# Courses of Study : Mathematics

Logical Reasoning
The validity of a statement or argument can be determined using the models and language of first order logic.
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
1. Represent logic statements in words, with symbols, and in truth tables, including conditional, biconditional, converse, inverse, contrapositive, and quantified statements.

Unpacked Content Evidence Of Student Attainment:
Students:
• Represent propositional statements using statement variables and logical operators.
• Construct a truth table for a statement.
• Construct conditional, biconditional, converse, inverse contrapositive, and quantified statements.
Teacher Vocabulary:
• Proposition
• Statement variables
• Logical operators
• Truth table
• Negation
• Conditional statement
• Hypothesis/antecedent
• Conclusion/consequent
• Converse statement
• Inverse statement
• Contrapositive statement
• Biconditional statement
• Equivalent statements
Knowledge:
Students know:
• How to determine if a simple statement is true or false.
Skills:
Students are able to:
• Construct a truth table for propositions with a variety of operators.
• Write a proposition using logical operators and statement variables such as p and q.
• Write the converse, inverse, contrapositive and biconditional of a conditional statement using logical operators and statement variables.
Understanding:
Students understand that:
• A conditional statement's validity is based on the validity of its components.
• Truth tables must contain all possible assignments of true and false for each component.
• A statement is either true or false.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
2. Represent logic operations such as and, or, not, nor, and x or (exclusive or) in words, with symbols, and in truth tables.
Unpacked Content Evidence Of Student Attainment:
Students:
• Determine the appropriate logical operators needed to write a compound statement with symbols and statement variables.
• Determine the truth value for each component of a compound statement and organize in a truth table.
Teacher Vocabulary:
• Compound statement
• Negation
• Conjunction
• Disjunction
Knowledge:
Students know:
• A statement is either true or false.
• A truth table must include every possible assignment of true and false for each component of a compound statement.
Skills:
Students are able to:
• Construct a truth table for a compound statement.
• Represent compound statements using statement variables and logical operators.
Understanding:
Students understand that:
• The validity of the simple statements that make up a compound statement determine the compound statement's validity.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
3. Use truth tables to solve application-based logic problems and determine the truth value of simple and compound statements including negations and implications.

a. Determine whether statements are equivalent and construct equivalent statements.

Example: Show that the contrapositive of a statement is its logical equivalent.
Unpacked Content Evidence Of Student Attainment:
Students:
• Solve an application-based logic problem using a truth table.
• Recognize when two statements have the same truth value.
Teacher Vocabulary:
• Equivalent statements or logical equivalence
Knowledge:
Students know:
• How to construct a truth table from a given logic statement.
Skills:
Students are able to:
• Represent an application-based logic problem as a statement(s) using logical operators and statement variables.
• Construct a truth table to determine a solution to a logic problem.
Understanding:
Students understand that:
• Complex situations including logic problems can be modeled using truth tables.
• Statements are logically equivalent if they have the same truth value for every possible assignment of true and false for each component.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
4. Determine whether a logical argument is valid or invalid, using laws of logic such as the law of syllogism and the law of detachment.

a. Determine whether a logical argument is a tautology or a contradiction.

Unpacked Content Evidence Of Student Attainment:
Students:
• Recognize if an argument is valid.
Teacher Vocabulary:
• Tautology
• Law of syllogism
• Law of detachment/modus ponens
Knowledge:
Students know:
• How to construct a truth table from a given logic statement.
Skills:
Students are able to:
• Construct valid arguments.
• Identify the validity of arguments.
Understanding:
Students understand that:
• Truth tables can be used to construct a valid argument or to determine the validity of an argument.
• In order for an argument to be valid, the form of the argument must be valid.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
5. Prove a statement indirectly by proving the contrapositive of the statement.

