Download Math Review Part 1

Math Review
8/28
• Get a calculator. Remember it’s address
and return at end of class. NO INSPIRES
ALLOWED TO BE USED IN CLASS!!!!
• Pick up note sheet at front. Work the
examples. Check example 2 on board
…radical covers 52 and 32!
• Have Math Inventory out.
• Safety sheet in drawer
• Goal: Math Review
• Quiz over reading on Friday
The student shall ___.
A. complete all written work using a standard
pencil and on notebook paper unless otherwise
instructed
B. place his/her name, calendar date, and class
period in the upper right-hand corner of the
paper
C. place the appropriate, specific title on all
exercises
D. answer in complete, properly constructed
sentences (8 word minimum per sentence)
where applicable
E. show all mathematical work being certain
to include all proper diagrams and steps.
Equations must be rearranged before
numerical substitutions.
F. arrange the prescribed written calculations
on paper from left to right, top to bottom, with
answers boxed
G. adequately space the written work apart, so
comments can be inserted as needed.
H. present papers in good repair with no
frayed edges
I. communicate proper lettering, numbering,
and decimals, otherwise no grade shall be
reported
J. record all numerical answers to 3
significant figures (unless otherwise
instructed)
K. make every effort to spell words correctly
L. not make use of these words: because,
cause, cuz, it,
• Why is math the language of physics?
• Quantitative vs Qualitative?
• What is the difference between precision
and accuracy?
Which is more precise?
A
B
Which is more precise?
A
B
Parallax
An apparent shift in position of an object
when it is viewed from different angles
Location, location, location! Align eyes
directly with indicator to increase
accuracy.
Precision
How long is the blue line?
0 ft
1 ft
2 ft
Always estimate one place beyond smallest
interval. Thus if the red line was
in millimeters, the length of the blue line would
be?
Order of Operations
• When there is more than one operation
involved in a mathematical problem, it must
be solved by using the correct order of
operations .
• Remember, calculators will perform
operations in the order which you enter them,
therefore, you will need to enter the operations
in the correct order for the calculator to give
you the right answer.
PLEASE EXCUSE MY DEAR AUNT
SALLY
or just
P.E.M.D.A.S.
Parentheses
Exponents
Multiplication
Division
Addition
Subtraction
Order of Operations Rules
1. Calculations must be done from left to right.
2. Calculations in parenthesis are done first. When you
have more than one set of brackets, do the inner
brackets first.
3. Exponents (or radicals) must be done next.
4. Multiply and divide in the order the operations occur.*
5. Add and subtract in the order the operations occur.*
• *Multiplication and division are transitive, so the order
does not matter for them. You can divide and then
multiple or vice versa. The same holds true for addition
and subtraction.
OOO Practice Problems
-3[4 – 2(32)] =
(62 + 4) √52 -32 =
OOO Practice Solutions
-3[4 – 2(32)] =
-3[4 – 2(9)] =
-3[4 – 18] =
-3[– 14] =
= 42
OOO Practice Solutions
(62 + 4) √52 -32 =
(36 + 4) √25 -9 =
(40) √16 =
(40) (4) =
= 160
Isolating Variables
• Work backwards to isolate a
variable.
• Perform order of operations
backwards too
Rules for Isolating Variables
1. Whatever is done to one side of the equation must be done on the
other side as well to remain balanced.
2. Do the opposite operation that is being done (ex: if the quotient is
being divided in the equation, multiply it on both sides).
3. Add or subtract all quotients not containing the variable to one
side of the equation.*
4. If the variable is in a denominator, get it out by multiplying by
the reciprocal on both sides.
5. Multiply or divide numbers not the variable. *
6. Exponents (or radicals) must be done next.
7. If the variable was in parentheses and that has been isolated, the
parentheses disappear and continue isolating using SADMEP.
*Multiplication and division are transitive, so the order does not
matter for them. You can divide and then multiple or vice
versa. The same holds true for addition and subtraction.
8/30
• Due today: 1-6 on Math notes
• Pick up: Trig Notes and Mathref 04
• Get a calculator. Numbers have not yet
been assigned.
Which is more precise?
A. 5 m
B. 5 cm
C. 5 km
Which is more precise?
