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Mosaic
Posted April
16, 2001 as a Math Forum EPoW
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Introduction: In this problem,
students use blocks of variable
length (fractional, not decimal)
to fill up a map of the USA like
a mosaic. They then use the size
of the blocks to estimate the area
of the map.
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Where’s the Math: This
problem deals with basic addition
of fractions (For a related problem,
see Fractris).
In addition, question #2 brings
up basic ideas of calculus, i.e.
using smaller and smaller rectangles
to approximate an irregular shape.
This question challenges students
to come up with that idea independently.
Standards: Measurement,
number
& operations, geometry
Role of Components: This
activity is composed of several
 PEN
components glued together in one
larger composite PEN component.
The individual components are (from
top to bottom) an image (US map)
overlaid with colored fractional
blocks, a current block indicator,
a fraction input widget, a color
chooser, and several java widgets
that control the placement of block
on top of the image.
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Try the applet!
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Sample submitted solution:
From: Nina, age 14
School: Taipei American School, Taipei,
Taiwan
1. What is the approximate area of the
USA map?
~ 4 5/18 sq. units
2. How could you get a better estimate
of the area of the USA map?
~ I could probably get a better estimate
of the area if I could just leave the smaller
bits of the USA map blank, and just fill
in with blocks for the bigger areas of the
map. For example, if the top edge of the
country would only fill up one cm of space,
I would just leave that piece and continue
to fill in the other larger areas of the
map.
3. Design your own image and use Mosaic
to create it. Describe it here, briefly
explaining how you made it.
~ I designed a butterfly and I used the
different rows and colors to show the wings
and the butterfly's body. For the first
1/3 of the row I used a light blue color
and then for the little head of the butterfly
I used the yellow, then I switched colors
to a pink. So the top left wing is blue
and the body will stay yellow, and the top
right wing is pink. For the rest of the
butterfly, I just changed the colors around
so that the bottom left wing is pink and
the body will stay yellow and the bottom
right wing is light blue.
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Reflections:
We didn't get many submissions for this
problem, and we're not sure why. It seems
some students didn't understand how to count
up the blocks to estimate the area of the
map in the mosaic. They were fractions of
the line, so the total number couldn't be
more than the number of lines.
Others didn't understand how to make a
mosaic to find the area of the underlying
map. Maybe this part was a little unclear
on our part. The reason different colors
were allowed was to allow the use of one
or more colors for the U.S. area and another
color for the area not in the US
In response to question 2, students didn't
think of the best ways to improve the estimate
on the map area. The idea of changing block
size and shape -- for example, the height
-- either didn't occur to them or they thought
they should have changed the way they did
the problem, for example, use more blocks
or only add some of the area.
The third question, design your own mosaic,
was actually the next toughest question
after question 2. I guess they just didn't
believe they were allowed to do anything
they wanted.
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