Summary
Students will graph translations (slides) and reflections (flips) on a coordinate plane.
Materials
Additional Resources
Book
- Visions of Symmetry: Notebooks, Periodic Drawings, and Related
Work of M.C. Escher, by Doris Schattschneider;
ISBN 0-7167-2126-0
Video
- The Fantastic World of M.C. Escher, directed by Michele Emmer;
ASIN: 6303146767
Additional Media
- M.C. Escher Sun and Moon Puzzle, by iproject (available from
http://www.iproject.com/, P.O. Box 101, 3740 AC Baarn, Holland,
Telephone: +31-35-5418041, Fax: +31-35-5411766);
Item# ES-1020
Background for Teachers
Students should understand the following vocabulary for this activity:
rotation (turn)The image of a figure that has been turned as if on
a wheel.
translation (slide)The image of a figure that has been slid to a new
position without flipping or turning.
reflection (flip)The mirror image of a figure that has been flipped
over a line.
transformationThe act of changing the form or appearance of an
object.
clockwiseIn the same direction as a clocks hands move.
counterclockwiseIn the opposite direction as a clocks hands move.
In Websters Dictionary, the word transform is defined as to
change or convert. Students should understand that rotations,
translations, and reflections are all types of transformations.
Maurits Cornelis (M.C.) Escher (1898-1972) was an architect and
graphic artist. He continually invented new visual constructions to
challenge the conventional perception of spatial relationships. Eschers
fascination with drawing figures that tile together perfectly led to
numerous tessellation drawings that involve the repetition of one or more
shapes that connect together in asymmetrical relationship.
Intended Learning Outcomes
2. Become mathematical problem solvers.
6. Represent mathematical situations.
Instructional Procedures
Invitation to Learn
(This activity requires adequate physical space.) Have students stand where they can spread their arms out and not
touch anyone. Give them the following directions:
- Slide one step to the right.
- Turn your body 1/2 turn to the left.
- Slide three steps to the left.
- Turn your body 1/2 turn to the right.
- Lie on your back on the floor.
- Flip over onto your stomach.
- Stand up and turn completely around once.
- Sit down in your seat.
Instructional Procedures
- Explain to students that just as they were transforming their
bodies by sliding, turning, and flipping them, they can also
transform geometric shapes in math.
- Introduce or review the following vocabulary: rotation,
translation, reflection, transformation, clockwise, and
counterclockwise (see background information).
- Discuss that a reflection (flip) is accomplished by graphing the
opposite of each coordinate (reflections across the x-axis change
each y-coordinate into its opposite).
- Distribute a Transforming Triangles worksheet to each student.
- Instruct the class to graph and label a triangle with the following
coordinate points in Grid 1: A(-5,4), B(-3,5), C(-2,1).
- Also in Grid 1, have students create triangle ABC by translating
(sliding) triangle ABC 7 units right and 6 units down. Label the
points. Translation = A(2,-2), B(4,-1), C(5,-5).
- Graph and label a triangle with the following coordinate points in
Grid 2: D(1,1), E(4,5), F(5,3).
- Reflect (flip) triangle DEF across the x-axis and label the points
in Grid 2. Reflection = D(1,-1), E(4,-5), F(5,-3).
- Graph and label a triangle with the following coordinate points in
Grid 3: G(1,-4), H(4,-2), I(4,-5).
- Reflect (flip) triangle GHI across the y-axis and label the points in
Grid 3. Reflection = G(-1,-4), H(-4,-2), I(-4,-5).
- Pair students up and pass out a privacy folder to each pair.
- Students draw and label triangle JKL in Quadrant IV of Grid 4
(they choose the coordinate points).
- Have each student create triangle JKL by sliding triangle JKL into
Quadrant II of Grid 4 (each students decides how many units to
move triangle JKL up and left).
- Each student takes turns giving his/her partner the coordinate
points so s/he can duplicate his/her partners triangles in Grid 5.
- Partners compare their triangles. (Grid 4 of one partner should
duplicate Grid 5 of the other partner.)
