Alignment of the National Mathematical Standards with the Materials Science Technology

Alignment of the National Mathematical Standards with the Materials Science Technology

Leonard Booth, MST Consultant

The following is an alignment of the National Mathematical Standards with the Materials Science Technology (MST) curriculum. Some of the portions of the mathematical standards have been abbreviated or omitted due to the lack of correlation between the two. Examples include selected areas where the standards (and expectations) are met or introduced. Since MST does not have a strictly followed curriculum, some of the standards may be taught in more detail in some classes due to the difference in approaches of one teacher as compared with another. Also, several other activities are often included by individual instructors. . This alignment is based primarily upon the experiments and demonstrations found in the 1995 publication: Materials Science and Technology Teachers Handbook by Pacific Northwest Laboratory. Most instructors supplement these activities with videos, brochures, and texts by ECI¹ or Jacobs & Kilduff². These National Mathematical Standards are covered even more thoroughly than listed below when these additional materials are included.

The following abbreviations are used:

EX = Students receive an EXPOSURE to the standard and expectation.

L.P. = Students receive LIMITED PRACTICE to the standard and expectation.

CM = Students achieve COMPETENCY in the standard and expectation.

The last column listed as "STANDARD CODE" gives an identification to the listed standard in a way that it can be cross-referenced more specifically with the specific MST activities.

Footnotes:

1: A series of handbooks based upon the Materials Science Technology program developed at the Pacific Northwest Laboratory. It was copyrighted by Energy Concepts, Inc. in 1996.

2: Jacobs, James A. and Tomas F. Kilduff. Engineering Materials Technology, 2^nd edition, Prentice Hall, Englewood Cliffs, New Jersey, 1994. Newer editions are in print.

NUMBER & OPERATIONS STANDARD	EXPECTATION	EX	L.P	CM	EXAMPLE	STAND. CODE
Understand numbers, ways of representing numbers, relationships among numbers, and number systems	Develop a deeper understanding of very large and very small numbers and of various representations of them				Introduced and used in many experiments including Glass Batching and Young’s Modulus - Beams. Also used with moles, sizes of atoms, tensile strength, and coefficient of thermal expansion.	N-1-a
	compare and contrast the properties of numbers and number systems, including the rational and real numbers, and understand complex numbers as solutions to quadratic equations that do not have real solutions.					N-1-b
	Understand vectors and matrices as systems that have some of the properties of the real-number system					N-1-c
	use number-theory arguments to justify relationships involving whole numbers					N-1-d
Understand meanings of operations and how they relate to one another	Judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities				Young’s Modulus - Beams is one example	N-2-a
Continued on next page	Develop an understanding of properties of, and representations for the addition and multiplication of vectors and matrices					N-2-b

NUMBER & OPERATIONS STANDARD	EXPECTATION	EX	L.P	CM	EXAMPLE	STAND. CODE
Continued from previous page	Develop an understanding of permutations and combinations as counting techniques					N-2-c
Compute fluently and make reasonable estimates	Develop fluency in operations with real numbers, vectors, and matrices, using mental computation of paper-and-pencil calculations for simple cases and technology for more-complicated cases.				Young’s Modulus - Beams, Night Light, RTV Rubber Mold, and Drawing a Wire are all experiments where this is incorporated	N-3-a
	Judge the reasonableness of numerical computations and their results.				Lost Wax Young’s Modulus - Beams	N-3-b

ALGEBRA STANDARD	EXPECTATION	EX	L.P	CM	EXAMPLE	CODE
Understands patterns, relations, and functions	generalize patterns using explicitly defined and recursively defined functions					A-1-a
	understand relations and functions and select, convert flexibly among, and use various representations for them					A-1-b
	analyze functions of one variable by investigating rates of change				This is introduced several places within the course—especially in metals where stress and strain relationships are studied.	A-1-c
	understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions					A-1-d
	understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions					A-1-e
	interpret representations of functions of two variables					A-1-f
Represent and analyze mathematical situations and structures using algebraic symbols	understand the meaning of equivalent forms of expressions, equations, inequalities, and relations					A-2-a
	write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases					A-2-b
Continued on next page.	use symbolic algebra to represent and explain mathematical relationships				Caloric Output of Al-Zn, Drawing A Wire, and Young’s Modulus - Beams are all experiments where this applies.	A-2-c

