Foundation of Elementary Mathematics Exam 1

The Penny of Death

If n= 1,4,7,10… make your opponent go first, and then after each selection by your opponent do the opposite.

If n= 2,5,8,11… go first and take 1. Then after each selection by your opponent do the opposite.

If n= 3,6,9,12… go first and take 2. Then after each selection by your opponent do the opposite.

Inductive Reasoning

Coming to a conclusion based on observing evidence, data, or specific examples. It often includes recognizing patterns and predicting future outcomes based on them.

Deductive reasoning

Coming to a conclusion based on logic.

Conjecture

A hypothesis or guess about what is true, often made on inductive (evidence) reasoning.

Counterexample

An example that shows that a conjecture is false

Mathematical Proof

A deductive argument that establishes the truth of a claim.

Recursive Thinking

Where one relates a value to it’s previous value

Functional Thinking

Where one relates a value to the value of another variable.

Base 10

The value of a digit depends entirely on its position within the number, with each position representing a specific power of 10.

Claim

Solution or results for a conjecture to be true

Scope of a Claim

Universal statements- ALWAYS true

Existential statement- Something exists, sometimes happens, or is possible.

How do I write a good proof?

Focus on explaining

Attend to the scope (specific examples and counter examples)

Make valid claims

Support all claims (every claim should be supported with an explanation)

Attend to the burden of proof (Identify a set of outcomes that would be sufficient to argue the case and provides an explanation that addresses each one)

=

Means equivalent, or balanced. Not a command to solve something. An equation is like a balanced scale

Graphic methods

To think of an equation as a function or type of relationship between values x and y.

Numeric amethods

Methods involving a technique to estimate the solution by ‘smart’ or iterative quessing and checking (like a table)

Algebraic Methods

Involve manipulating the equation to isolate the number represented by x using only the operations addition, subtraction, multiplication, division, and the extraction fo roots.

3x = y+x

2y=x+x

How many xs = z?

3x=z

6 = y+ t

2t=y

How much would it cost to buy a t?

How many to buy a y?

y=4

t=2

Natural Numbers

Numbers found in nature (1,2,3…)

Whole Numbers

Counting numbers (0,1,2,3….)

Integers

All numbers (…-2,-1,0,1,2…)

Elements

Parts of a set

Set

A collection of definite, distinguishable objects. Held within curly brackets {}

Empty Set

{} or ∅

What sets are ALWAYS subsets?

The empty set and the set itself.

Subset

Mini sets. Don’t have to include all elements of a set, but all elements of a subset must be in the original set.

When are two sets considered equal?

When they have exactly the same elements.

When are two sets considered equivalent?

When they both have the saem AMOUNT of elements.

When is a set considered ‘finite’?

When it has some whole number of elements.

When is a set considered infinite?

A set that is not finite.

Universal set (U)

A set of all the elements that are under conisderation at a particular atime.

set A complement or A^c

The set of all elements in U which are NOT in A.

Union

Think of the word “unite.” The union of two sets A and B ( A U B), is the set of all the elements in A together with all of the elements in B.

Intersection

Overlap. The intersection of two sets A and B (A n B), is the set of all elements that are in both A and B.

Cardinal

Describes the size of a set, basic counting numbers (1,2,3,4…)

Ordinal

Desbricribes the position, so first, second, third, fourth…

Vectors

On a number line, numbers are represented by vectors (distances with direction)

Set model

Where numbers are represented by a collection of objects.

Rational number

A number that can be written as a fraction, a “ratio” of two integers.

Criteria 1 of a Binary Operation

Defined- x*y must be defined for EVERY ordered pair of elements from A.

Ex: the operation of division is not defined on the set {0, 1, 2, 3} because 3/0 is not defined. (Has no output)

Criteria 2 of a Binary Operation

Uniquely Defined - Only ONE answer can be given for a X and Y in a given order. (Can’t have more than 1 output)

Criteria 3 of a Binary Operation

Closed - it must be an element of the set.

Ex: if the set is {1,2,3}, the operation of addition would not be closed as 2 + 3= 5 and 5 is not in the set.

Commutative Property

If x * y = y * x for all choices of x and y in the set.

Associative Property

If (x*y)*z = x*(y*z)

Identity element

if x*e = x and e*x=x

Y is the inverse x

Whatever number takes that number back to the identity. So y*x=e and x*y=e

How can you tell if a table is commutative from a table?

If the table is symmetrical.

Additive inverse

The number we need to add to x to get zero. Written as -x

Subtraction

A - B is defined to be the number you add to b to get a.