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Exploring
Shapes in Middle school ( Greeks, Mesopotamians )
Materials:
colored stones all in one pile, arrange into shapes,
one number at a
time. ( Note, this is NOT Dot-Math )
Introduce
certain algebraic facts, counting formulas and
∏.
Algebra:
Square numbers
4, 9, 16, 25, 36, etc. It is curious that 2 triangle numbers added together
results in a square. Now this isn’t just any two triangle numbers, but they
must be adjacent to each other. 3 and 6 and 10 are triangle numbers; 3 + 6 = 9,
a square, but 3 + 10 = 13, not a square. Determining all triangle numbers is
shown later. The important part of this lesson is to see both visually
and algebraically how to find successive square numbers.
The last image
above (bottom right) describes visually how to get the next square. Starting
with 4 stones, add 5 more stones to arrive at the next square,
i.e, 9. Repeat for 16 and for 25. After awhile, the student will physically see
the algebraic relation (a+1)*(a+1) = a*a + 2(a) + 1; SO, if I know that 7 * 7
is 49, say, then 8 * 8 must be 49 + 14 + 1, hmmm, rearranging I have 49 + 1
(easy!) which is 50 then add 14, hmmm, 50 + 10 (easy!) which is 60 + 4
remaining, that is, 64! BUT, this also means that, the pattern a2
+ 2a + 1 MUST be the same as (a +1)*(a+1) or better, (a+1) 2.
We have just factored (a+1) out of the expression a2 +
2a + 1 .
CAVEAT:
As fun as this is, it is IMPERATIVE that the squares of the numbers from 1 to
15 be memorized for immediate recall, as rapid as knowing that 9 follows 8.
Simply put, algebra is based on numbers, knowing basic number facts will make
algebra much easier for the student!
Triangle
numbers.
Note the
patterns above, the second triangle number has 2 dots at the base, the third
has 3, the fourth has 4, etc. The value of each triangle number is just the
number of dots you have for that number. So we have 1, 1+2, 1+2+3. 1+2+3+4.
1+2+3+4+5, so
the nth triangle number must be 1 + 2 + 3 + . . . + n, which thanks
to the brilliance of Gauss as a child, is n*(n+1)/2 (see Exploring Numbers 3),
so the 10th triangle number is 10*11/2 or 55.
Geometry:
All of the
regular polygons can be inscribed in a circle by dividing the circle into n
equal angles for the n sided polygon. Below, a hexagon is shown. All 6 sides of
this 6 sided polygon drawn this way are equal in length. Only a hexagon results
in its sides all being equal to the radius of the circle. The Mesopotamians
thought this was so interesting that they decided to make each interior angle
of each triangle equal to 60 degrees, and since there are six such angles,
moving from one to the next until you return to the point you started results
in 60 + 60 + 60 + 60 + 60 + 60 = 360 degrees around the circle!
Furthermore, it
was recognized that the distance around any circle compared to that circle’s
diameter was the same no matter the circle. This ratio we now call
∏. The
distance around the hexagon is 6r, or 3(2r) = 3d, where d is the diameter. This
distance is less than the distance around the circle, so 3 must be less than
the constant ratio
∏, that is,
∏
is no less than 3.
The lower left
figure above shows an estimate for the maximum value of
∏
.
The dark
blue hexagon
has the same
circle
inscribed
within it.
The black
triangle is
a 30-60-90
right
triangle and
with this it
can be shown
that length
of one of
the blue
edges of the
blue hexagon
is (2r)/(√3), giving a circumference no greater
than
(6 /
√ 3 ) * d
≈ 3.5 * d.
So, an upper bound for
∏
is 3.5
So, we have 3 <
∏ < 3.5.
There are ways to get better estimates for these bounds, see
Exploring Circles
, for example, for a hands on approach to estimate
∏ .
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