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					                                                                                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement
                                                     Ascension Parish Comprehensive Curriculum
                                                                 Concept Correlation
                                                         Unit 4: Geometry and Measurement

                                                                       Time Frame: 4 Weeks
Big Picture: (Taken from Unit Description and Student Understanding)
    Geometric figures can change position and maintain the same attributes on a coordinate plane.
    Geometric figures can change size and/or position while maintaining proportional attributes.
    The Pythagorean Theorem can be used to solve problems involving right triangles.
    Constructions are based on properties of geometric figures.

                                             Activities                                                                   Documented GLEs
     Guiding Questions              Essential Activities are denoted    GLE’s                                   GLEs                              Date and Method of
                                            with an asterisk                                                                          GLEs
                                                                                                            Bloom’s Level                             Assessment
Concept 1: Angle                   *Activity 46: The                                             Define and apply the terms
Relationships                      Bisection!                                                    measure, distance, midpoint, bisect,
                                                                       23                                                              23
                                   GQ 14                                                         bisector, and perpendicular bisector
14. Can students define and




                                                                                 DOCUMENTATION
                                                                                                 (G-2-M) (Application)
    apply the terms measure,       Activity 47: Chords and                                       Demonstrate conceptual and
    distance, bisector, angle      Triangles!
                                                                       24, 28                    practical understanding of
    bisector, and                  GQ 14                                                         symmetry, similarity, and
    perpendicular bisector
                                                                                                 congruence and identify similar and   24
    appropriately and use          Activity 48: Folding                                          congruent figures (G-2-M)
    them in discussing figures     squares
                                                                       23, 28                    (Analysis)
    synthetically and with         GQ 14
    reference to coordinates as
                                                                                                 Predict, draw, and discuss the
    well?                          *Activity 49: Angle                                           resulting changes in lengths,
                                   Relationships                                                 orientation, angle measures, and
                                                                       23, 28
                                   GQ 14                                                         coordinates when figures are
                                                                                                 translated, reflected across             25
Concept 2: Transformations         *Activity 50:                                                 horizontal or vertical lines, and
                                   Transformations!                                              rotated on a grid (G-3-M) (G-6-M)
                                                                       23, 24,
15. Can students use               GQ 15, 16                                                     (Application)
                                                                       25
    transformations
    (reflections, translations,
8th Grade Mathematics: Unit 4: Geometry and Measurement
                                                                                                       8th Grade Mathematics: Unit 4: Geometry and Measurement
    rotations) to match figures                                                   Predict, draw, and discuss the
    and note the properties of     Activity 51: Transform                         resulting changes in lengths,
    the figures that remain        Me!                           23, 24,          orientation, and angle measures that
    invariant under                GQ 15, 16                     25               occur in figures under a similarity        26
    transformations?                                                              transformation (dilation) (G-3-M)
                                                                                  (G-6-M) (Analysis)
16. Can students use the           *Activity 52: Dilations
    coordinate plane to            GQ 15                                          Apply concepts, properties, and
    represent models of real-                                    24, 26           relationships of adjacent,
    life problems?                                                                corresponding, vertical, alternate
Concept 3: Pythagorean             *Activity 53: Developing                       interior, complementary, and               28
Theorem                            the Theorem                   31               supplementary angles (G-5-M)
                                   GQ 17                                          (Analysis)
17. Can students state and         *Activity 54: The Theorem
    apply the Pythagorean          GQ 17                         31               Construct, interpret, and use scale
    Theorem and its converse                                                      drawings in real-life situations (G-
                                                                                                                             30
    in finding the lengths of      Activity 55: The Converse                      5-M) (M-6-M) (N-8-M) (Analysis)
    missing sides of right         of the Pythagorean
    triangles and showing                                        31
                                   Theorem                                        Use area to justify the Pythagorean
    triangles are right            GQ 17                                          theorem and apply the Pythagorean
    respectively?                  Activity 56: Rectangles                        theorem and its converse in real-life
                                   and Diagonals!                26, 30           problems (G-5-M) (G-7-M)                   31
                                   GQ 17                                          (Analysis)
                                   *Activity 57: Is This Table
                                   a Rectangle?                  30, 31
                                   GQ 17                                   Reflections:
Concept 4: Scale Drawings          *Activity 58: How Big is
                                   This Room Anyway?             30
18. Can students discuss           GQ 18
    similar and congruent          Activity 59: Scale
    figures, and make and          Drawings                      30
    interpret scale drawings of    GQ 18
    figures?                       Activity 60: Mapping my
                                   Way!                          30
                                   GQ 18

8th Grade Mathematics: Unit 4: Geometry and Measurement
                                                                   8th Grade Mathematics: Unit 4: Geometry and Measurement
                                   Activity 61: How Big was
                                   it Anyway?                 30
                                   GQ 18




8th Grade Mathematics: Unit 4: Geometry and Measurement
                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement


Unit 4 Concept 1: Angle Relationships

GLEs
*Bolded GLEs are assessed in this unit.

