Mathematics is a curse to some and a euphoric joy for
others. This article seeks to lessen the abrasive nature perceived by so many.
Some readers may wish to gather some materials and supplies to facilitate
visualizing what is written here. For those wanting to follow the visual path,
here are the things required: graph paper, a ruler, a compass (the device with
two hinged arms, one ending in a point and the other ending in a pencil),
scissors, a few pencils, a pencil sharpener, and an eraser.
Author’s request: As a leap of faith, please accept that a
circle has 360°. Also, please do all of the lessons while sitting comfortably at
a table.
Preparation work: Starting near the top left corner of a
piece of graph paper construct a 10x10 times table.
Lesson #1 – Squaring
the circle
Using a piece of graph paper, in each corner draw a circle
having a diameter of 3” (7.62 cm), use two intersecting graph lines for the
centre point of the of the circle. With the compass draw the first circle and
measure from one side of the circle to the opposite side by passing through the
centre point, this measurement better be 3” or you will need to get out the
eraser and try again.
The next step is mark the centre with the pencil by making a
small circle around the centre point ensuring the small circle reaches at least
the half point between the gridlines above, below and to each side. Now draw
two lines that pass through the centre point of the circle by following the
graph lines used to set the centre point. One line will have an up/down
orientation while the other line will be orientated from side to side based on
the grid. The next step is to cut out the circle and then by following the
lines cut the circle into four equal parts and reassemble the circle.
Finally, rotate each piece so that what was on the inside is
now on the outside and what was on the outside now forms the centre. On graph
paper place one corner on an intersection of two graph lines and place a dot
where the two graph lines cross and then mark the opposite corner with another
dot and remove the loose pieces of paper. Using only up and down lines
(vertical) and side to side lines (horizontal) connect the two dots, the reader
should now see a square.
How did this happen? Remembering the 360° circle rule
and that the full circle was cut into 4 equal pieces the reader can reference
the times table and turn it into a division table by simply going down till
they reach 4, then move sideways till they reach 36 and finally go to the very
top row where they will find a 9. The finale step is to bring back the zero (0)
and the degree sign (°) thus turning the 9 into 90°.
To further confirm what has happened measure the length of
two sides of the square that are touching. The two sides must be of equal
length, thus showing that when all of the inside angles are the same the
lengths of the lines must be the same.
Lesson #2 – NOT Squaring
the circle
Using the next circle from Lesson #1 add one horizontal line
to this circle ensuring the line touches opposite sides of the circle and
passes through the centre point. A second line is to be added to the circle,
make sure that the second line does not follow any graph lines but crosses many
graph lines and still passes through the centre. The next step is mark the
centre with the pencil by making a small circle around the centre point
ensuring the small circle reaches at least the half point between the gridlines
above, below and to each side. Now take a picture of the circle for later
referencing.
Repeat the same steps from Lesson #1 to remove the circle
and divide it in to four pieces by following the lines. Match up the new shapes
into two piles of two pieces each where each pile has two identical pieces.
Assemble the pieces on graph paper picking the next piece from the other pile
lining up the straight edges. Use the horizontal graph lines to line up the two
pieces that are touching that horizontal line or at least close to one. Finally
place a dot to mark the four points where the pointy bits are.
Remove the pieces of the circle and connect the dots with a
straight line with none of the lines crossing. If the work has been done
properly the four sided shape should look like a square that is leaning over to
one side.
How did this happen? Remembering the 360° circle rule
and that the full circle was cut into 4 pieces, though this time not all of the
pieces are the same though they did make two pairs of like shapes. Just a
reminder, in mathematics the word ‘like’ can be used as ‘the same as’.
Upon closer examination, or a harder look, the following
should be obvious or easily measured: opposite angles are equal and opposite
sides will never meet no matter how long the sides are extended for these are
the traits of a parallelogram.
Lesson #3 – Stretching
the Squaring
This lesson is more difficult and so the step will be
presented outside of paragraph form. Please read and understand the steps below
before proceeding.
- Using graph paper once again, place three dots on the same horizontal graph line with the same number of graph squares separating the outside dots from the centre dot.
- Using the compass draw a circle around each of the outer dots, use each outer dot as the centre of the circle and the middle dot as the edge of each circle. Draw a small square around the middle dot using the nearest gridlines.
- Using the ruler draw a straight horizontal line from the outside edge of one circle to the outside edge of the other circle passing through the three dots.
- Draw a straight line joining the top of the two circles and another joining the bottom of each circle.
- Draw a vertical line, using the graph line running through only the centre dot.
- Using the pencil, darken the top and bottom lines along with the outside half of each of the circles.
- Use the compass placing the point on the centre dot and the pencil at the outside edge of one circle where the center horizontal line passes through the circle and scribe a new larger circle.
The shape has now been set and so cut it out and then cut it
up as was done for Lesson #1, along the horizontal and vertical lines that pass
through the centre dot.
Repeat the process of reassembling the pieces with the centre
dot pieces now pointing outwards. Mark your four where the pieces of the centre
dot are found, remove the pieces of paper and connect those dots using the
ruler, once again without allowing any line to cross another line.
Congratulations as you have now made a rectangle.
