Mathematical Modeling of Minecraft – Using Mathematics to Model the Gameplay Of

Mathematical Modeling of Minecraft – Using Mathematics to Model the Gameplay of

Video Games

Thesis

Presented in Partial Fulfillment of the Requirements for the Degree Masters of Mathematical Sciences in the Graduate School of The Ohio State University

Raymond T Cox

Graduate Program in Mathematical Sciences.

The Ohio State University

2015

Committee:

Herb Clemens, Advisor

Azita Manouchehri Copyright by

Raymond T Cox

2015 Abstract

The use of video games to teach mathematics is decades old, but has been predominated by designing video games that directly teach basic skills to young children.

The rise of internet culture and modern gaming has created a population of older students that are emotionally and intellectually invested in digital entertainment. Simultaneously, changing educational standards favor problem solving and modeling behavior over basic skills. This paper explores the possibility of using commercial, popular video games as a

“real world problem” that can be modeled and solved using mathematics commonly taught to high school students. A specific example of this possibility is drawn from the extremely popular game Minecraft.

ii Dedication

This document is dedicated to my kitten, who was born after and died before.

iii Vita

2002...... Bridgewater-Raritan High School

2007...... B.S. Mathematics, York College of Pennsylvania

2011 to present...... M.M.S., The Ohio State University

Fields of Study

Major Field: Graduate Program in Mathematical Sciences.

iv Table of Contents

Abstract ...... ii

Dedication ...... iii

Vita ...... iv

List of Tables ...... vii

List of Figures ...... viii

Chapter 1: Modeling Behavior in Mathematics ...... 1

What are models? ...... 1

What are the phases involved in mathematical modeling? ...... 1

How do models develop? ...... 7

How are models used in teaching and learning? ...... 10

What is a good model? ...... 12

Why use video games to model mathematics? ...... 13

Why use Minecraft, specifically? ...... 14

Chapter 2: Modeling Activity: Working on the Railroad ...... 17

The World of Minecraft ...... 17

The Story ...... 18

v The Assignment ...... 22

The Transition ...... 31

Conclusion ...... 32

References ...... 35

Appendix A: Handouts ...... 36

Minecraft: Tunneling ...... 37

Minecraft-Minecart: Elevated Rail ...... 39

Minecraft-Minecart: Transition from Tunnel to Elevated Platform ...... 40

Minecraft-Minecart: Terrain Composition ...... 41

Minecraft-Minecart: Crafting Recipes ...... 42

At the Crafting Table ...... 42

At the Furnace ...... 42

Minecraft-Minecart: Tools ...... 43

vi List of Tables

Table 1: A Student's Progression Through the Modeling Activity...... 30

Table 2: Crafting Recipes...... 42

Table 3: Tool Recipes...... 43

vii List of Figures

Figure 1: Modeling Cycle...... 5

Figure 2: Jack's House...... 18

Figure 3: Forest and Extreme Hills Biomes...... 19

Figure 4: Rails don't work well in a mountain...... 20

Figure 5: Rails laid directly on the ground...... 20

Figure 6: Creepers obstruct the railway...... 21

Figure 7: A creeper explodes, destroying the rails...... 21

Figure 8: Rail built above the ground, with creepers safely below it...... 21

Figure 9: A player standing at (-23, 40, 317) as shown in-game...... 22

viii Chapter 1: Modeling Behavior in Mathematics

What are models?

Modeling can be used to describe how people interact with and learn from the world. It is a natural process wherein an individual forms conceptual objects, models, that describe his real-life experiences from his own perspective (Lesh & Doerr, 2003). This school of thought is derived from the constructivist that an individual's perspective of

“reality” is unique to that person (Lesh & Doerr, 2003). The modeling thought process is that models are used to interpret and organize the reality of their lives. Reality is objective to this school of thought, but the models are individual and thus every individual person interprets the same experience differently (Lesh & Doerr, 2003).

From this perspective, a “model” can be seen as the smallest complete unit of knowledge (Lesh & Doerr, 2003). Although it has many pieces, the entire model is needed for those pieces to have context, meaning, and understanding. A model will have internal and external components, with the external components consisting of representations that allow the individual to communicate the model to other people (Lesh

& Doerr, 2003).

What are the phases involved in mathematical modeling?

Mathematical modeling consists of multiple subprocesses that describe how the modeler defines the problem at hand, turns that definition into a mathematical construct, and interprets that mathematical construct to solve the original problem (Zbiek & Conner,

2006). As with any description of a method of thinking, the modeling cycle is an 1 abstraction of a continuous process into discrete pieces. As such, different researchers divide the cycle into similar but different cycles depending on their own needs. For this work, I have chosen to use the modeling cycle described by R.M. Zbiek. Refer to Figure

A working mathematical model is a gestalt of four distinct elements. These are a mathematizable situation, a mathematical object, a purpose or problem that spurred the modeling process, and the relationships between these three things (Zbiek & Conner,

2006). In simpler terms, these are the actual thing that has called the student's attention, the math that the student has used to describe the situation, the reason the student is thinking about the situation in the first place, and the way the student both turns raw information into math and then math into a solution. In the following activity, the mathematizable situation is the game Minecraft, the mathematical object is the collection of equations and formulas the student creates during the activity, the purpose is the scenario of the activity itself, and the relationships are the rules of Minecraft.

In this paper I will often refer to the “rules” of a game, particularly Minecraft.

When I speak of “rules” I refer not only to the allowed actions of the player but also the way objects within the game behave. For example, to create a stone pickaxe the player must place two sticks and three pieces of cobblestone into a crafting table in a specific pattern. The “rule” of the game is that the player must have those resources, and that those resources are consumed, in order to create his pickaxe. That pickaxe will only exist for a finite number of uses and must be repaired or replaced, that is another “rule.” An example of a “rule” that doesn't refer to the player's actions would be that blocks that are placed do not move, even if that would leave them floating in the air. In the coming 2 activity, the students are informed of the rules relevant to the scenario given to them and will need to mathematize them (as explained below) in order to solve the scenario.

The mathematizable situation is worked on by two processes, termed exploring and specifying. Exploring consists of any technique or approach by which the modeler learns more about the situation, such as asking questions or analyzing given information.

Mathematical observations are often made during this process, which helps to feed into the mathematical object. The second process, specifying, describes the decision making process of the modeler. When specifying, the modeler takes pieces of information and declares that they are important conditions or irrelevant trivia to his purpose. The important conditions will be drawn upon to construct the mathematical object, while the irrelevant trivia is largely ignored (but may become important through later cycles of exploring) (Zbiek & Conner, 2006). In the Minecraft activity that follows, both of these processes relate to the rules of Minecraft. In exploring, the students take the raw information available to them and define the rules of the game. In specifying, they decide which rules are important and which can be safely disregarded.