A. 5 m
B. 5.0 m
C. 5.00 m
IV Practice
1. 6 = 18a
a=
2. F = ma
a=
3. x/t = v
t=
4. 8 =16/x
F
Gm1m2
d2
5. d = 1/2at2
t=
6. F = Gm1m2
d2
If G = 6.67x10–11Nm2/kg2,
F = 5x10–5 N, m1 = 0.343 kg, and
m2 = 3500 kg, then d =
IV Practice Solutions
1. 0.33
2. F/m
3. x/v
4. 2
5. √d/.5a
6. 0.0400 m
Stop here 2011-12
Slope
Rise/Run
Δy
Δx
y2-y1
x2 –x1
Linear Equations and
Graphing
• Straight-line equations, or "linear"
equations, graph as straight lines,
and have simple variable
expressions with no exponents on
them. If you see an equation with
only x and y, then you're dealing
with a straight-line equation.
Equations of a line:
Slope-intercept form
y = mx + b
Where :
m is the slope
b is the y-intercept
This the most common way of writing a
straight-line equation
Slope-intercept practice
Write the slope-intercept form
equation for each line.
Identify the slope and y-intercept for
each line.
1)13x – 11y = -12
2)x– 3y = 6
3)11x – 8y = -48
4)4x –y =1 Sketch the graph
Slope-intercept practice solutions
Write the slope-intercept form
equation for each line.
Slope
y-intercept
1)y = 13/11x + 12/11
2)y = 1/3x - 2
3)y = 11/8x + 6
4)y = 4x -1
Equations of a line:
Standard form
Ax +By =C
Where:
-A/B is the slope
(C/A, 0) is the x-intercept
(0, C/B) is the y-intercept
4x-3y =12
•
•
•
•
What form is this?
What is the slope ?
What is the x-intercept?
What is the y-intercept?
4x-3y =12
•
•
•
•
•
•
•
•
•
What form is this?
standard
What is the slope ?
-A/B = -4/-3 or 4/3
What is the x-intercept?
(C/A, 0) = (12/4, 0) = (3,0)
What is the y-intercept?
(0, C/B) = (0,12/-3) = (0,-4)
Sketch the graph
8/27 Wow! It’s Friday!!!!!
•
•
•
•
Quiz rescheduled. We need to have done lab.
Pick up Note Sheet: Point-Slope Form
Goals:
Continue math review
– Point Slope Form
– Substitution Method to solve linear systems
• Cylinder Lab
If given a point and slope how do
you…..
• Find the equation of the line in the
slope-intercept form?
• Find the equation of the line in the
standard or general form?
• Graph the line?
When given a point and slope
•
•
•
•
•
Use point-slope form of a line.
y – y1 = m(x – x1)
Only substitute values for x1,y1, and m.
Do NOT substitute values for x and y.
This is a tool for finding the equation of
the line.
• The equation then can be written in either
the slope-intercept or standard form.
Point-slope form practice
• Given a line at point (4,5) with
slope of 2.
• Find the point-slope form of the
line.
• Write in slope-intercept form and
in standard form
• Graph the line
point (4,5) slope = 2
• y – y1 = m(x – x1)
• y – 5 = 2(x – 4)
y – 5 = 2(x – 4)
Change to slope-intercept form
• y – 5 = 2(x – 4)
• y-5 = 2x – 8
• y =2x -3
y – 5 = 2(x – 4)
Change to standard form
•
•
•
•
y – 5 = 2(x – 4)
y-5 = 2x – 8
-2x + y = -3
2x – y = 3
Systems of Linear Equations
• Two or more lines.
• Can solve graphically, algebraically, or by
substitution.
• Three possibilities:
1. The two lines intersect. Point of
intersection is the solution.
2. No solution; the lines are parallel.
3. Infinite solutions; they are the same line.
Substitution Method
• Write one equation so that it equals the variable.
• Substitute that equation for the variable in the
second equation.
• Solve for the variable
• Insert solution for variable into either equation and
solve for the other variable.
• Advisable to check in both equations.
Use the substitution method to find the
solution for the following system of
equations.
Ex. 1
A. y = 2x + 1
B. 2y = 3x - 2
Substitute one equation into the other. Solve
for the single variable.
A. y = 2x + 1
B. 2y = 3x – 2
2(2x + 1) = 3x – 2
4x + 2 = 3x – 2
x = -4
Insert solution for variable into either
equation and solve for the other variable.
Equation A:
x = -4
A. y = 2x + 1
y = 2(-4) + 1
y = -7
Insert solution for variable into either
equation and solve for the other variable.
Equation B:
x = -4
B. 2y = 3x - 2
2y = 3(-4 ) – 2
2y = -14
y = -7
Solution: (-4,-7)
The lines intersect at that point
•
Ex 2
A. 2y = 3x + 12
B. y = 5x - 1
Ex 3
A. y= 3x + 1
B. 4y = 12x + 4
Solutions
Ex 2
(2,9)
Ex 3
(12x + 4 = 12x + 4) Infinite solutions
Ex 4
(4 = 3) No solutions
Algebraic Method
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•
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Write equations in slope intercept form
Set the equations equal to each other.
