ISA 10th Grade
Math Performance Assessment
Geometry – A Day at the Museum
Researchers from Teachers College, Columbia University and the Institute for
Student Achievement (ISA) are working with your school to learn about how
students in the ISA schools are improving in math and writing.
It is important to try your best on this assessment. The scores from this
assessment will be used by your school to help improve its math instruction.
Look through the entire test before beginning. You may work on the problems in
any order you like. You may use a protractor, a ruler, and manipulatives during this
assessment.
When we score your test, we will award full and partial credit based on the work
you show. Please make sure to show all your work.
You should have plenty of time to complete the test. Your teacher will tell you
when half the time is over.
If you have any questions during the test, or if you need more paper, please raise
your hand.
Please try your hardest!
Please write your name below, and then turn the page to continue.
Name:
1
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A Day at the Museum
Your grandfather has decided that you play too many video games, so one
afternoon he brings you to the art museum.
As you walk around the European art room, you see an interesting drawing
by M.C. Escher.
Look closely at the drawing. Do you see the horses facing left? Do you see
the horses facing right?
Many people enjoy art like this because it has repeating patterns, symmetry,
and fills up all the space on the paper.
2
You ask your grandfather about this picture, and he says it is a
tessellation. You ask, “What is a tessellation?” He says, “A tessellation is
a group of shapes that completely fills a space with no gaps or overlaps. It
is like tiling – putting tiles on a kitchen or bathroom floor so that the whole
area is covered”
You see two more pictures.
#1
Is picture #1 a tessellation? Explain why or why not?
Is picture #2 a tessellation? Explain why or why not?
3
#2
As you and your grandfather look around, you see a tessellation made of
trapezoids.
Look closely. Do you see the individual trapezoids? Do you see how they fit
together?
4
You and your grandfather go into a room in the museum with tessellations
made from regular polygons. A regular polygon is a shape where all the
sides are the same length and all the angles have the same number of
degrees.
Your grandfather asks you, “Which of these regular polygons tessellate?”
-Equilateral Triangle (3 sides)
-Square (4 sides)
-Regular Pentagon (5 sides)
-Regular Hexagon (6 sides)
-Regular Heptagon (7 sides)
-Regular Octagon (8 sides)
Which of these shapes tessellate? Explain why some of these shapes
tessellate. Use mathematical and numerical evidence to explain.
5
Your grandfather asks you, “Why do some shapes tessellate and some
shapes do not?” Explain your answer using mathematical and numeric
evidence.
6
You and your grandfather go into another room where different shapes are
put together to create tessellations. You see an example of this from Islamic
Art.
Look closely. Do you see the octagons? Do you see the pentagons?
7
Your grandfather asks you, “Can you see the rotations, reflections, and
symmetry in this Islamic artwork?”
Look closely at the Islamic piece of art. Describe with as much detail as
possible how rotations, reflections translations and symmetry appear in this
work of art.
8
The museum has another piece of Islamic art, only this one is 800 years old
and broken into many pieces.
The pieces are all regular polygons.
-Equilateral Triangles
-Squares
-Regular Hexagons
Try to combine these 3 shapes to make a tessellation. How many ways can
you find to make them tessellate? Show each one of your attempts. Explain
why each one worked or did not work.
Attempt #1
Explain:
9
Attempt #2
Explain:
10
Attempt #3
Explain:
11