- Students draw and label triangle MNO in Quadrant III of Grid 6
(they choose the coordinate points).
- Each student reflects (flips) triangle MNO across the x-axis into
Quadrant II.
- His/her partner checks to see that s/he has reflected triangle MNO
correctly.
- Pass out a Rotating Triangles handout and brad to each student.
- Cut the paper on the dotted line, then cut out the two triangles.
- Push the brad through the black triangle, then the gray triangle.
- Next, push the brad (with the triangles on it) into the center point
of the square and secure.
- Instruct the students to rotate (turn) the black triangle 1/4 turn (90
degrees). Continue to give instructions for the students to rotate
the black triangle 1/2 turn, then 270 degrees, etc.
Give directions in both degrees and fractions of a turn.
- With a partner, have students take turns giving directions to their
partner to rotate one of the triangles (have them give directions in
both degrees and fractions of a turn).
Extensions
- Students look for letters in the alphabet that look the same when
reflected in a mirror. Try to find entire words that look the same,
for example, MOM or TOOT.
- The Kuba people of the Congo (Zaire) region of Africa use slides
and flips when making patterns on cloth. Examine some samples
of Kuba cloth.
- Tessellations (M.C. Escher)
- The famous Renaissance artist, Leonardo DaVinci, often wrote
words in mirror-image in his journals so that others could not read
them. Extend this concept into an art activity.
Family Connections
- Play with or discuss the popular Transformer toys. How do
they change or transform?
- Using an everyday object from home (such as a spoon), practice
sliding, turning, and flipping it on a flat surface.
- Play Simon Says and give instructions like Simon says slide two
steps forward, Simon says turn around two and a half times,
Lie down on your back and flip over three times.
Assessment Plan
Evaluate students using the following rubric: |
4
Full Accomplishment |
Student accurately graphs translations and reflections on the coordinate plane. |
3
Substantial Accomplishment |
Student graphs translations and reflections on the
coordinate plane, but not always accurately. |
2
Partial Accomplishment |
Student has difficulty graphing translations and reflections on the coordinate plane. |
1
Little Accomplishment |
Student does not accurately graph translations and
reflections on the coordinate plane. |
- Have each student write a paragraph telling how graphing a
translation is different from graphing a reflection.
- Have students draw and label a four quadrant grid. Then draw
triangle ABC with A(-1,2); B(-3,4); C(-2,0). Translate it 3 units
to the right and 1 unit down.
- Have students draw and label a four quadrant grid. Then draw
triangle DEF with D(-3,2); E(-2,4); F(-1,1). Reflect it across the
y-axis.
Bibliography
Research Basis
Dickinson, D. (1996). Learning Through Many Kinds of Intelligence. Seattle: New
Horizons for Learning.
It is important to encourage children to explore and exercise all of
their intelligences. Bodily/Kinesthetic Intelligence involves physical
coordination and dexterity, expressing oneself or learning through
physical activities. Logical/Mathematical Intelligence involves number
and computing skills, recognizing patterns and order, and the ability to
solve different kinds of problems through classifying and sequencing
activities, and solving various kinds of puzzles.
Marzano, R. J., Pickering, D. J., & Pollock, J. E. (2001). Classroom Instruction that Works:
Research-Based Strategies for Increasing Student Achievement. Alexandria: Association
for Supervision and Curriculum Development.
It has been shown that explicitly engaging students in the creation of
nonlinguistic representations stimulates and increases activity in the
brain. When students elaborate on knowledge, they not only understand
it in greater depth, but they can recall it much more easily.
Wahl, M. (1999). Math for Humans: Teaching Math Through 8 Intelligences. Seattle: New
Horizons for Learning.
With new pressures on teachers and students to meet higher
standards, this book offers tools that make concepts concrete and
understandable. It has many practical and creative methods that take into
consideration different learning styles and kinds of intelligences. Wahl
has developed strategies that all teachers can use to help their students
become successful in math.