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MTH 350 Learning Targets

Updated 2020-12-15

NOTES/TO DO

Make a list of "named theorems" for each Investigation Standard practice for definitions: State verbatim; give examples; give non-examples; use it to rephrase a statement

Big skills

  • Clear and correct mathematical writing
  • Comprehending definitions:
    • Stating verbatim
    • Construct examples
    • Construct non-examples
    • Use it to rephrase a statement or draw a conclusions
  • Theorems:
    • State verbatim, identify assumptions and conclusions
    • Use the result to draw conclusions give data, or determine that the theorem doesn't apply
  • Proofs:
    • Critical analysis (see Chad Wiley's proof writer workshop using Campuswire)
    • Writing one's own
  • Mathematical reasoning
    • Abstraction
    • Experimentation and conjecture
    • Is this just computational thinking?

Review Targets

Ted's book: https://scholarworks.gvsu.edu/books/24/

  • Given a conditional statement, identify the hypothesis and conclusion.
  • Given a conditional statement to prove, set up the framework (identify what to assume and what to prove) if doing a direct proof, proof by contrapositive, and proof by contradiction.
  • Do a critical analysis of a written proof of a conditional statement.
  • Explain what a "predicate" is.
  • Given a statement to prove by mathematical induction, state the predicate; the base case; the inductive hypothesis; and the proof step.
  • Set theory: Roster vs. set-builder notation; union, intersection, subset, set inclusion, empty set
  • Matrix multiplication and addition
  • Functions: Domain, range, codomain, injective, surjective, bijective
  • Error types: Logical, computational, semantic

Chapter-by-chapter microtargets

Investigation 1

  • State the contents of the sets of natural numbers, whole numbers, and integers and identify these sets by their symbols.
  • State the Axioms of Integer Arithmetic.
  • State the definition of an additive inverse of an integer.
  • Use the Axioms of Integer Arithmetic to prove basic results about integers (such as in Activity 1.6).
  • State the Ordering Axioms of the integers.
  • Use the Ordering Axioms to prove basic results about integers (such as in Theorem 1.7).

Investigation 2

  • State the definition of divides in the context of integers; use the definition to decide if one integer divides another and to rephrase a division relationship.
  • State the Division Algorithm; apply it to identify the quotient and remainder when dividing one integer by another.
  • State the Well-Ordering Principle.
  • Explain line-by-line the proof of the Division Algorithm.
  • Define what it means for two integers to be congruent modulo n; use the definition to decide if two integers are congruent mod n and to rephrase a congruence relationship.
  • State and apply Theorem 2.10.
  • Prove properties of congruence mod n (such as in Activity 2.12).

Investigation 3

  • State the definition of the greatest common divisor of two integers; use the definition of GCD to rephrase a statement about the GCD.
  • State and apply Theorem 3.4, Theorem 3.9 (Bezout's Identity), Theorem 3.10, Corollary 3.11, Theorem 3.14, and Theorem 3.15.
  • Prove results about the GCD of two integers (such as in Theorem 3.4).
  • Use the Euclidean Algorithm to find the GCD of two integers.
  • State the definition of a linear combination of two integers.
  • Explain line-by-line the proof of Theorem 3.10.

Proof of Bezout's Identity? Where is this exactly?

Investigation 4

  • State the definition of prime and composite numbers.
  • State and apply the Fundamental Theorem of Arithmetic.
  • State and apply Euclid's Lemma (both weak and strong forms) and Theorem 4.5.
  • Explain line-by-line the existence portion of the proof of the Fundamental Theorem of Arithmetic.
  • Explain line-by-line the uniqueness portion of the proof of the Fundamental Theorem of Arithmetic.
  • State the definition of the set of even integers and identify this set by its symbol.
  • Define what it means for a number to be "prime in $\mathbb{E}$".
  • Give reasonable definitions of "prime" numbers in number systems other than the integers (such as in Activity 4.10 and 4.11).

Investigation 5

  • Define the congruence class of an integer modulo n; use the definition to write out the congruence class of a specific integer given a specific value of n as a set in roster notation.
  • State and apply the properties of congruence classes listed on page 46.
  • Define the concept of an equivalence relation on a set; decide if a given relation on a set satisfies the definition or not.
  • Define the concept of the equivalence class of an element of a set with respect to a given equivalence relation; use the definition to write out the congruence class of a specific object given a specific equivalence relation.
  • State the definition of the integers modulo n and identify this set by its symbol.
  • Perform addition and multiplication of congruence classes within $\mathbb{Z}_n$$\mathbb{Z}_n$ and make addition and multiplication tables for specific instances of $\mathbb{Z}_n$.
  • Explain line-by-line the proof that addition in $\mathbb{Z}_n$ is associative.
  • Prove or disprove that the analogues of Axioms of Arithmetic hold for $\mathbb{Z}_n$.
  • Define the Cartesian product of two sets and a binary operation on a set; use the definition to list the elements of a Cartesian product.
  • Define what it means for a binary operation on a set to be well-defined; use the definition to determine if a specific binary operation is well-defined.
  • Define the concept of a zero divisor in $\mathbb{Z}_n$; use the definition to decide if an element in $\mathbb{Z}_n$ is a zero divisor and to list all the zero divisors in a specific instance of $\mathbb{Z}_n$.
  • Define the concept of a unit in $\mathbb{Z}_n$; use the definition to decide if an element in $\mathbb{Z}_n$ is a unit and to list all the units in a specific instance of $\mathbb{Z}_n$.
  • Define the concept of a multiplicative inverse of an element in $\mathbb{Z}_n$; use the definition to find the multiplicative inverse of a specific element in $\mathbb{Z}_n$ or decide that the inverse does not exist.
  • Find conditions on $a$ and $n$ that will guarantee that $[a]$ is a unit or that $[a]$ is a zero divisor in $\mathbb{Z}_n$.
  • State and apply Theorem 5.4 and Theorem 5.6.

