Mathematics Benchmarks, Grades K-12

Correlations to the Secondary Mathematics Benchmarks

KEY:

Middle School Course 1 = MS 1; Course 2 = MS 2; One-Year Advanced Course = MS A
High School Integrated Course 1 = Int 1; Course 2 = Int 2; Course 3 = Int 3
High School Traditional Algebra I = Alg I; Geometry = Geo; Algebra II = Alg II
Achieve's Algebra II End-of-Course Exam Core = EOC A2 Core; Module = EOC A2 Mod
ADP = American Diploma Project
Tasks = Tasks
part = partial; opt = optional

Discrete Mathematics

D.A.1 Sets and Boolean algebra.

Expectation	MS 1	MS 2	MS A	Int 1	Int 2	Int 3	Alg I	Geo	Alg II	EOC A2 Core	EOC A2 Mod	ADP	Tasks
a. Know the concepts of sets, elements, empty set, relations (e.g., belong to), and subsets, and use them to represent relationships among objects and sets of objects.				E.1			B.1
b. Perform operations on sets: union, intersection, complement.				E.1			B.1
c. Create and interpret Venn diagrams to solve problems.				E.1			B.1
d. Identify whether a given set is finite or infinite; give examples of both finite and infinite sets.				E.1			B.1

D.B.1 Permutations and combinations.

Expectation	MS 1	MS 2	MS A	Int 1	Int 2	Int 3	Alg I	Geo	Alg II	EOC A2 Core	EOC A2 Mod	ADP	Tasks
a. Determine the number of ways events can occur using permutations, combinations, and other systematic counting methods.				F.3			A.3				R1.a	L4.1
b. Interpret and simplify expressions involving factorial notation.				F.3			A.4				R1.a	L4.1

D.B.2 Discrete graphs.

Expectation	MS 1	MS 2	MS A	Int 1	Int 2	Int 3	Alg I	Geo	Alg II	EOC A2 Core	EOC A2 Mod	ADP	Tasks
a. Construct and interpret decision trees.				F.2				A.3			R1.c	L4.5
b. Create and interpret network graphs.				F.2				A.3					22
c. Construct and interpret flow charts.				F.2				A.3

D.B.3 Iteration and recursion.

Expectation	MS 1	MS 2	MS A	Int 1	Int 2	Int 3	Alg I	Geo	Alg II	EOC A2 Core	EOC A2 Mod	ADP	Tasks
a. Analyze and interpret relationships represented iteratively and recursively.						C.1	D.1 part, D.2 part				I2.a
b. Generate and describe sequences having specific characteristics.						C.2	D.2				I1.f, I2.a	J1.7*
c. Demonstrate the effect of compound interest, decay, or growth using iteration.						D.3	D.3				I1.f, I2.b	J5.6

D.C.1 Algorithms.

Expectation	MS 1	MS 2	MS A	Int 1	Int 2	Int 3	Alg I	Geo	Alg II	EOC A2 Core	EOC A2 Mod	ADP	Tasks
a. Identify and give examples of simple algorithms.	A.7		PK.A.8								I2.a
b. Analyze and apply algorithms for searching, for sorting, and for solving optimization problems.				C.3			B.4						27

D.C.2 Mathematical reasoning.

Expectation	MS 1	MS 2	MS A	Int 1	Int 2	Int 3	Alg I	Geo	Alg II	EOC A2 Core	EOC A2 Mod	ADP	Tasks
a. Use correct mathematical notation, terminology, syntax, and logic.				C.1			B.2					K1, MR3, MR4
b. Distinguish between inductive and deductive reasoning.				C.2			B.3					MR1
c. Explain and illustrate the role of definitions, conjectures, theorems, proofs, and counterexamples in mathematical reasoning.				C.1 part, C.2 part				B.1 part, B.2 part				K1, K1.1, MR3	1, 6
d. Make, test, and confirm or refute conjectures using a variety of methods.				C.2			B.3 part	B.2 part				MR1, MR3	21

D.C.3 Propositional logic.

