MAT156_Jaramillo
Order of Operations
Order of operations:
When you are performing order of operations always remember this: PEMDAS : Please Excuse My Dear Aunt Sally!
* PEMDAS means : Parenthesis, Exponents, Multiplication and Division, Addition and Subtraction.
Important Note:
This is the order that operations are solve in order to get the right answer. Mutiplication and division can be perfromed at the same time just make sure you do it from left to right, and do the same for addition and subtraction.
Examples:
a) (2+5)2 = b) 3(5-3) x 2 -1
(7)2 = 14 3(2) x 2 -1
6 x 2 -1
12 - 1 = 11
c) 12 + 2(3-2^2)(4-(2-5))
12 + 2(3-4)(4-(-3))
12 + 2 (-1)(-7)
12 + 2 (-7)
12 + (-14) = -2
Irrational Numbers
Irrational number: An irrational number is any real number that is not rational, that is not a/b.
Examples:
Pi, e, √2 (this are irrational numbers)
Square Roots:
√2 this is a square root.
√4 = 2 because 2^2=4
√36 = 6 because 6^2 =36
*When you look for the sqaure root of a number you are basically looking for a number that when multiplied by itself = that number inside the √
Sometimes you have square roots that are not the product of a number multiplied by itself ->
For example:
√32 *Even if there is not a number that when
multiplied by itself = 32 we can still simplify!
√32 = √8 x 4 (4 is a perfect sqaure root, 2)
2√8 = 2√4 x 2 ( after squaring the 4 factorize the 8)
2 x 2V2 = 4√2 (after factorizing the 8 and simplifying
you will achieve your final result)
Exponents & Scientific Notation
Scientific Notation
Check out the cool website to your right, and learn the importance of scientific notation!!!!
Exponent:
*The exponent represents the amount of times a number must be multiplied by itself. For example a^x. a has to be multiplied by itself x times.
Examples:
3^2= 3 x 3 = 9
2^4 = 2 x 2 x 2 x 2 = 16
Note:
*If the exponent is 0, then the problem equals 1.
6^0 = 1
* If the exponent is a - number this is what happens:
x^-2 = 1
x^2
3^-4 = 1
3^4
WARNING!!
You must know that -5^2 is different than (-5)^2 .
In -5^2 the "-" is separate from the 5 so the answer would
be = - 25.
In (-5)^2 the parenthesis is used to establish that the "-"
is part of the number 5 so (-5)^2 = 25
Ratios & Proportions
Proportions:
Proportions work much more deeper than ratios. Even though you can look at proprtions as comparisons as well, propertions plant questions that refer to expectations.
For example: We have 7 dogs and 3 cats, how many people
would be expected to have dogs if 12 people
have cats?
cats 3 = 12
dogs 7 x (x stands for "?" the unknown amount)
Perform:
You must apply cross multiplying:
3 = 12 = 3x = 12(7) ( divide both sides by 3 )
7 x 3 3 (12 can be divided into 3 4 times)
x = 4(7)
x = 28
Ratios:
Ratios are compaerisons between different things.
For example: There are 7 women and 5 men in a classroom
the ratio is:
# women to # men = 7:5 , you can also express it as 7/5
Integers
Integers are rational numbers such as : -5,-4,-3,-2,-1,0,1,2,3,4,5 and so on.
Dividing with The Charge Field Method:
When you think of count sets and count elements
a) -10 / 2 =
- - - - - -
- - - - (* How many elements can you put into
= 2 sets ?)
- -
- -
- - = you can put - 5 elements into 2 sets
- -
- -
so, -10 / 2 = -5
b) -10 / -2 =
- - - - - (* Now the question changes, into how many
- - - - - sets can we equally fit the ten elements?)
=
-- -- --
--> We can fit -2 elements into 5 sets.
-- --
So, -10 / -2 = 5
IMPORTANT:
c) 10 / -2 =
+++++
+++++ ( This equation cannot be done visually because
there is not (-) sets, you can perform it as a
multiplication; for example: )
-2 x ? = 10
-2 x -5 = 10 !!