     23 Define and apply the terms measure, distance, midpoint, bisect, bisector, and
        perpendicular bisector (G-2-M) (Application)
     24 Demonstrate conceptual and practical understanding of symmetry,
        similarity, and congruence and identify similar and congruent figures (G-2-
        M) (Analysis)
     28 Apply concepts, properties, and relationships of adjacent, corresponding,
        vertical, alternate interior, complementary, and supplementary angles (G-5-
        M) (Analysis)

Guiding Questions:                                  Vocabulary:
14. Can students define and apply the terms             Acute Angle
    measure, distance, bisector, angle                  Adjacent Angles
    bisector, and perpendicular bisector                Alternate Exterior Angles
    appropriately and use them in discussing            Alternate Interior Angles
    figures synthetically and with reference            Angle Bisector
    to coordinates as well?                             Bisect
                                                        Bisector
Key Concepts:                                           Complementary Angles
       Demonstrate conceptual and practical            Congruent
        understanding of symmetry,                      Corresponding Angles
        similarity, and congruence, and                 Distance
        identify similar and congruent                  Measure
        figures (for example, recognize                 Midpoint
                                                        Obtuse Angle
        reductions and expansions in similar
                                                        Perpendicular Bisector
        figures in two and three dimensions).
                                                        Right Angle
       Understand the terms distance
                                                        Segment Bisector
        (between two points, two lines, or              Similar
        from a point to a line) and midpoint,           Straight Angle
        bisect and bisector in regard to lines.         Supplemental Angles
       Apply concepts, properties and                  Transversal
        relationships of points, lines and line         Vertex
        segments, rays, planes, diagonals;              Vertical Angles
        right, acute, obtuse, supplementary,
        complementary, corresponding,
        vertical, and alternate interior angles;
        cube and rectangular prism.




8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                 59
                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement
Assessment Ideas:                                   Resources:
    See end of Unit 4                                  Chords and Triangles Handout
                                                        Folding Squares Handout
Activity Specific Assessments:                          Angle Relationship Handout
    Activity 48, 49                                    Square sheets of paper
                                                        Protractor
                                                        Graph Paper
                                                        Teacher-Made Supplemental
                                                          Resources


Writing Strategies
See the Teacher-Made Supplemental Resources for Daily Problems to use with journal writing.

                                      Instructional Activities
Note: The essential activities are denoted by an asterisk and are key to the development of student
understandings of each concept. Any activities that are substituted for essential activities must
cover the same GLEs to the same Bloom’s level.

*Activity 46: The Bisection! (LCC Unit 3 Activity 3)
(GLE: 23)
Materials list: grid paper, ruler, pencil, math learning log

Provide students with the coordinates of the end points of a horizontal line segment and have
them draw the line segment on a coordinate system. Next, have students determine the
coordinates of the point that bisects the line segment. Discuss the length of the line and have the
students determine how the coordinates can be used to determine the length of the segment. After
the midpoint is determined, discuss the coordinates of the midpoint and how these coordinates
relate to the coordinates for the endpoints of the segment. Have students draw a line
perpendicular to the line segment through the midpoint, thus illustrating a perpendicular bisector.
Repeat this activity with a vertical line segment and then line segments of positive or negative
slope. Have students verbalize a method for finding the coordinates of the midpoint of a segment
if the endpoints are known. (Average the x-coordinates and average the y-coordinates to find the
x and y coordinate of the midpoint).

As a real-life connection, have the students design a tile pattern for a rectangular room with
dimensions of 10 feet x 13 feet. The owner of the house has one request: a design in the floor
tiles should be in the center of the room. Students should use their understanding of finding
midpoint to determine where to place the design with the tile.

Students should record their method of finding the coordinates of the midpoint of a segment in
their math learning log (view literacy strategy descriptions). Remind the students that their math
learning log should reflect how they are thinking about the procedure so that they can use their
thinking later when reviewing the concept.


Note: Activity matches Textbook activity on page 271 Glencoe Course 3 (Eighth Grade)
Mathematics textbook.

8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                 60
                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement


Activity 47: Chords and Triangles! (CC Activity 8)
(GLEs: 24, 28)
Have students construct a circle on their paper. Instruct students to draw two chords through their
circle that intersect. Have students then connect endpoints of the two chords so that they form two
triangles. Challenge the students to use proportional measurements to determine whether the
triangles formed are similar. Repeat the activity with different chords to determine whether the
conjectures from the first circle hold true. Lead a discussion about properties and relationships of
angles formed with these chords. (See Teacher-Made Supplemental Resources)


Activity 48: Folding squares (LCC Unit 3 Activity 11)
(GLEs: 23, 28)
Materials list: paper cut into squares for each student, pencil, paper

Provide square sheets of paper to each student. Have the students fold the
paper in half with a horizontal fold (fold 1), make a good crease, and                              open
the paper up again. Then instruct students to fold each half in half again
                                                                                                     fol ds # 2
using a second horizontal fold (fold 2), make a good crease, and open the             fol d # 1

paper up again. Have students make a vertical fold (fold 3), make a good
crease, and open the paper up again. Ask students to make observations
about the relationships of length of the line segments formed by the folds.            fol d # 3 Have
students identify these as a bisector and a perpendicular bisector. Instruct students to take the top
right corner and fold it so that the vertex meets the intersection of their center folds, rotate their
paper 180 and repeat this fold with the opposite corner. Have them open their paper and in their
groups determine the measures of all angles formed by the different folds.