Unpacked Content Evidence Of Student Attainment:
Students:
• Construct the contrapositive of a statement.
• Construct a proof of a conditional statement by assuming that the negation of the consequent is true and deduce from that assumption that the negation of the antecedent is true.
Teacher Vocabulary:
• Contrapositive
• Proof by contrapositive
• Indirect proof
• hypothesis/antecedent
• Conclusion/consequent
Knowledge:
Students know:
• A contrapositive is formed by negating both the hypothesis/antecedent and conclusion/consequent and reversing the direction of inference.
• Proofs can be constructed by assuming that a hypothesis/antecedent is true and deducing that the conclusion/consequent is true.
Skills:
Students are able to:
• Write the contrapositive statement for a conditional statement such as a property of integers or other mathematical properties.
• Construct a logical argument to prove a statement (such as a property of integers) is true by proving the contrapositive.
Understanding:
Students understand that:
• The contrapositive of a statement is logically equivalent to a statement.
• A statement can be shown to be true by the laws of logic by proving that its contrapositive is true.
Diverse Learning Needs:
Complex counting problems can be solved efficiently using a variety of techniques.
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
6. Use multiple representations and methods for counting objects and developing more efficient counting techniques.
Note: Representations and methods may include tree diagrams, lists, manipulatives, overcounting methods, recursive patterns, and explicit formulas.
Unpacked Content Evidence Of Student Attainment:
Students:
• Use tree diagrams or other systematic listing methods to determine the number of total possible outcomes in an application-based problem.
Teacher Vocabulary:
• Tree diagram
• Recursive pattern
• Explicit formula
Knowledge:
Students know:
• Tree diagrams can be used to systematically list all possibilities for a given set of constraints.
Skills:
Students are able to:
• List all possible outcomes for a given set of constraints
Understanding:
Students understand that:
• Tree diagrams and other systematic methods can be used to count objects but may not be the most efficient method when counting large quantities.
• Recursive and explicit formulas can be developed from examining patterns in tree diagrams and systematic lists.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
7. Develop and use the Fundamental Counting Principle for counting independent and dependent events.

a. Use various counting models (including tree diagrams and lists) to identify the distinguishing factors of a context in which the Fundamental Counting Principle can be applied.

Example: Apply the Fundamental Counting Principle in a context that can be represented by a tree diagram in which there are the same number of branches from each node at each level of the tree.
Unpacked Content Evidence Of Student Attainment:
Students:
• Given a real-world context problem, determine if the the Fundamental Counting Principle can be applied, use various counting models to count using a variety of different context parameters.
Teacher Vocabulary:
• Fundamental counting principle
• Independent events
• Dependent events
• Tree diagram
• Branches
• Node
Knowledge:
Students know:
• How to construct a tree diagram.
Skills:
Students are able to:
• Count the number of events when given a variety of constraints/parameters when the Fundamental Counting Principle can be applied.
Understanding:
Students understand that:
• The Fundamental Counting Principle can be applied in contexts where an ordered list of events occur and there are a ways for the first event to occur, b ways for the second event to occur so the number of ways of the ordered sequence of events occuring is axb.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
8. Using application-based problems, develop formulas for permutations, combinations, and combinations with repetition and compare student-derived formulas to standard representations of the formulas.

Example: If there are r objects chosen from n objects, then the number of permutations can be found by the product [n(n-1) ... (n-r)(n-r+1)] as compared to the standard formula n!/(n-r)!

a. Identify differences between applications of combinations and permutations.

b. Using application-based problems, calculate the number of permutations of a set with n elements. Calculate the number of permutations of r elements taken from a set of n elements.

c. Using application-based problems, calculate the number of subsets of size r that can be chosen from a set of n elements, explaining this number as the number of combinations "n choose r."

d. Using application-based problems, calculate the number of combinations with repetitions of r elements from a set of n elements as "(n + r - 1) choose r."
Unpacked Content Evidence Of Student Attainment:
Students:
• Use formulas for permutations, combinations, and permutations with repetition in appropriate contexts.
• For student-derived or standard formulas, explain each part of the formula contributes to counting the number of permutations or combinations, with or without repetition.
Teacher Vocabulary:
• Permutations
• Combinations
Knowledge:
Students know:
• How to use tree diagrams or other counting models .
Skills:
Students are able to:
• Calculate the number of permutations or combinations for a real-world context.
Understanding:
Students understand that:
• Permutation is an ordered selection of r distinct objects from a set of n objects.
• A combination is a selection of a set of r distinct unordered objects from a set of n objects.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
9. Use various counting techniques to determine probabilities of events.

