Download Mathematical Formulas & Relationships

Mathematical Formulas & Relationships
Units: 1-5 Grade Level: 7-9
SCIENCE is a Content Area subject, and is very closely related to MATHEMATICS.
This Grade 7-9 Mathematical Formulas and Relationships PowerPoint is designed to compile
all the mathematical formulas and related concepts students will be introduced to and be
expected to understand by the end of Grade 9. ( Many of these are expanded on in High School )
The Grade Level where you will find each of the mathematical relationship concepts, and in which
unit it is introduced, is provided in the upper right hand corner of each slide.
Mathematical Formulas & Relationships
Unit: Mix and Flow of Matter
Grade Level: 8
Manipulating Formulas
When given a formula for a scientific principle, the variables within the formula are either known
or unknown. To know how to manipulate the formula you are given is important to be able to
solve the formula for the unknown variable. You could memorize the three formulas
independently or understand how to manipulate one formula you will need to memorize.
Another technique is to use an alternative shortcut method.
Scientific Formulas and Mathematical Equations
A mathematical equation is an equality between variables containing an unknown number as well
as other numbers which are known. Scientific Formulas with one unknown variable.
(Example: density is mass to volume ratio of a substance)
d=M/V
To "solve the equation" means to find which value of the unknown number.
If the equation is a scientific calculation, its solution tells what to expect from the design,
experiment or observation. A formula expresses a relationship between known numbers and an
unknown number. If you are given the values of all except one of the numbers, the unknown
number can be found by manipulating the formula.
d = M/V
m = dxV
V = M/d
Mathematical Formulas & Relationships
Unit: Mix and Flow of Matter
Grade Level: 8
Density
Density (d) = mass (m) / volume (V)
mass (m) = Density (d) X volume (V)
volume (V) = mass (m) / Density (d)
Density Calculations (Shortcut Memory Method )
This simple equation will help you figure out how to solve density problems:
Simply cover up whichever value you need to calculate
and the other two are shown in their proper placement,
either multiplied or divided.
_M_
dV
For example: cover up the M. This leaves you with dV or d X V
(ignore the fact that it is in the denominator).
Density times volume will give you mass.
You can also check it out by way of the units:
(g / cm3) x cm3 cancels out the volume unit leaving grams, the desired unit for mass.
Mathematical Formulas & Relationships
Force is a push or a pull on an object.
It is measured in Newtons (N).
Unit: Mix and Flow of Matter
Grade Level: 8
The upward force of a fluid on an object is called its buoyant force, which is also measured in
Newtons.
Calculation of Buoyant Force
Buoyant Force = Weight in Air - Weight in Liquid
Mathematical Formulas & Relationships
Photosynthesis
Unit: Interactions and Ecosystems
Grade Level: 7
Light Energy + Carbon Dioxide + Water  Food (Sugars and Starches) + Oxygen
Cellular Respiration
Food (Sugars & Starches) + Oxygen  Carbon Dioxide + Water + Energy (used by living things)
Unit: Structures and Forces
Units of Mass and Force
'Kilo' means one thousand (1000) and is equal to 1000 grams.
Very small masses are measured in milligrams (mg). 1000 mg. equal 1 g.
The standard unit for measuring force is called a Newton (N).
One Newton is the amount of force needed to hold up a mass of 100g.
Grade Level: 7
Mathematical Formulas & Relationships
Unit: Mix and Flow of Matter
Grade Level: 8
Flow Rate/Viscosity
Flow rate is measured as the volume movement of a fluid past a given point per second.
Units used, at this level in science, are ml divided by seconds.
Rate of fluid flow, is the volume of fluid which passes through a given surface per unit time. It is
usually represented by the symbol Q
Given an area A, and a fluid flowing through it with uniform velocity v with an angle θ away from
the perpendicular to A, the flow rate is: Q = A x V x cosθ
In the special case where the flow is perpendicular to the area A, that is, θ = 0, the volumetric
flow rate is: Q = A x V
We explore flow rate when we learn about,
examine and explore the concept of viscosity.
Mathematical Formulas & Relationships
Pressure is the amount of force applied in a given area.
Unit: Mix and Flow of Matter
Grade Level: 8
It is measured in pascals (Pa)
1 pascal (Pa) equals a force of 1 Newton (N) over a given area of 1m2
1Pa =1N / 1m2
The more force applied to a given area, the greater the pressure.
The formula to calculate pressure is P = F / A
The ‘ shortcut ’ formula is:
_F_
P A
Remember: cover the unknown part of the formula to determine what you have to do.
