale factor cards out so they can be
  easily seen by everyone in the group.
• The group then looks at the graphs and matches
  each one with the correct scale factor. When the
  group feels all graphs and scale factor cards are
  matched correctly, the checker determines if
  they match the answer key.
• First group to match all scale factor cards to
  their dilations, wins!

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17. Dilation Assessment (Indicators Rotation)

• Copy NCDPI Indicators on cards and place at
  different stations around the room.
• Students rotate at 1 min. intervals around the
  room, completing each indicator, writing their
  answer on an answer sheet.

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18. Project - Cartoon Dilation

• Find a simple cartoon figure about 3 or 4 inches
  tall.
• Trace the figure onto graph paper and list more
  than 50 coordinate points for the figure.
• Enlarge the figure by a scale factor or 2 or 3,
  making a new list of coordinate points.
• Graph the new coordinate points on new graph
  paper, enlarging the figure by dilation.
• Outline and color as a finished product.

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References

• Balanced Assessment Middle Grades
• Connected Mathematics, Looking for Pythagoras
• Exemplary Mathematics Assessments Tasks for
  Middle grades
• Mathscapes
• NCDPI Strategies
• NCTM Navigating Through Geometry Grades 6-8
• Real World Algebra
• Whats Your Angle, Pythagoras? by Julie Ellis