Department of Mathematics and Statistics
Transition Course Curricular Unit Descriptions
Unit 1: Modeling Change with Functions: Families of functions including linear, polynomial and exponential.
Sample Lesson 1.3 Exponential Functions
This unit extends and deepens student understanding of linear, polynomial, and exponential functions. These functions are used to model quantitative relationships and data patterns whose graphs are transformations of basic patterns. The families of these functions are developed using translations, reflections, stretching and compression of graphs with connections among symbolic representations of the functions, the corresponding graphical representations, and associated tables of values. Basic linear regression using the least squares criterion is developed with technology. Technology is also used to extend to modeling nonlinear relationships. Students review properties of relationships that are modeled by linear, polynomial, and exponential functions and learn to move fluidly from table to graphic to symbolic representations of these functions. They also learn how to modify symbolic representations of functions to produce new models for data patterns whose graphs are related to the basic functions by vertical and horizontal translations and reflections, and by vertical and horizontal compressing and stretching.
Sample Task
A table of pizza prices presents pizzas offered by two competing pizza companies. The task asks students to figure out how the pricing is determined and which is the better deal. The student has to develop a model for the pricing of pizzas by both companies. Students learn how to use finite differences to determine that the functions are both quadratic, and use the general quadratic to find an equation for each company. In addition students learn how to use technology and quadratic regression to find the graphical representations and compare to find which gives the better deal.
Unit 2: Interpreting Categorical Data: Introduction to probability, two-way frequency tables, conditional probability, and independence.
Sample Lesson 2.4.2 Conditional Probability
This unit develops student understanding of two-way frequency tables, conditional probability and independence, and using data from randomized experiments to compare two treatments. A summary statistic in analyzing categorical data is the frequency or proportion of the sample that falls into each category. Thus the basic mathematics of this unit is proportional reasoning that is critical for college-bound students. Students learn how to use proportional reasoning and percentages in real life situations.
Sample Task
Students are presented with three scatter plots for which they need to determine the strength of correlation. Three samples of student work to answer that question are given, and students have to determine problems with each method, then describe a better method for calculating the correlation for each graph, explaining why that method is better. The three methods include: 1/area of polygon surrounding all points; measuring the distance of each point from the line and summing all the distances, then taking the reciprocal; and, splitting the graph into four equal areas, finding the number of points in each rectangle, adding the numbers of points on each diagonal, and subtracting one diagonal from the other. Students learn how to evaluate potential methods for measuring correlation and the standard method used to calculate correlation.
Unit 3: Statistical Inference: Rules of probability and applications of analysis of data
This unit develops student understanding of discrete probability; expected value; testing a model; simulation; making inferences about the population based on a random sample; margin of error; and how randomization relates to sample surveys, experiments, and observational studies. Topics include review of basic rules and vocabulary of probability, including addition and multiplication rules; independent events and mutually exclusive events; expected value; design and analysis of surveys; and sample selection issues.
Sample Task
Students are provided data from a study showing pre and post cholesterol levels of participants who changed to a vegetarian diet. Students analyze the data, using a statistical package such as in Desmos, to answer the question as to whether change to a vegetarian diet was successful in reducing cholesterol levels. Students have to decide whether to recommend a vegetarian diet to reduce cholesterol levels.
Unit 4 : Voting and Apportionment
“Voting" happens in many contexts, other than politics, in which preferences are expressed (e.g. a job search). Does my vote matter? Are the votes for awards, scholarships or job hiring being counted fairly? How about the fairness of apportionment of representative seats and scholarships? This unit develops student understanding of how the methods of determining the results of a “vote” and apportionment can significantly impact the outcomes. Students learn appropriate mathematical structures to permit a systematic rather than an ad hoc analysis as to whether any election or apportionment rule reliably and appropriately produces outcomes that, arguably, are fair and represent the views of the voters. Students determine whether “paradoxical outcomes" are unlikely anomalies or reasonably prevalent behavior. Topics include determining the winner of an election using various methods such as plurality, Borda count, plurality-with-elimination, pairwise comparison, sequential pairwise and the approval voting methods. Additional topics to decide if an election violates criteria include Arrow’s Impossibility theorem, head-to-head comparison, the majority, the monotonicity, and the irrelevant alternatives criteria. Topics on apportionment of seats include standard, lower and upper quotes, geometric mean, and several methods such as Hamilton, Jefferson, Adams, Webster, Huntington-Hill, and their flaws.
This unit is intended to connect and extend students’ prior knowledge of voting and outcomes. What is the mathematics behind the various counting and apportionment methods? Why do different methods yield a different result? How do you determine which method is most appropriate?
Sample Assignment
A table shows the students ranking of three school trip options. The goal is to decide on the trip that reliably and fairly represents the views of the voters. Students are asked to determine the outcomes using plurality, Borda Count, Plurality-with-elimination and Pairwise comparison methods, compare them, explore their fairness, and make a decision. Students will learn how to use those methods and test if they violate any of the criteria for fairness.
