Technical Mathematics (Mathematics for Technology I)

Course Description:

Technical Mathematics uses problem situations, physical models, and appropriate technology to extend mathematical thinking and engage student reasoning. Problem solving situations will provide all students an environment that promotes communication and fosters connections within mathematics, to other disciplines, and to the technological workplace. Students will use physical models in a laboratory setting to represent, explore, and develop abstract concepts. The use of appropriate technology will help students apply mathematics in an increasingly technological world. The course includes: problem solving, reasoning, connections, communication, and representation and the relationship of these mathematical processes to applications of the content in the workplace.

Content Standard 1.0: Number and Operations

Students will recognize, represent, model, and apply real numbers and operations verbally, physically, symbolically, and graphically and will compute fluently and make reasonable estimates in problem solving related to the workplace.

Learning Expectations:

The student will:

• 1.1  demonstrate an understanding of the subsets, elements, properties, and operations of the real number system;
• 1.2  demonstrate an understanding of the relative size of rational and irrational numbers;
• 1.3  connect physical, graphical, verbal, and symbolic representations of real numbers;
• 1.4  informallydescribe and model the concept of inverse (e.g., opposites, reciprocals, and squares and square roots);
• 1.5  demonstrate an understanding of division involving zero;
• 1.6  apply number theory concepts (e.g., primes, factors, divisibility and multiples) in mathematical problem situations;
• 1.7  connect physical, graphical, verbal, and symbolic representations of absolute value;
• 1.8  use real numbers to represent real-world applications (e.g., rate of change, probability, and proportionality);
• 1.9  select and apply an appropriate method (i.e., mental arithmetic, paper and pencil, or technology) for computing with real numbers, and evaluate the reasonableness of results;
• 1.10  perform operations on simple algebraic expressions, and informally justify the procedures chosen;
• 1.11  use estimation to make predictions and determine reasonableness of computational results;
• 1.12  use mathematical notations appropriately.

 Student Performance Indicators:

At Level 1, the student is able to

• choose the correct prime factorization of a two-digit composite whole number;
• compare a fraction to a decimal using less than, greater than, and equals symbols;
• multiply a fraction by a multiple of its denominator (denominator less than or equal to 25);
• apply order of operations to evaluate numerical expressions (whole numbers only; no exponents or grouping symbols.

At Level 2, the student is able to

• identify the opposite of any rational number;
• select the best estimate for the coordinate of a given point on a number line (rationals);
• choose an equivalent exponential form of a one-variable monomial given in factored form (only first-degree variables with positive integral coefficients);
• multiply an integer by a one-variable binomial;
• select a reasonable solution for a real-world division problem in which the remainder must be considered;
• apply order of operations to evaluate numerical expressions containing whole numbers, exponents, and no more than two sets of grouping symbols (no power larger than two);

at Level 3, the student is able to

• select ratios and proportions to represent real-world problems such as scale drawings and samplings (all ratios are positive integers to positive integers).

Content Standard 2.0:  Algebra

Students will describe, extend, analyze, and create a wide variety of patterns and functions using appropriate materials and representations in real world problem solving.

Learning Expectations:  

The student will:

• 2.1  analyze, extend, and create mathematical patterns related to algebra in real-world problem solving;
• 2.2  communicate the meaning of variables in algebraic expressions and equation;
• 2.3  apply and interpret rates of change from numerical data;
• 2.4  apply the concept of variable to simplify algebraic expressions and solve equations;
• 2.5  model real-world phenomenon using graphs.

Student Performance Indicators:

At Level 1, the student is able to

• describe and extend geometric and numerical patterns in the workplace.

At Level 2, the student is able to

• translate a verbal expression into an algebraic expression in real-world problems;
• evaluate real-world formulas and algebraic expressions given values for one or more variables and grouping symbols;
• solve one- and two-step linear equations;
• justify correct results of algebraic procedures.

At Level 3, the student is able to

• apply the concept of rate of change to solve real-world problems;
• select the linear and non-linear graphs that model given real-world situations described in data sets and narratives.

Standard 3.0:  Geometry

The student will investigate, model, and apply geometric properties and relationship in work related situations.

Learning Expectations: 

The student will:

• 3.1  analyze and apply concepts and properties in the construction of lines, angles, and vertices when solving work-related problems;
• 3.2  synthesize and apply geometrical concepts, properties, and formulas of two-dimensional shapes when solving work-related problems;
• 3.3  synthesize and apply geometrical concepts, properties, and formulas of three-dimensional shapes when solving work-related problems.

Student Performance Indicators:

At Level 1, the student will be able to

• construct angles and vertices to solve work-related problems.

At Level 2, the student will be able to

• use geometric formulas to solve real-world problems (e.g. area, perimeter, surface area, volume and circumference);
• use scale and proportion in real-world situations (e.g. read a road map, scale drawings, and read blueprints);
• apply the Pythagorean Theorem in  real-world problems;
• determine the height of an object that is difficult to measure by using properties of similar triangles.

