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 instructedB. place his/her name, calendar date, and class period in the upper right-hand corner of the paperC. place the appropriate, specific title on all exercisesD. 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 correctlyL. 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

ParallaxAn 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.

PrecisionHow 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 SALLYor just

P.E.M.D.A.S.

Parentheses

ExponentsMultiplication

DivisionAddition

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

x =  

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 =

221

d

mGmF

221

d

mGmF

IV Practice Solutions1. 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 practiceWrite 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

Ex 4

A. y = 3x + 1

B. 4y = 12x + 3

Solutions

Ex 2

(2,9)

Ex 3

(12x + 4 = 12x + 4) Infinite solutions

Ex 4

(4 = 3) No solutions

Algebraic Method

• 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• 6 X 1024 kg and 9.11 X 10-31 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 10n

Adding or subtracting Sci Not

• If the exponents are the same, add or subtract the values of M and keep the same n

• 3 x 104 + 3 x 104 = 6 x 104

• 4 x 102 – 2 x 102 = 2 x 102

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 106

Multiplying Sci Not

• Multiply the M values then add exponents

• (3 x 106) (2 x 103) = 6 x 10 6 + 3 = 6 x 109

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 = Y

• Run X

= “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 + B2 = C2

• Pythagorean Theorem-right triangles only

• C2 = A2 + B2 – 2ABcos • Law of cosines- used on non-

right triangles