Have students outline the hexagon that is formed after the folds have been made (see diagram)
and use what they know about angle measures to determine the number of degrees in the angles
of a hexagon.

Have groups prepare a presentation to the class and justify their angle measurements (e.g.
complementary, supplementary, vertical angles, etc). (See Teacher-Made Supplemental
Resources)

Assessment
Have students turn in folded square with every angle measured and labeled.

Assessment
The teacher will provide the student with a list of vocabulary (bisector, perpendicular
bisector, complementary angles, supplementary angles, vertical angles, adjacent angles,
corresponding angles, corresponding angles, etc.) used in the unit. The student will write the
vocabulary on the folded square used in the activity.

*Activity 49: Angle Relationships (LCC Unit 3 Activity 10)
(GLEs: 23, 28)
Materials list: paper, pencil, protractor



8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                   61
                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement
Have students investigate the relationship among the angles that are formed by intersecting two
parallel line segments with a transversal. Have students determine pairs of angles that are
complementary, supplementary, congruent, corresponding, adjacent, and alternate interior. Using
a protractor, have students determine the measure of each of these pairs of angles. As an
application, pose the following problem to students: As a class project, you are going to build a
picnic table with legs that form an “X.” Of course, the top of the table must be parallel to the
floor. If one of the legs is attached so that it forms a 40o angle with the top of the table, what
measure should the leg form with the ground to ensure the tabletop is parallel to the floor? Ask
students to explain their reasoning.
Note: Teacher could choose to use this activity for assessment of Activity 48. (See Teacher-Made
Supplemental Resources)

Assessment
Provide the student with a sketch of an ironing board. In a math learning log entry, the
student will explain the relationships of the angles formed by the legs of the ironing board.
The teacher will provide the student with a sketch of an ironing board. In a journal entry, the
student will explain the relationships of the angles formed by the legs of the ironing board.




8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                 62
                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement
Unit 4 Concept 2: Transformations

GLEs
*Bolded GLEs are assessed in this unit.

      23 Define and apply the terms measure, distance, midpoint, bisect, bisector, and
         perpendicular bisector (G-2-M) (Application)
      24 Demonstrate conceptual and practical understanding of symmetry, similarity,
         and congruence and identify similar and congruent figures (G-2-M) (Analysis)
      25 Predict, draw, and discuss the resulting changes in lengths, orientation, angle
         measures, and coordinates when figures are translated, reflected across
         horizontal or vertical lines, and rotated on a grid (G-3-M) (G-6-M)
         (Application)
      26 Predict, draw, and discuss the resulting changes in lengths, orientation, and
         angle measures that occur in figures under a similarity transformation
         (dilation) (G-3-M) (G-6-M) (Analysis)
      28 Apply concepts, properties, and relationships of adjacent, corresponding,
         vertical, alternate interior, complementary, and supplementary angles (G-5-
         M) (Analysis)

Guiding Questions:                                  Vocabulary:
15. Can students use transformations                    Center of Rotation
    (reflections, translations, rotations) to           Congruent
    match figures and note the properties of            Dilation
    the figures that remain invariant under             Distance
    transformations?                                    Line of Reflection
16. Can students use the coordinate plane to            Reflection
    represent models of real-life problems?
                                                        Rotation
                                                        Symmetry
Key Concepts:
                                                        Similar
                                                        Translation
                                                        Vertex
Assessment Ideas:                                   Resources:
    See end of Unit 4                                  Transformations Handout
                                                        Transform Me Handout
Activity Specific Assessments:                          Graph paper
                                                        Index Cards
                                                        Rulers
                                                        Post-it Graph Paper
                                                        Teacher-Made Supplemental
                                                          Resources




Writing Strategies
See the Teacher-Made Supplemental Resources for Daily Problems to use with journal writing.


8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                 63
                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement
                                      Instructional Activities
Note: The essential activities are denoted by an asterisk and are key to the development of student
understandings of each concept. Any activities that are substituted for essential activities must
cover the same GLEs to the same Bloom’s level.

*Activity 50: Transformations! (LCC Unit 3 Activity 1)
(GLEs: 23, 24, 25)
Materials list: One Inch Grid BLM, Index Card Shapes BLM, ¼ Inch Grid BLM,
Transformations BLM, Transformation Review BLM, pencils, paper, scissors, ruler, unlined 3” x
5” index cards, large sheet of newsprint

Have students work in cooperative groups of 4. Give each student in the group a copy of One
Inch Grid BLM. Have students in each group cut off the edges around their grid paper and tape
the four sheets together to form a large coordinate grid. Tell students to draw the x and y axes in
the center of the large coordinate plane. Each sheet will represent one quadrant of the coordinate
plane.

Have students label the point at which all four sheets meet as the origin. Ask them to label both
the x- and y-axes, indicating the locations of –10 to 10 on each axis.