Mathematizing describes the process by which important conditions are turned into mathematical properties or vice-versa. This process is the means by which conceptual understanding of a situation is translated into mathematical language so that it can be used in a mathematical object (Zbiek & Conner, 2006). Mathematizing is not one- to-one. Many conditions may combine to form a single mathematical property, or a single condition may need to be expressed as many properties (Zbiek & Conner, 2006).

Mathematizing can seem trivial, particularly in a scholastic setting, because many of the scenarios presented to students build on skills and concepts the students are recently 3 exposed to and can easily recognize (Zbiek & Conner, 2006). In the following activity, mathematizing takes the form of representing the game's data and rules as variables and equations. Minecraft was chosen in part because the rules of the game are easily mathematized.

Once the conditions of the scenario have been mathematized into mathematical parameters, they must be acted upon by the mathematical entity. The process of choosing or creating an appropriate mathematical entity is called “combining” (Zbiek & Conner,

2006). In most cases, the student will not have a mathematical entity perfectly suited to the myriad properties he has mathematized. If such an entity exists, this process is trivial.

More likely, the student will take multiple mathematical objects he is familiar with and assemble them into a single entity (Zbiek & Conner, 2006). This completed entity is what is referred to as a “mathematical model.” Actually developing a workable model of a nontrivial scenario is a considerable accomplishment.

The process called “highlighting” deals with identifying mathematical properties that arise from the mathematical entity. Unlike parameters formed by mathematizing the scenario, these properties are formed from the mathematical entity (Zbiek & Conner,

2006). These properties can be solutions, interesting facts, relevant axioms of mathematics, or bits of trivia that later turn out to be useless.

Figure 1: Modeling Cycle Once properties have been highlighted, they are interpreted into the context of the scenario. Interpretation is the modeler asking “what do these properties mean to the scenario?” Like mathematizing, interpreting can often seem trivial, however it is a vital step when modeling is put into practical use. The modeler must understand the result of his mathematical efforts in order for his solution to have value. We call the results of interpretation “conclusions,” but this should not be confused with the conclusion of the modeling cycle (Zbiek & Conner, 2006).

Examining is the process which takes the interpreted conclusions and puts them into the scenario. This is a kind of testing. What do these conclusions mean to the scenario? Do they change anything? Do we learn anything from these new pieces of information? Do we have an answer to the problem, or are there questions still to ask?

This is often the step where a solution is declared, but the examining process feeds back into the scenario and just as often prompts new rounds of exploring and specifying. The modeling cycle is, after all, a cycle, and complicated scenarios often require multiple iterations of the process in order to generate a genuine solution (Zbiek & Conner, 2006).

Two processes exist outside this cycle, and are not shown on Figure 1. They are called “aligning” and “communicating.” Aligning is a constant process that could be termed a reality check in lay terms. The modeler is (or at least, should be) constantly watchful for signs of absurdity. For example, in a problem about relative speeds, a modeler may find that he has computed that an automobile has exceeded the speed of light. Aligning is the process where the student connects that mathematical result to the scenario and says, “No, that's not possible. Something has gone wrong” (Zbiek & Conner,

2006). Communicating deals with packaging the information in the model into usable 6 language. This could be to collaborate with others, report to a superior (or teacher), or simply to “talk through” the problem for the modeler's own benefit (Zbiek & Conner,

2006). Both of these processes can occur at any time during the cycle as a natural part of the problem solving experience.

How do models develop?

Mathematical modeling, also called model development, is a non-linear process that combines elements of treated-as-real-world problem solving and mathematical problem solving (Zbiek & Conner, 2006).

Models are constantly developed and refined as a natural process of thinking

(Lesh & Doerr, 2003). Models are by their nature incomplete; “Every model has some characteristic that the described system does not have; and, every model does not have some characteristic that the described system does have—otherwise the model and the described system would be the same” (Lesh & Doerr, 2003, p. 213). As an individual learns and discovers the ways his model does not accurately reflect the reality it is supposed to describe, he must adjust the model to account for potential existing discrepancies. This is the mechanism by which a person learns under the modeling school of thought (Lesh & Doerr, 2003).

Primitive model development of complex ideas begins prior to the formal introduction of those concepts in a curriculum. Students are natural modelers. However, the models formed by untrained, natural modeling techniques are consistently poor and incomplete (Lehrer & Schauble, 2000). For example, concepts like ratios and proportions begin to develop before students have a firm grasp of whole number arithmetic (Lesh &

Doerr, 2003). Children form models of fundamental concepts early and develop them 7 slowly over many years before the school curriculum addresses them. As the process is largely undirected, children demonstrate a highly inconsistent competency with problem solving (Lesh & Doerr, 2003). Students who face problems that happen to be well-suited to their intuitive models will show far more capability compared to when they face problems that are less in line with their primitive understanding. The goal of teaching is to guide students through learning how to develop better models (Lesh & Doerr, 2003).

Model development is usually spurred by “cognitive conflict.” In simple terms, a conflict is when two pieces of information don't work together (this is the process called

“aligning” in the modeling cycle) (Zbiek & Conner, 2006; Lesh & Doerr, 2003). There are three principle types of conflict in an educational environment: conflicts within a model, conflicts between a model and the scenario being modeled, and conflicts between different models (either models that an individual possesses or between models possessed by multiple individuals). This is subjective and belies the complexity of the thought process, but it is a useful tool to discuss how conflicts can arise (Lesh & Doerr, 2003).

Knowing how to generate a conflict is a valuable tool for a teacher to provoke model development.

Conflicts between a model and the scenario being modeled typically occur

“between” cycles of modeling; the modeler uses his model to predict the scenario, and the scenario does something else instead (Lesh & Doerr, 2003). As we generally do not assume that reality is wrong, such a failure to make an accurate prediction may lead to examining the model. For instance, a student may correct his process for solving a specific problem. For a grander example, one can look at the laws of motion laid down by

8 Aristotle and Newton (Lesh & Doerr, 2003). This leads to a second point: a more correct model is not, necessarily, a better one.

Conflicts within the model are often caused by the tools used in developing or using them. By tools, we refer to technological assistance (usually calculators) as well as representations (Lesh & Doerr, 2003). The modeler will interpret part of his model or data into symbols or graphs or some other entity and observe new properties, draw new conclusions. Conflicts also arise when entirely new elements are added to the model. This follows a similar path to a proof by contradiction – the new part of the model cannot coexist with one or more assumptions that already exist within the model (Lesh & Doerr,

2003).

