Math 157

Definition: A whole number a is divisible by a whole number b, if and only if, there exists a third whole number c such that a = bc.Often stated as b divides a and the notation is bIa

Assuming b divides a, then:- b is a factor of a- a is a multiple of b- b is a divisor of a- a is divisible by b.

What make a number even? A number is even if it is a multiple of 2. That is, it is 2 times some number or 2n.

An odd number is one more/less than an even number.That is, 2n+1 (2n-1 is also valid)

Discrete Sets - characterized by combining 2 sets of counted quantities i.e. blocks, markers.

Continuous sets - characterized by combining of 2 continous quantities, flowing quantities that we measure. i.e. time, distance, volume, area

Part to whole meaning - most commonDivision - in building math sophistication we remove the symbol. The fraction bar eventually becomes an alternative too for indicating divisionCopies of a unit fraction (supposed to accompany part/whole.Ration - involves comparing 2 separate things.

Any number that can be expressed as the quotient of 2 integers. Includes repeating decimals etc

Be careful of when you need to think about a different "whole" (yep, I still call it switching the whole.)

If a is an element of set x and b is an element of set x, then a plus b is an element of set xOf whole numbers:if a € W and b € W then a+b € W

If a € W and b € W then a+b = b+a

coming together, pair upif a € W, b € W, and c € W, then (a+b) + c = A+ (b+c) = (a+c) +b

If a ∈W, then a+ 0 = a = 0 + aThe identity element for addition is zero.

The traditional algorithm is efficient and saves space. It is suited to the resources available.

Structure/Patterns in Addition TablesDiagonally numbers appear in bands. All possible ways add 2 numbers and get 10.

The traditional algorithm is efficient and saves space. It is suited to the resources available.

converting mixed numbers:ADV: no regrouping of fractions needed.Looks like part-whole context Process is similar to mult/div.Dis: Larger numbers, more opportunity for math mistakesSpecficially when converting between 2 forms.Use mult/div within problem.leaving mixed numbersADV: no converstion, less work.Strengthens the idea of place valueMore consistentDis; Regrouping of fractions, particularly - what is the whole?

Know: Number of groups/partitionFind: size of partition

Know: Size of groupsFind: number of groups

characterized by repeatedly adding a quantity of continuous quantities. Measured quantity like time, distance, etc. a specified number of times.

characterized by a product of two numbers representing the sides of a rectangular region such that the product represents the number of unit sized squares within the rectangular region.

Context is characterized by finding all possible pairings between 2 or more sets of objects(cross product to come up with ordered pairs)

a ∈W, b ∈ W then a * b ∈ W

a ∈ W, b ∈ W then ab = ba(changes the order)

a ∈ W, b ∈ W, c ∈ W, then (a*b) * c - a * (b * c)(changes the group)

multiply by the same number, you get the identical thing (by 1) a ∈ W, then 1 * a = a1 is the identity element or multiplicative element

The effect of multiplying by zeroa ∈ W, then 0 * a = 0One of the numbersmust be zero

a ∈ W, b ∈ W, c ∈ W, then a * (b+c) = ab + ac

Equivalent (three bars) or congruentLook at remainder, not the divisor(scant notes/substitute)

3(five) + 4 (five) = 12 (five)3(m5) + 4 (m5) = 2 (only single digits 0,1,2,3,4)

Move or transpose3+4 = 4+3Mirror images

a + ____ (m5) = a

a+ b (m5) = 0

If a ∈W and b ∈ W, then (a-b) ∈ WLet a = 1, b = 3 then 1 - 3 ∈ W

If a ∈ W and b ∈ W, then (a-b) = (b - a)Let a = 1, b = 3 then a - b = -2 b - a = 2a - b ≠ b - a

If a ∈ W then a - 0 = a = 0 - alet a = 1 then 1 - 0 = 1 but 0 - 1 += -1 a - 0 = a 0 - a ≠ a

Characterized by distributing a given quantity amoung a specified number of groups (partitions) and determining the size (or amount) in each group ( partition)Know: Quantity we are starting withThe number of groupsFind:the size of each group (how many)

(Repeat subtraction)characterized by using a given quantity to creat groups (partitions) of a specified size (amount) and determining the number of partitions (groups) that are formed.Know:Quantity we are starting withSize of each groupFind:The number of groups