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Mark Anthony G. Arrieta BSEd Math 4 Math 116A Mr. Allen C. Barbaso Presentation 3 CHAPTER 4 Integers: Expanding a Mathematical System 4.3 The Absolute Value Function Introduction: A student will be asked to lead a prayer. Recall the previous topic being discussed by asking a student. Introduce the purpose of studying the lesson. Ask the students about their idea on the new topic being presented. Purpose: 1.) Introduce piecewise-defined functions. 2.) Introduce the notation for change in x, x. 3.) Investigate the absolute value function. 4.) Use the absolute value notation to state mathematical ideas, concepts and rules concisely. Investigation: College Tuition: The tuition policy at a large university allows students to take 12 or more semester hours for a flat charge of $1425. The charge is $98 per semester hour if a student takes less than 12 credit units hours and student fees and perks are not charged. 1.) Complete Table 1 for the given problem situation. TABLE 1 Tuition Charges Semester Hours Tuition ($) 6 9 11 12 15 18 2.) Table 1 displays a numeric representation of a function with input the number of semester hours and output the tuition. State the domain of the problem situation. 3.) Sketch a graph of the function using your answers to Investigation 2 to generate pairs of points on the graph. Clearly label the axes and the scale. 4.) Let h represent the number of semester hours and T represent the tuition charge. a.) Write an algebraic representation (equation) for the problem situation if a student takes fewer than 12 credit hours. b.) Write an algebraic representation (equation) of the problem situation if a student takes atleast 12 credit hours.

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Page 1: Math 116 pres. 3

Mark Anthony G. Arrieta BSEd – Math – 4 Math 116A

Mr. Allen C. Barbaso Presentation 3

CHAPTER 4 Integers: Expanding a Mathematical System

4.3 The Absolute Value Function

Introduction:

A student will be asked to lead a prayer.

Recall the previous topic being discussed by asking a student.

Introduce the purpose of studying the lesson.

Ask the students about their idea on the new topic being presented.

Purpose:

1.) Introduce piecewise-defined functions.

2.) Introduce the notation for change in x, x.

3.) Investigate the absolute value function.

4.) Use the absolute value notation to state mathematical ideas, concepts and rules concisely.

Investigation:

College Tuition: The tuition policy at a large university allows students to take 12 or more

semester hours for a flat charge of $1425. The charge is $98 per semester hour if a student takes

less than 12 credit units hours and student fees and perks are not charged.

1.) Complete Table 1 for the given problem situation.

TABLE 1 Tuition Charges

Semester Hours Tuition ($)

6

9

11

12

15

18

2.) Table 1 displays a numeric representation of a function with input the number of semester

hours and output the tuition. State the domain of the problem situation.

3.) Sketch a graph of the function using your answers to Investigation 2 to generate pairs of

points on the graph. Clearly label the axes and the scale.

4.) Let h represent the number of semester hours and T represent the tuition charge.

a.) Write an algebraic representation (equation) for the problem situation if a student

takes fewer than 12 credit hours.

b.) Write an algebraic representation (equation) of the problem situation if a student takes

atleast 12 credit hours.

Page 2: Math 116 pres. 3

Discussion:

The previous investigation revolved around a problem situation that required a split in the

domain. The function process depends on the input value. Such a function is called a

piecewise-defined function.

The domain of the College Tuition problem must include all possible numbers for total

credit hours taken in a semester. This is the set of all whole numbers less than some arbitrary

upper limit. To write the function, we must split the domain. The computation of output

depends on whether the input is less than 12.

If a student takes fewer than 12 credit hours, then the tuition charge is calculated by

multiplying $98 (the charge for one credit) by the number of credit hours taken. Using h as

the number of credit hours and T as the tuition charge, we can write

if h < 12 then T(h) = 98h.

If students take 12 or more hours, they pay a flat rate of $1425. Using the same variables we

write

if h 12 then T(h) = 1425.

This type of function requires a new type of function machine. The input must be evaluated

to determine which process to use. Upon entering the function machine, a decision about the

input is made. Based on the decision, we follow exactly one of the multiple paths.

Investigation:

Sarah’s Stock Portfolio. Sarah was a stockbroker who used a simple test to determine whether

to buy or sell a stock that was of interest. If the change in the stock from opening to closing

exceeded two points, she sold the stock if she owned it and bought the stock if she did not.

5.) Use the price of the stocks in Table 2 to decide which action (buy, sell, or wait) is determined

by Sarah’s rule. Sarah owns the stocks indicated by the asterisk (*).

