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(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 with
Size of each group
Find:
The number of groups
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 with
The number of groups
Find:
the size of each group (how many)
If a ∈ W then a - 0 = a = 0 - a
let a = 1 then 1 - 0 = 1 but 0 - 1 += -1
a - 0 = a 0 - a ≠ a
If a ∈ W and b ∈ W, then (a-b) = (b - a)
Let a = 1, b = 3 then a - b = -2 b - a = 2
a - b ≠ b - a
If a ∈W and b ∈ W, then (a-b) ∈ W
Let a = 1, b = 3 then 1 - 3 ∈ W
a+ b (m5) = 0
a + ____ (m5) = a
Move or transpose
3+4 = 4+3
Mirror images
3(five) + 4 (five) = 12 (five)
3(m5) + 4 (m5) = 2 (only single digits 0,1,2,3,4)
Equivalent (three bars) or congruent
Look at remainder, not the divisor
(scant notes/substitute)
a ∈ W, b ∈ W, c ∈ W, then a * (b+c) = ab + ac
The effect of multiplying by zero
a ∈ W, then 0 * a = 0
One of the numbersmust be zero
multiply by the same number, you get the identical thing (by 1)
a ∈ W, then 1 * a = a
1 is the identity element or multiplicative element
a ∈ W, b ∈ W, c ∈ W, then (a*b) * c - a * (b * c)
(changes the group)
a ∈ W, b ∈ W then ab = ba
(changes the order)
a ∈W, b ∈ W then a * b ∈ W
Context is characterized by finding all possible pairings between 2 or more sets of objects
(cross product to come up with ordered pairs)
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.
characterized by repeatedly adding a quantity of continuous quantities. Measured quantity like time, distance, etc. a specified number of times.
Know: Size of groups
Find: number of groups
Know: Number of groups/partition
Find: size of partition
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 mistakes
Specficially when converting between 2 forms.
Use mult/div within problem.
leaving mixed numbers
ADV: no converstion, less work.
Strengthens the idea of place value
More consistent
Dis; Regrouping of fractions, particularly - what is the whole?
Structure/Patterns in Addition Tables
Diagonally 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.
If a ∈W, then a+ 0 = a = 0 + a
The identity element for addition is zero.
coming together, pair up
if a € W, b € W, and c € W, then
(a+b) + c = A+ (b+c) = (a+c) +b
If a € W and b € W then a+b = b+a
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 x
Of whole numbers:
if a € W and b € W then a+b € W
Be careful of when you need to think about a different "whole" (yep, I still call it switching the whole.)
Any number that can be expressed as the quotient of 2 integers. Includes repeating decimals etc
Part to whole meaning - most common
Division - in building math sophistication we remove the symbol. The fraction bar eventually becomes an alternative too for indicating division
Copies of a unit fraction (supposed to accompany part/whole.
Ration - involves comparing 2 separate things.
Continuous sets - characterized by combining of 2 continous quantities, flowing quantities that we measure. i.e. time, distance, volume, area
Discrete Sets - characterized by combining 2 sets of counted quantities i.e. blocks, markers.
An odd number is one more/less than an even number.
That is, 2n+1 (2n-1 is also valid)
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.
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.
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