At Level 3, the student will be able to

• use parallel and perpendicular lines to solve work-related problems.

Standard 4.0:  Measurement

Students will become familiar with the units and processes of measurement in order to use various tools, techniques, and formulas to determine and estimate measurements in problem solving.

Learning Expectations: 

The student will:

• 4.1  select and use appropriate tools of measurement to determine length, area, angular measurement and volume with in given tolerances (i.e. vernier caliper, micrometer, machinist rule, graduated cylinders, protractors as well as rulers);
• 4.2  use measurements of length, area, angular measurement and volume to estimate and solve real-world problems;
• 4.3  apply measurement concepts and relationships in algebraic and geometric problem-solving situations;
• 4.4  demonstrate an understanding of rates and other derived and indirect measurements.

Student Performance Indicators

At Level 1, the student is able to

• select and apply appropriate tools and units to measure in real-world situations (e.g., manufacturing, construction, art);
• justify the selection of a unit of measure in specific situations (e.g., manufacturing);
• discover and explain formulas used to compute circumference, perimeter, area and volume (e.g., pool construction);
• apply the given formula to determine the area,  perimeter or volume of two dimensional objects;
• defend estimates of the perimeter and/or area of rectangles, triangles, trapezoids, and parallelograms. (e.g., flooring);
• estimate the area and volume of irregular geometric figures in work-related problems.

At Level 2, the student is able to

• apply the given formula to find the area of a circle, the circumference of a circle, or the volume and surface area of a rectangular solid, cylinder, and sphere;
• apply the concept of rate to determine ones such as miles per hour, cost per unit, and revolutions per minute;
• describe the procedure for determining the area of a composite shape in a real-world situation (e.g., surveying);
• defend an estimate for the volume of a container (e.g. bottling companies);
• compare various methods of measurement to estimated values (e.g., shadow of object vs. height of object);
• calculate a dimension of a geometric figure given the volume and other pertinent information (e.g. housing);
• determine if measurements are within given tolerance intervals;
• construct scale drawings to solve work related problems.

at Level 3, the student is able to

• discover the dimensions of a rectangle when given its area and the relationship between the length and width of the sides (e.g., art);
• explore the golden rectangle as it relates to measurement and proportions
• describe how changes in the dimensions of similar figures affect perimeter, area, and volume (e.g., construction).

Sample Task: 

Students will measure and make a scale drawing for a room and determine the amount of carpet or tile needed and the amount of paint needed for the walls.

Linkages: 

Mathematics – Geometry. Discuss connections to drafting and carpentry, agribusiness, marketing, consumer science, and industrial technology.

Standard 5.0:  Data Analysis and Probability

The student will collect, organize, represent, and interpret data; make and evaluate inferences and predictions; present and evaluate arguments based on data analysis; and model situations to determine theoretical and experimental probabilities.

Learning Expectations:

The student will:

• 5.1  read graphs, charts, and tables;
• 5.2  recognize if a problem needs more data and if so find a source for the data;
• 5.3  collect, organize and interpret data;
• 5.4  choose, construct, and analyze appropriate graphical representations for a data set;
• 5.5  interpret a set of data using the appropriate measure of central tendency;
• 5.6  interpolate readings on a graph as well as extrapolate to estimate values;
• 5.7  apply appropriate technology in data collection and analysis;
• 5.8  analyze the validity of statistical conclusions and the use, misuse, and abuse of data.

Student Performance Indicators:

At Level 1, the student is able to

• interpret bar graphs representing real-world data;
• interpret circle graphs representing real-world data;
• determine the measures of central tendency for a given set of data;
• determine the probability of a single event (i.e., spinning a spinner, rolling a die);
• collect data from a real-world situation and construct a graph (bar, circle, line) both by hand and using appropriate technology.

At Level 2, the student is able to

• choose the matching linear graph given a set of ordered pairs that represent real-world data;
• analyze student-collected data from a real-world situation to make predictions using appropriate technology;
• apply the appropriate measure of central tendency  (i.e., mean, median, and mode) to a real-world problem.

At Level 3, the student is able to

• select the measure of central tendency that best describes the given real-world situation;
• choose the matching scatter plot, bar graph, or histogram given a set of real-world data in table or chart form;
• choose the correlation of a scatter plot using real-world data; analyze the validity of statistical conclusions and the use, misuse, and abuse of data;
• use simulations to determine probabilities.

Sample Task: 

Students conduct a consumer preference survey and construct a graph to show the results and make any appropriate predictions.

Linkages: 

Students create spreadsheets and graphs on the computer to decide what type of graph best displays their data.  Economics – Census Bureau, Neilson Ratings.  Sports – game statistics.