Unpacked Content Evidence Of Student Attainment:
Students:
• Use combinations, permutations, tree diagrams, or other systematic listing methods to determine the number of total possible outcomes in an application-based problem.
• Use combinatorial reasoning to determine the probability of events.
Teacher Vocabulary:
• Tree diagrams
• Combinations
• Permutations
• Sample size
• Independent events
• Dependent events
• Mutually exclusive (disjoint) events
Knowledge:
Students know:
• Probability.
• Permutations and Combinations.
• Tree diagrams.
Skills:
Students are able to:
• Use a tree diagram or other systematic listing method to determine the number of possible outcomes in an application-based problem.
• Use combinations and permutations to count the number of possible outcomes in an application -based problem.
• Determine the probability of an event.
Understanding:
Students understand that:
• Solving probability in a discrete setting requires first applying combinatorial reasoning and counting techniques to determine the size of the event of interest.
• Some events consist of a sequence of (or partition into) smaller events that may be independent or dependent.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
10. Use the Pigeonhole Principle to solve counting problems.

Unpacked Content Evidence Of Student Attainment:
Students:
• Solve problems using the Pigeonhole Principle to solve problems such as determining the number of socks needed to be pulled out of drawer to guarantee a match if there are 5 distinct pairs of socks in the drawer.
Teacher Vocabulary:
• Pigeonhole principle
Knowledge:
Students know:
• How to construct counting models.
Skills:
Students are able to:
• Solve a combinatorial problem using the Pigeonhole principle.
Understanding:
Students understand that:
• If m>n and there are m pigeons (or any object) and n pigeonholes (or any position), there must be at least one pigeonhole with more than one pigeon.
Diverse Learning Needs:
Recursion
Recursion is a method of problem solving where a given relation or routine operation is repeatedly applied.
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
11. Find patterns in application problems involving series and sequences, and develop recursive and explicit formulas as models to understand and describe sequential change.

Examples: fractals, population growth
Unpacked Content Evidence Of Student Attainment:
Students:
• -Use inductive counting methods to find recursive patterns and explicit formulas.
Teacher Vocabulary:
• Difference equation
• Recursive process
• Recursive formula
• Sequences
• Series
Knowledge:
Students know:
• How to use inductive counting methods such as lists.
Skills:
Students are able to:
• Use inductive counting methods to collect data for conjecturing.
• Find recursive formulas from collected data.
• Develop explicit formulas.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
12. Determine characteristics of sequences, including the Fibonacci Sequence, the triangular numbers, and pentagonal numbers.

Example: Write a sequence of the first 10 triangular numbers and hypothesize a formula to find the nth triangular number.
Unpacked Content Evidence Of Student Attainment:
Students:
• Find the next term in a sequence such as the Fibonacci sequence, the triangular numbers, and pentagonal numbers.
• Describe the recursion pattern that builds a sequence.
• Hypothesize a formula to find the nth term in a sequence.
Teacher Vocabulary:
• Recursive process
• Recursive formula
• Triangular numbers
• Pentagonal numbers
• Fibonacci sequence
• Closed Formula
Knowledge:
Students know:
• How to recognize a pattern.
Skills:
Students are able to:
• Identify the pattern in a sequence.
• Explain why a pattern occurs.
Understanding:
Students understand that:
• The recursion process can be applied to many situations.
• A sequence lists the solutions of a set of related problems.
• Formulas can be hypothesized by identifying how the problems are related.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
13. Use the recursive process and difference equations to create fractals, population growth models, sequences, and series.

