

Summary: Students will graph translations (slides) and reflections (flips) on a coordinate plane.
Main Curriculum Tie: Mathematics Grade 6 Strand: THE NUMBER SYSTEM (6.NS) Standard 6.NS.8 Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same xcoordinate or the same ycoordinate. Materials:
Additional Resources
Book
 Visions of Symmetry: Notebooks, Periodic Drawings, and Related
Work of M.C. Escher, by Doris Schattschneider;
ISBN 0716721260
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: +31355418041, Fax: +31355411766);
Item# ES1020
Attachments
Web Sites
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.
transformation—The act of changing the form or appearance of an
object.
clockwise—In the same direction as a clock’s hands move.
counterclockwise—In the opposite direction as a clock’s hands move.
In Webster’s 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 (18981972) was an architect and
graphic artist. He continually invented new visual constructions to
challenge the conventional perception of spatial relationships. Escher’s
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 xaxis change
each ycoordinate 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 xaxis 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 yaxis 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 partner’s 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 xaxis 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 mirrorimage 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
yaxis.
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:
ResearchBased 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. Author: Utah LessonPlans
Created Date : Feb 27 2006 12:20 PM
