CHAPTER 6 Real Numbers: Completing a Mathematical System 6.6 Making the Connections: Is There REALly a Completion to the Number System?

1. Mark Anthony G. Arrieta BSEd – Math – 4 Math 116A
Mr. Allen C. Barbaso Presentation 5
CHAPTER 6 Real Numbers: Completing a Mathematical System
6.6 Making the Connections: Is There REALly a Completion to the Number System?
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.) Reflect on ideas explored in Chapter 6 (Real Numbers: Completing a Mathematical System).
2.) Explore the connections among number systems.
3.) Explore the connections among various features of linear and quadratic functions.
Investigation:
In this section you will work outside the system to reflect on the mathematics in Chapter
6: what you’ve done and how you’ve done it.
1.) State the most important ideas in this chapter. Why did you select each?
2.) Identify all the mathematical concepts, processes, and skills you used to investigate the
problems in Chapter 6.
Discussion:
 You might have listed a number of really important ideas including real numbers, square
roots, exponents, linear functions, quadratic functions, other basic functions, slope, vertical
intercepts, zeros of a function, the vertex of a parabola, and factoring.
Review:
 Use the following collection of numbers to answer Review Questions 1 – 4:
18 – 9
√ | | –√ –√ 0
√ 5-2
√
√ 0.82
√
1.) List the numbers that belong to
a.) the whole numbers.
b.) the integers
c.) the rational numbers
d.) the irrational numbers.
e.) the real numbers.
2.) Write a decimal representation for each of the numbers.

2. 3.) Rearrange the original list of numbers so they are written in order from the smallest number
to the largest number. Justify your ordering.
4.) Place all of the numbers on a number line.
5.) Compute the value of each of the following. If the result is not a real number, say so and
justify.
a.) √ e.) √
b.) √ f.) √
c.) √ g.) √
d.) √ h.) √
6.) Calculate the following. If you use the exponential notation on your calculator, be sure to put
the exponent in parentheses. If the result is not a real number, say so and justify.
a.) 31/2
e.) (-9) 1/2
b.) -641/2
f.) 641/2
c.) 191/2
g.) -61/2
d.) 1441/2
7.) Find the slope and length of the line segment for which the given ordered pairs define the
endpoints of the line segment.
a.) (-6, 7) and (9, 5)
b.) (4, -8) and (-7, -15)
8.) Consider the basic functions from Section 6.3. Find the value(s) of x for which:
a.) L(x) = 52 d.) opp(x) = -9
b.) Q(x) = 5 e.) rec(x) =
c.) abs(x) = 4 f.) sqrt(x) = 4
9.) Consider the function y(x) = 5x – 9
a.) Complete Table 1 for the output values:
TABLE 1 Input-Output Table
x -4 -3 -2 -1 0 1 2 3 4
y(x)
b.) Identify the vertical intercept and the horizontal intercept.
c.) Graph the function. What method did you use? Why are you sure your graph is
correct?
d.) Is this function increasing or decreasing? Defend your answer.
10.) Consider the function y(x) = x2
– 2x – 15.
a.) Complete Table 2 for the output values:
TABLE 2 Input-Output Table
x -5 -4 -3 -2 -1 0 1 2 3 4 5
y(x)
b.) Identify the vertical intercept and the horizontal intercepts.
c.) Graph the function. What method did you use? Why are you sure your graph is
correct?
d.) Is this function increasing, decreasing, or neither? Defend your answer.

3. For each linear function in Review Questions 11 – 14, identify the
a.) slope.
b.) output at the vertical intercept.
c.) input at the horizontal intercept.
d.) graph as increasing or decreasing.
11.) y(t) = 4t – 11
12.) y(x) = x + 5
13.) z(w) = – w
14.) y(x) = 0
Using the information in Review Questions 15 – 17,
a.) Graph the line.
b.) Write the equation of the line.
c.) Identify the input at the horizontal intercept.
15.) Slope = 9; vertical intercept = (0, 7).
16.) Slope = –4; vertical intercept = (0, 6).
17.) Slope = ; vertical intercept = (0, –3).
18.) For each of the following quadratic functions, how will the graph compare with the graph of
the basic quadratic function Q(x) = x2
?
a.) y(x) = –5x2
c.) y(x) = 0.3x2
+ 7
b.) y(x) = 4x2
– 3 d.) y(x) = –6x2
– 4
19.) Given the factored forms of the following quadratic functions, identify the zeros and the
values of x-values of the vertex.
a.) y(x) = (x – 5)(x + 5)
b.) y(x) = (x – 2)(x + 7)
c.) y(x) = (x + 1)(x + 9)
d.) y(x) = (x – 6)(x – 3)
20.) Given the following zeros of a quadratic function, write the equation on the function in
factored form. Then multiply the factors to obtain the expanded form. Verify your answers using
both a table and a graph.
a.) Zeros: 4, –7; a = 1
b.) Zeros: –8, –6; a = 1
c.) Zeros: –5, 5; a = 1
d.) Zeros: –6 only; a = 1
Concept Map:
 Construct a concept map centered on one of the following:
a.) real numbers e.) domain
b.) square root f.) linear function
c.) basic mathematical functions g.) quadratic function
d.) polynomial

4. Reflection:
The study of a basic function helps me analyze a more general function of the same
degree by knowing that there are various kinds of basic function. Base on my prior knowledge I
just know the functions based on their degree such as the linear, quadratic, polynomial etc., and
also I know about the constant functions, fractional functions and literal functions. However,
after studying this section it strengthen my understanding on functions and its different types. I
now become more knowledgeable about functions.
Reference:
De Marois, Phil; McGowen, Mercedes and Whitkanack, Darlene (2001). “Mathematical
Investigations”. Liceo de Cayagan University, Main Library. Jason Jordan Publishing.