Distribute four 3” x 5” index cards to each group. Make sure the students have assigned tasks as
they prepare these cards. Have students follow the steps listed below, the results of which are
shown on the Index Card Shapes BLM:                                                 A           B
1. Index card #1 - Label the vertices of the index card with A, B, C, and D.
                                                                                         C             D
2. Index card #2 - Mark the midpoint of one of the 3” sides. Draw segments
   connecting this midpoint to each of the vertices on the opposite side. Cut out the
   isosceles triangle that is formed. Label the vertices of the triangle E, F, and G.
3. Index card #3 - Put points on one of the 5” sides at 2” and 4” (i.e., 2 inches from
   one vertex and 1 inch from the other vertex). The segment between these two
   points forms the top of a trapezoid. Connect these points to the vertices the
   on the opposite side. Cut out the trapezoid. Label the vertices of the trapezoid
   H, I, J, and K.
4. Index card #4 – Measure 2 inches along one of the 5” sides and mark a          2 in
   point. Connect this point to the vertex on the opposite side to form an
   isosceles right triangle. Cut out this triangle. Label the vertices of the
   triangle formed L, M, and N.

After students have created shapes, the teacher could choose to have the students either do the
entire project after teaching each concept or complete each column of the chart immediately after
that specific concept is taught.

Post a large sheet of newsprint on the wall for the new vocabulary used. As each new geometry
term is discussed, have a student add the word to the word wall poster.




8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                 64
                                                      8th Grade Mathematics: Unit 4: Geometry and Measurement
Ask students to create a chart as shown below:
Shape       Original        Translation Rotation Reflection                       Reflection
            Coordinates Coordinates Coordinates coordinates                       coordinates
                                                 x-axis                           y-axis
rectangle   A:              A:            A:     A:                               A:
            B:              B:            B:     B:                               B:
            C:              C:            C:     C:                               C:
            D:              D:            D:     D:                               D:
isosceles   E:              E:            E:     E:                               E:
triangle    F:              F:            F:     F:                               F:
            G:              G:            G:     G:                               G:
trapezoid   H:              H:            H:     H:                               H:
            I:              I:            I:     I:                               I:
            J:              J:            J:     J:                               J:
            K:              K:            K:     K:                               K:
isosceles   L:              L:            L:     L:                               L:
right       M:              M:            M:     M:                               M:
triangle    N:              N:            N:     N:                               N:

Distribute Transformations BLM and ¼ Inch Grid BLM
    Have students place the rectangle in the first quadrant and record the coordinates of all
       four vertices of the rectangle in its original position in column one of the table.
    Have the students translate the rectangle up (or down) and right (or left), and then record
       the new coordinates in column two.
    Have students return the rectangle to its original                  8



       location and record coordinates of each vertex after                        B                     C
                                                                         6

       a 180 clockwise rotation. Discuss rotational
                                                                                              original
       symmetry as students begin to rotate their shapes. (If            4




       this is new to the students, it works well if students
                                                                                   A                     D



                                                                         2


       put a small piece of tape on the rectangle to hold the
       rectangle in its original place on the grid, trace the
                                                -10           -5
                                                                              tracing paper
                                                                                                    5        10




       figure, and then rotate the traced figure 180                    -2



       around the origin.) Have students discuss the new
       coordinates and identify the quadrant in which the                -4
                                                                                              rotation


       rotated rectangle lies.                                           -6




    Have students return the rectangle to its original
                                                                         -8

       location and then perform a reflection of the
       rectangle across the x-axis. Be sure to discuss line of symmetry as the rectangle is
       reflected. Model lifting the rectangle from the plane and flipping the triangle over the x-
       axis, if needed. Have students record coordinates of the four vertices.
    Have students return the rectangle to its original position, perform a reflection across the
       y-axis, and then record the new coordinates.
    Have the students complete the same actions using their trapezoid, right triangle, and
       isosceles triangle, recording all of the new coordinates on the chart. Remind them always
       to return their shapes to the original position before making a transformation.

After the class has had time to complete the transformations of all four shapes, have the groups
make some conjectures about how they might be able to determine the positions of polygons after
a transformation from the information in the chart. Have the groups share their conjectures with
8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                           65
                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement
the class. (See Teacher-Made Supplemental Resources)Have the groups share their conjectures
with the class by using the professor know-it all (view literacy strategy descriptions). The group
that is sharing conjectures will be selected by the teacher; therefore, all groups should be ready to
go first. The group will go to the front of the class, and using its conjectures, justify its thinking
and answer questions from the class about one of its conjectures. The teacher will then select a
second group to share another conjecture and continue until all conjectures and thinking are
clearly understood by the class.

Have the students use the Transformation Review BLM as a graphic organizer (view literacy
strategy descriptions) to guide them as they review the results of the different transformations.
Go through the example as a class. Allow students to discuss the answer with a partner. Students
should write that the result of reflecting a polygon across the y-axis is that the x-coordinates are
opposites of the originals and the y-coordinates stay the same. The BLM gives them either the
initial position with the transformation used or the result of a transformation, and the student
should give the other. As a result, some bridges have more than one solution. Problem 4 presents
a new situation for students.

