Study Tips
1- Read chapter BEFORE class
2- Do the end of the chapter exercises BEFORE
class.
3- To understand a concept: (a) Read the book
very carefully, (a) Come to class to hear the
professor’s explanation, (3) Talk to the TA, (4)
Check on the internet (only the good sources!)
Study Tips
1- As soon as you have a question, please send an
e-mail to your TA asap!
2- If you can’t solve a problem, take a picture of
what you did and send it to the TA.
3- Before coming to the office hour (Tuesdays 1:00
to 2:00 pm at BSB 402), please e-mail your TA to
let her know that you will be attending (as hours
will be extended when needed). Also, please email your questions in advance (to allow for
better preparation).
What should you bring to every
class
1- Calculator (not your phone)
2- The textbook
3- Paper and pen/pencil
4- A separate sheet with all the formulas you
have learned so far (including the ones you
will need for that day’s lecture!)
5- If you have an electronic copy of the book,
consider making a copy of the tables (as it will
make things much easier!)
Symbols and Notations
a- (5+3) X 2=
b- 5 + (3x2)=
c- (3+1)2 – 4 x 7/2=
d- 12/2 x 3=
e- 6 x (3+8)2 – 50=
f- (3+8) – 5= or 3 + (8-5)=
g- 2 x 32=
h- (2x3)2=
1- Parentheses
2- Squaring/Exponents
3- Multiplication/Division (left to right)
4- Addition/ Subtraction
Symbols and Notations
a- (5+3) X 2= 16
b- 5 + (3x2)= 11
c- (3+1)2 – 4 x 7/2= 2
d- 12/2 x 3= 18
e- 6 x (3+8)2 – 50= 676
f- (3+8) – 5= or 3 + (8-5)= 6
g- 2 x 32= 18
h- (2x3)2= 36
1- Parentheses
2- Squaring/Exponents
3- Multiplication/Division (left to right)
4- Addition/ Subtraction
Proportions: fractions, decimals and percentages
Fraction
¾
¼
½
1/10
Decimal
0.75
Percentage
75%
1- Fraction to decimal= Divide
2- Decimal to percentage = Multiply by 100
Proportions: fractions, decimals and percentages
Fraction
¾
¼
½
1/10
Decimal
0.75
0.25
0.50
0.10
Percentage
75%
25%
50%
10%
1- Fraction to decimal= Divide
2- Decimal to percentage = Multiply by 100
Finding equivalent fractions
Multiplying Fractions
Dividing Fractions
Invert second fraction and multiply:
1÷1= 1x4= 4 =2= 2
2 4
2 1 2
1
Dividing and Multiplying Fractions
9 ÷ 2=
10 3
1 x 7=
6
10
Dividing and Multiplying Fractions
9 ÷ 2 = 9 x 3 = 27
10 3
10
2 20
1 x 7= 7
6 10 60
Adding and Subtracting Fractions
Same denominator:
2+1= 3
5 5 5
Different denominator:
2 + 1 = 2 x 10
1 x 3 = 20 + 3 = 23
3 10 3 x 10
10 x 3
30 30
30
1+1= 1x1
1x2 = 1+2=3
4 2
4x1
2x2
4 4 4
7/22 + 2/3 =
7/8 – ½ =
1/8 + 2/3 =
Adding and Subtracting Fractions
Same denominator:
2+1= 3
5 5 5
Different denominator:
2 + 1 = 2 x 10
1 x 3 = 20 + 3 = 23
3 10 3 x 10
10 x 3
30 30
30
1+1= 1x1
1x2 = 1+2=3
4 2
4x1
2x2
4 4 4
7/22 + 2/3 = 65/66
7/8 – ½ = 3/8
1/8 + 2/3 = 19/24
Comparing the size of fractions
Converting Decimals to Fractions
0.1 = 1
10
0.05 = 5
100
0.32 = 32
100
0.001 =
0.5234 =
Converting Decimals to Fractions
0.1 = 1
10
0.05 = 5
100
0.32 = 32
100
0.5234 = 5234
10,000
0.001 = 1
1,000
Converting Percentages
Percentage to fraction (use 100 as nominator)
52% = 52
100
5% =
Percentage to decimal (divide by 100 or move decimal point
two places to left)
83% = 0.83
14.5% =
5% =
Converting Percentages
Percentage to fraction (use 100 as nominator)
52% = 52
100
5% = 5
100
Percentage to decimal (divide by 100 or move decimal point
two places to left)
83% = 0.83
14.5% = 0.145
5% = 0.05
Negative Numbers
How good were you at math before and after the
review with the TA?
Participant Before
001
6
002
7
003
2
004
9
After
7
6
4
6
Change
+1
Negative Numbers
How good were you at math before and after the
review with the TA?