Most measurements of pressure are in kilopascals (kPa)
Note: 1kPa = 1000 Pa
Pressure can also be measure in N/cm2
Mathematical Formulas & Relationships
Unit: Light and Optical Systems
The LAW OF REFLECTION states that:
Grade Level: 8
the angle of incidence equals the angle of reflection
The incident ray, the normal line and the reflected ray all lie in the same plane
(an imaginary flat surface)
Mathematical Formulas & Relationships
Unit: Mechanical Systems
Grade Level: 8
Mechanical Advantage
A machine makes work easier for you by increasing the amount of force that you exert on an
object. This produces a mechanical advantage, which is the amount of force that is multiplied by
the machine. The force applied to the machine (by you) is the input force. The force that is
applied to the object (by the machine) is the output force.
Calculating Mechanical Advantage
The mechanical advantage of a machine is the output force divided by the input force.
MA = Output Force
Input Force
The mechanical advantage is the force ratio of a machine.
MA = F input
F output
F = Force is measured in Newtons ( N )
The more a machine multiplies the force, the greater is the mechanical advantage of the machine.
Mathematical Formulas & Relationships
Unit: Mechanical Systems
Grade Level: 8
Speed Ratio
Speed measures the distance an object travels in a given amount of time. The measure of how a
machine affects speed is called the speed ratio. It is calculated by dividing the input distance by
the output distance.
SR = Input distance
Output distance
SR = d input
d output
d = distance ( m )
Using the formula provided, you can calculate the speed ratio of any device, like the gears provide
on a 21 speed bicycle.
Mathematical Formulas & Relationships
Unit: Mechanical Systems
Grade Level: 8
Efficiency
Efficiency is a measure of how well a machine or a device uses energy. The more energy that is
lost, the less efficient a machine is. Efficiency is represent in %.
Efficiency = Mechanical Advantage X 100
Speed Ratio
In complex machines, there are many subsystems that are affected by friction and other factors.
Because of this, most complex machines are not very efficient. The efficiency of any machine
measures the degree to which friction and other factors reduce the actual work output of the
machine from its theoretical maximum. A frictionless machine would have an efficiency of 100%.
A machine with an efficiency of 20% has an output only one fifth of its
theoretical output. The efficiency of a machine is equal to the ratio of its
output (resistance multiplied by the distance it is moved) to its input
(effort multiplied by the distance through which it is exerted); it is also
equal to the ratio of the AMA (actual mechanical advantage) to the IMA
(ideal mechanical advantage) . This does not mean that low-efficiency
machines are of limited use. An automobile jack, for example, must
overcome a great deal of friction and therefore has low efficiency, but it is
extremely valuable because small effort can be applied to lift a great
weight.
NO MACHINE CAN BE 100% EFFICIENT !!!!
Mathematical Formulas & Relationships
Unit: Mechanical Systems
Grade Level: 8
Calculating Work
The amount of work is calculated by multiplying the force times the distance the object moves.
The formula looks like this:
W=Fxd
Force is measured in Newtons and distance is measured in meters.
The resulting work unit is called a joule, named after the English scientist James Joule
Mathematical Formulas & Relationships
Unit: Mechanical Systems
Grade Level: 8
Work and Machines
The work done with a machine is the same as the work done without it. This can be shown by
calculating work input and work output.
Work input is the work needed to use,
or operate the machine
Work output is the work done
by the machine.
Work and Friction
Friction is the reason that work input does not equal work output in real situations. Friction
affects the machine’s efficiency.
Efficiency can be calculated
using work input and work output.
Mathematical Formulas & Relationships
Unit: Mechanical Systems
Grade Level: 8
The mechanical advantage in a hydraulic system comes from the fluid pressure in the system.
Calculating the input force and the output force will give you the Mechanical Advantage of the
system.
MA = Output force
Input force
MA = Fo X do
______
Fi X di
Mathematical Formulas & Relationships
Unit: Mechanical Systems
Grade Level: 8
Pressure and Mechanical Advantage
The reason for the large mechanical advantage in a hydraulic system is the ability of the fluid to
transmit pressure equally. It allows you to use a small force on the small piston to produce a
larger force on the large piston.
p=F
A
From Pascal’s law, we know that the pressure the small piston creates is the same everywhere in
the fluid. So the large piston has a larger area and is able to multiply the pressure because of its
larger area. The force and area at each piston act as ratios that have to be equal.
By solving this ratio you will find that the forces created within a hydraulic system provides very
large mechanical advantages - making them useful in many applications.