Activity 51: Transform Me! (CC Activity 2)
(GLEs: 23, 24, 25)
Have students chose one of the four polygons from Activity 50. Students will find the measure of
each angle, locate the midpoint of each side, and find the distance from vertex to vertex (i.e.,
length) for each side. (Teacher Note: Have students use rulers to measure lengths of sides which
are not vertical or horizontal.) Discuss which (if any) of the properties of the polygon changed
and which (if any) remained the same from Activity 50.

*Activity 52: Dilations (LCC Unit 2 Activity 2)
(GLEs: 24, 26)
Materials list: Dilations BLM, Quadrant I Grid BLM, protractor, pencil, paper, ruler

Discuss dilations as another transformation. Ask if anyone has an idea about what a dilation
might be. Students will relate to the eye doctor dilating their eyes, but very few of them relate a
dilation to being an enlargement or a reduction.

Provide students with copies of the Quadrant I Grid BLM and the Dilations BLM. Have students
plot the vertices of the polygon given on the Dilations BLM on a coordinate grid and then connect
the points to form the polygon. Have students find the measure of each angle, and find the
distance from vertex to vertex (i.e., length) for each side. (Teacher Note: Have students use rulers
to measure lengths of sides which are not vertical or horizontal.)

Next, have students use a ruler and draw a dotted line from the origin and extend the line through
Vertex A of the polygon, continue to do this by drawing                35




lines from the origin through each of the other four vertices          30
                                                                                    A'                 B'




                                                                                                                  C'




(see diagram).                                                         25




Instruct the students to follow the steps on the Dilations             20
                                                                                                  E'         D'




BLM and then discuss their conjectures about dilations and
                                                                            A            B

                                                                       15
                                                                                                   C




their effect upon angle measures, side lengths, and                    10       E
                                                                                              D




coordinates of the original figure. Make sure the students             5




understand that the dilation is different from the reflections,                              10             20         30   40   50




translations and rotations because it is the only one that
produces similar figures – the other transformations produce congruent figures.
8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                                               66
                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement


As a real-life connection, lead a discussion about when dilations that are used in everyday life:
using a projector to show an image to an entire class, enlarging a picture from the image stored in
a digital camera, projecting a video on large screens at sporting events, or making a scale drawing
of a large object.




8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                 67
                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement
Unit 4 Concept 3: Pythagorean Theorem

GLEs
*Bolded GLEs are assessed in this unit.

    26 Predict, draw, and discuss the resulting changes in lengths, orientation, and
       angle measures that occur in figures under a similarity transformation
       (dilation) (G-3-M) (G-6-M) (Analysis)
    30 Construct, interpret, and use scale drawings in real-life situations (G-5-M)
       (M-6-M) (N-8-M) (Analysis)
    31 Use area to justify the Pythagorean theorem and apply the Pythagorean
       theorem and its converse in real-life problems (G-5-M) (G-7-M) (Analysis)

Guiding Questions:                              Vocabulary:
17. Can students state and apply the                Area
    Pythagorean Theorem and its converse in         Hypotenuse
    finding the lengths of missing sides of         Legs
    right triangles and showing triangles are       Perpendicular
    right respectively?                             Pythagorean Theorem
                                                    Right Triangle
Key Concepts:                                       Square Root
     Graph on the coordinate plane to
        represent real-world problems,
        including graphing ordered pairs in
        all four quadrants.
     Predict the results of and perform
        transformations (translations,
        reflections, rotations) in problems set
        in a real-world context.
     Construct or use scale drawings.
     Understand and use the Pythagorean
        Theorem, including recognizing
        situations in which the theorem is
        relevant (with pictorial illustration).
Assessment Ideas:                               Resources:
     See end of Unit 4                             Developing the Theorem Picture
                                                    CC Sample Assessment with
Activity Specific Assessments:                        Developing the Theorem
     Activity 53, 54, 55                           CC Sample Assessment with The
                                                      Theorem
                                                    Rectangles and Diagonals Handout
                                                    Is this table a Rectangle Handout
                                                    Graph Paper
                                                    Teacher-Made Supplemental
                                                      Resources




8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                 68
                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement
Writing Strategies
See the Teacher-Made Supplemental Resources for Daily Problems to use with journal writing.


                                      Instructional Activities
Note: The essential activities are denoted by an asterisk and are key to the development of student
understandings of each concept. Any activities that are substituted for essential activities must
cover the same GLEs to the same Bloom’s level.

*Activity 53: Developing the Theorem (LCC Unit 3 Activity 4)
(GLE: 31)
Materials list: grid paper, straight edge, scissors, paper, pencil

Have students draw a right triangle on grid paper with the two perpendicular sides having lengths
of 3 and 4 units. Have students draw a square using one of the legs of the triangle as the side of
the square (i.e., draw a 3 x 3 square). Repeat using the other leg as a side of a square (i.e., draw a
4 x 4 square). Have students find the area of each square. Ask students to determine a method for
finding the area of the square of the hypotenuse of their right triangle and how the areas of the
three squares relate to one another. (Some students may remember the Pythagorean Theorem
from previous years and use that information to determine the length of the hypotenuse. Others
may compare the length of the hypotenuse to the units on the grid paper. The process used is not
important, but all students should eventually see that the hypotenuse length is 5 and the area of
the corresponding square is 25 square units.) Have students show that the sum of the areas of the
two smaller squares is the same as the area of the square formed by the hypotenuse by cutting and
rearranging the small squares inside the larger squares. (Many texts and websites show how to do
this. Two websites which use animations to develop the Pythagorean Theorem are:
http://www.nadn.navy.mil/MathDept/mdm/pyth.html
http://www.pbs.org/wgbh/nova/proof/puzzle/theorem.html.)