Conflicts between models are very much like conflicts within a model, but this type of conflict is the only one that can occur external to the modeler. In group work settings and partnerships, every individual develops his own model. The cooperative element generally makes these individual models similar, but every individual brings his own perspective (Lesh & Doerr, 2003). Conflict between models held by separate individuals can be very enlightening, as they necessarily expose the participants to different ways of thinking. A part of modeling is accepting that multiple representations of a single scenario or entity can and often do exist. Comparing these rival models gives new insights in the form of between-model conflict (Lehrer & Schauble, 2000; Lesh &

Doerr, 2003).

9 How are models used in teaching and learning?

Models are used in two primary fashions, which are not exclusive. They can be seen as knowledge or as tools for solving problems. The following activity focuses on developing mathematical models to solve a problem.

Modeling, when used in the classroom, is expected to cover multiple mathematical content areas, meaning many different kinds of mathematics should be used. This is unlike traditional “curricular mathematics” which focus almost exclusively on a single topic at a time (Zbiek & Conner, 2006). We define “curricular mathematics” as the type of mathematics taught in schools. It is formalized, generalized, adaptable, and versatile, but that is when used by an expert (Zbiek & Conner, 2006). To the layman, it is often arbitrary, rigid, and useless outside the classroom.

Curricular mathematics usually create situations with a narrow focus to ensure students use a specific mathematical entity (Zbiek & Conner, 2006). For example, all the scenarios and word problems in a textbook section on exponential growth will be written in a way where an exponential growth formula is the easiest and most obvious tool to solve the problem. In the same way, a modeling activity in the classroom has the purpose of the teacher wanting students to use a specific mathematical technique as part of their mathematical object (Zbiek & Conner, 2006). “Problems in school mathematics are often carefully mathematized and constructed to elicit very specific mathematical relations. The givens and goals of a problem situation are structured so that the irrelevant information has been filtered out. The structure of the problem maps onto a specific mathematical structure. What is problematic for the student in such situations is to find the nonobvious

10 path from the givens to the goals or to recall a lost tool to help solve the problem” (Lesh

& Doerr, 2003, p.224).

In contrast to curricular mathematics, “street mathematics” are “the mathematics that people develop for themselves, when they need it” (Devlin, 2011, p. 11). Street mathematics consist of techniques designed to solve specific, common problems in an individual's life. The techniques people develop are often sophisticated when expressed in formal language, but the individual's understanding of his own street mathematics are inherently tied to the context from which the techniques were developed. Repeated studies demonstrate these limits as a gap in mathematical ability in both children and adults (Devlin, 2011). Namely, the 'average person' faced with a math problem in a familiar context will consistently perform better than the same person, faced with the same problem, in formal mathematical language. Similarly, a person accomplished in street mathematics often has trouble with problems mathematically identical to those they solve readily but are expressed in a wholly different real world context (Devlin, 2011).

In unstructured problems, such as those most commonly found outside the classroom, instances of mathematical modeling do not come with explicit statements of the problem at hand or the mathematical entity that is best to use. Defining the problem and choosing the right method to solve the problem is the modeler's first task (Zbiek &

Conner, 2006). These choices are unsurprisingly influenced by the modeler's history and experiences with mathematical entities. In a school environment, a student making the same choice is influenced by the student's belief in the connection between the modeling task and its connection to curricular mathematics (Zbiek & Conner, 2006). In both cases, the choice of what tools to use is left up to the modeler. The modeling cycle has multiple 11 opportunities for a student to verify his progress, with himself or others (Zbiek & Conner,

2006). In a school environment, this places the teacher in an advisory role. Modeling requires making decisions, not merely following directions.

What is a good model?

Recall the definitions of “curricular mathematics” and “street mathematics” above

(Zbiek & Conner, 2006; Devlin, 2011). Curricular mathematics are difficult to understand but can be used in any context. Street mathematics are easy to understand but can only be used in specific contexts. Modeling mathematics strive to be as functional as curricular mathematics while being as accessible as street mathematics.

The “correctness” of a model is a function of two main factors: usefulness and generalizability (Lesh & Doerr, 2003).

By useful, we mean that the model starts from an easy to understand point, generates a significant result, and can be employed to solve the task that generated the model. By generalizable, we refer to the ease with which a model can be applied to other tasks (Lesh & Doerr, 2003). Turning again to the models of motion, Newton's laws are are generalized to most tasks, while Aristotle's theorems start from simpler assumptions and are much easier to apply to a subset of the tasks Newton's laws are capable of. Both models generate similar results, with Newton being more precise. The “correctness” of these models ultimately depends on the complexity of tasks the modeler expects to encounter.

Lehrer adds a third criteria, specific to models used in teaching and learning.

“Model-based explanations are typically judged by their explanatory coherence, by comparing the relative fit of models to data, and by the relative parsimony of description” 12 (Lehrer, 1998, p. 121) In other words, in addition to usefulness and generalizability, we are also concerned with coherence when using models as teaching tools. “Parsimony of description” refers to the model's length when turned into words or some other shared representation. More specifically, it implies that a student must be able to pay attention to the lesson and understand enough of the model simultaneously to appreciate the whole.

Newton's laws are more accurate and more useful, however they are also harder to apply and difficult to explain in full. Aristotle, on the other hand, created easily used theories that are only applicable in certain circumstances. These circumstances have a high correlation to those of everyday life, and so can be observed quickly (Lesh & Doerr,

2003). Under ordinary circumstances, Aristotle's theories generate a better model.

Why use video games to model mathematics?

The purpose of a modeling activity is to develop new or better models that can be applied to many tasks, not to solve the specific task in the activity itself (Lesh & Doerr,

2003). Thus, the specific context of the task can be chosen in a way that it would be beneficial to other goals, such as student engagement. Model development emphasizes generalization of models from a specific situation (Lesh & Doerr, 2003). We therefor choose contexts that are conductive to learning. Modeling activities motivate learning by connecting mathematics to situations that interest students, most commonly during the process of “exploring” (Zbiek & Conner, 2006).

Most games have some educational value simply in their construction. Dice games lend themselves to counting, card games to probability, games with money inevitably require basic arithmetic, so on. More complicated games lend themselves to more complicated mathematics. Games that use a computer often remove the need for a 13 player to perform calculations, but simultaneously are built around more complicated mathematical principles. Players will often examine a game's mechanics and rules in order to decide what actions, strategies, or game assets will provide the best results.

Often, such strategies are heavily influenced by personal taste. In some games, however, the rules are mathematizable and decisions can be made using mathematical techniques.

These games provide opportunities for modeling activities.

The dominant form of research on gaming as a learning environment focuses on developing games specifically for that purpose. In doing so, several principles for an ideal learning environment have been identified. The most important principle is that the game must provide a “real world environment” so that the student is able to connect the lesson to his own life and give it meaning outside the classroom or game (Devlin, 2011).

