Brief history about counting

1. On Counting
* Brief history of counting
* Multiples of 5’s, coins and symbols
* Using a lists,
the longest/the shortest list.
* Solving for coins.
Class exercise: Ask the students to pull out
coins, record them as a list. Exchange them
for coins that give the shortest list.
http://en.wikipedia.org/wiki/Lebombo_bone

2. On Counting

3. On Counting
A living organism must possess the ability to sense and track
quantifiable information such as “too large vs. too small,”
“too much vs. too little,” or “too hard vs. too mushy.”

4. On Counting
A living organism must possess the ability to sense and track
quantifiable information such as “too large vs. too small,”
“too much vs. too little,” or “too hard vs. too mushy.”
While animals such as pigeons and bees are able to track
directions and distance without solving equations, humans track
quantities and sizes using symbols, drawings, diagrams, tables,
etc.

5. On Counting
A living organism must possess the ability to sense and track
quantifiable information such as “too large vs. too small,”
“too much vs. too little,” or “too hard vs. too mushy.”
While animals such as pigeons and bees are able to track
directions and distance without solving equations, humans track
quantities and sizes using symbols, drawings, diagrams, tables,
etc. The symbolic system we build
to do this is called Mathematics.
1, 2, 3, 4, .. ∞, a, b c, ..α, β, γ ..
Some symbols used in
modern mathematics

6. On Counting
A living organism must possess the ability to sense and track
quantifiable information such as “too large vs. too small,”
“too much vs. too little,” or “too hard vs. too mushy.”
While animals such as pigeons and bees are able to track
directions and distance without solving equations, humans track
quantities and sizes using symbols, drawings, diagrams, tables,
etc. The symbolic system we build
to do this is called Mathematics.
The first task of mathematics is
to count, i.e. to track and record
quantities.
1, 2, 3, 4, .. ∞, a, b c, ..α, β, γ ..
Some symbols used in
modern mathematics

7. On Counting
A living organism must possess the ability to sense and track
quantifiable information such as “too large vs. too small,”
“too much vs. too little,” or “too hard vs. too mushy.”
While animals such as pigeons and bees are able to track
directions and distance without solving equations, humans track
quantities and sizes using symbols, drawings, diagrams, tables,
etc. The symbolic system we build
to do this is called Mathematics.
The first task of mathematics is
to count, i.e. to track and record
quantities.
Before numbers were invented,
human used matching methods,
where each whole item is recorded
by a corresponding physical mark,
to track quantities.
1, 2, 3, 4, .. ∞, a, b c, ..α, β, γ ..
Some symbols used in
modern mathematics

8. On Counting
Such a system serves as a
poignant warning to the visitors
at the Hanakapia Beach, Hawaii.
Hanakapia Beach HI (Wikipedia)

9. On Counting
Such a system serves as a
poignant warning to the visitors
at the Hanakapia Beach, Hawaii.
It’s more proper and effective as a
warning to the visitors since one
has to take the time to add up the
strokes to obtain the number “83.”
Hanakapia Beach HI (Wikipedia)

10. On Counting
Such a system serves as a
poignant warning to the visitors
at the Hanakapia Beach, Hawaii.
It’s more proper and effective as a
warning to the visitors since one
has to take the time to add up the
strokes to obtain the number “83.”
Note that the strokes are grouped
in 5’s.
Hanakapia Beach HI (Wikipedia)

11. On Counting
Such a system serves as a
poignant warning to the visitors
at the Hanakapia Beach, Hawaii.
It’s more proper and effective as a
warning to the visitors since one
has to take the time to add up the
strokes to obtain the number “83.”
Note that the strokes are grouped
in 5’s. In many cultures 5 strokes
are gathered as a group to match
the 5 fingers in ones hand.
Hanakapia Beach HI (Wikipedia)

12. On Counting
Such a system serves as a
poignant warning to the visitors
at the Hanakapia Beach, Hawaii.
It’s more proper and effective as a
warning to the visitors since one
has to take the time to add up the
strokes to obtain the number “83.”
Note that the strokes are grouped
in 5’s. In many cultures 5 strokes
are gathered as a group to match
the 5 fingers in ones hand.
Hanakapia Beach HI (Wikipedia)
The Chinese character
for “upright” is used as
a 5–count unit.

