* 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

On Counting

On Counting

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.”

On Counting

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.

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.”

On Counting

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.

Some symbols used in modern mathematics

1, 2, 3, 4, .. ∞, a, b c, ..α, β, γ ..

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.”

The symbolic system we build to do this is called Mathematics.

On Counting

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.

Some symbols used in modern mathematics

1, 2, 3, 4, .. ∞, a, b c, ..α, β, γ ..

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.”

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.

On Counting

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.

Some symbols used in modern mathematics

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 ..

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.”

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.

Hanakapia Beach HI (Wikipedia)

On CountingSuch 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)

On CountingSuch 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)

On CountingSuch a system serves as a poignant warning to the visitors at the Hanakapia Beach, Hawaii.

Note that the strokes are grouped in 5’s.

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)

On CountingSuch a system serves as a poignant warning to the visitors at the Hanakapia Beach, Hawaii.

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.

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)

On Counting

The Chinese character for “upright” is used as a 5–count unit.

Such a system serves as a poignant warning to the visitors at the Hanakapia Beach, Hawaii.

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.

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)

On Counting

The Chinese character for “upright” is used as a 5–count unit.

This is so because we used fingers to match and track quantities before numbers were invented.

Such a system serves as a poignant warning to the visitors at the Hanakapia Beach, Hawaii.

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.

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)

On Counting

The Chinese character for “upright” is used as a 5–count unit.

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.

Such a system serves as a poignant warning to the visitors at the Hanakapia Beach, Hawaii.

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.

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)

On Counting

The Chinese character for “upright” is used as a 5–count unit.

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.

Such a system serves as a poignant warning to the visitors at the Hanakapia Beach, Hawaii.

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.

Our present 10 based system came from matching the fingers on both hands. However, this matching method is inefficient in tracking large quantities.

Coin-SystemsOn Counting

Coin-SystemsOn Counting

To record “large” quantities, we bundle the counts into larger units, usually in multiples of 5, and record quantities using these larger units.

Coin-Systems

The Roman Numerals

On Counting

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.

Coin-Systems

The Roman Numerals

On Counting

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.

Coin-Systems

The Roman Numerals

Another example are U.S. coins wherewe bundle 5 pennies as a “nickel,” 10 cents as a “dime," 25 cents as a “quarter," 50 cents as “half a dollar," and 100 cents as a “dollar.”

On Counting

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.

Coin-Systems

The Roman Numerals

Another example are U.S. coins wherewe bundle 5 pennies as a “nickel,” 10 cents as a “dime," 25 cents as a “quarter," 50 cents as “half a dollar," and 100 cents as a “dollar.”

On Counting

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.

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.

The Roman number MMMMDC is 4,600 in our notation.

Coin-Systems

The Roman Numerals

Another example are U.S. coins wherewe bundle 5 pennies as a “nickel,” 10 cents as a “dime," 25 cents as a “quarter," 50 cents as “half a dollar," and 100 cents as a “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.

On Counting

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.

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.

On CountingExample 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.

On CountingExample 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?

On CountingExample 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.

On CountingExample 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?

There are 7 coins: a penny, a nickel, three dimes, a quarter and a dollar, their value is $1.61.

Arranging the coins from the smaller to the larger values, dNQDddP is PNdddQD.

On CountingExample 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?

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.

Arranging the coins from the smaller to the larger values, dNQDddP is PNdddQD.

On CountingExample 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?

There are 7 coins: a penny, a nickel, three dimes, a quarter and a dollar, their value is $1.61.

It’s much easier to answer various questions utilizing a coin-list than trying to manipulate the physical coins.

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.

Arranging the coins from the smaller to the larger values, dNQDddP is PNdddQD.

On Countingb. Write down another coin–list whose total value is also $1.61.

On Countingb. 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.

On Countingb. 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.

On Countingb. Write down another coin–list whose total value is also $1.61.

c. Write down the shortest list that gives $1.61?Describe the longest list that gives $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.

On Countingb. Write down another coin–list whose total value is also $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.

c. Write down the shortest list that gives $1.61?Describe the longest list that gives $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.

On Countingb. Write down another coin–list whose total value is also $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.

c. Write down the shortest list that gives $1.61?Describe the longest list that gives $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.

The shortest list, ordered from the smaller to the larger values, is PdHD.

On Countingb. Write down another coin–list whose total value is also $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.

c. Write down the shortest list that gives $1.61?Describe the longest list that gives $1.61.

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.

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.

The shortest list, ordered from the smaller to the larger values, is PdHD.

On Countingb. Write down another coin–list whose total value is also $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.

c. Write down the shortest list that gives $1.61?Describe the longest list that gives $1.61.

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.

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.

The shortest list, ordered from the smaller to the larger values, is PdHD.

On Countingb. Write down another coin–list whose total value is also $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.

c. Write down the shortest list that gives $1.61?Describe the longest list that gives $1.61.

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.

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.

The shortest list, ordered from the smaller to the larger values, is PdHD.

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.

On CountingExample 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?

On CountingExample 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.

On Counting

Subtracting that from $1.72, we deduce that XX must be $1.72 – $1.22 = $0.50 so the X must be a quarter.

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.

On Counting

Subtracting that from $1.72, we deduce that XX must be $1.72 – $1.22 = $0.50 so the X must be a quarter.

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?

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.

Rearrange these coins as PPNNdDXX. The portion PPNNdD gives $1.22.

On Counting

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?

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?

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.

Rearrange these coins as PPNNdDXX. The portion PPNNdD gives $1.22.

On Counting

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?

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 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.

Rearrange these coins as PPNNdDXX. The portion PPNNdD gives $1.22.

On Counting

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?

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 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.

Rearrange these coins as PPNNdDXX. 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.

On Counting

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?

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 PPNNdDXY. The portion PPNNdD gives $1.22.

Rearrange these coins as PPNNdDXX. 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.

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

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

On Counting

Hence place value systems, like the ones we use today, were invented and this is the topic of next section.

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

On Counting

Hence place value systems, like the ones we use today, were invented and this is the topic of next section.

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

Qn : What are other disadvantages for using a system like the Roman numerals or coin-system to track quantities?

On Counting

1. Complete the table by converting the nickels to pennies.

Exercise A.Nickel penny

1 5

2 10

3

4

5

6

7

8

9

10

Nickel penny

11

12

13

14

15

16

17

18

19

20

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?

On Counting

5. PNPN

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?

6. QNP 7. NPPQQ9. dDQNP 10. dPPPN 11. DQPddN 12. QPPNdN13. ddNQPdNPN 14. QdNPPdNDN 15. PPdNNDNQd

8. NPdQd

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?

On Counting19. XPHD = $1.52 what kind of coin is X?

21. XXddNDN = $1.40 what kind of coin is X?

20. XXdNdQ = $1.00 what kind of coin is X?

23. XXXddNQNQ = $2.30 what kind of coin is X?

22. XXXNNdH = $1.00 what kind of coin is X?

25. XYdddNDN = $2.00, X and Y are what kind of coins?

24. XYdNdQ = $0.65 what kind of coins are X and Y?

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?