The Distance Game develops numerous mathematics concepts in an enjoyable game format. The most obvious number concept is graphing in the coordinate plane. In order to determine the location of the mystery number, the player must select a point in the coordinate plane. More importantly using the distance away such as 5.4, new coordinates must be chosen. Students playing the Distance Game must make inferences and develop strategies based on both analytic and geometric information. For example, many students make a visual estimate to determine a likely second point based on the distance from the target point. After knowing the second distance, and using both as a guide, students then can know about where the mystery point is.. Students do not always use all the available information at first but tend to initially focus instead on the most recent data. Mathematical power is generated as students begin coordinating several distances.

In order to become an expert Distance Game player, one must have command of decimals, average, sums of squares, locus, the Pythagorean Theorem along with many other mathematical concepts. Rather than seeing these topics as prerequisites to be taught prior to playing the game, it is better to have students play the game with no initial explanations. The Distance Game is designed to help students learn this mathematics. As young students play the Distance Game, they come to give meaning to 6.3 in a rich way as the length of a diagonal segment in the coordinate plane, as a number which is more than 5.8 and as an approximation.

Decimals
In the Distance Game, the distance away from the chosen point is reported as a decimal. Rather than "teaching decimals" first, students can be introduced to the Distance Game and learn to think in decimals at a meaningful level. When a player has chosen a point and sees 6.4 their first reaction may be, "What does that mean?" but as she continues she will see other decimals, knowing she wants a "smaller" one. Arlishia first thought of 6.4 as the coordinates of a point (6,4). As she continued and saw 3 as a distance she revised her thinking and began giving a number meaning to 6.4 as a number between 6 and 7.

Plotting points in the coordinate plane
In a beginning algebra course or to use a graphing calculator, one needs to understand how to locate and read points in an x-y plane. As players engage in the Distance Game they soon, almost effortlessly, learn to plot points in the coordinate. Their action is goal oriented - they want to choose numbers which will score a hit so they think in coordinates to determine their next choice. Thus, x first and y second becomes second nature without drill

Distance in the coordinate plane
Central to the Distance Game is conceptualizing the distance between two points in the coordinate plane. The computer has chosen one of points in the grid. When a player chooses a point, he or she then learns how far away that point is from the mystery point. Players at first may operate at the intuitive level but with experience came to use strategies which utilize knowledge about distance between points. A player may choose (3, 7) and find that is 4.8 away. Learning to give meaning to 4.8 as a distance and devising an efficient strategy is at the heart of the Distance Game.

Concept of average (arithmetic mean)
Often, students learn a procedure for determining the average of a set of numbers but may not have a good sense of what an average is. In the Distance Game the average number of tries required to locate the chosen point is displayed. A goal of the game is to lower the average. Thus if a student has played eight games, she sees that the average changes more slowly than when only two or three games have been played. Also, if the average is high, such as 6.1 after five games, it is lowered dramatically by a score of two. The fundamental idea of central tendency becomes meaningful and clear.

Strategic Reasoning
While quite popular among beginners, the choice of a point near the grid’s center as a first guess is naive and results in too many possible second points. Using this strategy, at the third guess there are still two possible regions where the point might be located. As a sample play, consider a game in which the first guess is (9,6) with a distance away of 2.8. If a point is 2.8 away from (9,6) then it lies on a circle with radius 2.8 centered at (9,6). A second try of (2,5) with a distance away of 5.1 tells us that the mystery point must also lie on a circle of radius 5.1 centered at (2,5). The two circles intersect in two points so there are now only two possible locations for the mystery point. but one of these locations may not be on a grid point. Some mathematics calculations can show that indeed that is the case.

A more efficient strategy uses as a first guess some point near a corner of the array, e.g., (0, 0). The choice of (0, 0) determines an arc of radius 6.7 centered at (0,0) which is only one-fourth of a circle and thus fewer places for the mystery point to be. A second guess is (7,0) with a distance away of 7.2. Since the arcs only intersect in one point, using this strategy it is possible to always find the mystery point in three guesses.

Many initial strategies have been observed. Young children may guess randomly at first, later refining their strategies and becoming systematic in their work. As children play the Distance Game, they invariable progress in strategy development, their eventual level being a function of their mathematical sophistication and motivation. Sixth grade students have been observed using calculators to determine the Pythagorean triples satisfying the data points shown. In this way they were able to approach the theoretical minimum average of slightly less than 2.5 tries per game.

Since we want to provide a rich environment for students to construct their strategies, it is not advisable to teach strategies. While it may be useful to have students who have played the game discuss their methods, care must be taken to avoid short-circuiting student learning.

Units
The construction of units is central in doing mathematics. Many middle school students have not conceptualized distance along an oblique line and tend to think the distance between two adjacent diagonal points is 1 when it is actually about 1.4 (square root of 2). As students play the Distance Game, they have the opportunity to lay a metric on an oblique segment and think of its length as 2.8 units of the horizontal axis units. This more general construction of units is a major advance in mathematical reasoning.

Number theory: Sum of squares
A more advanced level of play uses the square of the distance, d, to determine possible pairs whose sum of squares is d. For example, a student may choose (1,2) as their guess and see that they are 8.6 away. The square of 8.6 is 74 and 74 can be decomposed into squares in only one way, 49 and 25. Thus the mystery point is at either (1+7, 2+5) = (8, 7) or (1 + 5, 2+7) = (6, 9). This approach develops an important concept in number theory, namely, Pythagorean triples.

Learning
A significant feature of the Distance Game is the nature of learning that occurs. Rather than being told how to compare decimals they give meaning to 8.6 is a rich way. In playing the game, students learn significant mathematics by constructing knowledge for themselves in an enjoyable setting.

Appeal
The game becomes addictive - children and adults have been observed playing the game for hours. Like chess it can be played at more and more advanced levels. It is accessible to elementary school students but still challenging for adults.