1.

INTRODUCTION

There have been several approaches suggested to simplify any given Boolean function to achieve gate level minimization. The most well-known method is Karnaugh maps (K-maps) introduced in 1953 by Maurice Karnaugh which was an improvement to the Veitch Chart that was presented by Edward W. Veitch’s in 1952 [1], [2]. This method exploits human’s pattern recognition capability to eliminate and group cells to represent the correct minimized logic expression. Solving any given Boolean expression requires the expression to be written first in its canonical form, which will result in having several minterms. These minterms are later on mapped on a Truth Table [3]. Afterwards, the minterms will be transferred to a 2-Dimentional grid, K-map. This grid utilizes grey code to order its cells, which are also referred to as minterms. Later on, those miterms will be grouped in an effective way to form the simplified Boolean expression.

The main reason to derive a simplified Boolean expression is to find the minimal number of Gates needed to implement any real-world logic requirement. Less number of gates means smaller size circuits. Having an expression with Sum-of-Product (SOP) will require AND gates that feeds into an OR Gate. On the other hand, having Product-of-Sum will require OR gates that feeds into an AND gate. K-maps can be used to reduce an expression with at most 4 variables. This method gets complex with an expression with more than 4 variables [3].

Another way to achieve gate level minimization was through applying Boolean Algebra method suggested by George Boole in 1854 [4]. The variables in the expression that utilizes Boolean Algebra are treated either as 0s or 1s (Binary Digits), or in other words, these variables are considered to be either True or False. Based on that concept, Boolean Algebra Laws and theorems are presented to assist in achieving a simplified expression. However, as the number of variables increase simplification process gets more challenging.

In 1952, another algorithm to minimize Boolean Functions was developed by Willard V. Quine and improved by McCluskey in 1956, called Quine–McCluskey algorithm (QMC) [5],[6]. This method is identical to K-maps method in functionality, but its tabular form makes it more appealing to be utilized in computer algorithms. However, this method has a high computational complexity.

From the simplifying methods listed above, K-maps is one of the prominent Digital Logic Design topics that has been given in many electrical, electronic information and computer curriculums. A teaching method was introduced in [7], which modified the K-maps teaching method in such a way to give the instructors an advantage while introducing the topic during the lecture, and as well in the grading process while assessing students. Also, this method gave students an importunity to go back through their notes and ask questions on how the maps were constructed. In all, this method enhances the learning and grading process of using K-maps.

In addition, other several teaching tools were developed by researches to facilitate teaching K-maps, such that students will grasp the concept more easily and increase their interest in the topic. In [8], proposed a teaching module in nano-technology course that uses K-maps to introduce undergraduate students to nano-technology logic design concepts. The main aim from introducing this teaching methodology is to increase students’ interest in the topic, thus enhancing their critical thinking, and further motivate them to investigate many emerging nano-technology devices.

In addition, many other forms of teaching approaches has been suggested to aid in introducing Elementary Digital Design course. In [9], authors have suggested a computer-aided software to aid students in solving any given Boolean expression using different minimizing techniques and build the correct combinational circuit. It provides a user-friendly environment due to its graphical interface. It is noted in [10] that adaptive and personalized learning technologies, open educational resources and learning analytics are the three most promising technologies that will impact engineering education.

In this study, a new teaching kit to be used by educators to teach Boolean functions representation will be introduced. The kit is a 3-D model referred as “The Cubes Model” to aid students in representing Boolean function minterms with 6 variables by sorting the cubes according to the grey code sequence. In addition, four different in-class exercises are proposed to aid educators in introducing this kit. The first exercise will lead students in finding the cube’s position. The second exercise will help students in finding the minterm expression from the cubes’ position, and as a result learn how the sides of the cube are labeled. The third one is an exercise that demonstrates how to sort all 64 cubes in the teaching kit. The last exercise will teach students how to find adjacent cubes and formulate the product term of the function Boolean logic expression.

2.

THE CUBES MODEL

This model in Figure 1 is used to represent all minterms of any given 6 variable Boolean expression G(A, B, C, D, E, F) in a 3D model. Where A is the most significant bit and F is the least significant bit. The model consists of three adjacent solid walls, where a total of 64 cubes will be placed on the model in an arrangement to construct the 3D model as shown in Figure 1. The solid walls can be made of wood (as suggested), plastic, acrylic or any other solid or transparent material.