“The learning environment needs to be sufficiently like the real world situations where ordinary people use mathematics – the market, shops, offices, factories, laboratories, sport facilities, recreational areas, etc. The mathematics to be learned has to arise naturally in that environment, and have meaning in it, and the learner in that environment has to be motivated to carry out the tasks that involve that mathematics” (Devlin, 2011, p.

25). I propose that commercial video games are one such “real world situation” where students employ mathematics. Popular video games with mathematizable problems are easily exploited modeling activities that students already have an invested interest in solving.

Why use Minecraft, specifically?

I considered three criteria when choosing a game for the current work. First, the game must be popular. Second, the game must be comprehensible to non-gaming learners 14 and educators. Third, the game must have at least one, preferably more, mathematizable problems that only require understanding of mathematics commonly taught in high school.

The selected game must be popular in order to appeal to as many students as possible. A commercial game that is unknown or actively disliked will not provide the motivation inherent in a game that students enjoy playing. The game Minecraft is among the most popular video games in history, reporting over fifty million copies sold across multiple platforms and devices.

The chosen game must be understandable to non-gaming learners and educators in order for the lesson to be achievable even to those students that are not targeted by the motivational aspect. A game that requires hours to understand is useless as an educational tool. Minecraft is a game about gathering and using resources, the vast majority of which are real-world substances and are represented accordingly. A non-gaming person should only need about fifteen minutes to understand Minecraft's interface and core concepts, such as gathering resources, using tools, and crafting. Most such concepts are will be specified and mathematized in the modeling cycle.

The chosen game must be easily mathematizable. Any computer game can be mathematized by simple virtue of being run on a computer that approaches all its instruction mathematically. The computer, however, has the luxury of performing extremely complicated mathematics in the blink of an eye. To approach these games from a human perspective we need to be able to model the results of those calculations with simpler mathematical principles. A useful game will have, as part of ordinary game play, challenges that can be expressed using simple mathematics, preferably algebraically or 15 geometrically. These are challenges that the player wants to solve for their own sake, challenges which can be solved by modeling them with mathematics.

Additionally, it is preferred that the game provide more than one activity. Even an easily understood game represents an investment in time and effort, and having multiple activities over the course of a grading cycle increases the return on that investment.

Minecraft offers a variety of resource use and optimization problems, such as the activity in the next chapter, which can be made more or less complex by simply adjusting variables and constraints. Additionally, the game offers geometric, trigonometric, and

Boolean logic problems in the various game mechanics and objects.

16 Chapter 2: Modeling Activity: Working on the Railroad

The following section is a single mathematical modeling activity using the game

Minecraft.

The World of Minecraft

Minecraft is a sandbox construction game. The player starts in a large, open world with no tools, no supplies, and no goals. Much like a person who has bought a collection of building blocks or a model set, Minecraft is a “toy” with which the player is given free reign to create structures and other objects, rather than a “game” in the usual sense which implies a beginning, an objective, and an ending.

Similar to a set of building blocks, Minecraft has rules that determine how the elements in the game can be assembles into new elements, structures, or tools. Unlike a set of building blocks, the player does not buy his materials from a hobby store but rather finds them in the game itself. He may cut down trees or mine for iron or cultivate a field of wheat, and in fact the majority of players will need to engage in multiple means of gathering supplies to build their chosen project.

The game world of Minecraft is made from cube shaped blocks. When the world is created, the game fills the space of the world with about sixty kinds of blocks in patters to create terrain features and objects. For example, a tree is made of wood and leaf blocks, while a river is made of dirt, stone, sand, gravel, clay, and water blocks. The game divides the world into large, irregular regions and assigns each region a single biome. The 17 biome determines what kind of land is found in the region – desert, ocean, swamp, and forest are all examples of biomes. The game engine prevents the most improbable combinations, such as a desert bordering a glacier, but biome selection is fundamentally random and has no relationship to any real geographical processes.

The Story

Jack has just bought Minecraft on the advice of his friend Ryan. Ryan runs a multiplayer server, allowing Jack and Ryan to play the game in the same world. Ryan created a new world using the 1.7.2 version of the game and hosted it on his server, allowing both friends to play in the same game at the same time. His server doesn't use any modifications to the game.

Jack and Ryan both enter the world at the master spawn location, which happens to be in a forest biome. Jack, being new to the game, chooses a spot near the master spawn to build a house (figure 2).

Ryan chooses to gather some wood and Figure 2: Jack's House seedlings and then heads east in search of a plains biome to build his own bunker.

Jack spends the early part of the game using wooden tools and parts to build his house and does not go below ground at all. Ryan, despite his delay in getting started, builds almost exclusively underground which gets him a large supply of useful resources, notably iron, gold, and a fictional mineral called redstone.

18 Jack complains that even though they're in the same world, he and Ryan almost never see each other in the game. Their homes are too far apart and are separated by an extreme hills biome, a rugged, mountainous region (figure 3). The journey is long, slow, dangerous, and tedious on foot. Ryan has horses that could solve that problem but is unable to find an easy path through the hills. He suggests instead that they build a minecart track connecting their homes and Figure 3: Forest and Extreme Hills Biomes defend it from monsters. He will supply the minecart rails if Jack builds the path to lay the track on. Jack agrees.

The minecart is essentially a simplified railroad. The minecart itself can carry one person but can only travel on minecart rails that have been placed in the game world.

Special kinds of rails and minecarts can provide propulsion as well as additional utility.

Jack studies his options.

Jack decides to use the most complicated to build, easiest to use option for minecart travel; powered rails. These special objects automatically give the minecart a large speed boost when it passes over them. They don't work as great going uphill, but once the track is built all Jack has to do is start the trip and the minecart will take care of the rest of the trip by itself.

Jack doesn't have the exact details of the terrain, but he knows that he is starting in a birch forest biome, passes through an extreme hills biome, and ends at a plains biome. A curving track through the trees is impractical; low hanging branches prove a 19 constant hazard and forests are often heavily populated by sheep, pigs, and zombies which can wander onto the track and stop the minecart. Likewise, a minecart rail cannot connect to a rail more than one block higher than itself, so the sheer cliffs of the extreme hills biome are an effective barrier (see figure 4 for an attempt to put rails through the extreme hills biome). The plains biome, however, presents only occasional trees in the path of the minecart. Jack decides to build the minecart track underground, in a self- contained tunnel, instead of above ground. This allows him to preserve the aesthetics of his house and avoids the complications of trees and cliffs.

Ryan rejects the tunnel idea. His home is a bunker, almost completely underground, and Jack's plan would cut right through his farm cave. Since Ryan doesn't want to move his cows out of the hole, Jack will need to build at least part of the Figure 4: Rails don't work well in a minecart track above ground. mountain Building the minecart track above ground

presents an additional complication.