Multiplying with The Charge field Method:
a) 3 x 4 = ( You mus have 3 stes of 4)
++++ ++++
++++ = 12 (the total of +)
b) 3 x -2 = (Must have 3 sets of -2)
- - - - - - = -6 (the total of -)
c) -3 x 2= ( We cannot have negative sets of something
so we must use the commutattive property
2 x -3 = to reverse the operation and make it look
the following way : )
(Now solve) ------> Refer to the previous example.
d) -3 x -2= (you must add zeros, the same way you
learned with subtraction)
+- +- +-
+- +- +-
You must cancel now the (-)three sets of (-)two.
You will be left will:
+ + +
+ + + = +6, so -3 x -2 = 6
In order to learn how to add and subtract integers using the chip method take a look at the video in order for you to learn visually.
In order to test your abilities on how to add and subtract with the chip method, as you learned in the previous video, go to the website on your right!
7 Decimals
Rational Numbers have 3 forms:
-Fractions
-Decimals
-Percentage
Decimals, Fractions and Percentages
0.1 ( the 0= ones but the 1= tenth)
Note: Always remebeber that after the period numbers start of Tens not Ones.
EXAMPLES:
0 . 1 2 3 4 5
- 0 = Ones
- 1 = Tenths
- 2 = Hundreths
- 3 = Thousandths
- 4 = Ten Thousandths
- 5 = Hundred Thousandhts ( and so on)
Decimal as a Fraction
12
1000 = 0.012 ( make sure the last number end in
Thousanths that is the purpose of
the 1000 )
It is very important to know how to convert decimals into fractions and percents and vice-versa.
Example: Terminated Numbers:
0.4 ----> 0.4 x 100 So, 40 which is 40%
100=(percent) 100
0.4 is the same as four tenths so 4 = 2
10 5
Repeated Numbers:
- Repeated numbers are numbers that dont terminate such as 0.33333
0.3(r) [ (r)= Reepated]
10n = 3.3(r)
- n = 0.3(r)
9n = 3.0 = 1 (1. Now divide 9 in both sides)
9 9 3 (2. Simplify)
1 x 100
3 = 100 = 33 1
100 3 3
6 Rational Numbers as Fractions
Rationals Numbers: a a: Numerator
b b: Denominator
Equivalent or Equal Fractions:
1 2 4
2 6 12
This are all equivalent fractions
Simplying Fractions:
60 = 6 x 10 = 6
210 21 x 10 21
TEST YOURSELF BY GOING TO THE LINK TO YOUR RIGHT!
Do you want to learn how to multiply and divide fractions!!!?
CHECK THE VIDEO OUT!!
www.youtube.com/watch?v=B7MtFQW7i_I
Subtracting Fractions:
- With the same denominator:
2 - 1 = 1
2 2 2
- With different denominators:
3 - 2
2 4 ( (look for the LCM of the denominator
or also known as Least Common
Denominator LCD)
2 = 2, 4
4= 4
3 (2) - 2 = 6 - 2 = 4 1 = 1
2 (2) 4 4 4 4 (simplify) 1
Adding Fractions:
-With the same denominator:
2 + 3 = 5
6 6 6
-With different denominator:
1 + 2
2 4 (look for the LCM of the denominator
or also known as Least Common
Denominator LCD)
2 = 2, 4
4= 4
1 (2) + 2 = 2 + 2 = 4
2 (2) 4 4 4 4
= 4 = 4 x 1 = 1
4 (simplify) 4 x 1 1
5 Integers and Number Theory
Greatest Common Factor and Least Common Multiple
Greatest Common Factor:
_____ ______
6 rods 8 rods
_ _ _ --------
2 rods 2 rods so, therefore the gratest common facto of
6 and 8 is 2
GCF of 12: 1, 2, 3, 4, 6, 12
Least Commom Multiple: of 6: 6, 12, 18, 24, 30, 36
Prime and Composite Numbers
- Prime Numbers: 2,3,5,7,11 ....
- Composite Numbers: 4, 6, 8, 9, 10.....
Prime Factorization: 260= 26 x 10= (12 x 13)(2 x 5)= 2 x 2 x 5 x 13 = 2^2 x 5 x 13
Factor Tree: 260 260
/\ /\
26 10 or 2 52
/\ /\ /\
2 13 2 5 2 26
/\
2 13
GO TO THE LINK FOR MORE EXAMPLES!
