the Technology Interface / Spring 97

Design of an Integrated Algebra and Chemistry Curriculum Based on Manufacturing Applications

Mukasa E. Ssemakula, Ph.D. Div. of Engineering Technology Wayne State University Detroit, MI 48202 Chorng S. Houh, Ph.D. Dept. of Mathematics Wayne State University Detroit, MI 48202

Mary L. Mayer, Ph.D. GelmanSciences 600 South Wagner Road Ann Arbor, MI 48103 (formerly with Dept. of Chemistry, WSU)

Vladimir Sheyman, Ph.D. Div. of Engineering Technology Wayne State University Detroit, MI 48202


This paper describes an integrated Algebra and Chemistry course that has been developed as part of the requirements for a Manufacturing Engineering education curriculum being implemented by the NSF funded Greenfield Coalition for New Manufacturing Education. The philosophy behind the development of the course was the need to teach basic mathematics and chemistry to students pursuing degrees in the manufacturing engineering area, with the manufacturing shop floor serving as the motivating force for the instruction. The goal was to teach the fundamentals of algebra and chemistry in a manufacturing context. The need to understand the chemistry of the various processes served as the backdrop to motivate the teaching of the basic algebra relevant to the analysis. The student body addressed by this project was mainly inner city minority students, and they were mostly first generation college students. They were all working full time in the manufacturing facility at Focus:Hope and this working experience was integrated into the educational curriculum being developed.


The Greenfield Coalition for New Manufacturing Education, also called the Greenfield Coalition; is made up of Focus:Hope's Center for Advanced Technologies (CAT); academic partners University of Detroit-Mercy, Central State University, Lawrence Technological University, Lehigh University, University of Michigan, and Wayne State University; as well as industrial partners Chrysler, Ford, General Motors, Detroit Diesel, and Cincinnati Milacron; and the Society of Manufacturing Engineers. The CAT serves as the primary implementation platform for the Coalition's curricula, enabling integration of hands-on manufacturing experiences with an interdisciplinary engineering education within an applications context. The degree candidates are full-time employees at the CAT, and the educational goal is to provide these candidates with a diverse technological education utilizing the manufacturing resources available on the shop floor.

The Coalition's educational model entails hands-on training in the programming, operation, maintenance, and repair of manufacturing equipment; interdisciplinary study of pertinent mathematics, science, engineering, business, and general education courses; as well as structuring and delivery of knowledge within a production environment to provide context. The curriculum is being developed in a modular fashion, with each module being delivered in an innovative instructional process. Modularization makes it possible to use a mix of sub-discipline experts to design and deliver modules within a given knowledge area, thus enhancing overall learning.

The Coalition has identified desirable competencies for the manufacturing engineer of the future that would enable the engineer to achieve leadership in the field. Working with the industrial partners, competencies appropriate to various stages of the manufacturing engineer's career have been specified. A person meeting with these competencies has been named the Renaissance Manufacturing Engineer. The competencies are then related to individual knowledge areas requiring mastery, which are in turn developed into individual modules. In implementing the modules, developers in general begin with practice and end in theory in order to instill an intuitive sense of physical phenomena to which candidates can then turn their attention.

The development and implementation of the Algebra/Chemistry curriculum described in this paper involved team teaching by faculty from the disciplines of chemistry, mathematics and engineering technology; all working to demonstrate the practical relevance and applications of the theoretical material covered. Out of class assignments were given and group projects were used to encourage collaborative learning. There was also a wide range of computer-based experimentation utilizing commercially available software to motivate learning.

Curriculum Model

The combined Algebra and Chemistry curriculum was developed within the context of the guiding principles discussed above. The knowledge area was divided into four modules designed to link mathematics to chemistry, together with applications and case studies, using the manufacturing experiences in the CAT to provide context. The focus was on the study of Chemistry and Algebra through experimentation, with particular attention to chemical processes that candidates witness on the manufacturing floor on a regular basis. The material covered in this knowledge area was equivalent to four semester credits in a traditional setting, with two credits assigned to Algebra and two credits assigned to Chemistry. Table 1 below shows the structure of the four modules.