Unpacked Content Evidence Of Student Attainment:
Students:
• Create fractals using technology or other tools based on a recursive formula.
• Use population growth models to find population sizes at various times.
Teacher Vocabulary:
• Difference equation
• Recursive process
• Recursive formula
• Fractals
• Population growth models
• Sequences
• Series
Knowledge:
Students know:
• How to recognize a pattern.
Skills:
Students are able to:
• Apply recursive formulas in real world contexts.
Understanding:
Students understand that:
• Models such as population growth should be recognized as recursively developed models.
• The recursion process can be applied to many situations.
• A sequence lists the solutions of a set of related problems.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
14. Use mathematical induction to prove statements involving the positive integers.

Examples: Prove that 3 divides 22n - 1 for all positive integers n; prove that 1 + 2 + 3 + ... + n = n(n + 1)/2; prove that a given recursive sequence has a closed form expression.
Unpacked Content Evidence Of Student Attainment:
Students:
• Use the process of mathematical induction to prove simple statements about positive integers.
Teacher Vocabulary:
• Proof by mathematical induction
Knowledge:
Students know:
• How to find equivalent expressions.
Skills:
Students are able to:
• Show that a statement is true for the first case, generally n=1.
• Show that a statement is true for n=k+1 if it is assumed that the statement is true for n=k.
Understanding:
Students understand that:
• Proof by induction is a way of proving statements that includes two steps.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
15. Develop and apply connections between Pascal's Triangle and combinations.

Unpacked Content Evidence Of Student Attainment:
Students:
• Identify the connection between Pascal's triangle and combinations
Teacher Vocabulary:
• Pascal's Triangle
• Recursion
• Combinations
Knowledge:
Students know:
• How to calculate combinations.
Skills:
Students: Use recursive pattern to construct Pascal's triangle.
• Compare combinations to each row of Pascal's triangle to identify each row as the set of all combinations for a given set of objects.
• Understanding:
Students understand that:
• Each row in Pascal's triangle is the number of combinations of N take r where N is the row of the triangle starting with N=0 and r is the entry in the row from left to right.
Diverse Learning Needs:
Networks
Complex problems can be modeled using vertex and edge graphs and characteristics of the different structures are used to find solutions.
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
16. Use vertex and edge graphs to model mathematical situations involving networks.

a. Identify properties of simple graphs, complete graphs, bipartite graphs, complete bipartite graphs, and trees.

Unpacked Content Evidence Of Student Attainment:
Students:
• Model a real-world situation using a vertex and edge graph.
Teacher Vocabulary:
• Graph
• Vertex
• Edge
• Network
• Complete graph
• Bipartite graph
• Tree
Knowledge:
Students know:
• How to construct a vertex and edge structure
Skills:
Students are able to:
• Determine what a vertex and an edge would represent in modeling a real-world problem.
• Construct simple graphs, complete graphs, bipartite graphs, complete bipartite graphs, and trees..
Understanding:
Students understand that:
• Both the vertex and edge is used to represent some part of a real-world problem.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
17. Solve problems involving networks through investigation and application of existence and nonexistence of Euler paths, Euler circuits, Hamilton paths, and Hamilton circuits. Note: Real-world contexts modeled by graphs may include roads or communication networks.

Example: show why a 5x5 grid has no Hamilton circuit.

a. Develop optimal solutions of application-based problems using existing and student-created algorithms.

b. Give an argument for graph properties.