Solve for x
Insert solution for x into either equation and
solve for y.
Find the solution for the
following system of equations.
y = 2x + 1
y = 4x - 1
Set the equations equal to each other. Solve
for x.
2x + 1 = 4x - 1
2 = 2x
x=1
Insert solution for x into either equation and
solve for y.
y = 2x + 1
y = 2(1) + 1
y=3
OR
y = 4x - 1
y = 4(1) - 1
y=3
Point of intersection is the solution.
Solution: (1,3)
The lines intersect at that point
Find the solution for the
following system of equations.
3y = 6x + 3
2y = 3x - 2
Solution: (-4,-7)
The lines intersect at that point
Math Review Part 1
• Significant figures (digits)• Significant digits are all the
digits of a measurement
that you are sure of plus an
estimated digit
Rules of Sig figs
• 1) Non zeros are always
significant
• 2) All final zeros after the
decimal are significant
• 3) Zeros between 2 other sig
figs are always significant
• 4) Zeros used for spacing the
decimal are not significant
?????????????????
• Why do we give a rat’s
patoot about sig figs?
2 Reasons
• 1) The final, overall
measurement of an object
cannot be more precise
than the least precise
measurement involved.
Example: volume
• 2) We want to be able to
write answers in scientific
notation. Sig figs tell us
what numbers to write
when using scientific
notation
Scientific Notation
• 600000000000000000000000 kg
• This is the mass of the Earth
• .000000000000000000000000000000911 kg
• This is the mass of an electron
24
-31
• 6 X 10 kg and 9.11 X 10
kg are much easier to write
Rules for Scientific Notation
• 1) All non-zeros must be
expressed in your final number
• 2) Zeros between two sig figs
must be expressed
• 3) Zeros used to space the
decimal may be “exponentized”
• 4) Final answer has only one digit
to the left of the decimal
Working with scientific notation
• We will use the imaginary
“scientifically notated”
expression
•M x
n
10
Adding or subtracting Sci Not
• If the exponents are the same,
add or subtract the values of
M and keep the same n
4
4
4
• 3 x 10 + 3 x 10 = 6 x 10
2
2
2
• 4 x 10 – 2 x 10 = 2 x 10
Adding or subtracting Sci Not
• If the exponents are different,
move the decimal around until
the n values are the same then
use the previous rule
• 4 x 106 + 3 x 105 =
• 4 x 106 + 0.3 x 106 =
• 4.3 x
6
10
Multiplying Sci Not
• Multiply the M values then
add exponents
6
3
6
+
3
• (3 x 10 ) (2 x 10 ) = 6 x 10
9
= 6 x 10
Dividing Sci Not
• Divide the values of M and
subtract the exponent of the
divisor from the exponent of
the dividend
• 8 x 106 = 4 x 10 6-3 = 4 x 103
2 x 103
Dimensional Analysis
• We use DA to convert. That’s
it.
Things to remember in DA
• 1) Any number divided by itself
or its equivalent is 1
• 2) Any number multiplied by 1
does not change in value
• 3) When making final
calculations, make sure all units
cancel
Basic algebraic problem solving
• 1) Isolate your variable
• 2) Whatever you do to one
side of the equation you
MUST do to the other
• 3) Solve the problem step by
step
Graphing
• We will be using line graphs in
this course. When plotting a
line graph follow these steps:
Line Graph Steps
• 1) Independent variable goes on
the X axis and dependent is
plotted on the Y axis.
• 2) Time is always plotted on the
X axis
• 3) Decide if the origin is a
valid data point
• 4) Number and label the X
and Y axis and include
units
• 5) When the data points are
plotted use a line of best fit
and not a “connect the dots”
• 6) Give the graph a very
boring, very explanatory title
Slope
• Slope tells the relationship
between the X and Y axis.
• Slope = Rise = DY
•
Run
DX
D = “change in”
Another formula
• Y = mx + b
• Y = the Y coordinate
• m = slope
• X = the X coordinate
• b = Y intercept
Trig (eeeewww)
• Sine = opposite/hypotenuse
• Cosine =
adjacent/hypotenuse
• Tangent = opposite/adjacent
• SOH CAH TOA
Geometry
• A2 + B 2 = C 2
• Pythagorean Theorem-right
triangles only
• C2 = A2 + B2 – 2ABcos q
• Law of cosines- used on nonright triangles