Expectation	MS 1	MS 2	MS A	Int 1	Int 2	Int 3	Alg I	Geo	Alg II	EOC A2 Core	EOC A2 Mod	ADP	Tasks
a. Use and interpret relational conjunctions ("and," "or," "not"), terms of causation ("if . . . then") and equivalence ("if and only if").				E.2			B.2
b. Describe logical statements using terms such as assumption, hypothesis, conclusion, converse, and contrapositive.				C.1				B.2
c. Recognize and avoid flawed reasoning, including, but not limited to, "Since A → B, therefore B → A."				C.2				B.3
d. Recognize syllogisms, tautologies, and circular reasoning and use them to assess the validity of an argument.				C.2				B.3

D.E.1 Quantitative applications.

Expectation	MS 1	MS 2	MS A	Int 1	Int 2	Int 3	Alg I	Geo	Alg II	EOC A2 Core	EOC A2 Mod	ADP	Tasks
a. Identify and apply the quantitative issues underlying voting, elections, and apportionment.
b. Know and use methods of fair division and negotiation strategies.				C.3

D.E.2 Sequences and series.

Expectation	MS 1	MS 2	MS A	Int 1	Int 2	Int 3	Alg I	Geo	Alg II	EOC A2 Core	EOC A2 Mod	ADP	Tasks
a. Know and use subscript notation to represent the general term of a sequence and summation notation to represent partial sums of a sequence.						C.3 opt					I1.a, I1.b	J1.7*, MR4
b. Derive and apply the formulas for the general term of arithmetic and geometric series.						C.3 opt					I1.a	J1.7*, MR4
c. Derive and apply formulas to calculate sums of finite arithmetic and geometric series.						C.3 opt					I1.b	J1.7*
d. Derive and apply formulas to calculate sums of infinite geometric series whose common ratio r is in the interval (-1, 1).						C.3 opt					I1.c	J1.7*
e. Model, analyze, and solve problems using sequences and series.						C.3 opt					I1.f, I2.b	J5.6

D.E.3 Recursive equations.

Expectation	MS 1	MS 2	MS A	Int 1	Int 2	Int 3	Alg I	Geo	Alg II	EOC A2 Core	EOC A2 Mod	ADP	Tasks
a. Convert the recursive model for discrete linear growth (A₁ is given and A_n+1 = A_n + d for n > 1, d a constant difference) to a closed linear form (A_n = a+d(n – 1)).											I1.d	J1.7*
b. Convert the recursive model of discrete population growth (P₁ is given and P_n+1 = rP_n, for n > 1, r a constant growth rate) to a closed exponential form (P_n = ar^n-1).											I1.c	J1.7*
c. Analyze, define and calculate sequences that are neither arithmetic nor geometric using recursive methods.											I2.a

D.E.4 Digital codes.

Expectation	MS 1	MS 2	MS A	Int 1	Int 2	Int 3	Alg I	Geo	Alg II	EOC A2 Core	EOC A2 Mod	ADP	Tasks
a. Interpret common digital codes (e.g., zip codes, universal product codes (UPCs), and ISBNs on books) and identify their special characteristics.
b. Understand, evaluate, and compare how error detection and error correction are accomplished in different common codes.
c. Identify characteristics of common forms of data compression (e.g., mp3, jpeg, and gif).
d. Analyze the concepts underlying public-key encryption and digital signatures that enable messages to be transmitted securely.

D.E.5 Mathematical induction.

Expectation	MS 1	MS 2	MS A	Int 1	Int 2	Int 3	Alg I	Geo	Alg II	EOC A2 Core	EOC A2 Mod	ADP	Tasks
a. Analyze and describe how mathematical induction rests on the definition of whole numbers and explain how proof by mathematical induction establishes a proposition.												MR1
b. Identify common theorems that can be proved by mathematical induction and explain why this method of proof works for these theorems.
c. Use mathematical induction to prove simple propositions.

D.E.6 Proof by contradiction.

Expectation	MS 1	MS 2	MS A	Int 1	Int 2	Int 3	Alg I	Geo	Alg II	EOC A2 Core	EOC A2 Mod	ADP	Tasks
a. Analyze and explain how proof by contradiction can be used to establish a proposition.												MR3
b. Identify examples of theorems for which an indirect argument is useful and assess whether an indirect argument is useful to prove a particular theorem.
c. Use an indirect argument to prove a result.