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Symbolic Models

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Symbolic Models

Problem Solving with Computers

- Use symbols to convey information
- Highly structured
- Symbols
- Words
- Grammar (syntax)

- Languages are symbolic models
- So is mathematics (algebras)

- Variable
- A value that can change

- Constant
- A value that does not change

- Addition: 2 + 32 + 3
- Subtraction: 5 â€“ 25 - 2
- Multiplication: 10 ï‚´ 410 * 4
- Division: 12 ï‚¸ 312 / 3
- Exponentiation: 323^2

- Arithmetic expressions are evaluated according to the following order of operations
- At each level, operations are evaluated left to right
- (1) Parenthesis, Functions, Exponentiation
- (2) Multiplication, Division
- (3) Addition, Subtraction

- Parenthesis are used to alter the order with which operations are evaluated
- Ex.
- 4 + 5 * 2 equals 14
- (4 + 5) * 2 equals 18

- Why?

- An expression that sets two expressions equal to one another
- Example
- Final Cost = Price â€“ Discount

- Note the use of words for each value
- These are called identifiers

- Use common words and/or abbreviations
- Should reflect the nature of the represented value
- Bad example
- A1 = XY â€“ Z + Q

- What does that mean?

- Consists of values, formulae and labels
- Arranged into rows and columns
- Each row is numbered
- Each column is lettered
- The intersection of a row and a column is called a cell

Columns

Cell

Rows

- Creating a worksheet:
- Develop a solution
- Write the solution as a set of equations
- Design a layout
- Convert the equations into formulas
- Enter labels, values and formulas into worksheet
- Ta Da!

- Develop a method for determining the net pay for an employee based upon the employeeâ€™s gross pay, federal income tax rate and union dues
- Your method should include names (identifiers) for all known and unknown values

- Gross pay
- Tax rate
- Union dues

- Tax
- Net Pay
- How to calculate the above

- Tax istax rate times gross payTax = Tax Rate ï‚´ Gross Pay
- Net pay is gross pay less all deductionsNet Pay = Gross Pay - (Tax + Union Dues)

- Which are the cells where you plan to place your formulas?

- Tax = Tax Rate ï‚´ Gross Pay
- B4 = D2 x B3
- Net Pay = Gross Pay - (Tax + Union Dues)
- B6 = B3 - ( B4 + B5 )

- Develop a worksheet to calculate the cost of a meal consisting of an appetizer, a beverage, a main entrÃ©e, and a desert.
- Assume a sales tax rate of 7% and an automatic gratuity of 15%.

- Examples
- F(x) =2x+x2
- F(3) =2(3) + 32
- =6+9
- =15

2

11

- Maps inputs to outputs
- Ex. F(x) = 7x - 3

F(x)

F(2)

7(2)-3

14-3

- Math
- SQRT(x)
- FLOOR(x), CEILING(x)
- MIN(list), MAX(list)
- SUM(list), COUNT(list)

- Trigonometric
- SIN(x), COS(x), TAN(x) â€¦

- And lots more!

- Problem: At a restaurant, patrons can use a coupon to get 3 entrees for the price of 2. To do this, the entrÃ©e with the lowest price is free.
- How do we do this?

Hint:

Youâ€™ll need to use one of the functions previous mentioned

- Know
- There are 3 entrees
- One will be free

- Need
- A way to determine the lower price
- A way to calculate the final bill

- Do:
- Develop set of equations (algorithm)
- Create layout
- Convert equations to formulas
- Put it all together

- Set of equations (algorithm)
- Use MIN( )
- Lowest = MIN(First, Second, Third)
- Cost = First + Second + Third - Lowest

- Layout

- Formulas
- Lowest = MIN(First, Second, Third)
- D2 = MIN(B2, B3, B4 )
- Cost = First + Second + Third â€“ Lowest
- D4 = B2 + B3 + B4 - D2

- Put it all together!

