The beginning chapter in the UCSMP Geometry book introduces students to four definitions of a point - dot as a point, location as a point, ordered pair as a point, and a network node as a point. A lot of information is presented to students but not necessarily a lot of physical experience with each idea. The Artistic Expression Institute in the Educator's Science and Mathematics Institute Series brought home the point that different students think different. Some need a physical representation of an idea, preferably one that they have created themselves. This unit will present each idea of a point, accompanied by an exercise that allows students to represent the idea on paper, and/or link the idea to a practical example of where this idea is used.

This unit will be used at the beginning of the Geometry course. It will represent the material in the first half of Chapter 1 in the UCSMP text and take approximately 1 ½ weeks to cover. Each of the four representations will be addressed, appropriate vocabulary will be introduced, and students will take a pre and post quiz to access their understanding. In order to give further visual reinforcement to the ideas in this unit students will be encouraged to bring in pictures (from magazines, newspapers, etc.) that illustrate the chapter's vocabulary words ( Figure 1). These will be labeled and displayed on a classroom bulletin board.

The material in this unit is included in the Michigan Curriculum Framework Standards. Strand II Geometry and Measurement Section 1- Shape and Shape Relationships includes using shapes as a descriptive tool, identifying characteristics and defining shapes, and identifying properties and describing relationships among shapes. Strand II Section 2- Position includes identifying location of objects. Strand VI Probability and Discrete Mathematics Section 2- Discrete Mathematics includes investigation of practical situations such as routing and networking.

Information in this section includes such practical skills as measuring distance on a map and determining if a route on a network is possible. It also includes basic mathematical concepts such as determining the distance between two points, plotting points on a coordinate system, determining points of a graph from it's equation. This unit will serve as a review for some algebraic concepts, put some old ideas into a new context with new notation, and present some ideas that the students have not yet encountered.

Before proceeding with the content students will take to preteaching assessment tools. One is provided by the ESMIS institute and is a measure of student attitudes towards math. The other is a prequiz ( Figure 2) on the content to establish a measure of student's knowledge. This second assessment tool will be used as a quiz for the chapter after the activities that are outlined below are completed. A comparison of the results will be done.

Students will be provided with 1.5 cm grid paper. They will create a picture on this grid using dots. The picture can be of their own choosing, perhaps a letter of their name. We will then discuss the difficulties of drawing this way and what could make it easier. Pixels and resolution of computer and television screens will be defined. The character of the dots and lines formed in their pictures will be considered. The difference between oblique, horizontal and vertical lines will be review, the idea of discrete points which have size will be introduces, and these ideas will be tied to the concept of discrete geometry. Examples of pointillism as an art form will be included.

Students will refer to a mileage chart (Figure 3). They will be asked to determine distances between various cities using the mileage chart - examples: How far is it from Detroit, Mich. to New York City?; How much farther is it from Milwaukee, WI to Los Angeles, CA if you must drive through Houston, TX?.

Students will then look at a set of maps of the Western Upper Peninsula. Figures 4-6 show the Ontonagon - White Pine area with each of the maps at a different scale. Using Figure 4 students will calculate distances between a variety of places using the map scale and comparing measured value to the value given on the map. Figures 5 and 6 will be used to determine the distance between White Pine and Ontonagon. Discussion items will include the use of appropriate scales, choice of landmarks to measure from, comparison of measurements, reasons for differences in measurements, and the idea of location as point with no size.

Measuring between two points will then be applied to number lines. An exercise, Figure 7, will be completed. The following concepts will be covered: denseness of a number line, absolute value, distance notation and the classification of this idea of a point into synthetic geometry.

Students will plot the following points and connect the points in order: (4,4); (0,0);(-3,0); (-2,2 ½ ); (4,4); (-3,7); (-3 ½, 9 ½); (0,10), (4,4); (3,9); (5,11); (6,9); (4,4); (9,9); (11,8); (10,6); (4,4); (12,4); (14,2); (11,0); (4,4); (8,-2); (7,-5); (4,-3); (4,4); (-1,-11); (-4,-6); (-7,-5); (-6,-8); (-1,-11); (3,-21); BREAK, Begin (1,-16); (4,-11); (8, -10); (7,-13); (1,-16).

This exercise results in a cute flower design. It requires students to review graphing ordered pairs and scaling on a coordinate system. ( Activity from “ Getting the Point!”, Marcia Snyder, Activities from the Mathematics Teacher, 1981)

Students will then draw a picture of their own and list the coordinates required to reproduce it. It is the teacher's hope that these two activities will cure students of coordinate reversal for the rest of their mathematical careers.

A review of line equations, slopes, finding points, and graphing of lines will be conducted. An activity in which students points for lines and then graph them will be used (Figure 7). Another activity, which requires calculating slopes of lines, will also be used (Figure 8). (Activities from Algebra with Pizzazz!, Book C, Steve and Janis Marcy, Creative Publications,1983)

Network geometry is a discrete math topic that is briefly touched upon in the UCSMP Geometry text. It is a topic that has some practical applications in route planning. The first activity students will do on this topic is Graphs and Games (Activities from the Mathematics Teacher, 1981, Christian Hirsch), Figure 9. The terminology used in the activity varies a bit from the books so the equivalent terms will be discussed: node=vertex, edge=arc, network=graph. The concept of a traversable graph will be discussed along with the use of even and odd nodes as indicators.

Students will then extend this study with an application problem. The first two activity sheets from the “ Flying through Graphs: An Introduction to Graph Theory”, (Amy Roth McDuffie, Mathematics Teacher, November 2001) will be used if an airline's route map can be obtained (Figure 10). A search of the Internet has so far proved futile - lists of cities are available but the routes are not indicated on the maps. Otherwise “Directed Graphs”, ( Student Math Notes, NCTM, September 1988) will be used (Figure 11). Either of these activities will provide the students with a practical application of network geometry.

The four view points on a point pack a lot of information into a short time frame. The ideas from discrete geometry and network geometry are not addressed again in the UCSMP Geometry text except in review problems. This serves as context, background, interesting information for the students. The coordinate geometry and synthetic geometry encompass the traditional definition of a point as a zero dimensional object. This is the concept used in Euclidean geometry. The ideas from these two parts are used throughout the rest of the text.

The unit has addressed a number of points in the Michigan Curriculum Framework Standards. It has attempted to present information in a couple of different formats to engage students with different learning styles. Throughout this period text questions will be discussed, and if necessary assigned, to reinforce the ideas. The assessments given at the beginning of the unit will again be given at the end. Comparisons will be made in the unit evaluation.