ALGEBRA STANDARD	EXPECTATION	EX	L.P	CM	EXAMPLE	CODE
Continued from previous page.	use a variety of symbolic representations, including recursive and parametric equations, for functions and relations					A-2-d
	judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology					A-2-e
Use mathematical models to represent and understand quantitative relationships	identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships					A-3-a
	use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts				The relative sizes of atoms and the comparison of temperature scales are two examples of where these are introduced	A-3-b
	draw reasonable conclusions about a situation being modeled				The packing of atoms are discussed throughout the Metals unit and when crystals are studied.	A-3-c
Analyze change in various contexts	approximate and interpret rates of change from graphical and numerical data				Several areas including the relative sizes of atoms, stress vs strain, and interpretation of binary phase diagrams. Experiments include Alloying Tin and Lead and Making Glass from Soil.	A-4-a

GEOMETRY STANDARD	EXPECTATION	EX	L.P	CM	EXAMPLE	CODE
analyze characteristics and properties of two- and three- dimensional geometric shapes and develop mathematical arguments about geometric relationships	analyze properties and determine attributes of two- and three-dimensional objects				These are covered when studying how atoms pack together in solids, in bonding, and amorphous vs crystalline polymers as well as other areas.	G-1-a
	Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them and solve problems involving them				Drawing A Wire is an experiment where this is introduced. It is also introduced when studying fiber loading in composites.	G-1-b
	Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others					G-1-c
	Use trigonometric relationships to determine lengths and angle measures					G-1-d
Specify locations and describe spatial relationships using coordinate geometry and other representations systems	Use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations				This is introduced when studying the biasing of fibers in the Composite unit.	G-2-a
	Investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates.					G-2-b
Apply transformations and use symmetry to analyze mathematical situations Continued on next page	Understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices.					G-3-a

GEOMETRY STANDARD	EXPECTATION	EX	L.P	CM	EXAMPLE	CODE
Continued from previous page	Use various representations to help understand the effects of simple transformations and their compositions					G-3-b
Use visualization, spatial reasoning, and geometric modeling to solve problems	Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools				Young’s Modulus—Beams and RTV Rubber Mold are two experiments. This is also taught when learning how crystalline solids are packed on the atomic level.	G-4-a
	Visualize three-dimensional objects from different perspectives and analyze their cross sections					G-4-b
	Use vertex-edge graphs to model and solve problems					G-4-c
	Use geometric models to gain insights into, and answer questions in, other areas of mathematics				This is taught when learning how crystalline solids are packed on the atomic level.	G-4-d
	Use geometric ideas to solve problems in, and gain insight into, other disciplines and other areas of interest such as art and architecture				Included in several experiments including Drawing A Wire, RTV Rubber Mold, Lost Wax Casting, and Young’s Modulus—Beams.	G-4-e

MEASUREMENT STANDARD	EXPECTATION	EX	L.P	CM	EXAMPLE	Code
Understand measurable attributes of objects and the units, systems, and processes of measurement	Make decisions about units and scales that are appropriate for problem situations involving measurement				Young’s Modulus—Beams is one experiment. This is also dealt with when tensile strength is studied in metals and polymers.	M-1-a
Apply appropriate techniques, tools, and formulas to determine measurements	Analyze precision, accuracy, and approximate error in measurement situations				Young’s Modulus—Beams and Polymer ID are two experiments. Measurements and approximations are use throughout the course.	M-2-a
	Understand and use formulas for the area, surface area, and volume of geometric figures, including cones, spheres, and cylinders				Drawing A Wire and the studying of how atoms pack together in solids are two examples.	M-2-b
	Apply informal concepts of successive approximation, upper and lower bounds, and limit in measurement situations					M-2-c
	Use unit analysis to check measurement computations				This is introduced in several areas including while working with tensile strength and calculating density. The experiment Young’s Modulus—Beams also incorporates this.	M-2-d