Animals and hostile monsters spawn in the

game world under the proper conditions, and

the plains biome is appropriate for both of

them. If a minecart hits a creature it will

bounce off and go backwards at a reduced Figure 5: Rails laid directly on the ground speed, putting an effective halt to the 20 journey. In addition to this inconvenience, hostile monsters will then attack and could kill the player's character, damage the minecart track, or both (see figures 5 through 7). To protect themselves, Jack wants to build a platform above the ground to put the rails on.

Figure 7: A creeper explodes, destroying the

Figure 6: Creepers obstruct the railway rails

Jack decides that Stone Brick is the most aesthetically pleasing material available.

He further decides to use up as much Cobblestone as possible from his tunneling because he doesn't want to have chests filled with rocks cluttering up his house. In addition to using up stone for his elevated rail, Jack wants to use stone tools for his mining and levers to turn on the powered rails, all of which consume cobblestone. He needs to figure out when he should stop digging his tunnel and start building his elevated rail in order to use up as much stone as he can without running out before the end of the Figure 8: Rail built above the ground, with build. creepers safely below it

21 The Assignment

As a point of clarity, Minecraft uses an atypical coordinate system. The three coordinates are labeled X Y and Z and will be written vector-style as (X,Y,Z). Each block is unit-length on each edge, so each position in Minecraft can be uniquely defined by an integer value (the lower-northwest vertex of the cube). The coordinate axises can be drawn as mutually perpendicular vectors; X increases to the eastern direction, Z increases in the southern direction, and Y increases with altitude. To be entirely clear with the matter of altitude, this paper will refer to the Figure 9: A player standing at (-23, 40, 317) coordinates available to the player. If a as shown in-game player's coordinates read (0,64,0) then the block at (0,64,0) is the empty air the player's legs are in, the block (1,64,0) is waist-high to his east, and (0,63,0) is the block he is standing on.

Jack has already excavated a basement which will serve as his “train station” for the track. He has chosen a piece of the east side wall to start digging his tunnel in, located at (-23, 40, 317). Ryan has supplied the coordinates (993, 67, 104) to be the last piece of track to be laid, and is building his own “train station” to be ready once Jack gets there.

Jack's house and stash of materials starts him with a crafting table, a furnace, 30 blocks of cobblestone, 20 blocks of wood, 6 pieces of coal, and other irrelevant resources such as dirt and gravel. Ryan will supply the necessary gold, redstone, and iron for the build.

Table 1 (following pages) runs through the activity with an emphasis on the algebraic models the students will need to produce and use in order to solve the activity.

See the appendices for handouts that contain all the information given to students. 22 Continued experienced that Minecraft players may feel account pressured for to everything. This complete is a misdirect. Students commonly will apply familiar equations that to seem be relevant when they even not. are Minecraft's coordinates are unusual a that for students instantly hurdle but values, Y use familiar X and worse that the than geometry core used the by activity is not euclidean. The will right examine column the educational value of those mathematics. and are Furnace Table The Crafting required for many critical in-game actions this in activity not are but relevant of the any to mathematics. They mentioned are for 27 213 1016 = = = 1038.09 Y 1016.36 Z X = Δ = Δ Δ d d from those equations as Jack builds an increasingly complicated model. If Jack uses the Z coordinate instead is (which Y of more correct) This column will record values the 2 Y Δ + 2 X Δ √ = d These areThese initial conditions that will toneed accounted for in be later equations. distance metricThe commonly most used is This column willThis the highlight actions,mathematics those used by eliminating the game's context.

25 basement starts track the at (- has 23, 40, 317) while Ryan supplied coordinates the of the as (993, 67, points end 104) 2) Jack needs f to igure how the out long entire track is before can he determine the length of any of its His components. Cobblestone 20 blocks, 6 Coal and blocks, Wood 1) Jack starts Crafting a with Furnace, a 30 Table, The describe left column will Jack's game. actions the in These use actions will game-related largely language. Table 1: A Student's Progression Through the Modeling Activity the Modeling Through Progression Student's A 1: Table 23 Continued factors can be added. This is theThis factors be can added. initial model, vague inaccurate and but and foundation a for revision improvement. Taxicab geometry is extremely geometry Taxicab simple butunfamiliar is most to it The properties of a students. minecart track (see handout 1) are for students discover to sufficient the for the properties on their own, They will purposes activity. of this need equation invent this to from their understanding of this geometry. Like the initial distance this metric, it step However, is inadequate. provides basis new on which the 2 1 2 = = = 409.67 819.33 1229 E R T = = = A B d 1 A assuming the that d + − R d Z = = Δ A B + X 2 1 E T Δ = = = = E T R d is the elevated length of the rail. is the amount of Cobblestone needed is the length rail of elevated that can is the length of the is the tunnel. is the amount of Cobblestone A B Students startStudents defining this variables at following: define the We stage. T digging by unitproduced a length of tunnel. E buildto unit a length rail. of elevated R builtbe for every of tunnel. unit The distance metricThe Minecraft in is With inchanges are elevation gradual. of thistaxicab geometry the kind, distance path two points of a between is independent of the of that shape largely path. building the elevated platform. 3) Jack needs decide to how much tunnel how to much elevated rail. He a figures 1:2 do the ratio will job, since dig as much he'll twice in tunnel that the he'll need 2.1) Jack needs2.1) Jack plan to out not lengths layout. only but The track turn to needs from going going to at east north least once. Table 1: Continued 1: Table 24 Continued and the percentages while elevated rail is more exact deals and with fractions. This first student is the the has time to change and go back equation, an rather than variables. modifying The student's model is increasing in complexity it and becomes important track for him keep not to only be needs to of what but done how has he accounted for existing complications. This first interpretingis the in step raw numerical data create useful to information. case, this In the student percentages is given for everything thatis not stone. In thisstudent the needs to step think of the about design the elevated track create and numbers from 32 that Every description. blocks, is removed and block one see also We three are added. multiple representations cropping up – the tunnel with deals decimals 1.893 1.752 1.702 446.55 782.45 437.65 791.35 1.0625 454.88 774.12 1.8082 1.8082 0.03125 ======R T R A R A B E A B L R T ) 3 32 + L 2 ( 1 E − 32 ∗ T = = L 0.9041 R ∗ 34 32 2 0.9041 be Cobblestone the used on L = = = E T T Let perLevers length of track. unit 4.2) Building Levers the in tunnel drain is a on Cobblestone (see handouts1 and 5) 4.1) The rail elevated won't 4.1) use only BrickStone one block unit per length (see handout 2) 4) Jack starts and digging realizesproblems with many his model. the First, ground is not entirely made of Stone. (see handout 4) Table 1: Continued 1: Table 25 Continued The part hardest of calculating a section problem of a in unique a is getting thatway piece tointo fit There a are the overall model. number techniques to of possible but use of varying complexity, students rarely the are given chance to or practice develop those This that step is a techniques. encourages creativity and discussion among students. In most cases, to small adjustments the tunnel length will call of the not Sometimes simplest methods the are best. is a the Counting powerful tool its and apply don't limitations This teaches students in case. this pieces of a to apart break difficult problem up their to and mix techniques, in falling by case this backbasic very on a approach. seeMany students “simple” as a of callingway process a “wrong” – this of the part activity reinforces the value complicating over of not an already complicated problem. 98 74.91 931.50 863.59 = 92.235 402.87 798.13 436.39 764.62 = = S ======D O A B M F A B Theis transition too this for detailed space will and be detailed at end the of this section. A F − 1 − + 28 28 R − − d d = = B A A 1 − 28 68 + 28 − S E R + d − be the be material amount of random = be and amount of Dirt the Gravel. d 30 = represent Cobblestone the F M D S A = = B S is the length rail of elevated that can Let dug through. Let Let othergenerated or used things by than thedigging tunnel building the and rail.elevated F this builtbe from excess Cobblestone. The transitionThe of is small, composed parts, in is unique and the integer-valued fastest dealThe way to overall activity. issue is to this outwith count every constructcomponent, model not a based andon formulas equations. Cobblestone his building in 6) Jack has accounted for everything that uses from transitional the section. He needs to account for both. 5.1) Jack has initial stockpiles of Cobblestone expectedand an now gain 5) The between transition the 5) tunnel the and elevated rail is either than very different structure (see handout 3) Table 1: Continued 1: Table 26 Continued mechanic Minecraft in means each tool will slightly longer last than the because first, first had the tool no tool it repair before to with. These equations that by simulate the between adding difference the first and other to tool tools all the There are needed.number of uses other algorithms that can used as be Again, the tests step this well. students' ability develop to a process deal to with unusual an circumstance in problem. a for another tool. This realization is This for another tool. critical to usage; handling tool the extra to uses to up stone needed dig build dig the will to stone tools the not recursive a cause nightmare needing limit a function. Thiscalls to also step back handout 4, getting percentages of Dirt the The and particular. in Gravel numbers transitional the from section are step. this used also in This is completely step trivial in most however the respects, repair 1 7 = = P Sh ) 7 + ) ) 88 0.9041 + ) + ) ) 3 ) 0.0329 32 6 6 + 0.013 + + + 2 138 + 138 D ( O ( ∗ ( 0.04 ( A 0.01 ( ∗ ( = ∗ M ( ceiling be the of Shovels be number ceiling M M be and amount of Stone the Ores. be the of Pickaxes be number = = = P Sh O = D P Sh O Let needed. Let needed. Let 6.1) Jack needs figure to out how many and Pickaxes Shovels he need. will tools. design continues and digging until his first is Pickaxe about break. to He builds another realizes and he's spending on Cobblestone Table 1: Continued 1: Table 27