Unit 5: Financial and Business Decision Making: Financial mathematical models
The aim of this unit is to acquire mathematical approaches and tools to make better decisions for real life contexts. A "Problem-Scenario Approach" is used to foster applied learning and understanding of the mathematical concepts. This unit gives students an edge with strong analytic and quantitative skills that are highly in demand nowadays to make good and fair decisions
Decision-Making is central to human activity, and is the process of optimally achieving a given objective. Mathematical modeling is at the heart of the financial and business decision-making process. This unit offers practical concepts on this process with a focus on mathematical model-driven decision support techniques to identify, analyze, interpret, predict, and present results, so as to transfer quantitative information into decisions and explain objectively the reasons for the decision. Techniques include applications of linear and exponential functions, decision trees without probabilities, and expected value. Students implement those tools in practical financial and business scenarios that involve: Understanding the Problem; Constructing a Mathematical Model; Finding a Good Solution; and Communicating the Results.
Sample Assignment
A company conducted a preliminary study to build a condominium complex with three alternative projects. The financial success of the project depends upon the size/number/price of the units and the chance event concerning the demand. The task is to select the size of the new complex that will lead to the largest profit given the uncertainty concerning the demand for the condominiums. Students will learn how to set up a payoff matrix with three decision alternatives and two states of nature, and determine and analyze the best decisions using optimistic, conservative and the minimax regret approaches. In addition students will learn how to develop and use a decision tree and compute expected values for a probabilistic approach
Unit 6: Counting Methods
This unit extends students’ ability to count systematically and solve enumeration problems, developing student understanding of the principles of counting; permutations and combinations; and the Binomial Theorem. Students apply basic principles of counting, including the multiplication rule and the addition rule, to solve complex counting problems. They develop formulas for counting permutations and combinations and analyze the relationships between them. Finally, they investigate Pascal’s triangle and explore connections between Pascal’s triangle and binomial coefficients, leading to the Binomial Theorem
Sample Task
Combinations, binomial coefficients, Pascal’s triangle, and flipping a coin come from seemingly different contexts, but are closely related mathematically. In this task, students work on one of four jigsaw activities – computing combinations of 4 objects, expanding the binomial (x+1)4, finding the 4th row of Pascal’s triangle, or flipping a coin 4 times and counting heads. When they return to their original groups, students see that the numeric answers to each task are the same. Students discuss the connections between the context, which leads to a statement of the Binomial Theorem.
Unit 7: Graph Theory Applications
Sample Lesson 7.2 Graph Theory
This unit develops student understanding of graphs and their properties, with a focus on using graphs to model and solve problems about real world phenomena. Topics include the language and structure of graphs; graph coloring and chromatic number; paths and circuits; Eulerian and Hamiltonian circuits; trees; and minimal spanning trees. Students find the chromatic number of graphs and apply graph coloring to solve scheduling and map coloring problems. They investigate the existence of Eulerian and Hamiltonian circuits in graphs to solve path-finding problems. And they explore the graph structure of trees and find minimal spanning trees to solve optimization problems.
Sample Task
A graph representing the map of a town is given in which the edges of the graph represent roads. The task asks students to determine whether there is a route through town for a snow plow such that the snow plow traverses every road exactly once. Students find an appropriate path through the graph (an Eulerian circuit), but show that no such path exists when one of the roads is removed. Students learn the conditions under which a graph contains an Eulerian circuit and how Eulerian circuits can be used to solve real world problems
Unit 8: Informatics
This unit develops student understanding of the mathematical concepts and methods behind information processing, particularly on the Internet, focusing on issues of access, security, accuracy, and efficiency. Topics include elementary number theory and modular arithmetic; cryptography; and error-detecting and error-correcting codes. Students use modular arithmetic to encrypt and decrypt messages and investigate the security of various encryption schemes. They explore the mathematics behind common error-detecting codes (including ZIP, UPC, and ISBN). And they examine how computer networks use error-correcting codes to increase accuracy and efficiency.
Sample Task
In cryptography, many ciphers rely on modular arithmetic to encrypt and decrypt messages. In this task, students use modular arithmetic to encrypt using an Affine cipher. They investigate which numbers have a multiplicative inverse mod 26 and discover that certain multipliers cannot be used in an Affine cipher. Finally, they use their knowledge of modular arithmetic and multiplicative inverses to decrypt messages that were encrypted using an Affine cipher.
Unit 9: Representations in 3-D
This unit deepens students’ understanding of three-dimensional shapes including cubes, spheres, cylinders, tetrahedra and cones; and explores their real-life applications. The unit investigates foundations of mathematical modeling in today’s world using 3D shapes and their representations. Dynamic geometry software like GeoGebra is used to represent and visualize 3 dimensions. Finally, the unit explores the connection between the representation of 3D objects in 2D and 3D printing, and the connection between the visualization of 2D objects in 3D and animation for movies and video gaming.
Sample Task
This lesson discusses 3D printing and connects it to 2D representation of 3D objects. During the lesson, students learn how to match representations of three-dimensional objects with two-dimensional cross-sections. Students create a prototype of printing their 3D object using their 2D cross-sections
Unit 10 : Symmetries and Tessellations
Sample Assignment
Students explore different shaped tiles to investigate which regular polygonal shapes can be used to tile the plane. Then students will produce a conjecture for the best regular polygon for bees to use to create a honeycomb and compare that conjecture with the real honeycomb. Students investigate properties of hexagons and tessellations to learn why this shape is the most efficient design for a honeycomb compared with other regular polygons.