Table 1: Structure of the Algebra/Chemistry Modules:
Module 1
CAT Applications
Gauging and inspection Manipulations with real numbers Scope of Chemistry
Precision and accuracy Algebraic manipulations States of matter
Preferred numbers Exponential expressions Solutions, compounds, mixtures
Standard sizes Use of scientific notation Scientific numbers
Mass production SI and English systems of units Basic Chemistry measurements
. Significant digits Chemical reactions
. Series of numbers Oxidation states

Module 2
CAT Applications
Graphing applications Relationships between sets A history of chemistry
Mechanical properties of metals Graphing functions Moles as a measure of matter
Metals vs. plastics Manipulating functions Mass - mole - mass conversions
Comparison of various metals Equations and function values The periodic table
.Linear functions Chemical nomenclature
.Equation of a line Oxidation states
.Exponential functions .
.Significance of the exponent .

Module 3
CAT Applications
Estimating answers Recognize graph of f(x) = k/x Aqueous solutions
Solving manufacturing problems Polynomial functions Chemical equations
Verifying solutions Shapes of curves of polynomials Balancing chemical equations
Corrosion Relate length, area and volume Limiting reagents
Rusting Solving mathematical problems Stoichiometry
..Acid-base solutions
..Oxidation and Reduction

Module 4
CAT Applications
Turning diameters Perform algebraic manipulations Atomic and molecular shapes
Bolt-circle diameters Recognize shapes Orbits and electron shells
Tool positioning Know properties of polygons Heat and enthalpy
Geometry calculations for CNC Law of sines and Law of cosines Heat of reaction
Crystalline structure Parametric equations Molecular bonding
.Trigonometric identities The octet rule


The following discussion will focus on the content of the first module. We will describe how the content was designed to satisfy the development goals and how course delivery was implemented. This same model was followed in the development of all the other modules.

Module Integration

The students that are targeted for all Coalition courses are experienced machinists. They are familiar with the measurement and inspection of machined parts. This module was designed to introduce formerly the concepts of tolerances, accuracy and precision in part inspection. Since this was an activity that the students are involved with daily, they could see the significance right away. Topics in systems of units, scientific notation and significant numbers were then introduced from the mathematical standpoint. But keeping applications in measurement in mind, the practical applications could be easily demonstrated. The measurement that candidates do in the manufacturing context was used as a bridge to introduce measurements that are relevant in Chemistry. Since Chemistry was a new science to the students, the opportunity was taken to introduce its scope but with particular attention to its relevance in a discrete parts manufacturing environment. This entailed discussion of what matter is, the various possible states of matter and introduction of the concept of a chemical reaction. Within the module, the mathematical topics were developed to include the manipulation of real numbers as well as algebraic manipulations. The treatment of mathematical series was anchored on the use of preferred sizes in manufacturing, which are themselves derived from geometric series. Finally, standardization of parts which makes mass production possible was detailed. Below, we describe how this helps to round out the module content.

Module Contextualization

The basic property about a series of numbers is that there is a fixed relationship between the members of the series. If one member of the series is known as well as the functional relationship between the members of the series, then all other members of the series can be determined easily. This fact in algebra can be used to illustrate the meaning of functions. For this course however, we go beyond the simple illustration to actual applications. The use of preferred numbers in product standardization serves as a basis to illustrate series in a way that manufacturing engineers and technologists can readily associate with.

Preferred Numbers and Standard Sizes:

Many mass produced articles such as bolts, and semi-finished products such as steel bars, can be made in a wide range of sizes. In most cases, it is technically feasible to make an infinite range of sizes for these products. From a practical consideration however, it would be too difficult and costly to manufacture, distribute, and stock all the different possible sizes. It is therefore preferable to make only a select set of sizes chosen so as to serve the needs of the majority of customers. A good range of sizes for any product should start with small increments at the bottom of the range, with gradually increasing step sizes towards the top of the range. The range of sizes normally made is referred to as standard. By standardizing on a small select set of sizes, it is possible to take advantage of economies of scale and make large quantities at a relatively low cost and avoid frequent changes of setup that would otherwise be required. The range of sizes typically used in manufacturing involve a geometric series. A geometric series has the important property that the ratio of any one number to the previous (or next) number in the series, called the common ratio, is always constant.