Free Meal Example

- Problem:
- At a diving competition you plan to attend, a diverâ€™s score is the sum of the scores for all the judges, less the low and high scores, and multiplied by a difficulty factor.
- Create a worksheet to compute diverâ€™s score based on the scores for seven judges and a difficulty factor

- Relational
- Logical

Traditional

Less than

A < B

Greater than

A > B

Equal to

A = B

Excel

A < B

A > B

A = B

Traditional

Not less than

A ï‚³ B

Not greater than

A ï‚£ B

Not equal to

A ï‚¹ B

Excel

A >= B

A <= B

A <> B

Traditional

A ïƒ™ B

A ïƒš B

ïƒ˜ A

Excel

AND(A, B)

OR(A, B)

NOT(A)

Q: What is a decision?

- Something that represents a branching point in a solution
- Outcomes are often dependent on initial conditions

- Problem:
- You have a numerical grade for a student
- 65 or better is passing
- Youâ€™d like to see a â€˜Pâ€™ for passing or a â€˜Nâ€™ for not passing in the worksheet
- How do you do this?

- Starting numerical grade
- 65 or better is passing

- Final letter grade (P or N)
- Method for performing conversion

- If the grade is greater than or equal to 65, then the students passes, else the student does not pass.

- Yes!
- Use Pseudocode
- Wait!
- What the CENSORED is Pseudocode?

- Looks like a programming language
- Has all the structure of a programming language
- Has a verrrrrry loose syntax

- Example:get xresult <- x2 + 5x + 7print result
- Thatâ€™s it!
- Sloppy, ainâ€™t it?

- If Grade ï‚³ 65Then â€˜Pâ€™Else â€˜Nâ€™

- The condition is a Boolean expression
- When the condition is True, the then-action is executed
- When the condition is False, the else-action is executed

- IF(condition, then, else)
- condition is a Boolean expression(same as for If statement)
- Function evaluates to then expression when condition is true
- Function evaluates to else expression when condition is false

Numerical Grade

Formula

- Problem:
- Youâ€™d like to go see a movie.
- The movie costs $8.00, a soda costs $2.50 and a large popcorn costs $4.50.
- Based on the amount of money in your pocket, determine whether you could...
- (a) Just see the movie,
- (b) See the movie and buy a soda, or
- (c) See the movie, and buy soda and popcorn.

- Movie costs $8.00
- Soda costs $2.50
- Popcorn costs $4.50
- How much money I have in my pocket

- Cost of movie and soda
- Cost of movie, soda and popcorn
- Way to select one of the three options(that is, make a decision!)

- Option (a) costs $8.00
- Option (b) costs $10.50
- Option (c) costs $15.00
- Is there a hidden option?
- Yes! Stay home!
- So, what next?

Money < $8

- Say, isnâ€™t this a flowchart?

Money < $10.50

Stay home

Money < $15.00

Movie

Movie & soda

Movie, soda & popcorn

Money < $8

- Boxes represent actions

Money < $10.50

Stay home

Money < $15.00

Movie

Movie & soda

Movie, soda & popcorn

Money < $8

- Diamonds represent decision points

Money < $10.50

Stay home

Money < $15.00

Movie

Movie & soda

Movie, soda & popcorn

Money < $8

- Arrows show flow

Money < $10.50

Stay home

Money < $15.00

Movie

Movie & soda

Movie, soda & popcorn

Money < $8

- The arrow at the top tells us there were previous steps
- The arrow at the bottom tells us there are subsequent steps

Money < $10.50

Stay home

Money < $15.00

Movie

Movie & soda

Movie, soda & popcorn

Condition

Then-action

Else-action

If (Money < $8) ThenStay homeElse If (Money < $10.50) ThenMovie

Else If (Money < $15.00) Then

Movie & soda

Else Movie, soda & popcorn

- Using Microsoft Excel...
- Wait (again) !
- Donâ€™t you need to design a layout first?

- Problem: Your bowling team (five members) wishes to create a worksheet to keep track of its performance. The worksheet will show, for each team member as well as the entire team, scores, number of games, average, low score and high score.
- Question: How would you determine who on the team has the highest score?