"The only way to be totally free is through education" - Jose Marti

Study Journal Notes

Measurement:

Definitions you should know:

• referent

• metric system (SI)

• imperial system

• radius

• diameter

• surface area

• volume

• composite object

Lesson 1:

• A referent is an item or object that is concrete (does not change) that is used to estimate the size of another object or item

• There are two prevalent systems of measurement:

  • the Metric system - we use this in Canada​​​

  • the Imperial system - this is used in the United States

• The mnemonic device used to remember the order of the units in the metric system is: King Henry Danced Merrily Down Country Meadows (kilo, hecto, deca, main unit, deci, centi, milli)

• To move among the units in the metric system you just have to move the decimal according to the number of spaces moved between the units in the metric system (this is called a decimal system)

• Ensure that you are converting your units to the unit that you are aiming to finish the question in before you complete any of your calculations.

Lesson 2:

• The imperial system is used in the United States

• The imperial system is NOT a decimal system (you cannot just move the decimal to go to a different unit)

• Imperial measurements must be stated in fractions while you are working with them, not decimals

• Use cross multiplication to convert between units with a known conversion ratio (see page 8 of your notes for common ratios)

• You can also use cross multiplication to convert between metric and imperial (see page 9 for the conversion ratios)

• Ensure that you are converting your units to the unit that you are aiming to finish the question in before you complete any of your calculations.

Lesson 3:

• Area is stated in square units (2-dimensional), and volume is stated in cubed units (3-dimensional)

• Ensure that you are converting you units when they are in one dimension (for example: from m to ft, but never from square meters to feet)

Lesson 4:

• Surface area is the total area of all of the surfaces on a three-dimensional object

• To find the surface area of any prism, you can use the formula SA = 2B + Ph (B is the area of the base, P is the perimeter of the base, and h is the distance from one base to the other matching base)

• The surface area of a pyramid is the area of the base plus the area of each triangular side (recall that the area of a triangle is found using the formula 1/2 base x height)

• The surface area of a cone is found using pi x r x (r + s) where s is the slant-height of the cone

• The surface area of a sphere is found using 4 x pi x r x r

• When you find the surface area of a composite object, remember to subtract the surfaces that are not exposed (the ones that are touching each other)

Lesson 5:

• Volume is the amount of space that a three-dimensional object occupies

• To find the volume of any prism, multiply the area of the base by the height

• To find the volume of any cone or pyramid, multiply the area of the base by the height and divide by 3.

• To find the volume of any sphere, use 4/3 x pi x r x r x r

• To find the volume of a composite object, add up the volume of each of the individual objects (you do not ahve to subtract anything)

• Know how to rearrange a formula so that you can isolate any of the variables (the volume, radius, height, length, width, etc.)

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Solving Systems Algebraically:

Lesson 1:

• The TANGENT ratio (looks like tan on your calculator) compares the length of the side opposite an angle to the length of the side adjacent to an angle. 

• Remember, a ratio is just a type of fraction used to compare two things with the same units

• Side are simple - just make two fractions (trig function over 1 on one side, fractions of sides on the other) and cross multiply

• Angles are a little complex - take the inverse trig function of the fraction of the sides

Lesson 2:

• The SINE ratio (looks like sin on your calculator) is used to compare the length of the side opposite an angle to the hypotenuse of a right triangle

• The COSINE ratio (looks like cos on your calculator) is used to compare the length of the side adjacent an angle to the hypotenuse of a right triangle.

• ALWAYS draw a diagram FIRST

• ALWAYS label your triangle (side opposite the angle, side adjacent the angle, hypotenuse)

  • Use your labels to identify the information that you will be working with, and use that information to determine which trigonometric function you will be using​

• The angle of elevation is from horizontal looking up

• The angle of depression is from horizontal looking down

Lesson 3:

• Remember: solving means finding ALL of the angles and ALL of the sides on a right angled triangle

  • if you have 2 sides you can use the Pythagorean theorem to solve for the last side​

  • if you have 2 angles you can use the sum of angles in a triangle (180) to solve for the last angle

• An angle of elevation is like looking up from horizontal (think of what happens to the angle of your head when you look up from straight ahead to look at a bird in the sky)

• An angle of depression is like looking down from horizontal (think of what happens to the angle of your head when you look down from straight ahead to look at an ant on the ground)

• The sum of all of the angles in a triangle is 180 degrees - in order to solve all of the angles in a triangle, you need to find one angle using a trigonometric ratio, and then subtract that angle and the right angle from 180 to find the last angle

Lesson 4:

• When you solve a problem involving multiple triangles, make sure that you solve for the triangle that has the most information first (at least two pieces of information)

• Solving problems with multiple triangles should not be seen as intimidating or difficult...you are just solving one triangle at a time, and you already know how to solve just one triangle!

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