Concurrent Enrollment Mathematics 1030 requires students to use advanced mathematical concepts to make decision and communicate ideas within five domains: statistics, logic, probability, mathematical modeling, and financial management. The course necessitates students to reason abstractly and quantitatively to make decisions about situations and then communicate their decisions using a mathematical argument. Further, students will use appropriate tools, including technology, to model their mathematical thinking, and use structure and regularity to describe mathematical situations and solve problems.

These materials consist of 8 modules, each one of which representing between 3 and 6 weeks of work. See the list under Course Modules below. Note that Module M has four lessons that can be separated in several ways into two sets of two lessons, each of which can be considered an independent module.

The target audience for this course consists of high school students who have completed the mathematics requirements for graduation (Secondary Mathematics I, II, III) and who desire to take a senior level mathematics course to meet any one of these purposes:

- to meet college level QL (at the University of Utah, QA) requirements and be done with mathematics requirements;
- to consolidate mathematical knowledge, particularly in the context of applications in order to move forward in mathematics-related fields;
- to get a sense of how mathematics is used in decision making in life
- to provide an online curriculum for the USOE Course, Mathematics for Decision Making in Life.

Instructional Videos

The course is based on videos that present and explain the course content, complete with illustrative examples. Each module consists of two to four lessons, each of which contains one to three Instructional Videos. Each lesson starts with an overview, describing in detail the content, providing suggestions of places where the video can be stopped (if viewed in class) to enable clarification and discussion. The overview also contains ideas establishing connections and extensions of the material of the lesson, followed by a brief summary of new vocabulary, formulas and algorithms.

Video PDFs

So that the teacher can quickly go through the content of a video, we have provided, for each video, two pdf documents: one is the text before the video is made, and the second includes the working of problems and side comments that are in the completed video.

Discussion Problems

Included in each lesson is a collection of problems that can serve either as homework or for class work and discussion. Solutions are provided.

Homework Problems

This is a bank of problems available at MyOpenMath that can be used for homework assignments and examinations. When students do the work at home, they can get instant feedback on their work from the MyOpenMath program.

- The student is able to apply mathematics-based skills used in college and career, including reasoning, planning and communication, to make decisions and solve problems in applied situations.
- The student is able to analyze numerical data using a variety of quantitative measures (tables, graphs, statistics) and numerical processes.
- The student analyzes and evaluates risk and return in the context of everyday situations, making decisions based on understanding, analysis and critique of statistical information.
- The student can communicate methods and results in statistical studies and reports.
- The student can model data in a variety of ways, generate predictions and evaluate their validity.
- The student uses mathematical models to represent, analyze and solve problems involving change.
- The student uses mathematical models and analyses to make decisions related to earning, investing and borrowing.
- The student can generate network models to organize data, make decisions and solve problems.

Module L: Logic (Numbers, Sets, Logical Thinking And Counting)

Mathematical Foundation:

Inductive and deductive reasoning

Logic statements and definitions and critical thinking

Relationships in sets and counting

Set operations & Venn diagrams

Dimensional analysis

Application of Mathematics:

Objective 1: Understand and interpret numerical arguments.

- Use Venn Diagrams to represent relationships among sets of objects.
- Use Venn Diagrams to solve counting problems.
- Demonstrate facility in conversion of units in complex situations (i.e., miles per hour to seconds per yard).
- Solve problems by applying multiple unit conversions.

Objective 2: Understand and interpret logical arguments.

- Identify whether an argument is inductive or deductive. Evaluate inductive arguments using strength; evaluate deductive arguments using validity and soundness.
- Use Venn diagrams to demonstrate the relationships among sets in categorical propositions and to determine an argumentâ€™s validity.
- Given a premise and conclusion, identify the logic statement behind the argument (affirming the hypothesis, affirming the conclusion, denying the hypothesis, denying the conclusion) and determine the argumentâ€™s validity.

Mathematical Foundation:

Sequences, arithmetic and geometric, closed form for the general term; Series, arithmetic and geometric, closed form for the sums of terms.

Applications of Mathematics:

Objective 1: Understand the concept of recursive in the context of process.

- Understand that an additive process develops by adding the same new addend.
- Distinguish between the recursive form of the evolution of the process and the closed form.
- Understand that a multiplicative process develops by multiplying by the same new factor.
- Distinguish between the recursive form of the evolution of the process and the closed form.

Objective 2: Know how to sum elements in a sequence, and understand in what contexts this is important.

- Understand and apply the formula for adding any set of consecutive terms of an arithmetic series.
- Understand and apply the formula for adding any set of consecutive terms of a geometric series.

Module LE: Modeling with Functions

Mathematical Foundation:

Linear, exponential, logarithmic and logistic models Sequences, series, and recursive relations

Applications of Mathematics:

Objective 1: Use various mathematical models to understand, explain, and make decisions about real world situations

- Identify whether a context is modeled by linear or exponential change.
- Identify whether a sequence is arithmetic or geometric.
- Given a few members of a sequence (arithmetic or geometric), find a recursive definition of the sequence.
- Find a formula for the general term of a sequence (arithmetic or geometric).
- Find the sum of a finite sequence (arithmetic or geometric).

- Model and solve problems about contexts that have discrete and continuous structure using algebraic tools like tables, algebraic expressions, graphs and charts.
- Use models to predict future or past values and evaluate the effectiveness of the models.

Mathematical Foundation:

Fundamentals of Probability

Independent and Conditional Probability

Calculating Probabilities

Probability & Odds

Expected Value

Introduction to counting

Application of Mathematics:

Objective 1: Use the rules of probability to calculate probabilities of independent and dependent events.