Investigation 6

  • Define the sets of rational numbers, complex numbers (and the real and imaginary parts of a complex number), and $2 \times 2$ and $n \times n$ matrices over the real numbers; and identify these sets by their symbols.
  • Define the power set of a set and the symmetric difference of two sets; find the power set of a finite set and the symmetric difference of two sets.
  • Explain line-by-line the proof that the symmetric difference operation is associative.
  • Prove or disprove the analogues of the Axioms of Arithmetic for the alternative number systems introduced in this Investigation (as in Activity 6.3).

Investigation 7

  • State the definition of a ring and the seven Ring Axioms; use the Axioms and the definition to decide whether a set with two binary operations is or is not a ring.
  • State the definition of a commutative ring; use the definition to decide whether a ring is commutative or not.
  • State the definition of an identity of a ring and a ring with identity; use the definition to decide if a ring has an identity or not.
  • Give examples and non-examples of rings with different combinations of commutativity and having an identity.
  • State the definition of a multiplicative inverse of a ring element; use the definition to find a multiplicative inverse for a specific ring element or determine that one doesn't exist.
  • State the definition of a unit in a ring; use the definition to determine whether a specific ring element is or is not a unit, and to find all the units of a specific ring.
  • State the definition of a zero divisor in a ring; use the definition to determine whether a specific ring element is or is not a zero divisor, and to find all the zero divisors of a specific ring.
  • Explain line-by-line the proofs of Theorems 7.16, 7.20, 7.21
  • State the definition of an integral domain and a field; use the definitions to decide whether a specific ring is an integral domain or a field (or both or neither).
  • Theorems to state and apply: Theorem 7.3, Theorem 7.5, Theorem 7.10, Theorem 7.11, Theorem 7.15, Theorem 7.16, Theorem 7.17, Theore, 7.19 , 7.20, 7.21, Corollary 7.24, Theorem 7.26

Investigation 8

  • State the recursive definition (8.2) of what it means to add multiple ring elements together and the definition of an integer multiple and integer exponent of a ring element; use the definition to add specific ring elements together and to multiply and exponentiate them by an integer.
  • State the definition of what it means to multiply or exponentiate a ring element by zero or a negative integer; use the definition to compute zero and negative multiples and exponents of specific ring elements.
  • Explain line-by-line the portions of the proof of Theorem 8.5 built in Activity 8.6; the proof of Theorem 8.7; and the proof of Lemma 8.8.
  • Define the characteristic of a ring; use the definition to compute the characteristic of a specific ring.
  • Theorems to state and apply: Theorem 8.5, Theorem 8.7, Lemma 8.8, Theorem 8.12.

Investigation 9

  • State the definition of a subring of a ring; use the definition to decide if a specific subset of a specific ring is a subring of that ring.
  • Use Theorem 9.4 (the Subring Test) to decide if a specific subset of a specific ring is a subring of that ring.
  • Determine which properties of the parent ring are "inherited" by a subring and which ones are not. (Activity 9.7)
  • State the definition of a subfield of a field; an extension of a field; the field extension generated by a subset of a field; and a simple extension of a field; use the definitions to decide wither a subset of a field is a subfield and to list the elements of a field extension.
  • Explain line-by-line the proof of Theorem 9.10.
  • State the definition of the direct sum of two rings; use the definition to list the elements of a direct sum of two specific rings and create addition and multiplicaton tables for the direct sum.
  • Theorems to state and apply: Theorem 9.3, Theorem 9.4, Theorem 9.10, Theorem 9.15, Theorem 9.16

Investigation 10

  • State what it means for a function between two rings to preserve an operation.
  • State the definition of an isomorphism between two rings and what it means for two rings to be isomorphic to each other; use the definition to decide whether a specific function between two specific rings is an isomorphism and whether two specific rings are isomorphic.
  • State the steps necessary to prove that a two rings are isomorphic and that a function between two rings is an isomorphism; also state what it means to say that two rings are not isomorphic.
  • Define the concept of an isomorphism invariant and give a list of isomorphism invariants.
  • Given two rings, decide whether they are isomorphic and then prove it by constructing an isomorphism between them.
  • Theorems to state and apply: Theorem 10.13,

Assessable Learning Targets

Coming as soon as I review the microtargets

Big picture

  • Why do we care about rings?
  • Why should teachers care about this stuff?
  • Practical applications?