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Mark Anthony G. Arrieta BSEd Math 4 Math 116A Mr. Allen C. Barbaso Presentation 4 CHAPTER 5 Rational Numbers: Further Expansion of a Mathematical System 5.2 Rational Numbers, Rational Expressions, and Proportional Reasoning 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.) Use rates of change to explore proportional reasoning. 2.) Introduce the rational number system. 3.) Explore multiple representations of rational numbers. 4.) Introduce the reciprocals of the whole numbers. Investigation: Okimbes Tee-Shirt Purchase: Do you remember Okimbe, our tee-shirt salesperson? Okimbe is negotiating with a tee-shirt wholesaler to buy more tee-shirts (business is very good). The wholesaler charges $30 for a box of 8 tee-shirts. Okimbe wants to analyze the situation mathematically. 1.) To analyze the situation numerically, complete Table 1. Fill in the total cost column first, then compute the successive finite differences n and c. TABLE 1 Buying Tee-Shirts from a Wholesaler 1 n Number n of Tee-Shirts Purchased Total Cost c of Tee-Shirts ($) c 0 8 30 16 24 32 40 2.) What is , the rate of change of cost per tee-shirt purchased? Why? Is this rate of change a constant? 3.) Using the number of tee-shirts purchased as input and total cost as output, sketch a graph of the points listed in Table 1. Do the points lie in a straight line? How could you determine that from the information in Table 1? 4.) How much will Okimbe pay for 184 shirts? What did you do to answer this question? 5.) Write five other ratios that are equivalent to the original rate of change. How many other ratios are equivalent to the original rate of change?

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

Mr. Allen C. Barbaso Presentation 4

CHAPTER 5 Rational Numbers: Further Expansion of a Mathematical System

5.2 Rational Numbers, Rational Expressions, and Proportional Reasoning

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.) Use rates of change to explore proportional reasoning.

2.) Introduce the rational number system.

3.) Explore multiple representations of rational numbers.

4.) Introduce the reciprocals of the whole numbers.

Investigation:

Okimbe’s Tee-Shirt Purchase: Do you remember Okimbe, our tee-shirt salesperson?

Okimbe is negotiating with a tee-shirt wholesaler to buy more tee-shirts (business is very good).

The wholesaler charges $30 for a box of 8 tee-shirts. Okimbe wants to analyze the situation

mathematically.

1.) To analyze the situation numerically, complete Table 1. Fill in the total cost column first, then

compute the successive finite differences n and c.

TABLE 1 Buying Tee-Shirts from a Wholesaler 1

n Number n of

Tee-Shirts

Purchased

Total Cost c of

Tee-Shirts ($) c

0

8 30

16

24

32

40

2.) What is

, the rate of change of cost per tee-shirt purchased? Why? Is this rate of change a

constant?

3.) Using the number of tee-shirts purchased as input and total cost as output, sketch a graph of

the points listed in Table 1. Do the points lie in a straight line? How could you determine that

from the information in Table 1?

4.) How much will Okimbe pay for 184 shirts? What did you do to answer this question?

5.) Write five other ratios that are equivalent to the original rate of change. How many other

ratios are equivalent to the original rate of change?

6.) In response to customer demand, the wholesaler now sells tee-shirts packaged 4 shirts to the

box for $15. Complete Table 2 in which the n = 4 instead of 8.

TABLE 2 Buying Tee-Shirts from a Wholesaler 2

Number n of Tee-Shirts

Purchased Total Cost c of Tee-Shirts ($)

0

4 15

8

12

16

20

7.) What happens to the entries in the cost column if n is smaller than 4?

8.) Write an equation for the function whose input is the number of tee-shirts purchased and

whose output is the total cost. What role does the rate of change of cost per tee-shirt purchased

play in the equation?

9.) What if Okimbe were interested in the rate of change of the number of tee-shirts purchased

per cost? What is this rate? How is this rate related to the original rate?

Discussion:

The set of integers is no longer sufficient to represent rates of change. The integers are closed

under addition, subtraction, and multiplication but not under division, since integers do not

include fractions or decimals. The integers are primarily used to answer questions that ask

“how many?” Once we have fractions or decimals, we can answer questions about

measurement rather than counts. We can also do multiplicative comparisons, such as finding

a unit price or a speed.

The fraction

deserves a special note. This number can also be expressed as 25%. The

word percent means “per 100”. So

is the ratio 25 per 100 or 25 percent. Any fraction

with a denominator of 100 can be expressed as a percent.

To conclude this discussion, let’s return to simplifying rational expressions. To simplify

rational expressions, write numerator and denominator in a factored form and divide out

common factors. The first step, factorization, has already been done on

. The fraction

can be simplified to

by dividing the numerator and denominator by common factor 3. In a

similar fashion, the numerator and denominator of

share a common factor, a. Notice that

the denominator is in a factored form, since the last (only) operation indicated in the

denominator is multiplication.

To simplify

, divide numerator and denominator by a. Note that either a nor b can be zero.

So the simplest equivalent form of

is

with a 0 and b 0. Note that the restriction on a

should be stated. In the original expression, it is implicit that both a and b are not zero. In the

simplified form, we have no way of knowing that a cannot be zero unless it is explicitly

stated.

Notice that

cannot be simplified (though the fraction can be written as

). While 3 and 4

are terms in the denominator, they are not factors. Likewise,

cannot be simplified. The

denominator is not in a factored form, thus prohibiting us from simplifying the expression.

can be factored

, so it simplifies to

. This happens only when a and b have

common factors or a is a factor of b.

Explorations:

1.) Write the simplest form of each of the following ratios.

a.) There are 32 apples sold for every 20 oranges sold.

b.) For every 33 men, there are 51 women.

c.) There are 12 dogs for every 8 cats in the neighborhood.

d.) I was able to drive 190 miles in 4 hours.

e.) The pressure was 36 pounds for every 8 square inches.

f.) There are x motorcycles for every 7 cars.

g.) We drove 200 miles in y hours.

h.) For every 2x men, there were 5y women.

i.) The restaurant ordered 16 cases of soda for every 9 cases of canned ice tea.

2.) Write the reciprocal.

a.) 6 e.) y – 3

b.) –5 f.) 3x + 2y

c.) x g.)

d.) 7x h.)

3.) For every two students who attend college full time, there are seven students who are part-

time.

a.) What is the ratio of full-time students to part-time students?

b.) Write at least five other ratios that are equivalent to the given ratio of full-time

students to part-time students.

Explorations: (Answers)

1.) Write the simplest form of each of the following ratios.

a.) 8:5

b.) 11:17

c.) 3:2

d.) 95:2

e.) 9:2

f.) x:7

g.) 200:y

h.) 2x:2y

i.) 16:9

2.) Write the reciprocal.

a.)

e.)

b.)

f.)

c.)

g.)

d.)

h.)

3.) For every two students who attend college full time, there are seven students who are part-

time.

a.) 2:7

b.) 4:14, 6:21, 8:28, 10:35 and 12:42

Reflection:

Studying ratios is very important in our day-to-day living. It is one of the fundamentals of

mathematics. If not all, almost all economic activities involve ratios especially when you are

going to purchased something. For instance, you are planning to buy a chocolate worth P20.00

per bar and suddenly you have realize that you have only P97.00 in your pocket, and by using

ratio you can easily get the maximum number of chocolate bar that you can purchased. Another

important thing about ratio is that you can easily solve the profit (if you are selling something)

based on the rate of change or the change of the amount per output.

Reference:

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

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