13. On Counting
Such a system serves as a
poignant warning to the visitors
at the Hanakapia Beach, Hawaii.
It’s more proper and effective as a
warning to the visitors since one
has to take the time to add up the
Hanakapia Beach HI (Wikipedia)
strokes to obtain the number “83.”
Note that the strokes are grouped
The Chinese character
for “upright” is used as
in 5’s. In many cultures 5 strokes
a 5–count unit.
are gathered as a group to match
the 5 fingers in ones hand.
This is so because we used fingers to
match and track quantities before numbers
were invented.

14. On Counting
Such a system serves as a
poignant warning to the visitors
at the Hanakapia Beach, Hawaii.
It’s more proper and effective as a
warning to the visitors since one
has to take the time to add up the
Hanakapia Beach HI (Wikipedia)
strokes to obtain the number “83.”
Note that the strokes are grouped
The Chinese character
for “upright” is used as
in 5’s. In many cultures 5 strokes
a 5–count unit.
are gathered as a group to match
the 5 fingers in ones hand.
This is so because we used fingers to
match and track quantities before numbers
were invented.
Before numbers were
invented, one way to
track a quantity is to
match our fingers to the
tracked items.

15. On Counting
Such a system serves as a
poignant warning to the visitors
at the Hanakapia Beach, Hawaii.
It’s more proper and effective as a
warning to the visitors since one
has to take the time to add up the
Hanakapia Beach HI (Wikipedia)
strokes to obtain the number “83.”
Note that the strokes are grouped
The Chinese character
for “upright” is used as
in 5’s. In many cultures 5 strokes
a 5–count unit.
are gathered as a group to match
the 5 fingers in ones hand.
This is so because we used fingers to
match and track quantities before numbers
were invented. Our present 10 based
Before numbers were
system came from matching the fingers on invented, one way to
both hands. However, this matching method track a quantity is to
match our fingers to the
is inefficient in tracking large quantities.
tracked items.

16. Coin-Systems
On Counting

17. On Counting
Coin-Systems
To record “large” quantities, we bundle the counts
into larger units, usually in multiples of 5, and record
quantities using these larger units.

18. On Counting
Coin-Systems
To record “large” quantities, we bundle the counts
into larger units, usually in multiples of 5, and record
quantities using these larger units.
For example, the Roman Numerals is such a system.
The Roman Numerals

19. On Counting
Coin-Systems
To record “large” quantities, we bundle the counts
into larger units, usually in multiples of 5, and record
quantities using these larger units.
For example, the Roman Numerals is such a system.
The Roman number MMMMDC is 4,600 in our
notation.
The Roman Numerals

20. On Counting
Coin-Systems
To record “large” quantities, we bundle the counts
into larger units, usually in multiples of 5, and record
quantities using these larger units.
For example, the Roman Numerals is such a system.
The Roman number MMMMDC is 4,600 in our
notation. Another example are U.S. coins where
we bundle 5 pennies as a “nickel,”
The Roman Numerals
10 cents as a “dime," 25 cents as a “quarter,"
50 cents as “half a dollar," and 100 cents as a “dollar.”

21. On Counting
Coin-Systems
To record “large” quantities, we bundle the counts
into larger units, usually in multiples of 5, and record
quantities using these larger units.
For example, the Roman Numerals is such a system.
The Roman number MMMMDC is 4,600 in our
notation. Another example are U.S. coins where
we bundle 5 pennies as a “nickel,”
The Roman Numerals
10 cents as a “dime," 25 cents as a “quarter,"
50 cents as “half a dollar," and 100 cents as a “dollar.”
Let’s use the following symbols to represent these coins:
P = Penny, N = Nickel, d = dime,
Q = Quarter, H = Half a dollar, and D = Dollar.

22. On Counting
Coin-Systems
To record “large” quantities, we bundle the counts
into larger units, usually in multiples of 5, and record
quantities using these larger units.
For example, the Roman Numerals is such a system.
The Roman number MMMMDC is 4,600 in our
notation. Another example are U.S. coins where
we bundle 5 pennies as a “nickel,”
The Roman Numerals
10 cents as a “dime," 25 cents as a “quarter,"
50 cents as “half a dollar," and 100 cents as a “dollar.”
Let’s use the following symbols to represent these coins:
P = Penny, N = Nickel, d = dime,
Q = Quarter, H = Half a dollar, and D = Dollar.
Suppose we have a penny, a dollar, a nickel, and a dime,
we may list these coins, from the smaller to the larger values,
as PNdD. The coin-list “PNdD” indicates that there are four
coins, what they are, and we may total their value as $1.16.

23. On Counting
Example A. We pull out a handful of coins and recorded them
as dNQDddP where P = Penny, N = Nickel, d = dime,
Q = Quarter, H = Half a dollar, and D = Dollar.