In 1879, a Frenchman by the name of Renard proposed a set of geometric series that could be used as the basis for determining sizes for use in a wide variety of applications. There is a total of five of these geometric series, also called R-series, represented by the codes R5, R10, R20, R40 and R80. For each series designated Rn, the common ratio n is given by 101/n. Thus the R5 series has a common ratio 101/5, the R10 series has a common ratio 101/10, and so on. The raw numbers in this set of series have been rounded to convenient values that have since been accepted internationally as a basis for determining sizes for a wide variety of applications. These resulting rounded values are used in preference to other numbers and they are referred to as preferred numbers. The values of the preferred numbers are given in the US national standard ANSI Z17.1-1973 as well as the international standard ISO R286. Table 2 shows the members of the R5, R10 and R20 Renard series and the corresponding preferred numbers in the range 1 - 10. These values can be used to derive other members of the series by multiplying by an a power of 10.

Table 2: Renard Series and Preferred Numbers

In all cases where one has a choice of sizes to use, the preferred numbers should be used provided they meet functional requirements. The R5 series of numbers are referred to as the first choice preferred numbers and where possible, these should be used in preference to all other numbers. The R10 series are second choice preferred numbers, R20 are third choice preferred numbers and so on. It is left as an exercise to the students to derive all the members of the R40 and R80 series.

The derivation of preferred sizes from the raw Renard series numbers is a perfect example of the application of significant figures, rounding-off of numbers and standardization - all important issues in today's global manufacturing environment. Looking at the preferred numbers and comparing them to the raw members of the Renard series, students can see how approximation and rounding of numbers are applied in this case. Students can also see that in the interest of standardization, some of the rules of rounding are not used in generating the preferred numbers. Indeed some values are rounded up while others are rounded down. Application of preferred numbers in Chemistry are also included in the course for completeness.

The complete curriculum developed for the candidates at Focus:Hope was first offered to the candidates in Spring of 1995. The material was delivered primarily as stand up delivery with provisions for videotaping all lectures and demonstrations. The class typically met once a week for three hours per session. Course delivery entailed team teaching of the course by faculty in the three areas of Chemistry, Mathematics and Engineering Technology. Each session was planned to provide an integrated experience involving each of these three areas. In-class demonstrations were utilized to showcase the principles being covered in the lectures, while discussions of shop floor experiences that students were familiar with served to ground the material in an applications context. Student learning was evaluated through traditional written tests.


Due to the non-traditional approach used to deliver the course, a number of challenges had to be faced. The students were not used to the idea of team teaching and it took them a while to adjust to a course being taught by multiple instructors. There are inevitable differences in teaching styles. For an approach such as this to be successful, careful attention has to be given to the need to coordinate not only course content, but also teaching styles overall pedagogy.

It was found that whereas the students had a very good understanding of the principles of manufacturing, their level of preparation in mathematics varied widely. Although they had all passed pre-tests, applying the mathematical concepts in unusual (non-canned type) settings was a major leap. Some could not overcome this hurdle and they ended up dropping out of the course. This might have been exacerbated by the fact that this was a not a campus setting and accessibility of faculty to the students was limited. Other than the weekly meetings, students relied primarily on using the telephone and e-mail which did not work quite as well as personal contact. Those who were able to complete the course however were very positive and enthusiastic about the course design.

Finally, one practical problem peculiar to the setting was the need to accommodate changing production schedules that would sometimes lead to students missing learning sessions. This is an issue that is still being grappled with in the delivery of courses in other knowledge areas.


This work was funded by the National Science Foundation through a subcontract under the Greenfield Coalition, cooperative agreement number EEC-9221542.