5-2 Divisibility Test
Check the Website!
3 Whole Numbers and Their Operations
3-5 Estimation
Computational Estimation
1) Front-end with adjustment:
423 1) Add front-end digits:
338 4 + 3 + 5 = 12
+561
2) Place value: 1200
3) Adjust:
61 + 38 ≈ 100 and
20 + 100 = 120
4) Adjust Estimate:
1200 + 120 = 1320
2) Grouping to nice numbers:
23 - 23, 32, 64 make about 100
39 - 64 and 39 make about 100
32
64 Therefore, the sum is about 100 + 100 = 200
+49
3) Clustering:
6200 - Estimate the "average" about 6000
5842
6512 - Multiply the avergae by the number of the
5521 values (5) so, 5 x 6000 = 30,000
+6319
4) Rounding:
4724 ---> 5000
+3192 ---> +3000
8000
5) Using the range:
Problem Low Estimate High Estimate
378 300 400
+524 +500 +600
800 1000
3-3 Algorithms for Mulitiplication and Dvision
Check the link for a better explanation
3-3 Multiplication and Division of Whole Numbers
Visualization of Division
1) Count Elements / Partition Model:
How many dots fit on the lines evenly?
o o o
o o o
o o o
o o o
__ __ __
12 dots , divided in to 3 groups, equals 4 dots for each group
12/3 = 4
2) Count sets :
How many cans does O does Karah need to place 15 dots?
o o o o o o o o o o o o o o
o o o o o ---->
o o o o o o o o o o o
After separating the dots evenly I found
out i need 5 O to puto the dots in
o o o --> O
o o o --> O
o o o --> O So I need 5 Boxes to place the 15 dots
o o o --> O evenly. In each box 3 dots fit, so
o o o --> O 15 / 3 = 5
3) Array Division:
o o o o o
3 o o o o o = total dots 15 , so 15 / 5 = 3
o o o o o
5
Visualization of multiplication
1) Repeated addition:
- 3 + 3 + 3 + 3= 12
four 3's
so, 4 x 3 = 12
2) Array:
o o o o o
o o o o o
4 o o o o o
o o o o o
by 5,
so 4 x 5 = 20
3) Cartesian Product:
Anna has 3 shirts 2 hats and 2 pants,
how many outfits can she wear?
H1 P1
P2
S1
H2 P1
P2
H1 P1
P2
S2
H2 P1
P2
H1 P1
P2
S3
H2 P1
P2
So, 3 x 2 x 2 = 12
3-2 Algorithms for Whole Numbers Additions and Substractions
Algorithm for Additions:
Expanded Algorithm: 14
+ 23
-------
7 =4+3 (add ones)
+ 30 =20 + 10 (add tens)
--------
37
Standard Algorithm: 14
+23
------
37
Left to right: 568
+757
------
a) 500+700=1200
b) 60 + 50 = 110
c) 7 + 8 = + 15
------
1325
Expanded notation:
3 tens and 4 ones
+ 1 ten and 7 ones
--------------------------
1 ten 11 ones
3 tens 1 ones ( bring 1 ten to the tens)
1 ten
----------------------------
5 tens and 1 ones
3-1 Addition and Substraction of Whole Numbers
Definition of Substracyion of Whole Numbers: For any whole numbers a and b such that a ≥ b, a - b is the unique whole number c such that b+c=a
Visual Substraction Methods:
1) Take away model: works by removing objects from one set to make the set smaller.
EX: Vanessa had 4 flowers, she gave 2 to her mom. How many flowers does vanessa have left?
2) Missing Adden Model: works by starting with a small set and finding out how many more objects are needed in order to create a specific larger set, altohugh this method seems to illustrate and addition it is really a substraction.
EX: 2 + ? = 5
5 - 2 = 3 so 2 + 3 = 5
3) Comparison Model: is the comparison of how much larger is one set then another.
EX: Betty: 2 2 2 2
Carla: 3 3 3 3 3 3 3
How many more numbers does Carla have than Betty?