24. On Counting
Example A. We pull out a handful of coins and recorded them
as dNQDddP where P = Penny, N = Nickel, d = dime,
Q = Quarter, H = Half a dollar, and D = Dollar.
a. Rearrange and list them according to their values.
How many coins, and how many of each kind, do we have?
What’s their total value?

25. On Counting
Example A. We pull out a handful of coins and recorded them
as dNQDddP where P = Penny, N = Nickel, d = dime,
Q = Quarter, H = Half a dollar, and D = Dollar.
a. Rearrange and list them according to their values.
How many coins, and how many of each kind, do we have?
What’s their total value?
Arranging the coins from the smaller to the larger values,
dNQDddP is PNdddQD.

26. On Counting
Example A. We pull out a handful of coins and recorded them
as dNQDddP where P = Penny, N = Nickel, d = dime,
Q = Quarter, H = Half a dollar, and D = Dollar.
a. Rearrange and list them according to their values.
How many coins, and how many of each kind, do we have?
What’s their total value?
Arranging the coins from the smaller to the larger
values, dNQDddP is PNdddQD.
There are 7 coins: a penny, a nickel, three dimes, a quarter
and a dollar, their value is $1.61.

27. On Counting
Example A. We pull out a handful of coins and recorded them
as dNQDddP where P = Penny, N = Nickel, d = dime,
Q = Quarter, H = Half a dollar, and D = Dollar.
a. Rearrange and list them according to their values.
How many coins, and how many of each kind, do we have?
What’s their total value?
Arranging the coins from the smaller to the larger
values, dNQDddP is PNdddQD.
There are 7 coins: a penny, a nickel, three dimes, a quarter
and a dollar, their value is $1.61.
The list dNQDddP above represents a specific collection of
coins. Rearranging the list dNQDddP as PNdddQD
corresponds to physically repositioning the coins from the
penny to the dollar.

28. On Counting
Example A. We pull out a handful of coins and recorded them
as dNQDddP where P = Penny, N = Nickel, d = dime,
Q = Quarter, H = Half a dollar, and D = Dollar.
a. Rearrange and list them according to their values.
How many coins, and how many of each kind, do we have?
What’s their total value?
Arranging the coins from the smaller to the larger values,
dNQDddP is PNdddQD.
There are 7 coins: a penny, a nickel, three dimes, a quarter
and a dollar, their value is $1.61.
The list dNQDddP above represents a specific collection of
coins. Rearranging the list dNQDddP as PNdddQD
corresponds to physically repositioning the coins from the
penny to the dollar.
It’s much easier to answer various questions utilizing a coin-list
than trying to manipulate the physical coins.

29. On Counting
b. Write down another coin–list whose total value is also $1.61.

30. On Counting
b. Write down another coin–list whose total value is also $1.61.
The question is asking us to come up with $1.61 of coins in
another manner.

31. On Counting
b. Write down another coin–list whose total value is also $1.61.
The question is asking us to come up with $1.61 of coins in
another manner. There are many ways to do this, one possible
answer is PdHHH.

32. On Counting
b. Write down another coin–list whose total value is also $1.61.
The question is asking us to come up with $1.61 of coins in
another manner. There are many ways to do this, one possible
answer is PdHHH.
c. Write down the shortest list that gives $1.61?
Describe the longest list that gives $1.61.

33. On Counting
b. Write down another coin–list whose total value is also $1.61.
The question is asking us to come up with $1.61 of coins in
another manner. There are many ways to do this, one possible
answer is PdHHH.
c. Write down the shortest list that gives $1.61?
Describe the longest list that gives $1.61.
The “shortest list of symbols” translates to the “least number
of coins," i.e. how to make $1.61 with the fewest number of
coins.

34. On Counting
b. Write down another coin–list whose total value is also $1.61.
The question is asking us to come up with $1.61 of coins in
another manner. There are many ways to do this, one possible
answer is PdHHH.
c. Write down the shortest list that gives $1.61?
Describe the longest list that gives $1.61.
The “shortest list of symbols” translates to the “least number
of coins," i.e. how to make $1.61 with the fewest number of
coins. The shortest list, ordered from the smaller to the larger
values, is PdHD.

35. On Counting
b. Write down another coin–list whose total value is also $1.61.
The question is asking us to come up with $1.61 of coins in
another manner. There are many ways to do this, one possible
answer is PdHHH.
c. Write down the shortest list that gives $1.61?
Describe the longest list that gives $1.61.
The “shortest list of symbols” translates to the “least number
of coins," i.e. how to make $1.61 with the fewest number of
coins. The shortest list, ordered from the smaller to the larger
values, is PdHD.
The “longest list” means to use “as many coins as possible.”
That would be 161 pennies “PP...P," i.e. the list with 161 P’s.