- Construct and interpret tree diagrams and probability distribution tables and use these tools to calculate probabilities.
- Apply addition and multiplication rules (assisted by Venn Diagrams for the addition rules), and the probability of the complement to calculate probabilities.
- Use permutations and combinations to calculate possible numbers of outcomes and find probabilities.

Objective 2: Understand and use probability in a variety of contexts.

- Use predicted outcomes to make decisions about real life situations.
- Given a set of facts, be able to make a reasonable argument about the likelihood of an event.
- Analyze risk and return by using expected values to weigh the possible outcomes of a decision.
- Identify whether the probability in a situation is theoretical, experimental, or subjective probability.

Mathematical Foundation:

Describing data: mean and median, variation and standard deviation

Sampling and measures of central tendency

Bivariate data: correlation, best fit, and causality

Comparing distributions in context

Application of Mathematics:

Objective 1: Use and communicate data in written and quantitative forms.

- Calculate a mean, median, and/or mode of data, describe its spread, and identify outliers.
- Communicate the significance of the mean, median, mode, spread, and/or outliers in a data set to make an argument or decision about the data and its use.
- Discuss shape of the distribution: peaks, symmetry, and skewness.
- Calculate and use measures of variation (standard deviation, box-and- whisker plots, five-number summaries) in order to interpret and use a data set.
- Use and interpret bivariate data to make an argument or decision.
- Given a table, make a scatter diagram, and determine the shape of the data (linear, nonlinear, clustered, random).
- If the data appear to be linear, try to eyeball a best fitting line (using a computer-based tool if available).

Objective 2: Understand and communicate statistical information.

- Report results of statistical studies in both oral and written form including graphical representations.
- Describe strengths and weaknesses of sampling techniques, data and graphical displays, and interpretations of summary statistics.
- Describe the relationship between various distributions and the contexts and or processes to which they relate best.
- Identify uses and misuses of statistical analyses.

Module F: Financial Management

Mathematical Foundation:

Increments as addends or percentages Simple Interest Compound Interest Processes defined by recursion Annuities and Installment Loans

Application of Mathematics:

Objective 1: Determine, represent, and analyze mathematical models for various types of income calculations.

- Understand the terms and conditions related to investments
- Create and analyze mathematical models that represent projected income or growth.
- Select the most advantageous investment strategy given two different sets of conditions.

Objective 2: Determine, represent and analyze mathematical models of various types of loan calculations.

- Understand terms and conditions of loans and how they may vary.
- Create mathematical models to represent loan repayment and use these models to answer questions, in particular, the total amount repaid at any stage during the loan.
- Understand loan payment and/or prepayment/principal payment strategies and how they affect the total interest paid over the lifetime of a loan.
- Use and create amortization tables for a variety of situations.

Mathematical Foundation:

Polygon, circle, segments and sectors Area, perimeter, volume Euclidean motions and scaling

Applications of Mathematics:

Objective 1: Determine, represent and analyze area and perimeter of planar figures.

- Understand the basic properties of area and perimeter of polygons and circle sectors.
- Explore, in context, areas and perimeters and understand the relations to dimension.
- Calculate area and perimeter of polygons, using the definition (in terms of rectangles) and properties of area.
- Make scale drawings of polygonal and circular figures and understand the proportional relation of length and area in the scaled drawing to the length and area in the actual figure.

Objective 2: Understand the role and basics of triangle geometry in planar geometric analysis (in context).

- Obtain facility in the decomposition of complex planar objects into triangles and circle sectors in order to calculate area and perimeter.
- Use triangles in real-life contexts to determine distances in general (neither horizontal nor vertical) directions.
- Estimate, by triangulation, lengths of inaccessible line segments (such as the height of a tree by measuring its shadow and comparing to a known height).
- Explore and use the concepts of pitch, grade and slope.

Objective 3: Understand the relation of volume, area and perimeter in the context of three dimensional objects.

- Understand perimeter as the one-dimensional measure of the boundary of a planar region; understand surface area as the two-dimensional measure of the boundary of a region in 3 dimensions.
- Understand the properties of volume: that they are the same as the properties of area based on squares, but now based on cubes.
- Understand how to calculate the volume, surface area and lengths of edges of various polyhedral objects.

Objective 4: Understand the effect of scaling: in particular, how it affects perimeter, area and volume.

- Understand that rescaling an object amounts to a change in the definition of the unit.
- Realize that if the one-dimensional unit is rescaled by a factor of r, then the 2D unit is rescaled by the facto r 2 and the 3D unit by the factor r 3.

Mathematical Foundation:

Matrices and Matrix Operations

Networks and Adjacency Matrix

Linear Processes

Transition Matrix and Probability Matrix

Algorithms

Applications of Mathematics:

Objective 1: Understand matrix multiplication in context.

- Find paths in a directed graph using the adjacency matrix.
- Use matrices to find paths and Costs in a weighted directed graph.

Objective 2: Understand the use of matrices in linear processes.

- Represent transfer of commodities in a closed system by a matrix, and understand the use of multiplication to determine stability (for example, in a closed economy, the (i,j) entry is the percentage of the wealth of country j that is passed to country i through trade in a given period of time).
- Understand the concept of transition matrix and its use in economics, ecology, scheduling.
- Use probabilities of matrices to calculate probabilities in a sequence of events.

Objective 3: Understand the use of matrices in encryption.

- Understand the use of matrices in encryption: a word consists of a column vector of length n, for some positive integer n; the message is encrypted by multiplying each word by an nxn matrix A, and the encryption is decoded by multiplying by the inverse of A.

Objective 4: Use matrices in Operations Research

- Understand how to use linear programming to optimize a linear objective function on a polygonal figure.
- Use matrices as convenient number arrays to make decisions, as in assignment and ranking situations.