Example: Explain why a graph has a Euler cycle if and only if the graph is connected and every vertex has even degree. Show that any tree with n vertices has n - 1 edges.
Unpacked Content Evidence Of Student Attainment:
Students:
• Identify a path or circuit (Euler or Hamilton) in a graph.
• Identify characteristics of graphs that cannot contain a Hamilton circuit or path.
• Identify characteristics of graphs that cannot contain an Euler circuit or path.
• Develop an algorithm for finding an Euler path and Euler circuit.
• Develop optimal solutions of application-based problems.
Teacher Vocabulary:
• Degree of a vertex
• Graph
• Bipartite graph
• Grid (a type of bipartite graph)
• Vertex
• Edge
• Circuit (Euler, Hamilton)
• Path (Euler, Hamilton)
• Algorithm
Knowledge:
Students know:
• How to make systematic lists to solve problems.
Skills:
Students are able to:
• Create a graph that models a given situation.
• Apply an algorithm to find a Hamilton or Euler circuit or path in a graph.
Understanding:
Students understand that:
• Graphs can be used to model real world problems and Hamilton and Euler circuits and paths can provide solutions to such problems.
• An Euler circuit cannot exist in a graph with any odd degree vertices.
• An Euler path cannot exist in a graph without exactly two odd degree vertices.
• No known good algorithm has been established for finding a Hamilton path or circuit since no necessary and sufficient conditions for the existence of a Hamilton path or circuit have been identified.
• The graph must be connected in order for a Hamilton or Euler path or circuit to exist.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
18. Apply algorithms relating to minimum weight spanning trees, networks, flows, and Steiner trees.

Example: traveling salesman problem

a. Use shortest path techniques to find optimal shipping routes.

b. Show that every connected graph has a minimal spanning tree.

c. Use Kruskal's Algorithm and Prim's Algorithm to determine the minimal spanning tree of a weighted graph.
Unpacked Content Evidence Of Student Attainment:
Students:
• Use shortest path techniques to find optimal shipping routes.
• Find a minimum weight spanning tree using various algorithms including Kruskal's and Prim's.
• Identify properties of Steiner points.
• Find the Steiner points that produce a Steiner tree using various tools including technology.
Teacher Vocabulary:
• Spanning tree
• Minimum weight spanning tree
• Network
• Flow
• Kruskal's algorithm
• Prim's algorithm
• Steiner tree
• Steiner points
Knowledge:
Students know:
• Graphing procedures and properties.
Skills:
Students are able to:
• Model a problem using flows in networks.
• Use technology or other tools to construct Steiner points.
• Apply minimum weight spanning tree algorithms.
Understanding:
Students understand that:
• A spanning tree of a graph is the smallest subgraph.
• There are n-1 edges in a spanning tree of a graph with n vertices.
• Various algorithms are efficient methods for finding minimum weight spanning trees of a graph and shortest paths in a graph.
• Steiner points of a graph are vertices added to create a shortest spanning tree which connects the original vertices, using Euclidean distance as edge weights.
• Steiner points have degree 3, and the 3 edges form angles of 120 degrees.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
19. Use vertex-coloring, edge-coloring, and matching techniques to solve application-based problems involving conflict.

Examples: Use graph-coloring techniques to color a map of the western states of the United States so that no adjacent states are the same color, determining the minimum number of colors needed and why no fewer colors may be used; use vertex colorings to determine the minimum number of zoo enclosures needed to house ten animals given their cohabitation constraints; use vertex colorings to develop a time table for scenarios such as scheduling club meetings or for housing hazardous chemicals that cannot all be safely stored together in warehouses.
Unpacked Content Evidence Of Student Attainment:
Students:
• Use vertex coloring and edge coloring techniques to solve application-based problems modeled using a graph.
• Provide explanations for why no fewer colors may be used to color a graph.
Teacher Vocabulary:
• Vertex coloring
• Matching techniques
• Conflict graphs
• Odd wheel graph
• Proper coloring
Knowledge:
Students know:
• Graphing procedures and properties.
Skills:
Students are able to:
• Model application-based problems that may be solved using graph colorings.
• Color the edges or vertices of a graph using the least number of colors so that no two adjacent vertices or edges are colored the same.
• Interpret the coloring of the graph in terms of a solution for an application-based problem, such as scheduling committee meetings (vertex colorings) or class scheduling (edge-colorings).
• Identify structures in a graph that require a minimum number of colors for a proper coloring.
Understanding:
Students understand that:
• -Techniques are used to minimize colors needed to color the vertices (edges) of a graph so that no two adjacent vertices (edges) are colored the same. -Real-world problems such as scheduling and conflict can be modeled with graphs and solved using the minimization of the number of colors.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
20. Determine the minimum time to complete a project using algorithms to schedule tasks in order, including critical path analysis, the list-processing algorithm, and student-created algorithms.