36. On Counting
b. Write down another coin–list whose total value is also $1.61.
The question is asking us to come up with $1.61 of coins in
another manner. There are many ways to do this, one possible
answer is PdHHH.
c. Write down the shortest list that gives $1.61?
Describe the longest list that gives $1.61.
The “shortest list of symbols” translates to the “least number
of coins," i.e. how to make $1.61 with the fewest number of
coins. The shortest list, ordered from the smaller to the larger
values, is PdHD.
The “longest list” means to use “as many coins as possible.”
That would be 161 pennies “PP...P," i.e. the list with 161 P’s.
Suppose we have $1.06 of coins PNXX, where XX are two
coins of the same type.

37. On Counting
b. Write down another coin–list whose total value is also $1.61.
The question is asking us to come up with $1.61 of coins in
another manner. There are many ways to do this, one possible
answer is PdHHH.
c. Write down the shortest list that gives $1.61?
Describe the longest list that gives $1.61.
The “shortest list of symbols” translates to the “least number
of coins," i.e. how to make $1.61 with the fewest number of
coins. The shortest list, ordered from the smaller to the larger
values, is PdHD.
The “longest list” means to use “as many coins as possible.”
That would be 161 pennies “PP...P," i.e. the list with 161 P’s.
Suppose we have $1.06 of coins PNXX, where XX are two
coins of the same type. We may deduce that the value of XX
must be $1.00, so the XX must be two half dollars or that the
list must be PNHH.

38. On Counting
Example B. a. We have $1.72 of coins XdPNXDPN where the
X’s are the same type of coins. What kind of coin is X?

39. On Counting
Example B. a. We have $1.72 of coins XdPNXDPN where the
X’s are the same type of coins. What kind of coin is X?
Rearrange these coins as PPNNdDXX. The portion PPNNdD
gives $1.22.

40. On Counting
Example B. a. We have $1.72 of coins XdPNXDPN where the
X’s are the same type of coins. What kind of coin is X?
Rearrange these coins as PPNNdDXX. The portion PPNNdD
gives $1.22. Subtracting that from $1.72, we deduce that XX
must be $1.72 – $1.22 = $0.50 so the X must be a quarter.

41. On Counting
Example B. a. We have $1.72 of coins XdPNXDPN where the
X’s are the same type of coins. What kind of coin is X?
Rearrange these coins as PPNNdDXX. The portion PPNNdD
gives $1.22. Subtracting that from $1.72, we deduce that XX
must be $1.72 – $1.22 = $0.50 so the X must be a quarter.
Note that if we use X to designate one specific type of coin,
then we must use X and Y, two different symbols,
to represent two different types of coins.

42. On Counting
Example B. a. We have $1.72 of coins XdPNXDPN where the
X’s are the same type of coins. What kind of coin is X?
Rearrange these coins as PPNNdDXX. The portion PPNNdD
gives $1.22. Subtracting that from $1.72, we deduce that XX
must be $1.72 – $1.22 = $0.50 so the X must be a quarter.
b. We have $1.28 of coins XdPNYDPN where the X and Y are
two different kind of coins. What kind of coin are X and Y?
Note that if we use X to designate one specific type of coin,
then we must use X and Y, two different symbols,
to represent two different types of coins.

43. On Counting
Example B. a. We have $1.72 of coins XdPNXDPN where the
X’s are the same type of coins. What kind of coin is X?
Rearrange these coins as PPNNdDXX. The portion PPNNdD
gives $1.22. Subtracting that from $1.72, we deduce that XX
must be $1.72 – $1.22 = $0.50 so the X must be a quarter.
b. We have $1.28 of coins XdPNYDPN where the X and Y are
two different kind of coins. What kind of coin are X and Y?
Rearrange these coins as PPNNdDXY. The portion PPNNdD
gives $1.22.
Note that if we use X to designate one specific type of coin,
then we must use X and Y, two different symbols,
to represent two different types of coins.