How did this happen? Remembering the 360° circle rule
and that the full circle was cut into 4 pieces in Lesson #1. The reader might
be tempted to, and probably will, proclaim – ‘but wait, we started with two
circles and so the total must be 720°’and the reader would not be wrong as they
are two circles. What may have been missed is how much of each circle is being
used. The two horizontal lines that start and stop at the tops and bottoms of
each circle means that only half of each circle is being used; thus holding the
number of degrees to 360 as show by the larger circle from step #7.
Lesson #4 – Halving what
you have
Have on hand the cutout pieces from Lesson #1 and Lesson #3.
On a fresh piece of graph paper, reassemble the pieces with Lesson #1 near the
top of the page and Lesson #3 near the bottom and label each location with the
lesson number near the hand side of the page. When positioning each shape near
the centre of the page, ensure that bottom left piece is positioned at the
intersection of both a horizontal gridline and a vertical gridline and
construct the shape in a clockwise manner. Remove the top right piece of the paper
and mark the three remaining pieces of what was the centre dot.
For each of the lesson shapes connect the dots with out
crossing the lines. Using a light touch and the compass do the following, place
the compass point on the bottom left hand dot and the pencil on the right hand
dot, now move the pencil up and around until you reach the same horizontal
gridline you started from. Using a light touch extend the horizontal line to
the left until it pass where the curved line stopped. Replace all of the pieces
of the previous lessons and mark the top right corner, which must be outside of
the curved line.
Now that the work is done we will move on to the reasoning
or logic.
What do we have now? Remembering the 360° circle rule
and how when that circle was cut up into pieces the inside corners still have
to add up to 360°. With Lesson #1 and Lesson #3 there was four pieces of the
same shape and size and so each inside angle was the same. For Lesson #2 there
were two different shapes and so the end result was seeing two different inside
angels with the same angle being opposite from one side to the other, yet the
inside angles still must add up to 360°.
Returning to the graph paper it can be seen that the inside
angles of the shape are inside a half circle and so the inside angles of the
shape need only add up to half of 360° or 180°. Another ‘must be’ realization
is for the area of the triangle. It is known that in Lesson #1 and Lesson #3 a square
and a rectangle were created. By removing just one corner, our new shape became
half of the original, and this halving applies to the area as well.
Lesson #5 – Adding it
all up
Return to the graph paper with the times table then using a
fresh page, create three new times tables. The first table need only end at the
number 3 in both directions, the second table ends at 4 in both directions and
the final table ends at 5 in both directions. Mark the bottom row of each table
place a dot on the left hand side of the ones column and the right hand side of
the furthest right hand column. Take a note of the lowest far right box; for
example for the 3 table the number would be a 9 or simply count the squares in
the multiplication area, when that level of work is preferred.
Task: take the bottom right hand corner of the two
smaller squares and add them tighter and compare that the bottom right hand
corner of the largest square.
Cut out theses three tables and arrange them on a fresh
piece of graph paper getting the pairs of dots as close to each other without
over lapping. It should be noted that when done corrected where the 3 table and
the 4 table meet, the inside angle looks like a single corner of a square or a
rectangle.
By now there should be plenty of scrap bits of paper around
to use for hand done long multiplication. Extend the master times table up to
13 only focusing on the squares where the number is multiplied by itself, for
example 11×11. Write down the row or column number, both are the same anyways,
and next to it write down the answer found in the graph square under that
number. Example: column 6 would show 36 and column 8 would 64. Returning the
graph paper worksheet make a triangle using the columns 5, 12 and 13.
Hint: To get things right the 5 line and the 12 line will
look like a single corner of a square or rectangle. Connect the far points of
the 5 line and the 12 line and then take a scrap piece of graph paper and see
if that line is 13 graph squares.
Task: refer to the master time table and take what
would the bottom right hand corner of the two smaller squares and add them
tighter and compare that the bottom right hand corner of the largest square.
What has been learned? Both physically and mathematically it has been shown that certain
combinations of lengths can be used to make a triangle where one corner looks
like a single inside angle of a square. Additionally, the side opposite the
largest angle is the side that is the longest.
If the work for Lesson #5 was done accurately, the adding up
the bottom right corner of the short sided squares equals the bottom right
corner of the longest sided square.
Conclusion:
The Ionian Greek mathematician and philosopher Pythagoras of
Samos (570–490 BC) is credited for figured out Lesson #5 all on his own and
that was almost 3,000 years ago at the time of this writing. The earlier
lessons provided here were also sorted out by people working without the help
of a teacher who already had the knowledge.
Some of the hard lessons learned here:
-
On a times table when a number is multiplied by itself
the bottom right number forms a square when all of the graph-squares are the
same size.
-
When a four sided enclosed shape is created from a
circle the 360° can not be lost.
-
When a three side enclosed shape is created from a half
circle the 180° can not be lost
-
When a three sided figure contains an inside angle that
is 90° the other two inside angles MUST add up to 90°
-
The inside angle that is largest is opposite the side
that is longest
-
By using the squares made up within the times table, the
results of the two short sides equals square made by the longest side. Assuming
the sides of a triangle are label by length as A,B and C, and that the symbol
for the times table square is ² then
to find the length of any side when the other two are given can be written as
A² + B² = C²
This article has been presented for anyone who has basic
knowledge of, or the acceptance of numbers and rudimentary addition and
multiplication. For beginner learners this article can be used by those who
instruct to better describe and demonstrate the ‘why’, while for middle
learners this article can show the ‘why’ and for out of practice adults this
article has been written to give a well guided review.
Future articles similar to this one are anticipated for
other mathematical topics.
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