Definition of Addition of Whole Numbers: Let A and B be two disjoint finite sets. If n(A)=a and n(B)=b, then a+b= n(A ∪B)
Visual Addition methods:
1) Unite sets: Unite sets is to combine two different sets of objects and put them together into one set.
EX: o o
o + o = ?
o
2) Count forward: To count forward is to add objects to an existing set.
EX: o o o + o o (1,2,3....4,5)
Whole Number Addition Properties
Theorem 3-1 Closure Property of Addition of Whole Numbers : If a and b are wholes numbers , then a+b is a whole number.
Theorem 3-2 Commutative Property of Addition of Whole Numbers: If a and b are any whole numbers, then
a+b = b+a
Theorem 3-3 Assosiative Property of Addition of Whole Numbers: If a, b and c are whole numbers, then
(a+b)+c = a+(b+c)
Theorem 3-4 Identity Property of Addition of Whole Numbers: There is a unique whole number 0, the additive identity, such that for any whole number a, a +0 = a = 0+a
2 Numeration Systems and Sets
2.2 Set
The Venn Diagram
The venn diagram was named after John Venn who used such diagram diagrams to illustrate ideas in logic. Venn diagrams are great visual organizers.
Check out the fun website! Press the animations for a little challenge!
Subsets: When we define a set, if we take pieces of that set, we can form what is called a subset.
For exammple: A{1,2,3,4,5,6,7} is a set. Then B{1,2,4} is a subset of the set.
Set: A set is understood to be any collection of objects. Elements or members are known as the individual objects of a set.
Sets hacve important symbols one should consider learning. Press the link to your right to see some examples!
2.1 Numeration System
Number base system: The base number is the number that is going to be raised cartain power.
For example: 10^2 , 10 is the base number.
Its very important to be able to understand how to solve the base-ten system which is consider to be the simplest one, in order to continue.
It is very important to know how to understand different number bases. Follow the link for examples. Learn how to solve base-five systems!!
1 An Introduction to Problem Solving
1.2 Exploration with Patterns
Pattern: A repeated decorative sign. Number Pattern: is a list of numbers that follow a certain rule.
Geometric Sequence
A Geometric Sequence is formed by multiplying a particular value each time to the value just before it.
Fibonacci Sequence
The Fibonacci Sequence starts with 0 and the sequence is formed by adding the two previous numbers.
EX: 1,1,2,3,5,8,13,21...
Arithmetic Sequences
An Arithmetic Sequence is formed by adding a particular value each time to the value just before it.
EX: 0,5,10,15,20,25... RULE: n+5
Inductive Reasoning
Inductive reasoning is the method of making generalizations based on observations and patterns.
CHECK THE LINK FOR EXAMPLES!!!
Look back
· Does your answer make sense? Did you answer all of the questions?
· What did you learn by doing this?
· Could you have done this problem another way – maybe even an easier way?
Looking back means to refelct on the problem. Make sure that the answer is correct and if its not find out what is missing. Make sure you LEARNED how to enact the problem and check if the most effective strategy was used. AFTER FOLLOWING ALL THE STEPS YOU WILL BE READY TO GO!!!!
Carry out the plan
· Carrying out the plan is usually easier than devising the plan
· Be patient – most problems are not solved quickly nor on the first attempt
· If a plan does not work immediately, be persistent
· Do not let yourself get discouraged
· If one strategy isn’t working, try a different one
Make sure to have a complete procedure when you carry out the plan. It is very important to have the solution organized step by step and have it as clear as possible for later corrections.
Devise a plan
When devising a plan remember the list of Problem Solving Strategies includes:
-Guess and check -Solve a simpler problem
-Make an organized list -Experiment
-Draw a picture or diagram -Act it out
-Look for a pattern - Work backwards
-Make a table -Use deduction
-Use a variable -Change your point of view
Press the link to your right and find great examples of the strategies listed above!!!
Understand the question
· What are you asked to find out or show?
· Can you draw a picture or diagram to help you understand the problem?
· Can you restate the problem in your own words?
· Can you work out some numerical examples that would help make the problem more
clear?
This is a very important step!!! You better have the question clear in order to proceed!!
It is very important to understand the queston asked in the problem solving. Always remember the questions above in order clear any doubt you may encounter.