Mathematical Formulas & Relationships
Unit: Matter and Chemical Change
Grade Level: 9
Chemical Formulas and Equations
A chemical formula tells how many of each type of atom is present in the molecule.
A compound made from two elements is called a binary compound.
There are many chemical formulas within the Matter and Chemistry unit. Most of these formulas
are used to help us understand the process involved, when matter changes from one form to
another; either with energy, or giving off energy.
To understand chemical formulas, it is necessary to be acquainted with and be able to interpret
information about, various elements that form together as compounds and interact in chemical
equations.
In a chemical reaction, the total mass of the reactants,
is always equal to the total mass of the products.
Acids and bases react together when they are mixed.
This type of reaction is called neutralization.
When the acid and the base are used up, salt and water are produced.
Hydrochloric Acid + Sodium Hydroxide -------» Salt + Water
HCl
+
NaOH
------ »
NaCl
+
H2O
Mathematical Formulas & Relationships
Unit: Environmental Chemistry
.Grade Level: 9
ppm – parts per million
Concentration is the measure of how much of a given substance is mixed with another
substance. This usually applies to homogeneous solutions, where it refers to the amount of
solute in a solvent.
Parts Per Million, or ppm represents the concentration of the solute in the solvent.
The grams of a solute, in the ml of solvent.
Solute (g)
Solvent (ml)
X
106
To concentrate a solution, add more solute, or reduce the amount of solvent. (higher ppm)
To dilute a solution, add more solvent, or reduce the amount of solute. (lower ppm)
Less ppm
More ppm
Mathematical Formulas & Relationships
Unit: Electrical Principles & Technologies
.Grade Level: 9
Ohm's Law
Georg Simon Ohm, a mathematician, proved a link between voltage (V), current (I) and
resistance (R). The unit of resistance was named after him, the ohm.
Ohm's Law states:
as long as temperature stays the same
the resistance of a conductor stays constant,
and the current is directly proportional to the voltage applied
OHM'S LAW EQUATIONS
OHMS
VOLTS ÷ AMPERES
VOLTS ² ÷ WATTS
WATTS ÷ AMPERES ²
VOLTS
SQRT ( WATTS × OHMS )
WATTS ÷ AMPERES
AMPERES × OHMS
AMPS
VOLTS ÷ OHMS
WATTS ÷ VOLTS
SQRT ( WATTS ÷ OHMS )
WATTS
VOLTS ² ÷ OHMS
AMPS ² × OHMS
VOLTS × AMPS
R=V/I
Mathematical Formulas & Relationships
Unit: Electrical Principles & Technologies
Power is the rate at which a device converts energy.
The unit of power is the watt (W), which is equal to 1 joule per second.
For an electrical device the power is the current multiplied by the voltage.
.Grade Level: 9
(P) Power in watts (I) current in amperes (V) voltage in volts
P=IxV
I=P/V
Shortcut
V=P/I
_P_
IV
(E) Energy in joules (P) Power in watts (J/s) (t) time in seconds
E=Pxt
P=E/t
Shortcut
_E_
Pt
t=E/P
Mathematical Formulas & Relationships
Unit: Electrical Principles & Technologies
.Grade Level: 9
The efficiency of an electrical device …
is the ratio of the useful energy that comes out of a device to the total energy that went in.
The more input energy converted to output energy, the more efficient the device is.
% Efficiency = Joules of useful output x 100%
Joules of input energy
USC researchers are at work refining and miniaturizing a new kind of fuel cell, one that uses
biological processes to continuously generate electricity.
A basic working design exists. Now underway is a systematic effort by a large interdisciplinary
team of biologists, earth scientists and engineers to improve every aspect of this “bactery”– from
the mechanical components to the nutrient medium to the bacteria themselves.
The goal is to increase the device’s efficiency 1000-fold.
Mathematical Formulas & Relationships
Unit: Electrical Principles & Technologies
.Grade Level: 9
Triangulation
Triangulation is based on the geometry of a triangle. By measuring the angles between the
baseline and a target object, you can determine the distance to that object.
To measure the distance indirectly, you need
to know the length of one side of the triangle
(baseline) and the size of the angles created
when imaginary lines are drawn from the ends
of the baseline to the object.
Parallax
Parallax is the apparent shift in position of a nearby object when the object is viewed from two
different places. Astronomers use a star’s parallax to determine what angles to use when they
triangulate the star’s distance from the Earth. The larger the baseline, the more accurate the
result. The longest baseline that astronomers can use is the diameter of Earth’s orbit.
Measurements have to be taken six months apart to achieve the diameter of the orbit.