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

Mr. Allen C. Barbaso Presentation 2

CHAPTER 3 Functional Relationships

3.3 Geometric Representation

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 graphs as a method of communicating information geometrically.

2.) Develop the ability to interpret graphs that display relationships between two quantities.

3.) Develop a geometric representation of functions.

Discussion:

Mayan Mix-Up:

Background: Two archaeologists, Dr. Art Fact and Dr. Barry Treasure, each have acquired

writings describing the location of an ancient Mayan city. Each archaeologist is aware of the

other’s planned search for Mayan treasures. Both Art and Barry have planted spies among

the persons working for the other archaeologist. Each plans to search for the other’s city first,

delaying the excavation of his own site to avoid having the other get both treasures first.

Each man hopes the other will run out of funds, get tired, and leave for home.

Rules:

Each player begins with a Reference Map (Figure 1) of his or her city by recording the

location of four sites as follows:

The Site of Worship intersects 4 consecutive vertices on the Reference Map,

horizontally, vertically, or diagonally.

The Site of Ancient Secrets intersects 3 consecutive vertices on the Reference Map,

horizontally, vertically, or diagonally.

The Site of Mathematical Writings, each intersecting 2 consecutive vertices on the

Reference Map, horizontally, vertically, or diagonally.

Vertices are labeled by recording the number of horizontal units from the origin followed

by the number of vertical units. The origin (0, 0) is at the lower left corner, the vertex (10,

10) is at the upper right corner. to mark the vertex (3, 7), we move 3 units to the right

from (0, 0) followed by 7 units up.

Player 1 names 7 locations where s/he plans to start digging by 7 vertices on the Search

Plan (Figure 1). Player 1 announces the 7 locations by naming the ordered pair

corresponding to the vertices selected.

Player 2 marks each of Player 1’s locations on the appropriate vertices his/her Reference

Map. After the 7 locations are named, Player 2 announces how many sites were located

and how many times each site was selected, saying “You located the Site of Worship

once” or “You located the Site of Ancient Secrets twice.” Player 2 does not name the

vertex or vertices of the sites discovered by Player 1.

When all the vertices of a given site have been located by a player, the site is considered

excavated and the other player loses location attempts: 3 attempts are lost when the Site

of Worship has been located; 2 attempts when the Site of Ancient Secrets is located, and

1 attempt when each Site of Mathematical Writings is located. For example, if Player 1

has located all 3 vertices of Player 2’s Site of Ancient Secrets, Player 2 gets only 5 rather

than 7 location probes in the next round.

After Player 1 completes 7 location probes, Player 2 has 7 attempts to locate Player 1’s

sites, unless Player 2 has excavated one or some of Player 1 sites. The game continues

until one player has located all of the opponent’s sites.

Investigation:

1.) Begin the game by creating a Reference Map for your city. Don’t show this to anyone else.

Record the vertices where each site is located.

Site of Worship:

Site of Ancient Secrets:

Math Site 1: Math Site 2:

Play the game. Keep track of all attempts and successful probes in each round, using the Search

Plan and Reference Map. Record vertices in boxes.

Figure 1

Reference Map

Search Plan

10

9

8

7

6

5

4

3

2

1

0

0 1 2 3 4 5 6 7 8 9 10

Discussion:

In the Mayan Mix-Up game, a vertex indicating the location of a site is given by supplying

two pieces of information: a horizontal component and a vertical component. Given these

two components, a point on the graph is specified.

An ordered pair has the form (a, b), where a and b are numbers in which the first number,

a, is called the first component, and the second number, b, is called the second component.

The notation (a, b) implies that order is important. This means that (a, b) and (b, a) are

different unless a = b.

We have a correspondence between an ordered pair of numbers and a point in the plane. This

is a powerful idea since it allows the application of geometry (a graph) to algebra and vice

versa. The idea of creating a correspondence between pairs of numbers and points in the

plane is credited to the French mathematician Rene Descartes in 1637.

Points to Ponder:

Point in the plane are indicated by choosing a starting point, called the origin.

The origin is the point that corresponds to the ordered pair (0, 0).

We identify a point by listing how far we must travel horizontally and then vertically from

the origin to reach the point.

For a given ordered pair (a, b):

o The first component, a, indicates a horizontal position of a point in the plane.

o The second component, b, indicates a vertical position of a point in the plane.

The input axis is the horizontal number line in the plane.

The output axis is the vertical number line in the plane.

Consider a possible placement of sites on the Reference Map (Figure 2). The vertices of each

site are identified by ordered pairs, as follows:

Site of Worship: (3, 3), (4, 3), (5, 3), (6, 3).

Site of Ancient Secrets: (2, 6), (3, 7), (4, 8).

Mathematical Writings Site I: (6, 5), (6, 6).

Mathematical Writings Site II: (7, 9), (8, 9).

Figure 2

Reference Map

Explorations:

1.) Given the Reference Map (Figure 3), identify the locations of each site by recording the

ordered pairs of vertices for each site.

Figure 3

Reference Map

2.) Solve for the variable.

a.) 4x = 28

b.) 4t – 9 = 11

c.) y + 5 = 17

d.) 3x + 5 = 5

Math Site II

Site of Ancient Secrets

Math Site I

Site of Worship

Math Site II

Site of Ancient Secrets

Math Site I

Site of Worship

3.) Given the graph in Figure 4, identify the point on the graph where

a.) Input is zero. Write an ordered pair for each point identified.

b.) Output is zero. Write an ordered pair for each point identified.

Figure 4

Explorations: (Answers)

1.) Given the Reference Map (Figure 3), identify the locations of each site by recording the

ordered pairs of vertices for each site.

Site of Worship: (4, 10), (4, 9), (4, 8), (4, 7)

Site of Ancient Secrets: (7, 6), (8, 6), (9, 6)

Math Site I: (5, 1), (6, 1)

Math Site II: (1, 3), (2, 2)

2.) Solve for the variable. (Answers)

a.) 7

b.) 12

c.) 5

d.) 0

3.) Given the graph in Figure 4, identify the point on the graph where

a.) Input is zero. Write an ordered pair for each point identified.

Answer: (0, 2)

b.) Output is zero. Write an ordered pair for each point identified.

Answer: (5, 0) and (8, 0)

Reflection:

Using different types of representation help me become more knowledgeable in

mathematical representation as well as it improves my visual analysis. Using them, it is easy for

us to understand graphical representations and problems that are need to be represented

graphically. As a future math teacher in the future it is very important for me to know how to

represent problems using graphical representation because there are problems that easily be

understood when represented graphically.

Reference:

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

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