Have students draw a triangle on the grid that is not a right triangle and determine whether they
get the same results. Discuss conjectures that students develop about the results of their
explorations. (See Teacher-Made Supplemental Resources for a picture)

Using a modified version of reciprocal teaching (view literacy strategy descriptions), have
students brainstorm predictions as to whether or not the Pythagorean theorem will work when
finding side lengths of triangles that do not have a right angle. Reciprocal teaching is used to
move instruction from delivery to discovery. Have groups write their predictions about the use of
the theorem in these other triangles on paper. The prediction is the first part of a reciprocal
teaching lesson.

Assign the roles of questioner, clarifier, predictor and conjecturer to groups of four students as
they experiment with these other triangles. The „questioner‟ will begin by asking the group to
restate how it thinks its prediction relates to the triangles without right angles. The clarifier should
make sure that the answers that the questioner gets to the questions are clear and understood by
all group members. Have students draw a triangle on the grid that is not a right triangle, and have
the questioner ask the group questions that will help it determine whether it gets the same results.
The „clarifier‟ will offer input, and the group will then work with the „conjecturer‟ to write its
summary statement. The predictor might make other predictions as other triangles are drawn to
test the conjectures made by the conjecturer. As a class, discuss conjectures that students develop
about the results of their explorations.
8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                 69
                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement


Assessment
The student will work these problems as journal prompts and explain the answers.
   a) Washington, DC, is 494 miles east of Indianapolis, Indiana. Birmingham, Alabama is
       433 miles south of Indianapolis. Determine the distance from Birmingham to
       Washington D.C.
   b) The ladder of a water slide is 8 ft. high, and the length of the slide is 17 ft. Determine
       the length of the horizontal base of the slide. Justify all of your thinking using valid
       mathematical reasoning. (See Teacher-Made Supplemental Resources)

*Activity 54: The Theorem (LCC Unit 3 Activity 5)
(GLE: 31)
Materials list: The Theorem BLM, pencils, paper, calculators,
graph paper

Provide students with the side lengths of several right triangles.
Have students compute the square of each measure and then add
the squared lengths of the two smaller sides and compare this sum with the square of the longest
side (the hypotenuse). Make sure that students understand that in a right triangle, the sum of the
squares of the two perpendicular sides is the same as the square of the hypotenuse. Extend this
activity to include real-life situations that require students to find the length of one of the sides of
a right triangle with situations like the ones that follow:
 James has a circular trampoline with a diameter of 16 feet. Will this trampoline fit through a
    doorway that is 10 feet high and 6 feet wide? Explain your answer.
 A carpenter measured the length of a rectangular tabletop he was building to be 26 inches, the
    width to be 12 inches and the diagonal to be 30 inches. Explain whether or not the carpenter
    can use this information to determine if the corners of the tabletop are right angles.

Provide students with the side lengths of several right triangles missing the length of one of the
sides. Discuss the use of the formula as it applies to the missing lengths in the triangles. Extend
this activity to include real-life situations that require students to find the length of one of the
sides of a right triangle with situations by distributing The Theorem BLM. Have students verify
their solutions to the BLM by comparing answers with another student and discussing any results
that differ.

Assessment
       Assign students these problems as journal prompts and have them explain the
         answers.
            a) Washington, DC, is 494 miles east of Indianapolis, Indiana. Birmingham,
            Alabama is 433 miles south of Indianapolis. Determine the distance from
            Birmingham to Washington D.C.

                b) The ladder of a water slide is 8 ft. high, and the length of the slide is 17 ft.
                Determine the length of the horizontal base of the slide. Justify all of your
                thinking using valid mathematical reasoning.

       The teacher will provide the student with a list of number triples that represent the
        side lengths for triangles. The student will determine which triples represent the side
        lengths of a right triangle.
8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                 70
                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement


Activity 55: The Converse of the Pythagorean Theorem (LCC Unit 3 Activity 9)
(GLE: 31)
Materials list: grid paper, protractors, pencil, paper

Have student pairs cut out squares from grid paper that are 9, 16, 25, 36, 49, 64, 81, 100, 121,
144, and 169 square units. Then have them create triangles using the sides of any three squares.
Have students use a protractor to determine the measures of each angle in the triangles formed.
Next, have students determine the relationship between the sum of the areas of the two smaller
squares and the area of the largest square (i.e., are they the same or different?) Have students
make a conjecture about the relationship between the areas of the squares when one of the angle
measures of the triangle is 90 degrees. Remind students that these relationships are those of the
Pythagorean Theorem and its converse (studied in earlier activities). Lead a discussion of
applications of the converse of the Pythagorean Theorem to real-life situations. For example, a
carpenter goes to the corner of a frame wall that he is building and marks off a 3 foot length on
one board and a 4 foot length on the adjacent board. He then nails a 5 foot brace to connect the
two marks. What is the purpose of his work? (He is making sure that the two boards are
perpendicular (that his wall is ‘ square’) because a triangle with sides of 3-4-5 is a right
triangle.)