Unpacked Content Evidence Of Student Attainment:
Students:
• Model the tasks of a project with a graph and use the graph to find the minimum amount of time (critical path) needed to complete the project.
• Use various algorithms such as critical path analysis, the list-processing analysis, and student-created algorithms to find a critical path.
Teacher Vocabulary:
• Graphs
• Critical paths
• List
• processing algorithm
Knowledge:
Students know:
• Graphing procedures and properties.
Skills:
Students are able to:
• Model tasks of a project in a graph.
• Identify critical paths using various algorithms.
Understanding:
Students understand that:
• Graphs can be used to model sequential tasks in a project.
• Critical paths identify the tasks that must be performed as soon as possible in order to minimize the time taken to complete the project.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
21. Use the adjacency matrix of a graph to determine the number of walks of length n in a graph.
Unpacked Content Evidence Of Student Attainment:
Students:
• Use a matrix to represent the number of walks of different lengths in a graph.
Teacher Vocabulary:
• Walk
• Matrix
Knowledge:
Students know:
• How to form graphs.
• How to determine walks and paths.
• How to multiply matrices.
Skills:
Students are able to:
• Use a graph to create a matrix that shows the number of walks between any two vertices.
• Use matrices to determine the number of walks of various lengths.
Understanding:
Students understand that:
• Adjacency matrices can be used to determine the number of walks between any two vertices of varied lengths and is especially useful for calculating the number of walks when simple counting becomes too cumbersome.
Diverse Learning Needs:
Fairness and Democracy
Various methods for determining a winner in a voting system can result in paradoxes or other issues of fairness.
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
22. Analyze advantages and disadvantages of different types of ballot voting systems.

a. Identify impacts of using a preferential ballot voting system and compare it to single candidate voting and other voting systems.

b. Analyze the impact of legal and cultural features of political systems on the mathematical aspects of elections.

Examples: mathematical disadvantages of third parties, the cost of run-off elections
Unpacked Content Evidence Of Student Attainment:
Students:
• Compare election methods and identify strengths and weaknesses of each.
Teacher Vocabulary:
• Ranked choice voting or preferential ballot voting
Knowledge:
Students know:
• Basic understanding of election methods.
Skills:
Students are able to:
Understanding:
Students understand that:
• There are a variety of voting systems other than those most frequently used systems and may provide advantages or disadvantages as compared to our current system.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
23. Apply a variety of methods for determining a winner using a preferential ballot voting system, including plurality, majority, run-off with majority, sequential run-off with majority, Borda count, pairwise comparison, Condorcet, and approval voting.

Unpacked Content Evidence Of Student Attainment:
Students:
• Determine the outcomes of using various voting methods such as Borda, Runoff, Majority winner, Plurality Winner, and Condorcet
Teacher Vocabulary:
• Ranked choice voting or preferential ballot voting
• Plurality winner
• Majority winner
• Runoff method
• Simple majority
• Sequential runoff (instant runoff) method
• Borda count
• Condorcet method
• Approval voting
Knowledge:
Students know:
• Basic voting methods such as single choice ballots.
Skills:
Students are able to:
• Interpret data from ranked-choice voting ballots or summarized in preference schedules.
• Use data to determine an election winner using a variety of methods.
• Compare and contrast the methods and their results.
Understanding:
Students understand that:
• Various election methods can be used to achieve a group decision from the preferences of the individuals of the group.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
24. Identify issues of fairness for different methods of determining a winner using a preferential voting ballot and other voting systems and identify paradoxes that can result.