Sh Continued and and P changes response in M . The roundingThe on . A the actual usefulness of the result. A the actual usefulness of the result. quick of the eyeball can numbers give estimate easy an axes, of two with leeway enough that more calculation necessary. not is just aFrom mathematical pure standpoint, on the hand, other these of portion calculations force large a the build enumerated be to and This be can step easy or very very hard on how the depending student The presented modeled 4.4. equations provide to easy way an slip tool “lost” stone to in construction. Note potential as the for recursion the variable to stops that the potential without need limit. for a At this the point amount of calculation outweighs drastically 1 = 76 Sh 76.05 ; 71.53 947.25 0 410.39 790.71 878.20 = 7 76 = = S ======D F A B O M P Ax S ) 3 ) ∗ 20 Ax − 6 + ) 3 3 + ∗ ∗ 138 P P Ch + + + 1 Sh Sh W ( ( ( − − ) ) 68 68 + + ceiling be the of Charcoal be number be the number of Stone Axes the of Stone be number = 30 30 be number needed. the of Sticks ( ( St Ch Ax Ax = = is used as a subtotal for Wood used Wood is used as subtotal a for 1 S S Let needed. Let Let needed. W below. need a second Axe? A third? A Axe? need second a 7) When Jack runs outWhen of Jack runs 7) makes he blocks a Wood chop Axe to down Stone There go another more trees. he Will three Cobblestone. 6.2) Jack needs6.2) Jack a find to way to account Cobblestone for spent on tools. Table 1: Continued 1: Table 28 Continued used.the If extra in practice dealing of variables is systems with large step this a turn into unnecessary, discussion of the practical use of estimation problem solving. in Sticks almost used in are everything. Recursion up shows again, in but casethis severe the rounding iterations eliminates any this step from Notably, happening. has of the some longest calculations but the all impact does not model at because of the rounding. extreme 0 18 45 122 143 1184 = = = 1 = = = Ax Pr St W Tr Rl ) L ∗ ) 9 A ( Rl + 16 ) ( B 32 ) ) floor ( ) 4 8 St Tr + d ceiling ( 10 ( 2 Pr ( floor ∗ + ) ) − + d ) 6 Ax Pr floor = ceiling ceiling ( A 32 + = + ( = Rl 1 Sh Tr W + floor P ceiling be the number of Power the of be Rails number be number the of Rails be the number of Torches the of be number ( + = = Rl Pr Tr Pr St Let Let Let 7.1) How many sticks are sticks 7.1) How in many Jack's How construction? blocks are Wood many needed? Table 1: Continued 1: Table 29 is the 0 C existence of The

The needs student use the to calculated construct to a values plan minecart for the track and further prepared be explain why to that plan achieve will stated the goals problem. of the Charcoal is the time first resource a is recursive a genuinely in way that influences the outcome. Charcoal needed a the allows supply of Coal recursion quickly and end to cleanly. One trick here recipe is that for the Stone Cobblestone requires Bricks so to groups smeltedin of four, be the needs amount Bricks of Stone to the up to rounded be nearest multiple of four. 1 = 2 2 3 19 17 16 140 70 Sh 136 145 = = = 76.37 = 65.88 ; = = 2 951.55 = = 412.44 788.56 882.18 1 = 1 = 7 0 = = C Ax S Ax = = C = = Cl St W C Ch = F D A B O M P Cl 4 > ∗ 1 ] − ) n + 6 ) 1 4 Cl C + ( 4 − if Tr ∗ ) ) ( ) n , the length of the, the 1 0.01 − C B 34 n 8 8 ∗ ∗ 32 C ∑ ( B M ( and and ceiling ( = = A [ = 0 Cl Ch C ceiling be Coal the gained mining. from be an iterative an be series of variables x = ceiling Cl C n C answer are tunnel elevated and the rail, also and transitionlength of the them between These numbers next(see section). all toshould rounded be closest the integers. The onlyThe variables for the that matter Let that calculatethe of Charcoal number needed. Let : A Student's Progression Through the Modeling Activity the Modeling Through Student's Progression A : 1 8) Jack has numbers that he thinks right. are 7.2) How much charcoal7.2) How much will Jack need? Table Table Table 1: Continued 1: Table 30 The Transition

The transitional piece (step 5, above) is best treated as an independent construction. The limitations on Rails and Powered Rails set by handout 3 demand that the transition be as long as it is tall, and one piece of track must be placed on every elevation from 40 through 67, to connect to the elevated rail at 68. Thus the length of the transition is 28 blocks, which must be removed from the total length of the track when calculating A, B, and other related variables.

