The bottom row of (gold) beads may be
associated with groups of elements from a set. Each count or
touch of these beads notes an element in the bead group. Each
column in the yellow array is a bead group.
Press space bar (PSB) or click up arrow
(CUA) to display a subset defining numeral in the upper left
corner of dark gray area. Then touch the leftmost bottom gold
bead to input subset defining numeral in yellow array. To display numerals click small square
under (X) box. Move mouse carefully over
beads in order to input more numerals in yellow array.
To explore CUA until (0) displays, click leftmost gold bead and click right arrow
(CRA) to push beads against right post. CUA until the numeral 1
displays
To change subset CUA again. When zero
(0) displays no element inputs. Click on element to remove it
from array. Click (X) to reset array.
Steps 1a:
Starting at the leftmost gold
bead count three beads (touching each with mouse), click third bead and click left arrow
(CLA) to push beads against left post.
Step 1b:
Note the upper leftmost spaces in
array have inputs of three 1's
representing the count of beads.
Step 2a: CUA until 2 displays. Starting at the leftmost gold bead
against the right post count five beads, click the fifth bead and
CLA to push beads to the middle of frame.
Step 2b:
The next five spaces will have
inputs of five 2's representing the second count of beads.
CUA until (0) displays. Click leftmost gold bead of triangle in middle of frame CRA
to reveal rectangle of brown beads. As indicated by the
arithmetic statements to the left of the frame the count of
brown beads in the rectangle is the solution to the demonstrated
math operation 3 X 5 = 15.
The solution is also equal to the count of unique pairs of
elements (1,2) possible between the two subsets articulated in
the array. This count is termed the outer product of the unions
of the subset groups
indicated by the bottom number of the addition statement. Each
Brown bead represents a pair of elements.
Step 3a: Click the rightmost bead of the
rectangle, found on the 6 rod counting from the bottom rod, and
CLA to push rectangle against triangle on left post. CUA until
the numeral 2 displays again. Then,
starting with the gold bead against the left post count after
five to seven. CUA until (0) displays. Click the next (third bead of triangle) and CRA
to push it a little to the right.
Video
Demonstrations
Demonstration Application
Step 3b:
The first two spaces of the second array row will have
inputs of 2's representing the continuation of the second count of beads.
As indicated by the multiplication arithmetic statements
the solution to the demonstrated math operation is 3 X 7 = 21.
The solution though is represented in a different base than
10, because the count of the second number was continued with
the leftmost gold bead before all 10 gold beads were included in
the counting. Since we only included 8 beads in the counting the
base of the solution must be 8. So the two beads of the three on
the left post counted to make seven are multiples of the base, 2
X 8 =16 indicated by the top number of the addition statement.
The third gold bead separated from the two, with the five gold
beads to its right, are the factors of the line of brown beads
above the third bead. The count of these brown beads is
indicated by the bottom number of the addition statement.
Therefore the elements of the set are included in 8 groups
(columns in the array). The count of unique pairs of
elements (1,2) found in groups between the subgroups is termed the inner product. These products are the
indicated by the red row of numbers above the groups or columns.
The sum of these inner products equal the multiple of the base.
See this sum at the right end of the red row of numbers (inner
products). The outer product equals the count of unique pairs of
elements (1,2) found between groups included in the two
extensions of groups that have equal
counts of elements. In this case a single group with an element
designated by a 1 and five groups with elements designated by
2's.
Step 4a: Click the topmost bead of the brown
line and CLA to push beads back into the triangle against the
post.
Step 4b:
Place mouse over the third (rightmost) bead against the left
post and see the input a third 2 in the second row of the
array.Now there are three groups with an inner product. Note the
sum of inner products is now 3 and 3 times the base 8 equals 24,
the top number of the addition statement. There is now no outer
product which is indicated by the bottom number being 0.
Step 5a:
Now CUA until (2) displays. Continue the second count over the
five beads in the middle by touching three of the beads. CLA to
push beads a little to the left. The brown rectangle will
include nine beads indicated by the bottom number of the
addition statement. 24 + 9 = 3 X 11 the demontrated math
operation.
Step 5b:
Note there are two extentions of groups that have 2 elements in
a group. The groups include the elements (1,2) and (2,2). The
count of unique pairs of found between these group produces the outer product of (9). The
inner products of the groups with elements (1,2) produce the
multiple of the base, giving 3 X 8 = 24.