Advanced II Geometry, Coordinate Geometry, Trigonometry, and Measurement (10) Flashcards
SI Base Units
You can use the SI base units to come up with all the SI related units. For example, time can also be measured in minutes by taking the definition of a second and using a new definition based on the second to establish the minute.
Many units in SI use factors of 10 to go between units, so generally conversions can be made by moving a decimal left or right. But this is not true of all SI units such as the second when trying to convert to minutes. However, if you see the prefixes, then you can use the factor of 10 idea to convert between units with the same base.
If you are interested, there is a huge amount of information available on SI units, including how we define the second. Feel free to investigate if you find that interesting, but it is not required knowledge.
Imperial units of measurement
These are the imperial units you are likely to still see still in use today. They are mostly used in trade with the USA since they are on a modified imperial system.
Canada started the conversion from imperial to metric in 1970, so that is why some things in Canada have not yet made the switch to metric entirely, purely based a slow switch over.
Imperial units do not use factors of 10 to move from one unit to the next, so you will need to use conversion factors to convert between units.
SI unit prefixes
Ensure you know from kilo to milli. Learn more for specific fields depending on if you are using them or not.
Observe that from one place to the next you do not always multiply or divide by 10. Sometimes you must multiply or divide by 1000.
Try saying each of these with the word “meter” after it and you will notice that you do recognize some of them such as kilometer, metre or centimeter.
Then try saying each with the word “gram” in order to get kilogram, gram, or milligram.
Try saying each with the word “litre” in order to get millilitre, or litre
This works with seconds too to get second, millisecond
Generally it is good practice to have the prefixes and their position relative to other prefixes memorized for the units you are seeing in your class. Pay attention to which ones your teacher is using and make sure you know those well.
conversion factor and unit analysis
Show a converision of 55 km/h to m/s using unit analysis
In order to convert between one unit and another unit, you need a conversion factor. A conversion factor can be a fraction, where the unit you are trying to convert to is on the top of the fraction, and the unit you are trying to no longer have is on the bottom of the fraction (this is assuming that the unit you are trying to remove is in the numerator to begin with, and that you wish for the new unit to be in the numerator; see the picture for both cases).
The key is to make the top and bottom of your conversion factor fraction equal to 1. You do this by making the top of the fraction equal to the bottom of the fraction (so you could have 3 feet on top, and 1 yard on the bottom since 3 feet is equal to one yard).
If you multiply anything by 1 you are not changing the original measurement, so this is why it works to multiply by 1 (which is your conversion factor).
The image shows how to convert through many units at once so that you can get an idea for how effective this method is to convert when you must use several conversion factors. This method works in all situations and should help you a lot in your science classes as well.
In general, when working with varying units in math class, make sure you are always paying attention to units since this is a form of ensuring that you are only working with like terms (such that you cannot perform any addition without having the same units first). So you will need to recognize that a conversion must be made and you may not be told to convert specifically. You would then get to choose what units to convert to in some cases. Just try to make the units the same since you need that to occur in order to add the numbers together.
I purposely don’t bother with proportional reasoning of one fraction equaling another in order to solve these problems because it is challenging to solve all problems with that method. The setup takes longer and it is not necessary to use proportional reasoning in that way. There is a card on proportional reasoning in the 7-9 math that will help you to see what that method is if you are interested.
Notice that the method presented on this card feels like the scale factor equation:
original ( scale factor ) = new
Instead of this we have:
original ( conversion factor ) = new
where the conversion factor is equal to 1, so the original and new are the same measurement but just expressed in different units. So there is no enlargement or reduction, simply restating the same size with different units.
Conversion factors are supplied in formula sheets so you do not have to memorize them. However, there is an expectation to memorize the SI prefixes and thus conversions such as 1000 m = 1 km is not going to appear on your formula sheet.
What is the surface area of a right pyramid?
pyramids come to a point, but can have different bases.
total surface area = area of the base + area of all the triangles
Make sure to use symmetry so that you are not doing more math than necessary. For example, use 6(area of a triangle) if you notice that all 6 triangles have the same measurements.
You may be given the height but not the slant height. Use Pythagorean theorem to get the slant height and then use that as the triangle height.
If you have a base that is more challenging, such as a hexagon, then see it as a composite shape (such as 6 triangles in the base), and add the smaller areas to get the larger area.
Some teachers will supply a formula for these but it really is not necessary since the surface areas are based off rectangles and triangles and those formulae should already be memorized.
Surface area and volume formulae that are likely on your formula sheet since they require calculus to derive
circle related formulas such as:
area of a circle
volume of a sphere
surface area of a sphere
Anything that comes to a point or is curvy such as:
volume of anything that is not a box shape
surface area of anything that is curvy such as a cone
Make sure that you understand your specific formula sheet and what the variables mean. You also need a solid understanding of solving for a variable by using preservation of equality (including preservation of equality with powers and roots!). See cards in the lower grades to learn more about preservation of equality with addition, subtraction, multiplication, and division.
What are the primary trigonometric ratios?
sine –> sinΘ = opposite / hypotenuse
cosine –> cosΘ = adjacent / hypotenuse
tangent –> tanΘ = opposite / adjacent
Only for use in a right triangle, where the hypotenuse is the longest side length, adjacent is the leg of the triangle that is touching the angle of concern, and opposite is the leg of the triangle that is not touching the angle of concern.
some people use this to memorize the above information:
SOH CAH TOA
See the image for a fun way to label the sides of a triangle so that you know what is opposite, adjacent, and hypotenuse in the right triangle.
How do you solve for the angle if you are given two side lengths of a right triangle?
If you are given opposite and hypotenuse, choose the sine ratio.
sinΘ = opposite / hypotenuse
Θ = sin^-1(opposite / hypotenuse)
There is a button on your calculator that will have the exponent -1 and that will be used to calculate the angle where you can input the sine ratio.
You would do something similar for cosine and tangent, just make sure you use the trigonometric ratios based on what side lengths you are given in your triangle.
It is important to properly label your triangle before you begin to choose a ratio so that you are absolutely sure if you have adjacent vs. opposite since these are confused often.