Most of the transition is underground and requires three blocks of vertical clearance rather than two. Given elevations 63 and 62 are guaranteed Dirt, only elevations 40 through 59 dig through 3 blocks of random material. 60 and 61 dig through

2 and 1 blocks respectively, while the remainder is Dirt. 40, 43, 46, 49, 52, 55, and 58 will need to have additional digging to plant a Lever, as will 61 (which digs through two blocks of material and one Dirt). The transition therefor produces

(20∗3)+2+3+(7∗3)+2=88 blocks of material that need to be added to variable M.

Additionally the transition digs through 7 blocks of Dirt that will need to be added to the variable D.

Elevations 65 and 66 will need to have Stone Brick blocks forming a staircase up to the elevated rail at height 68. 64 and 67 use a Block of Redstone for power. The

Redstone on 67 is further supported by a column of 2 stone brick blocks while the

Redstone on 64 needs to dig out a block of Dirt to place the Redstone – this is flush with the expected surface of the world. The stone consumption of the transition is therefor

8 Le v er s+4 S t one B r i c k s=12 C obbl e s t on e while the expected gain of 88 31 blocks of random material generates approximately 80 blocks of stone. Thus the transition will net an expected gain of 68 blocks of Cobblestone which is added to the variable S.

When it is time to calculate construction of Powered Rails, this construction adds

9 to Pr.

Conclusion

By its nature, this Minecraft activity is a large investment of classroom time far in excess of the typical classroom problem. This complexity and investment engages the students' attention and promotes the development of a mathematical thought process above and beyond the skills-based curriculum that dominates contemporary mathematical education. Students that are used to applying formulas and procedures by rote will not be able to apply those techniques to this activity – instead they must develop the necessary equations themselves from mathematical principles.

This activity is not designed to teach those basic principles that it employs. A student with no comprehension of ratios, distance metrics, or probabilities will struggle with this activity. Instead, this activity focuses on creating a scenario where multiple principles must be used on concert, without specific formulas to call upon, to require students to apply problem solving techniques to the scenario and synthesize the specific mathematical tools, the equations and variables, from the general skills taught earlier in the curriculum.

Adding the difficulties of problem solving and model building enhances the student's existing knowledge in a number of ways. Most of a student's mathematical knowledge is theoretical, rooted in procedures and rote use of formulas as dictated by the 32 chapter and section of a textbook. A common tactic in test taking is to choose the “right” equation or procedure based on the words used in a test problem. Activities like the one above use many of the mathematical principles of such problems, but no textbook will tell a student how to create an equation for the number of uses of a stone pickaxe in a modern video game, nor will the student have a sheet of equations to draw upon. As with real-world scenarios, the student is forced to understand the activity and create or choose which mathematical tools to use in order to complete it.

This activity provides a context and a goal that, while trivial and grounded in recreation, the student can relate to and understand as relevant, perhaps to his own hobby or to that of a friend's. The personal engagement allows the student to approach the scenario as something real, rather than just another math problem. Likewise the time investment prevents a frustrated student from employing the tactic of giving up and waiting for the teacher or his classmates to run through the problem for him. The student has no option but to think about the problem and invest the time and attention to it.

The use of a video game for context is an avenue largely unexplored by similar activities. While these games seem complicated and foreign to those that do not play them, the actual complexity is not very different from a sport, a board game, or a card game. Those older games have the advantage of being culturally significant, however recent generations will often spend more time at a computer or a console than they will with a football. Further, by their nature video games have an abundance of mathematical calculations available to exploit as modeling activities, and the rules of the game help focus students away from too many “what if” conundrums that can stymie early modelers. With this activity of digging a tunnel, a possible real-world complication is the 33 kind of rock the tunnel is going through and if it is even suitable for construction.

Minecraft does not have “kinds” of stone at all, and so that complication is ignored.

34 References

Devlin, Keith J. Mathematics Education for a New Era: Video Games as a Medium for

Learning. Natick, Mass.: K Peters, 2011. Print.

Lehrer, R. (1998). Models as Explanations. Issues in Education. (4, 1, 121-123).

Lehrer, R. and Schauble, L. (2000). Developing Model-Based Reasoning in Mathematics

and Science. Journal of Applied Developmental Psychology. (21(1) pages 39-48).

Lesh, R. and Doerr, H.M. (2003). Beyond Constructivism. Mathematical Thinking and

Learning. (5(2&3), 211-233).

Zbiek, R.M., and Conner, A. (2006). Beyond Motivation: Exploring Mathematical

Modeling as A Context for Deepening Students' Understandings of Curricular

Mathematics. Educational Studies in Mathematics. (63(1), 89-112).

35 Appendix A: Handouts

The following pages are handouts for students to provide exact information that may or may not be needed to complete the described activity. The handouts are organized by the relevant game topic or player goal, meaning mathematical questions will often be answered by connecting raw data from two or more handouts. This is intentional; the students learn to extract useful data from a presentation that has not been optimized for their specific goals.

36 Minecraft: Tunneling Handout 1

As Jack digs his tunnel, he keeps in mind the following features.

1. The tunnel is exactly large enough for Jack to walk through or ride a minecart through, no larger.

A. The player is two blocks tall and one block wide.

i. A player riding in a Minecart is also two blocks tall and one block wide.

B. The tunnel's floor is Y=40.

2. The Minecart can only travel on Rails. One Rail must be placed on every block.

A. Rails are placed on the tunnel floor but do not prevent the player from walking over them.

B. Rails automatically bend to make 90 degree turns when placed next to other Rails or Power Rails.

3. The Minecart will need Power Rails at regular intervals to maintain its speed.

A. One Power Rail every 32 blocks maintains the highest possible speed the game will allow.

B. Power Rails take the place of normal Rails.

i. A single block needs either Rails or Power Rails, not both.

C. Power Rails need to be turned on by another object or they act as a brake instead of an accelerator.

i. Jack uses Levers in the tunnel. Each Power Rail needs a Lever.

ii. The Lever must be placed on the side of the same block the Power Rail is on (the block at Y=39)

4. The tunnel needs a light source.

A. One Torch every 10 squares is sufficient. 37 B. These torches can be placed on the walls at Y=41 and do not interfere with player movement.

