Midterm 1 Flashcards
What are Polyas 4 steps
- Understand the problem- read it, pick out what you know, figure out what its asking
- Devise a plan- draw diagram, look for patterns, guess & check, solve a simpler problem, work backwards, use variable
- Carry out the plan
- Look back- make sure your answer answered the question being asked, check your work, work backwards, evaluate your strategy
- come up with a related problem
Set definition
a group of objects, called elements
what does A = B mean regarding sets
the sets are the same
1-1 correspondance
each element in sat A can be matched with one, and only one element in set b
subset
every element of A is also an element of B
finite
can be counted with an end
infinite
never ends
union (U)
the set of elements that bring to A or B or Both
Intersection
the elements common to A and B
Complement
The elements of the universal set that aren’t in A
Difference of sets
the set of all elects in A that are not in B
Cartesian Product of sets
set of all ordered pairs made from elements in each
N
the number of all ordered pairs made. Multiply the total number of elements in each set by each other
Roman Numerals
I=1 V=5 X=10 L=50 C=100 D=500 M=1000 Can only subtract from 2 letters up the ladder (I,C,X) 4=IV, 9=1X, 99=XCIX, 40=XL, 90=XC, 400=CD, 900=CM
Hindu-Arabic Number system
Ours- 0,1,2,3,4,5,6,7,8,9
4 Properties of addition
- Closure property- the sum of any 2 whole numbers is a whole number
- Commutative property- a+b=b+a
- Associative property- (a+b)+c=a(b+c)
- Identify property- unique whole number such that a+0=0+a=a
Closure Property (+)
the sum of any two whole numbers is a whole number
Commutative Property (+)
a+b = b+a
Associative Property (+)
(a+b)+c = a+(b+c)
Identify Property (+)
There is a unique whole number (usually zero) such that a+0=0+a=a
Addition of whole numbers
let a and b be any whole numbers. if A and B are disjoint sets with a=n(A) and b=n(B), then a+b=n(AUB)
The Take away approach
let a & b be whole numbers and A & B are sets such that, a=n(A), B=n(B) then a-b=n(A-B)
Missing addend approach
let a & b be any whole numbers. then a-b=c if and only if a=b+c
2 Subtraction approaches:
The take away approach (If I had 5 cookies and Mum ate 2 how many do i have now?)
The Missing addend approach (I have 4 dollars but need 9 dollars to buy a sandwich how much more do i need?)
Repeated addition approach (x)
axb=b+b+b+……b+b (a amount of times)
Rectangular Array Approach (x)
axb is the number of elements in an array having a as rows and b as collums
Cartesian Product Approach (x)
if a=n(A) and b=n(B) then axb=n(AxB) (tree diagram)
3 Multiplication Approaches:
- Repeated addition approach
- Rectangular array approach
- Cartesian product approach
Closure Property (x)
the product of two whole numbers is a whole number
Commutative property (x)
ab = ba
Associative property (x)
(ab)c = a(bc)
Identity Property (x)
The number 1 is a unique whole number, 1xa = ax1 = a
Distributive property of multiplication over addition
a(b+c) = ab + ac
Property Zero (x)
ax0 = 0xa = 0
Distributive property of multiplication over subtraction
a(b-c) = ab - ac
7 Properties of Multiplication
- Closure
- Commutative
- Associative
- Identity
- Distributive multiplication over addition
- Distributive multiplication over subtraction
- Zero
Sharing Division
make a specified number of groups and share items between them. Ex. 20 kids, 4 teams, how many in each group?
Measurement Division
make groups of a specified size and see how many groups you get. ex. 20 kids, 4 per team, how many groups?
Division missing factor approach
if a and b are whole numbers and b does not = 0, then a divided by b =c if and only if a = b x c
Dividend
a (being divided)
Divisor
b (doing the dividing)
quotient/missing factor
c
Property of Zero (division)
if a does not = 0, then 0 divided by a = 0
The division algorithm
if a and b are any whole numbers with b not = 0, then there exist unique whole numbers q (quotient) and r (remainder) such that a = bxq + r, where 0 is < or = to r which is < b
Division Repeated-substraction approach
keep repeating the division number how ever many times possible until you’re left with a remainder
Transitive Property of less than
if a<b></b>
Addition property of less than
if a<b></b>
Multiplication property of less than
if a<b></b>
3 properties of less than
Addition, Multiplication, Transitive
Compatible (friendly) numbers
can reorder, look for numbers that add together to make ‘easy’ numbers (10’s)
Additive compensation
borrow from one number to give to another to make the addition easier
equal additions method
increase each number by the same amount to make subtraction easier
multiplicative compensation
regroup the multiplication to get easier numbers
