The document presents a lesson plan on rational numbers, focusing on proportional reasoning and the rational number system. It includes examples and investigations involving the purchase of tee-shirts to illustrate rates of change, equivalent ratios, and the simplification of rational expressions. The importance of studying ratios in everyday life and economic activities is also emphasized.

1. 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?

2. 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

3. 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

4. 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.