38 Minecraft-Minecart: Elevated Rail Handout 2

Instead of placing Rails directly on the ground, Jack decides to build an elevated platform so monsters and animals will not interfere with the Minecart.

1. The elevated rail should have three blocks of clearance over the ground

A. The usual height of the ground in the Plains biome is Y=64, meaning the topmost solid block is Y=63

B. Three blocks of clearance means the platform must be built at Y=67

2. The platform must be made of Stone Brick blocks.

A. The platform is only one block wide.

3. The Minecart needs Rails and Power Rails as described on Handout 1, points 2 and 3.

A. Exception: instead of Levers, Jack will use Blocks of Redstone as power sources.

i. Blocks of Redstone are solid blocks, not objects like Levers.

ii. Every Block of Redstone will replace a Stone Brick block.

iii. Every Powered Rail will require one Block of Redstone. No other Blocks of Redstone are needed.

4. This bridge must be well-lit at night.

A. Torches placed one every 10 blocks are sufficient.

B. The torches can be hung on the side of the platform and do not need additional support.

39 Minecraft-Minecart: Transition from Tunnel to Elevated Platform Handout 3

The hardest part of Jack's build is getting from the tunnel to the elevated platform. The following rules govern the means by which that is accomplished.

1. When a Rail is placed next to a Rail no more than one block higher than itself, it connects to the higher Rail. A. A single piece of Rail cannot turn horizontally and connect vertically. Any turns must be made using level pieces of Rail. B. Powered Rails can be placed on a slope and behave as normal Rails. i. Going up an incline is very hard. Every third Rail needs to be replaced with a Powered Rail. ii. Powered Rails still need to be powered. Underground Powered Rails use Levers, above ground Powered Rails use Blocks of Redstone. C. The Minecart with Jack still takes up a two block tall space, but that is measured from the top of the incline of each piece of Rail, not the bottom. i. The ceiling must be three blocks above the floor. 2. The top two layers of the ground (Y=63 and Y=62) are made entirely of Dirt. 3. The transitional space must conform to the other pieces of the track. Specifically A. The transitional needs a torch every ten blocks for light. B. Any Blocks of Redstone must be supported by a pillar of Stone Brick blocks.

40 Minecraft-Minecart: Terrain Composition Handout 4

The Minecraft world is composed of blocks of raw and processed materials that are randomly distributed by a complicated software engine. For this activity we are assuming that Jack will not encounter caves, dungeons, ravines, or other structures which are not possible to model in this way. As such the only substances Jack will encounter are Stone,

Dirt, Gravel, Coal Ore, and Iron Ore.

• The world building engine starts by making everything from Y=61 to Y=2 Stone. Stone must be mined by using a Pickaxe and will drop one Cobblestone item when mined. ◦ The engine then replaces Stone with patches, veins, or clumps of other materials. These groupings vary between four and twelve blocks of total volume. • Coal Ore replaces 1% of Stone. Coal Ore is mined with a Pickaxe and will drop one Coal item. Note that the Coal item is not the same as the Coal Ore item. ◦ Iron Ore replaces 1.3% of Stone. Iron Ore is mined with a Pickaxe and will drop one Iron Ore item. • Dirt replaces 4% of Stone. Dirt is mined with a Shovel and will drop one Dirt item when mined. ◦ Y=62 and Y=63 are entirely composed of Dirt • Gravel replaces 3.29% of Stone. Gravel is mined with a Shovel and will drop either one Flint item (10% chance) or one Gravel item (90% chance). ◦ Gravel is the only block that simulates gravity: if Jack mines under Gravel then it will fall into the newly emptied space. In addition to the material of the tunnel, Jack will need to harvest a great deal of Wood. • Jack's house is in a Birch Forest biome. He has a large supply of birch trees. ◦ A birch tree is composed of 5 to 7 Wood blocks in a column and many Leaves blocks. ◦ Jack plants new trees as he goes. This is not relevant to the activity but ensures there are always more trees. • Each Wood block is mined with an Axe and drops one Wood item. 41 Minecraft-Minecart: Crafting Recipes Handout 5

This handout lists all items needed to build the minecart track that Jack will make for himself.

At the Crafting Table

Note that many crafting table recipes create a stack of items. Torches, for example, are always made in batches of four. The number of items will be written next to the name for such cases: “Torches (4).”

Produced Items Ingredients Produced Items Ingredients Wood Planks (4) One Wood Stick (4) Two Wood Planks Torches (4) One Stick Stone Bricks (4) Four Stone One Coal OR Charcoal Lever One Stick Block of Redstone Nine Redstone One Cobblestone Minecart Rail (16) One Stick Powered Rail (6) One Stick Six Iron Ingots One Redstone Six Gold Ingots Stone Shovel Two Sticks Stone Pickaxe Two Sticks One Cobblestone Three Cobblestone Stone Axe Two Sticks Three Cobblestone Table 2: Crafting Recipes

At the Furnace

The Furnace always turns one item into one other specific item; this is referred to as smelting. For example, smelting 5 Iron Ore will always return 5 Iron Ingots. Smelting an item also requires fuel. Jack always uses Coal and Charcoal for fuels, both of which smelt 8 items per piece of Coal or Charcoal. Jack only needs to smelt two items in this activity. Cobblestone is smelted into “Stone” and Wood blocks (not planks) are smelted into Charcoal.

42 Minecraft-Minecart: Tools Handout 6

Jack uses Stone Pickaxes, Shovels, and Axes. The crafting recipes for each of these can be found on Handout 5 and are repeated here.

Stone Pickaxe Stone Shovel Stone Axe Two Sticks Two Sticks Two Sticks

Three Cobblestone One Cobblestone Three Cobblestone Table 3: Tool Recipes

Each of these tools has a specific use

• The Pickaxe is used to mine Stone (which drops Cobblestone), Coal Ore (which drops Coal), and Iron Ore (which drops itself) • The Shovel is used to mine Dirt and Gravel, which drop themselves (Gravel can drop Flint) • The Axe is used to mine Wood by cutting down trees found on the surface. A tree is made of Wood blocks and Leaf blocks. Tools only last a certain number of uses. A newly-created stone tool can mine 132 blocks.

Immediately after the 132nd block is mined, the tool breaks and is removed from the game. If Jack stops using a tool before exhausting all uses, however, the game allows a limited amount of repair with a special crafting recipe.

The repair recipe takes two identical tools and combines them into a single tool with their combined durability, plus a bonus of 5% the maximum durability of the tool (rounded down). The repaired tool's durability cannot exceed the durability of a brand-new tool.

Jack repairs tools whenever he can, but never in a way that he would lose durability.

(Meaning he never tries to repair a tool to above 132 uses.)