TABLE 2 Sarah’s Stock Choices

Stock Open Close Change Action

*IBM 70 72

AMOCO 53 49

Apple 17 24

*Casio 20 18.5

Sharp 13 13

*TI 19 –3

HP 43 40.75

*SPC 11 7

DMW 62 3

Motorola 59 –2

Discussion:

When computing an amount of change, the question of order of subtraction is an important

one. Since we are interested in change, we want to know the amount (magnitude) of change,

as well as the direction of the change. If the quantity gets larger, the change should be

positive, but if the quantity gets smaller, the change should be negative. This computation

can be accomplished by subtracting the initial value of the quantity from its final value.

Page 3: Math 116 pres. 3

If p represents the price of a stock, let

p1 represent the initial value,

p2 represent the final value, and

p denote the change in p.

The change in p is the final value minus the initial value, or p = p2 – p1.

Points to Ponder:

The Greek letter delta, represented by the symbol , is used in mathematics for the phrase

change in. So the symbol p means the change between two values of p.

Notice that we use small numbers below and to the right of the variable, such as 1 in p1.

These numbers, called subscripts, are used by mathematicians when they use the same

variable in several different settings.

We are using the variable p to represent both the initial and final price. The subscripts 1 and

2 on p1 and p2 represent a symbolic way to distinguish between the two values. When you see

p1 and p2, read these as “p sub 1” and “p sub 2”. The word sub indicates subscript. In this

case, the subscript 1 is used to designate the initial value and the subscript 2 used to designate

the final value.

Investigation:

6.) a.) Complete Table 3 where the input is x and the output is the absolute value of x,

written as | |. b.) For each answer in part (a), how does the output (answer) compare to the input (the

original value of x)?

TABLE 3 Absolute Value

x | |

5

–7

0

–3

8

2

–17

Discussion:

As you noticed in the preceding investigation, unless you know where a quantity is positive

and where it is negative, you can’t determine the absolute value of that quantity. Absolute

value is a unary operation that has one input and one output. Before the output is determined,

we must know the sign of the input. If the input is positive or zero, then the output is the

same as the input. However, if the input is negative, then the output is the opposite of the

input.

Explorations:

1.) If x represents the money in my petty cash fund, what notation would be used to designate the

a.) amount of money at the beginning of the day (initial value)?

b.) amount of money at the end of the day (final value)?

c.) the change in cash during the day?

Page 4: Math 116 pres. 3

2.) Complete Table 4 by finding x. Assume that x1 represents the initial value and x2 represents

the final value.

TABLE 4 Daily Change in Petty Cash Account

Initial Value x1 in $ Final Value x2 in $ Change in Petty

Cash x in $

5 3

3 5

3 3

0 7

7 0

0 –2

–2 0

–3 4

4 –3

1 –2

–3 3

3.) What are the solutions to the following equations?

a.) | | = 7

b.) | | = 0

c.) | | = –3

d.) | | = 11

Explorations: (Answer)

1.) If x represents the money in my petty cash fund, what notation would be used to designate the

a.) amount of money at the beginning of the day (initial value)?

Answer: x1

b.) amount of money at the end of the day (final value)?

Answer: x2

c.) the change in cash during the day?

Answer: x

Page 5: Math 116 pres. 3

2.) Complete Table 4 by finding x. Assume that x1 represents the initial value and x2 represents

the final value.

TABLE 4 Daily Change in Petty Cash Account

Initial Value x1 in $ Final Value x2 in $ Change in Petty

Cash x in $

5 3 –2

3 5 2

3 3 0

0 7 7

7 0 –7

0 –2 –2

–2 0 2

–3 4 7

4 –3 –7

1 3 –2

–6 –3 3

3.) What are the solutions to the following equations?

a.) | | = 7 Solutions: –2 and 12

b.) | | = 0 Solutions: –5 only

c.) | | = –3 Solutions: No Solution

d.) | | = 11 Solutions: –8 and 14

Reflection:

Explaining the concept of absolute value is a great way to examine your skills and

knowledge on number system and integers. If a person will ask me what is an absolute value I

will just show an example using a number line for easy understanding. Also, I will emphasize

that absolute value is always positive and if that person tries to solve for the solution of an

absolute value equation and it happen that the right side equation is a negative number, therefore

that equation has no solution because as what I have said absolute value is always positive.

Understanding problems that involve the rate of change is necessary for most people especially

those works are in the field of business because a minimal amount of change has already a big

impact on their life. As a future math teacher, I will really point out the importance of studying

rate of change and absolute value including its absolute value notations.

Reference:

De Marois, Phil; McGowen, Mercedes and Whitkanack, Darlene (2001). “Mathematical

Investigations”. Liceo de Cayagan University, Main Library. Jason Jordan Publishing.