# Explain applications of mathematics in the engineering design process.

## Definition

Explanation should include- reasons mathematics is essential in the engineering design process
- summary of types of mathematics commonly used by engineers
- reasons why solving for unknown variables may be necessary before finding a solution
- ways to determine the formulas needed for a solution to an engineering design problem.

## Process/Skill Questions

- What is the definition of
*algorithm*? - What are some common examples of algorithms?
- In what kinds of engineering design problems should one establish and use a mathematical algorithm to solve the problem?
- How can one determine whether one's solutions to unknown variables are feasible?
- Why should one not depend exclusively on computer-generated results?

## Related Standards of Learning

## English

### 10.5

The student will read, interpret, analyze, and evaluate nonfiction texts.

- Analyze text features and organizational patterns to evaluate the meaning of texts.
- Recognize an author’s intended audience and purpose for writing.
- Skim materials to develop an overview and locate information.
- Compare and contrast informational texts for intent and content.
- Interpret and use data and information in maps, charts, graphs, timelines, tables, and diagrams.
- Draw conclusions and make inferences on explicit and implied information using textual support as evidence.
- Analyze and synthesize information in order to solve problems, answer questions, and generate new knowledge.
- Analyze ideas within and between selections providing textual evidence.
- Summarize, paraphrase, and synthesize ideas, while maintaining meaning and a logical sequence of events, within and between texts.
- Use reading strategies throughout the reading process to monitor comprehension.

### 11.5

The student will read, interpret, analyze, and evaluate a variety of nonfiction texts including employment documents and technical writing.

- Apply information from texts to clarify understanding of concepts.
- Read and correctly interpret an application for employment, workplace documents, or an application for college admission.
- Analyze technical writing for clarity.
- Paraphrase and synthesize ideas within and between texts.
- Draw conclusions and make inferences on explicit and implied information using textual support.
- Analyze multiple texts addressing the same topic to determine how authors reach similar or different conclusions.
- Analyze false premises, claims, counterclaims, and other evidence in persuasive writing.
- Recognize and analyze use of ambiguity, contradiction, paradox, irony, sarcasm, overstatement, and understatement in text.
- Generate and respond logically to literal, inferential, evaluative, synthesizing, and critical thinking questions about the text(s).

### 12.5

The student will read, interpret, analyze, and evaluate a variety of nonfiction texts.

- Use critical thinking to generate and respond logically to literal, inferential, and evaluative questions about the text(s).
- Identify and synthesize resources to make decisions, complete tasks, and solve specific problems.
- Analyze multiple texts addressing the same topic to determine how authors reach similar or different conclusions.
- Recognize and analyze use of ambiguity, contradiction, paradox, irony, overstatement, and understatement in text.
- Analyze false premises claims, counterclaims, and other evidence in persuasive writing.

## Mathematics

### A.1

The student will

- represent verbal quantitative situations algebraically; and
- evaluate algebraic expressions for given replacement values of the variables.

### A.4

The student will solve

- multistep linear and quadratic equations in one variables algebraically;
- quadratic equations in one variables algebraically;
- literal equations for a specified variable;
- systems of two linear equations in two variables algebraically and graphically; and
- practical problems involving equations and systems of equations.

### A.7

The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including

- determining whether a relation is a function;
- domain and range;
- zeros of a function;
*x*- and*y*-intercepts;- finding the values of a function for elements in its domain; and
- making connections between and among multiple representations of functions including concrete, verbal, numeric, graphic, and algebraic.

### A.9

The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of linear and quadratic functions.

### AFDA.1

The student will investigate and analyze linear, quadratic, exponential, and logarithmic function families and their characteristics. Key concepts include

- domain and range;
- intervals in which the function is increasing or decreasing;
- absolute maxima and minima;
- zeros;
- intercepts;
- values of a function for elements in its domain;
- connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs;
- end behavior; and
- vertical and horizontal asymptotes.

### AFDA.3

The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems using models of linear, quadratic, and exponential functions.

### AFDA.4

The student will use multiple representations of functions for analysis, interpretation, and prediction.

### AFDA.5

The student will determine optimal values in problem situations by identifying constraints and using linear programming techniques.

### AII.3

The student will solve

- absolute value linear equations and inequalities;
- quadratic equations over the set of complex numbers;
- equations containing rational algebraic expressions; and
- equations containing radical expressions.

### AII.4

The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically.

### AII.5

The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve practical problems, including writing the first

*n*terms, determining the*nth*term, and evaluating summation formulas. Notation will include*Ʃ*and*a*._{n}### AII.6

For absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic functions, the student will

- recognize the general shape of function families; and
- use knowledge of transformations to convert between equations and the corresponding graph of functions.

### AII.7

The student will investigate and analyze linear, quadratic, absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic function families algebraically and graphically. Key concepts include< /br>

- domain, range, and continuity;
- intervals in which a function is increasing or decreasing;
- extrema;
- zeros;
- intercepts;
- values of a function for elements in its domain;
- connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs;
- end behavior;
- vertical and horizontal asymptotes;
- inverse of a function; and
- composition of functions algebraically and graphically.

### AII.10

The student will represent, create, and solve problems, including practical problems, involving inverse variation, joint variation, and a combination of direct and inverse variations.

### COM.4

The student will design an algorithm to solve a given problem.

### G.11

The student will solve problems, including practical problems, by applying properties of circles. This will include determining

- angle measures formed by intersecting chords, secants, and/or tangents;
- lengths of segments formed by intersecting chords, secants, and/or tangents;
- arc length; and
- area of a sector.

### G.12

The student will solve problems involving equations of circles.

### MA.4

The student will determine the limit of an algebraic function, if it exists, as the variable approaches either a finite number or infinity.

### MA.7

The student will perform operations with vectors in the coordinate plane and solve practical problems using vectors.

### PS.8*

The student will describe the methods of data collection in a census, sample survey, experiment, and observational study and identify an appropriate method of solution for a given problem setting.

### PS.12*

The student will determine probabilities (relative frequency and theoretical), including conditional probabilities for events that are either dependent or independent, by applying the Law of Large Numbers concept, the addition rule, and the multiplication rule.

## Science

### PH.1

The student will plan and conduct investigations using experimental design and product design processes. Key concepts include

- the components of a system are defined;
- instruments are selected and used to extend observations and measurements;
- information is recorded and presented in an organized format;
- the limitations of the experimental apparatus and design are recognized;
- the limitations of measured quantities are recognized through the appropriate use of significant figures or error ranges;
- models and simulations are used to visualize and explain phenomena, to make predictions from hypotheses, and to interpret data; and
- appropriate technology including computers, graphing calculators, and probeware is used for gathering and analyzing data and communicating results.