Math 110 Final Exam
5 NCTM Process Standards -problem solving
-connections
-communication
-reasoning & proof
-representation
What does NCTM stand for and when was it National Council of Teachers of Mathematics (1989) (2000)
made/updated?
Problem Solving Have students try multiple different strategies; open ended problems, non-routine
problems
Connections Show why it's useful in the real world. Why do they need to know this?
-connections within math
-connections with other subjects
-connections to the real world
Communication Student discussion, have students write sentences;
How would you explain today's concept to a student that was absent? (students
answer)
Reasoning & Proof WHY we do problems the way we do; have students make conjectures; have
students discover something; have students support what they say
Representation Represent ideas in as many different ways that you can think of
Inductive Reasoning specific -> general
ex. eyeglasses some people wear eyeglasses, everyone wears eyeglasses
(conjecture is false)
Deductive Reasoning general -> specific
ex. if right triangle, then a2+b2=c2 (general proven fact)
5^2+12^2=c2 -> 13=c
Conjecturing Inferring or predicting from incomplete evidence; making a statement
Investigation uses reasoning to find answers
ex. odd+odd=even
even+even=even
even+odd=odd
Counterexample an example that disproves a statement
Negation ~A; NOT; S: Today is Thursday; N: Today is NOT Thursday
conditional statement If A, then B (-_>) (double bar arrow means implies)
Stuvia 2026-2027
, Math 110 Final Exam
converse If B, then A
ex. S- If today is TH, then tomorrow is F.
Converse- If tomorrow is F, then today is TH.
Inverse If not A, then not B
ex. S- If today is TH, then tomorrow is F.
Inverse- If today is not TH, then tomorrow is not F.
Contrapositive If not B, then not A
ex. S-If today is TH, then tomorrow is F.
Contrapositive- If tomorrow is not F, then today is not TH.
everyday language to conditional language Every dog is a mammal -> If it is a dog, then it is a mammal
not all vs. all not S- All Americans are Pennsylvanians
Negation- Not all Americans are Pennsylvanians
WRONG negation- all Americans are not Pennsylvanians
Roster method Listing each item in a set within braces {0,1,2,3,4}
Informal word description {whole #s less than 5}
set builder notation {x| xEW, x<5}
set a collection of objects called elements
element an object in the set
E (symbol not really an E) is an element of
cardinality the number of objects in a set shown as n(A)=4
Empty set set with no objects; {} ; always a subset of a set; zero with the line- null
equal sets {q,r,s}={q,r,s}={r,q,s} SAME SET
Equivalent sets sets with the same cardinality (same # of elements) ex. {a,b,c} ~ {dog, cat, fish}
Stuvia 2026-2027
5 NCTM Process Standards -problem solving
-connections
-communication
-reasoning & proof
-representation
What does NCTM stand for and when was it National Council of Teachers of Mathematics (1989) (2000)
made/updated?
Problem Solving Have students try multiple different strategies; open ended problems, non-routine
problems
Connections Show why it's useful in the real world. Why do they need to know this?
-connections within math
-connections with other subjects
-connections to the real world
Communication Student discussion, have students write sentences;
How would you explain today's concept to a student that was absent? (students
answer)
Reasoning & Proof WHY we do problems the way we do; have students make conjectures; have
students discover something; have students support what they say
Representation Represent ideas in as many different ways that you can think of
Inductive Reasoning specific -> general
ex. eyeglasses some people wear eyeglasses, everyone wears eyeglasses
(conjecture is false)
Deductive Reasoning general -> specific
ex. if right triangle, then a2+b2=c2 (general proven fact)
5^2+12^2=c2 -> 13=c
Conjecturing Inferring or predicting from incomplete evidence; making a statement
Investigation uses reasoning to find answers
ex. odd+odd=even
even+even=even
even+odd=odd
Counterexample an example that disproves a statement
Negation ~A; NOT; S: Today is Thursday; N: Today is NOT Thursday
conditional statement If A, then B (-_>) (double bar arrow means implies)
Stuvia 2026-2027
, Math 110 Final Exam
converse If B, then A
ex. S- If today is TH, then tomorrow is F.
Converse- If tomorrow is F, then today is TH.
Inverse If not A, then not B
ex. S- If today is TH, then tomorrow is F.
Inverse- If today is not TH, then tomorrow is not F.
Contrapositive If not B, then not A
ex. S-If today is TH, then tomorrow is F.
Contrapositive- If tomorrow is not F, then today is not TH.
everyday language to conditional language Every dog is a mammal -> If it is a dog, then it is a mammal
not all vs. all not S- All Americans are Pennsylvanians
Negation- Not all Americans are Pennsylvanians
WRONG negation- all Americans are not Pennsylvanians
Roster method Listing each item in a set within braces {0,1,2,3,4}
Informal word description {whole #s less than 5}
set builder notation {x| xEW, x<5}
set a collection of objects called elements
element an object in the set
E (symbol not really an E) is an element of
cardinality the number of objects in a set shown as n(A)=4
Empty set set with no objects; {} ; always a subset of a set; zero with the line- null
equal sets {q,r,s}={q,r,s}={r,q,s} SAME SET
Equivalent sets sets with the same cardinality (same # of elements) ex. {a,b,c} ~ {dog, cat, fish}
Stuvia 2026-2027