 Assessment
 Provide the student with a list of number triples that represent the side lengths for triangles.
 Challenge students to determine which triples represent the side lengths of a right triangle.


Activity 56: Rectangles and Diagonals! (CC Activity 4)
(GLEs: 26, 30)
Provide students with grid paper and straight edges. Have the students use the grid paper to sketch
a scale model of a 10 foot by 16 foot room. Have students draw a diagonal through the scale
model of the room. Ask the students to draw a smaller rectangle inside the first rectangle, using
the diagonal drawn through the scale model of the room as the diagonal of the new rectangle.
Have students work in small groups to determine whether the rectangles are similar and repeat the
actions with different sized rectangles to determine if their conjectures hold true. Have students
find the actual dimensions of the new rooms, using the scale established using the original
rectangle. Have groups share and justify their conjectures. Note: The two rectangles should share
the diagonal.

*Activity 57: Is This Table a Rectangle? (CC Activity 7)
(GLEs: 30, 31)
Present the following scenario to students:
        Jason‟s dad is a carpenter. He asked Jason to find out if the rectangular table he was
        building had square corners. Jason said he had learned something in math class that would
        help him find out. The table had measurements of 14 feet x 10 feet. Jason measured the
        diagonal and found the diagonal to be 16 feet.
Have the students determine if the table has square corners, explain how they know, and provide
a scale drawing as part of the explanation. The website http://www.tpub.com/builder2n3/65.htm
is an architectural website that gives some examples of using the Pythagorean Theorem to make
sure beams are perpendicular.

8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                 71
                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement



Unit 4 Concept 4: Scale Drawings

GLEs
*Bolded GLEs are assessed in this unit.

    30 Construct, interpret, and use scale drawings in real-life situations (G-5-M)
       (M-6-M) (N-8-M) (Analysis)

Purpose/Guiding Questions:                         Vocabulary:
18. Can students discuss similar and                   Blueprint
    congruent figures, and make and                    Diagonal
    interpret scale drawings of figures?               Dimensions
                                                       Length
                                                       Model
                                                       Scale
                                                       Width



Assessment Ideas:                                  Resources:
    See end of Unit 4                                 Graph Papers
                                                       How Big is this Room Anyway grid
Key Concepts:                                          Mapping My Way Handout
    Construct or use scale drawings                   Poster Board
                                                       Ruler / Straight Edge
                                                       Index Cards
                                                       Teacher-Made Supplemental
                                                         Resources




Writing Strategies
See the Teacher-Made Supplemental Resources for Daily Problems to use with journal writing.




8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                 72
                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement
                                      Instructional Activities
Note: The essential activities are denoted by an asterisk and are key to the development of student
understandings of each concept. Any activities that are substituted for essential activities must
cover the same GLEs to the same Bloom’s level.


*Activity 58: How Big is This Room Anyway? (LCC Unit 3 Activity 6)
(GLE: 30)
Materials list: meter sticks or tape measures, newsprint or other large paper for blueprint, rulers,
scissors, pencil, paper

Assign different groups of students the task of measuring the classroom dimensions. Have the
class determine a scale that would fit on a piece of newsprint or poster board and then have
someone draw the room dimensions to scale on the poster. Tell students that the class will make a
classroom blueprint.

Divide students into groups of three to five. Assign each group a different object in the classroom
to measure (file cabinets, book shelves, trash can, etc. - remember only length and width of the
top of the object is needed for the blueprint). Have students convert actual measurements using
the scale measurements determined earlier. Instruct students to measure, draw and cut out models
from an index card. Have each student measure his/her own desktop and make a scale model for
the classroom blueprint. Remind students to write their names on the desktop model. Ask, “What
is the actual area of your desktop? What is the scale area of your desktop? What comparisons do
you see as you make observations of the areas of your room and desktop?” List your
observations.

Have groups submit their scale models of the classroom objects (not desks at this time) for the
blueprint. Discuss methods used to determine the measurements of the models and then glue the
models in the correct position on the classroom blueprint. Have students, one group at a time
place their desktop models on the classroom blueprint, working so that those who sit in the center
of the room can add their models first. Post blueprints/scale models on the wall for all classes to
compare.

Using the class scale model of the classroom, have students make predictions about distance from
various points in the room (i.e., If the distance from the teacher’s desk to the board is 5 inches on
the scale model and the scale is 1 inch represents 4 feet, then the that the actual distance is 20
feet.) Have student measure the actual distance(s) to check for accuracy of the scale model of the
classroom.