Figure 1

The structure of the 3D cubes model

The 3D cubes are numbered from 0 to 63 as shown in Figure 2 (a). Each cube has the position and the equivalent decimal minterm labeled on four out of the cube’s six sides. The used as sides are shown in Figure 2 (b). The minterm decimal equivalent is labeled in the bottom side. The other three sides are labeled either with one of the combinations of AB, CD, and EF or their complements to represent the correct combination of a minterm. In other words, the side of the cube will be labeled with one of the four combinations Ā B̄, ĀB, AB or AB̄, the front side of the cube will marked with one of the four combinations C̄ D̄, C̄D, CD or CD̄ and top side of the cube will denote one of the four combinations Ē F̄, ĒF, EF or EF̄. Figure 2 (c) and (d) are showing minterm number 38 as an example. Two sides are left not labeled.

Figure 2

The labeling scheme of each Cube.

TABLE I presents all 64 minterms numbered from 0 to 63 showing each side of the cube with what letter combinations it will be labeled with.

TABLE I

– CUBES’ LABELS LIST

The 64 cubes will be arranged utilizing the grey code sequence, such that adjacent cubes will differ by only one bit. Figure 3 (a) and (b) represents how cubes will be arranged.

Figure 3

Cubes arrangement

3.

CLASSROOM EXERCISES

This model is proposed to students as a teaching tool to aid them in representing Boolean Logic Functions with 6 variables. To achieve this objective, several exercises should be presented to the students during the lecture. Those exercises are listed below.

3.2

Labeling Cubes’ Sides

With this exercise, students will learn how to label the side, front and top of the cube by only giving them the number in the bottom of the cube which denotes the cube’s position. The given number is in in decimal format. First students will convert the number given to binary and then will represent the obtained value to its actual minterm. Whenever the number is a binary 0 it means that the letter value will be expressed with a bar , where X can be any letter from A to F. The minterm founded will label the sides of the cube in the previously mentioned order.

For example, given the cube in Figure 2 (b). Consider that the side, front and top are unknown for the students. Students are provided with the bottom value which is 38. Students will convert the decimal value 38 into binary by doing the following process in Figure 4. The obtained converted value from Figure 2 (b) is 100110. This means the minterm represented by this cube is: AB̄ C̄DEF̄.

Figure 4

Decimal to binary conversion

Like in the previous exercise, each student can be provided with more than one cube during the session to help them practice more in labeling the sides of the cube in individual and group work settings.

4.

CONCLUSION

This work presented the teaching kit “The Cubes Model” as a 3D tool to aid students in representing minterms of Boolean logic functions. The structure of the model consists of three solid walls, were 64 cubes arranged in a way to represent a 6 variable Boolean function. Students will use different senses and their recognition capabilities to sort and arrange the cubes. Four different exercises were proposed in this study. The first two exercises will lead students in knowing how to find the cubes’ positions and minterms expressions, and thus help them to better understand how cubes’ sides are labeled. The third exercise will teach students how to sort and fix up the cubes in the correct way on the model according to grey code arrangement. The fourth exercise will give students a better understanding in realizing adjacent cubes, thus learning how to express the simplified expression of those adjacent cubes. All exercise can be done in individual and group work settings.

Since Boolean logic function is a common topic found in several engineering curriculums, this kit has an important impact in facilitating education in several engineering fields, where both educators and students will benefit from using it. Educators will find it as an easy teaching tool to be introduced to students. Thus, resulting in an increase student’s interactivity, which is the main objective of every educator. On the other hand, students will use more than one sense which will positively impact their learning process. Thus, this will lead students to better understand the topic. In conclusion, programs that will benefit from using this kit, but not limited to, are electrical, computer and communication engineering programs.

For future work, a computer-based software version of this kit is suggested to aid educators in the teaching process and help students better understand Boolean logic functions. Moreover, additional features maybe added to the software such as auto grading tools to help students in practicing and do self-assessments. Besides, additional interactive features can be added to the suggested 3D model, such as different sounds or lightning effects, to notify students if any incorrect positioning of any cube is present. Thus, leading them to learn independently.