Participant Before
001
6
002
7
003
2
004
9
After
7
6
4
6
Change
+1
-1
+2
-3
Negative Numbers
Adding negative numbers
3 + (-2) + 5 = 6
-1 + 3 + (-4) + 3 + (-6) + (-2)
Positive sum = 6 Negative sum = 13
Subtract negative from positive (i.e. positive first)
6-13 = -7
3 + (-8) + 5 + 7 + (-1) + (-3) =
5 – (-9) + 2 – (-3) – (-1) =
3 – 7 – (-21) + (-5) – (-9) =
Negative Numbers
Adding negative numbers
3 + (-2) + 5 = 6
-1 + 3 + (-4) + 3 + (-6) + (-2)
Positive sum = 6 Negative sum = 13
Subtract negative from positive (i.e. positive first)
6-13 = -7
3 + (-8) + 5 + 7 + (-1) + (-3) = 3
5 – (-9) + 2 – (-3) – (-1) = 20
3 – 7 – (-21) + (-5) – (-9) = 21
Negative Numbers
Subtracting negative numbers
4 – (-3) =
Multiplying negative numbers
3 x (-2) =
-4 x (-2) =
Dividing negative numbers
-6 ÷ 3 =
8 ÷ (-4) =
-8 ÷ (-4) =
Negative Numbers
Subtracting negative numbers
4 – (-3) = 4 + 3 = 7
Multiplying negative numbers
3 x (-2) = -6
>>> -2 + (-2) + (-2)
-4 x (-2) = +8
>>> - (-2) - (-2) – (-2) – (-2)
Dividing negative numbers
-6 ÷ 3 = -2
8 ÷ (-4) = -2
-8 ÷ (-4) = 2
Basic Algebra: Solving Equations
Equation: Math statement indicating two
quantities are identical
12 = 8 + 4
It may have an unknown value
12 = 8 + X
To solve equation:
1- Isolate unknown value
2- Keep both sides of equation balanced
Basic Algebra: Solving Equations
Value added to X:
X+3=7
X+3–3=7–3
X= 4
Value Sub. From X:
X – 8 = 12
X – 8 + 8 = 12 + 8
X = 20
Value multiplied:
4 X = 24
4 X/4 = 24/4
X=6
Value divided:
X/3 = 9
3 (X/3) = 9 x 3
X = 27
Basic Algebra: Solving Equations
A- 3X = 18
B- X – 4 = 18
C- X = 5
9
D- X = -5
5
Basic Algebra: Solving Equations
A- 3X = 18
B- X – 4 = 18
X=6
X = 22
C- X = 5
9
X = 45
D- X = -5
5
X = -25
More Equations
3X + 7 = 22
3X + 7 – 7 = 22 – 7
3X = 15
3X/3 = 15/3
X=5
2X = 12
3
X+1=3
3
X+3=2
4
4 (X+3/4) = 2 x 4
X+3=8
X+3–3=8–3
X=5
More Equations
3X + 7 = 22
3X + 7 – 7 = 22 – 7
3X = 15
3X/3 = 15/3
X=5
2X = 12
3
X+1=3
3
X = 18
X=6
X+3=2
4
4 (X+3/4) = 2 x 4
X+3=8
X+3–3=8–3
X=5
Exponents
53 =125 (5x5x5) (“3 cubed”)
42 =
(“4 squared”)
24 =
(“2 raised to the fourth power”)
61=
90 =
XY2 =
X2Y2 =
Exponents
53 =125 (5x5x5) (“3 cubed”)
42 = 16 (“4 squared”)
24 = 16 (“2 raised to the fourth power”)
61= 6
90 = 1
XY2 = XYY
X2Y2 = XXYY
Exponents
(-3)2 =
(3 + 5) 2 =
32 + 52 =
32 =
42
(3/4)2 =
(-6)3 =
Careful! Not the same!!
Same result
Exponents
-32 = 9 (-3) (-3) ///// (-6)3 = -216 (-6)(-6)(-6)
(3 + 5) 2 = 64
32 + 52 = 34
Careful! Not the same!!
32 = 9/16
42
(3/4)2 = (3/4) x (3/4) = 9/16
Same result
Square Roots
What value multiplied by itself gives you the
value underneath the radical?
√4 = 2
√16 =
√32 =
(√64)2 =
√9 + 16 =
Careful! Not the same!!
√9 + √16 =
Square Roots
What value multiplied by itself gives you the
value underneath the radical?
√4 = 2
√16 = 4
√32 = √9 = 3
(√64)2 = 82 = 64
√9 + 16 = 5
Careful! Not the same!!
√9 + √16 = 7
Square Roots
√16 =
√4
Same result
√16/4 =
√9 x √16 =
Same result
√9 x 16 =
Square Roots
√16 = 4 = 2
√4 2
Same result
√16/4 = √4 = 2
√9 x √16 = 3 x 4 = 12
Same result
√9 x 16 = √144 = 12