Note: Possible modifications including having each group create a blueprint for the room. (See
Teacher-Made Supplemental Resources)

Activity 59: Scale Drawings (LCC Unit 3 Activity 12)
(GLE: 30)
Materials list: Scale Drawings BLM, pencil, paper

Provide the students with the problems to practice scale drawing problems by distributing Scale
Drawing BLM. Give students time to work through these situations and then divide students into
groups of four to discuss these situations. Have students in groups come to consensus on the
solutions to these problems and then have them prepare for a discussion using professor know-it-
8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                 73
                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement
all (view literacy strategy descriptions). With this strategy, the teacher selects a group to become
the “experts” on scale drawing required in the situation that is selected. The group should be able
to justify its thinking as it explains its proportions or solution strategies to the class. All groups
must prepare to be the “experts” because they are not told prior to the beginning of the strategy,
which group(s) will be the “experts” and ask questions about scale drawings.

Have students measure the side lengths of their bedrooms, the length and width of their beds, and
length and width of any other bedroom furniture. Give students a sheet of grid paper and have
them construct a scale drawing of their bedrooms. Instruct students to determine the appropriate
scale to make their rooms fit on the grid paper. Lead a discussion to generate questions that
students can solve using their scale drawings (e.g., distance between objects, proportions
involving the length of sides of their scale drawing to actual objects).

Activity 60: Mapping my Way! (CC Activity 16)
(GLE: 30)
Provide the students with the following information: Sandy was given the assignment during a
summer job to draw a map from the city recreational complex to the high school. Sandy started
from the recreational complex and walked north 3.5 miles, west 10 miles, north 5.3 miles, and
then east 3 miles. Sandy was given a space 3 1 inches x 4 inches to sketch the route on a brochure
                                             2
being made by the staff at the complex. Determine a scale that Sandy will be able to use and draw
a map that can be used in the space provided. Explain how the scale was determined. (See
Teacher-Made Supplemental Resources)

Activity 61: How Big was it Anyway? (CC Activity 17)
(GLE: 30)
Provide groups of four students with situations like the ones below. Lead the class in a discussion
of these scale-drawing situations after the groups have had time to complete the problems.
1. Draw a diagram of a rectangular bedroom with dimensions of 24 feet by 15 feet. Use a scale
    of 1 inch = 6 feet.
       2

2. The picture of the amoeba at the right shows a width of 2 centimeters. If
    the actual ameba‟s length is 0.005 millimeter, what is the scale of the
    drawing?




8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                 74
                                                    8th Grade Mathematics: Unit 4: Geometry and Measurement
                                      Unit 4 Assessment Options

General Assessment Guidelines
    Whenever possible, the teacher will create extensions to an activity by increasing the
      difficulty or by asking “what if” questions.
    The student will create a portfolio containing samples of experiments and activities.
    The teacher will provide the student with unlined paper and rulers. The student will design
      a stained-glass window to show understanding of the terms midpoint, bisector,
      perpendicular bisector, symmetry, similar, complementary, supplementary, vertical
      angles, corresponding angles, and congruent angles. The student will label the different
      components of his/her stained-glass window to assure that examples have been included
      for each of the vocabulary words from the unit. The student will present his/her stained-
      glass sketch to his/her group and justify examples to the group members. The teacher will
      provide the student with a rubric to self-assess his/her work prior to presentations and
      teacher evaluation.
    The teacher will provide the student with a sketch of a baseball diamond showing that
      there are 90 feet between the bases. The student will prepare a presentation explaining
      how to determine the distance the catcher must throw the baseball to the 2nd baseman if he
      needs to get the runner on second base out.
    The teacher will give the student a piece of grid paper which shows a polygon and a
      transformation of the polygon (the second polygon). The student will determine a
      transformation or transformations that would produce the second polygon.
    The teacher will provide the student with several right triangles that have a missing side
      measure. The student will find the lengths of the missing sides. (See APCCSM)
    The teacher will provide the students with paper and the scale of 0.25 inches to represent 2
      feet. The student will a) draw a model of a rectangular swimming pool measuring 16 feet
      by 36 feet; b) draw a 2 foot by 6 foot diving board so that it bisects one of the short ends
      of the pool; c) find the perimeter and area of the pool; and d) put a walk around the
      perimeter of the pool with a width of 4 feet and find the area and the outer perimeter of the
      walk.
    The student will create a scale drawing. A rubric that assesses the appropriateness of the
      scale factor, as well as the accuracy of the drawing, will be used to determine student
      understanding.



Activity-Specific Assessments
    Concept 1 Activity 48, 49
    Concept 3 Activity 53, 54, 55




8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                 75
                                                                      8th Grade Mathematics: Unit 4: Geometry and Measurement
    Name/School_________________________________                                             Unit No.:______________

    Grade         ________________________________                                    Unit Name:________________


                                                       Feedback Form
                 This form should be filled out as the unit is being taught and turned in to your teacher coach upon completion.



Concern and/or Activity                              Changes needed*                                          Justification for changes
       Number




    * If you suggest an activity substitution, please attach a copy of the activity narrative formatted
    like the activities in the APCC (i.e. GLEs, guiding questions, etc.).




    8th Grade Mathematics: Unit 4 -Geometry and Measurement                                                                        76

				
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