DATA ANALYSIS AND PROBABILITY STANDARD	EXPECTATION	EX	L.P	CM	EXAMPLE	STAND. CODE
Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them	Understand the differences among various kinds of studies and which types of inferences can legitimately be drawn from each other					D-1-a
	Know the characteristics of well-designed studies, including the role of randomization in surveys and experiments					D-1-b
	Understand the meaning of measurement data and categorical data, of univariate and bivariate data, and of the term variable					D-1-c
	Understand histograms, parallel box plots and scatterplots and use them to display data				Paper Clip Destruction	D-1-d
	Compute basic statistics and understand the distinction between a statistic and a parameter					D-1-e
Select and use appropriate statistical methods to analyze data	For univariate measurement data, be able to display the distribution, describe its shape, and select and calculate summary statistics				Paper Clip Destruction	D-2-a
	For bivariate measurement data, be able to display a scatterplot, describe its shape, and determine regression coefficients, regression equations, and correlation coefficients using technological tools				This is introduced in demonstrations and while studying stress vs strain.	D-2-b
Continued on next page	Display and discuss bivariate data where at least one variable is categorical					D-2-c

DATA ANALYSIS AND PROBABILITY STANDARD	EXPECTATION	EX	L.P	CM	EXAMPLE	STAND. CODE
Continued from previous page	Recognize how linear transformations of univariate data affect shape, center, and spread					D-2-d
	Identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled.					D-2-e
Develop and evaluate inferences and predictions that are based on data	Use simulations to explore the variability of sample statistics from a known population and to construct sampling distributions					D-3-a
	Understand how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference				Paper Clip Destruction	D-3-b
	Evaluate published reports that are based on data by examining the design of the study, the appropriateness of the data analysis, and the validity of conclusions					D-3-c
	Understand how basic statistical techniques are used to monitor process characteristics in the workplace				This is introduced in the experiment Paper Clip Destruction as well as when quality control is discussed in several units.	D-3-d
Understand and apply basic concepts of probability	Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases					D-4-a
	Use simulations to construct empirical probability distributions					D-4-b
	Compute and interpret the expected value of random variables in simple cases					D-4-c
	Understand the concepts of conditional probability and independent events;					D-4-d
	Understand how to compute the probability of a compound event					D-4-e

PROBLEM SOLVING STANDARD	EXPECTATION	EX	L.P	CM	EXAMPLE	STAND. CODE
Build new mathematical knowledge through problem solving					Lost Wax Casting, Young’s Modulus—Beams, and Glass Batching are all experiments where this is done.	P-1
Solve problems that arise in mathematics and in other contexts					Young’s Modulus--Beams	P-2
Apply and adapt a variety of appropriate strategies to solve problems					Lost Wax Casting and Young’s Modulus—Beams are two examples.	P-3
Monitor and reflect on the process of mathematical problem solving						P-4

REASONING AND PROOF STANDARD	EXPECTATION	EX	L.P	CM	EXAMPLE	STAND. CODE
Recognize reasoning and proof as fundamental aspects of mathematics					Lost Wax Casting	R-1
Make and investigate mathematical conjectures						R-2
Develop and evaluate mathematical arguments and proofs						R-3
Select and use various types of reasoning and methods of proof					Paper Clip Destruction	R-4

COMMUNICATION STANDARD	EXPECTATION	EX	L.P	CM	EXAMPLE	CODE
Organize and consolidate their mathematical thinking through communication						Com-1
Communicate their mathematical thinking coherently and clearly to peers, teachers, and others					This is done in several demonstrations and discussions.	Com-2
Analyze and evaluate the mathematical thinking and strategies of others						Com-3
Use the language of mathematics to express mathematical ideas precisely						Com-4

CONNECTIONS STANDARD	EXPECTATION	EX	L.P	CM	EXAMPLE	CODE
Recognize and use connections among mathematical ideas						Con-1
Understand how mathematical ideas interconnect and build on one another to produce a coherent whole						Con-2
Recognize and apply mathematics in contexts outside of mathematics					Young’s Modulus—Beams, RTV Rubber Mold, and Lost Wax Casting	Con-3

REPRESENTATION STANDARD	EXPECTATION	EX	L.P	CM	EXAMPLE	STAND. CODE
Create and use representations to organize, record, and communicate mathematical ideas					Alloying Tin and Lead and Making Glass from Soil are two experiments where this is done.	Rep-1
Select, apply and translate among mathematical representations to solve problems						Rep-2
Use representations to model and interpret physical, social, and mathematical phenomena					This is introduced in several areas including when the tensile strength of materials is studied.	Rep-3