Example: Arrow's Theorem
Unpacked Content Evidence Of Student Attainment:
Students:
• Compare results after using a variety of voting methods and identify paradoxes or other issues related to fairness.
• Form arguments for or against methods and results based on issues of perceived fairness.
Teacher Vocabulary:
• Arrow's fairness conditions
• Arrow's Theorem
Knowledge:
Students know:
• How to determine election winners using a variety of methods.
Skills:
Students are able to:
• Discuss the fairness of a voting method based on Arrow's conditions (Identify how a method may violate one or more of Arrow's conditions using specific voter preference data)
Understanding:
Students understand that:
• Commonly used group ranking methods are flawed.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
25. Use methods of weighted voting and identify issues of fairness related to weighted voting.

Example: determine the power of voting bodies using the Banzhaf power index

a. Distinguish between weight and power in voting.
Unpacked Content Evidence Of Student Attainment:
Students:
• Identify situations in which some voters may have more power than others and discuss how power might be distributed more equitably.
Teacher Vocabulary:
• Weighted voting
• Power index
• Voting Coalition
• Winning coalition
• Simple majority
Knowledge:
Students know:
• How to determine a simple majority.
Skills:
Students are able to:
• Determine each voting body's power based on their voting weight using the Banzhaf power index.
• Identify situations that result in unequal distribution of power among voters.
Understanding:
Students understand that:
• Power distribution can vary in weighted-voting situations.
• High individual voting strength may not result in a high power index.
Diverse Learning Needs:
Fair Division
Methods used to solve non-trivial problems of division of objects often reveal issues of fairness.
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
26. Explain and apply mathematical aspects of fair division, with respect to classic problems of apportionment, cake cutting, and estate division. Include applications in other contexts and modern situations.

Unpacked Content Evidence Of Student Attainment:
Students:
• Develop student-invented methods for dividing objects including objects that cannot be meaningfully divided.
• Use established methods for dividing objects such as estates.
Teacher Vocabulary:
• Fair division
• Continuous division
• Discrete division
Knowledge:
Students know:
• How to divide.
Skills:
Students are able to:
• Explain and use ways to divide objects and argue how their proposed method could be considered fair.
Understanding:
Students understand that:
• In some cases of division, fairness can be achieved through agreement on method by parties involved.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
27. Identify and apply historic methods of apportionment for voting districts including Hamilton, Jefferson, Adams, Webster, and Huntington-Hill. Identify issues of fairness and paradoxes that may result from methods.

Unpacked Content Evidence Of Student Attainment:
Students:
• Use population data to apportion objects using a variety of methods.
• Identify paradoxes that result in some methods of apportionment.
Teacher Vocabulary:
• Ideal ratio
• Quota
• Hamilton method
• Jefferson method
• Arithmetic mean
• Geometric mean
• Webster method
• Huntington-Hill method
• Quota rule
Knowledge:
Students know:
• Apportionment is a method of dividing based on population.
Skills:
Students are able to:
• Calculate an ideal ratio for diving objects based on populations.
• Use the ideal ratio to determine exact district quotas and apply a variety of apportionment methods to determine apportionment of discrete objects such as representative seats.
• Use a variety of apportionment methods to adjust the divisor and calculate subsequent quotas.
• Identify specific examples of paradoxes or violation of the quota rule.
Understanding:
Students understand that:
• Some methods result in unfair paradoxes and may favor larger or smaller districts.
• Apportionment methods are used to divide discrete objects when they cannot be divided exactly proportional to the populations.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
28. Use spreadsheets to examine apportionment methods in large problems.

Example: apportion the 435 seats in the U.S. House of Representatives using historically applied methods
Unpacked Content Evidence Of Student Attainment:
Students:
• Compare and contrast apportionment methods with larger datasets efficiently using spreadsheets.
Teacher Vocabulary:
• Truncate
• Quota
• Arithmetic mean
• Geometric mean
Knowledge:
Students know:
• How to apply various methods of apportionment with smaller data sets.
Skills:
Students are able to:
• Determine and use formulas for spreadsheet cells for determining parts of the apportionment process including calculating quotas, truncated quotas, mean and geometric mean of upper and lower quotas, and final apportionments using various methods (Hill, Webster, Hamilton, etc.)
Understanding:
Students understand that:
• Apportionment methods can be applied efficiently on large data sets using spreadsheets.
Diverse Learning Needs:
Information Processing
Effective systems for sending and receiving information include components that impact accuracy, efficiency, and security.
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
29. Critically analyze issues related to information processing including accuracy, efficiency, and security.