44. On Counting
Example B. a. We have $1.72 of coins XdPNXDPN where the
X’s are the same type of coins. What kind of coin is X?
Rearrange these coins as PPNNdDXX. The portion PPNNdD
gives $1.22. Subtracting that from $1.72, we deduce that XX
must be $1.72 – $1.22 = $0.50 so the X must be a quarter.
b. We have $1.28 of coins XdPNYDPN where the X and Y are
two different kind of coins. What kind of coin are X and Y?
Rearrange these coins as PPNNdDXY. The portion PPNNdD
gives $1.22. Subtracting that from $1.28, we deduce that
XY must be $1.28 – $1.22 = $0.06
so the XY must be a nickel and a penny.
Note that if we use X to designate one specific type of coin,
then we must use X and Y, two different symbols,
to represent two different types of coins.

45. On Counting
Example B. a. We have $1.72 of coins XdPNXDPN where the
X’s are the same type of coins. What kind of coin is X?
Rearrange these coins as PPNNdDXX. The portion PPNNdD
gives $1.22. Subtracting that from $1.72, we deduce that XX
must be $1.72 – $1.22 = $0.50 so the X must be a quarter.
b. We have $1.28 of coins XdPNYDPN where the X and Y are
two different kind of coins. What kind of coin are X and Y?
Rearrange these coins as PPNNdDXY. The portion PPNNdD
gives $1.22. Subtracting that from $1.28, we deduce that
XY must be $1.28 – $1.22 = $0.06
so the XY must be a nickel and a penny.
In the above context of coins, we pose a question using various
coin symbols and use the symbol “X / Y” to represent a specific
“unknown coin.” Then by a series of back-track reasoning,
we recover what X is. This procedure is referred to as
“solving for X / Y” and it’s one of the main purpose of algebra.

46. On Counting
As human civilizations progress, the necessity of recording
ever larger and more precise quantities arises.

47. On Counting
As human civilizations progress, the necessity of recording
ever larger and more precise quantities arises.
Coin-systems, such as the Roman numerals or our coinlists are inefficient in recording large quantities.

48. On Counting
As human civilizations progress, the necessity of recording
ever larger and more precise quantities arises.
Coin-systems, such as the Roman numerals or our coinlists are inefficient in recording large quantities.
Hence place value systems, like the ones we use today, were
invented and this is the topic of next section.

49. On Counting
As human civilizations progress, the necessity of recording
ever larger and more precise quantities arises.
Coin-systems, such as the Roman numerals or our coinlists are inefficient in recording large quantities.
Hence place value systems, like the ones we use today, were
invented and this is the topic of next section.
Qn : What are other disadvantages for using a system like
the Roman numerals or coin-system to track quantities?

50. On Counting
Exercise A.
1. Complete the table by
converting the nickels to pennies.
2. How many nickels are there
in one dime? four dimes?
five dimes? eight dimes?
3. How many nickels are there
in one quarter? three quarters?
five quarters? six quarters?
seven quarters? ten quarters?
fourteen quarters?
4. How many nickels are there
in a half–dollar?
one and a half–dollar?
one dollar and a quarter?
two dollars and three quarter?
Nickel
penny
Nickel
1
2
5
10
11
12
3
4
5
6
7
8
9
10
13
14
15
16
17
18
19
20
penny

51. On Counting
Exercise B. In each of the following problems, we pulled out a
handful of coins and recorded them where
P = Penny, N = Nickel, d = dime,
Q = Quarter, H = Half a dollar, and D = Dollar.
Rearrange and list them according to their values.
How many coins, and how many of each kind, do we have?
What’s the total value of each?
5. PNPN
6. QNP
7. NPPQQ
8. NPdQd
9. dDQNP
10. dPPPN 11. DQPddN
12. QPPNdN
13. ddNQPdNPN
14. QdNPPdNDN
15. PPdNNDNQd
Suppose X is one type of coin and Y is another type of coin,
given the information about the value of each coin–list,
answer each of the following question.
16. XPN = $0.16 what kind of coin is X?
17. XNdN = $0.25 what kind of coin is X?
18. XdNQ = $0.45 what kind of coin is X?

52. On Counting
19. XPHD = $1.52 what kind of coin is X?
20. XXdNdQ = $1.00 what kind of coin is X?
21. XXddNDN = $1.40 what kind of coin is X?
22. XXXNNdH = $1.00 what kind of coin is X?
23. XXXddNQNQ = $2.30 what kind of coin is X?
24. XYdNdQ = $0.65 what kind of coins are X and Y?
25. XYdddNDN = $2.00, X and Y are what kind of coins?
26. XXYQ = $0.50, X and Y are what kind of coins?
27. XYYQ = $0.45, X and Y are what kind of coins?
28. XYYQ = $0.45, X and Y are what kind of coins?
29. Is it possible to have XQ = $0.45 where X is another coin?
30. Is it possible to have XYQ = $0.50 where X and Y are two
other coins?