Unpacked Content Evidence Of Student Attainment:
Students:
• Identify examples of data processing in real-world contexts.
• Identify problematic data processing issues that exist in the real-world contexts such as digital security breaches.
Teacher Vocabulary:
• Cryptography
• Error-detecting codes
• Error-correcting codes
• Data compression
Knowledge:
Students know:
• Electronic transfer of information such as email are susceptible to breaches.
Skills:
Students should be able to: Give examples of information processing where accuracy, efficiency or security may be an issue.
Understanding:
Students understand that:
• Cryptography is used in online settings to keep information secure.
• Error-detecting and error-correcting codes can be used to detect and correct errors that may occur when information is read or transmitted electronically.
• Data compression methods are used to transmit large amounts of data efficiently.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
30. Apply ciphers (encryption and decryption algorithms) and cryptosystems for encrypting and decrypting including symmetric-key or public-key systems.

a. Use modular arithmetic to apply RSA (Rivest-Shamir-Adleman) public-key cryptosystems.

b. Use matrices and their inverses to encode and decode messages.
Unpacked Content Evidence Of Student Attainment:
Students:
• Use a cryptosystem to encrypt and decrypt small messages.
Teacher Vocabulary:
• Modular arithmetic
• Cipher
• Encryption
• Decryption
• Symmetric-key cryptography
• Public-key cryptography
• RSA cryptosystem
Knowledge:
Students know:
• How to multiply matrices and find inverses.
Skills:
Students are able to:
• Carry out modular arithmetic procedures.
• Use a variety of ciphers to encrypt and decrypt moderate-sized messages.
Understanding:
Students understand that:
• Symmetric-key cryptography such as substitution ciphers or public-key cryptosystems can be used to encrypt and decrypt messages.
• Cryptosystems are used to ensure the privacy and authenticity of information.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
31. Apply error-detecting codes and error-correcting codes to determine accuracy of information processing.

Unpacked Content Evidence Of Student Attainment:
Students:
• Determine if an error in a code can be detected or corrected.
• Use a method such as check digit to determine if a code contains an error.
Teacher Vocabulary:
• Check digits
• Information digits
• Barcodes
• UPC codes
• Binary digits or bits
• Substitution error
• Transposition error
• Maximum likelihood decoding
• Hamming distance, minimum distance
Knowledge:
Students know:
• How to perform modular arithmetic.
• Binary numbers
Skills:
Students are able to:
• Determine if a code such as a zip code or a UPC code contains an errors.
• Determine if a code can be corrected.
• Identify situations where errors should be corrected or just detected.
Understanding:
Students understand that:
• Data that is transferred can contain errors and codes can be used to detect errors.
• Error correcting codes can be used to increase the likelihood of accuracy.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Applications of Finite Math All Resources: 0
32. Apply methods of data compression.

Example: Huffman codes
Unpacked Content Evidence Of Student Attainment:
Students:
• Use a method of data compression to recode information with fewer bits.
Teacher Vocabulary:
• Variable-length codes
• Prefix (or prefix-free) codes
• Huffman codes
• Huffman trees
Knowledge:
Students know:
• How to construct a binary tree with vertices and edges
Skills:
Students are able to:
• Use variable length codes to recode data with shorter codes for the more frequently used characters.
• Use a data compression code to decode.
• Construct a Huffman code for a given set of characters and their frequencies.
Understanding:
Students understand that:
• Data compression is accomplished by using shorter binary strings for commonly used characters.
Diverse Learning Needs: