Unit 1: Numbers
Key points:
The entire basis of chapter 1 is knowing how to find the prime factor of numbers. This allows you to find the gcf and lcm of numbers, and help solve some entire and mixed radical questions.
to find the prime factorization of a number, you first need to know the list of prime numbers up to about 50. Here is a list up to 100,
now what you do is take the number you are factoring, and try dividing by a prime number. Here are some tricks to save time looking for a number, if the number is even, it will go into two. If it ends in 0 or 5, it will go into 5. If it is a 2 digit number and both digits are the same, it will go into 11. If the number does not fit into any of the guidelines, start with 3 and work your way up. keep trying numbers until you are left with 1. That set of prime numbers you are left with is the prime factor of that number!
here is an example,
Unit 2: Exponents
In my opinion, the most key part of the exponents unit is the rule “Flower Power”. The mistake I most commonly find myself making out of all of math 10 is when I am a number with a fraction as an exponent. To do this you must first draw a square root sign with the base going to the radicand in the square root, the denominator going to the index in the square root and the numerator going to the exponent of the radicand in the square root. This may seem a bit wordy, so here is a visual to help,
Also, if the denominator is 2, there is no need to write that in the index spot since all regular square root have the index of 2 already there.
Where the flower power part comes in is when you are looking at the fraction and cant remember what goes where in the square root, think of the denominator as the bottom part or root of the flower, and put the denominator in the root part of the square root.
Unit 3: Measurement
An important part of measurement that people often make mistakes of in converting inits that are cubed or squared. It is a fairly simple switch from doing it normally, but can often be forgotten. When converting, you MUST do the multiplication of that certain measurement the number of the dimensions of the unit. ex. squared = 2x, and cubed = 3x. A simple way to remember this is if you see an exponent, do whatever number it is that many times for the conversion. If you were converting 53cm to cm, you would multiply 53 by 100/1 or 100 to get 5300cm³. If you were converting 53m³ to cm³, you would multiply 53 by 100/1 or 100 3 times to get 53000000cm³. See how big a difference 1 small change can make?
Unit 4: Trigonometry
A big mistake in unit 4 is regarding the angle of depression. Here is the triangle we will be going off of,
The angle of elevation is angle bac or cab, which is on the inside of the triangle and facing upwards. Because of this, you would assume that the angle of depression is angle abc or cba which is also on the inside of the triangle but facing downwards. Howevery the angle of depression is actually the angle on the outside of point abc or cba facing downwards. Here is a visual, x = angle of depression
If we are given the measurements ca = 5m, bc = 4m, and angle cab = 34°, we can find the angle of depression. Here is an equation solving for cba or y,
Now, since the angle of a straight line = 180°, we can find the angle of depression by 90 – 51.3 = 38.7
this makes all the angles add up to 180°, so the angle of depression = 38.7°
Unit 5: Polynomial operations
When simplifying polynomials, I find the most used step is FOIL. It is used when distributing a bracket with 2 or more terms into another bracket with 2 or more terms. The letters FOIL stand for First, Outer, Inner, Last. First means multiply the terms which occur first in each binomial. Then Outer means multiply the outermost terms in the product. Inner means multiply the innermost two terms. Last means multiply the terms which occur last in each binomial. Then simplify the products and combine any like terms which may occur.
Here is an example of me going through the steps
- (x + 5) (x + 5) Multiply the first terms in each bracket (x), → (x + 5) (x + 5) = x²
- (x + 5) (x + 5) = x² Now take x and multiply it with 5, the outer most terms in each bracket (x and 5) →(x + 5) (x + 5) = x² + 5x
- (x + 5) (x + 5) = x² + 5x Next multiply the two innermost terms together (x and 5) →(x + 5) (x + 5) = x² + 5x + 5x
- (x + 5) (x + 5) = x² + 5x + 5x Now multiply the two last terms in the brackets together (5 and 5) →(x + 5) (x + 5) = x² + 5x + 5x + 25
- The last step is to collect any like terms. The only two like terms in this question is 5x and 5x which when collected will get you the final answer of x² + 10x + 25
Unit 6: Factoring polynomial expressions
When factoring an polynomial in the x² + bx + c, it is important to remember that the factors of the 2 last terms in the equation must multiply to = c, and must add to equal bx. For example if I am factoring x² + 3x + 2, the x² will be the first terms in each bracket (x + ) and (x + ). Now, we list all the factors of the outermost term. For 2 it is -2 x -1, and 2 x 1. Now we pick the set that also adds to = 3x. -2 + -1 + -3, and 2 + 1 = 3, so the right set of numbers to use is 2 and 1. Now we plug them into the bracket to get (x + 2) (x + 1). You can check your answer by re-expanding the equation.
Unit 7: Relations and functions
An important part of relations and functions is finding the x and y intercepts of equations. If we were to find the x intercept of the equation, we know that the coordinates for an x intercept must be (?,0). The y is always zero, same for the y intercept, except the x is zero. Now you plug the 0 into whatever equation you are doing and solve for the variable. Here is an example,
y = 4x + 8
to find the x intercept we plug zero into then y spot
0 = 4x + 8
next we begin to isolate x by removing 8. Minus 8 on both sides
-8 = 4x
now divide 4x by 4 to completely isolate x. Do it on the other side as well
-8/4 = x
finally, simplify the equation
-2 = x-int
Unit 8: Characteristics of linear relations
The main point of unit 7 is finding the slope with 2 sets of 2 given coordinates. Slope is used to find the equation of a line, and finding things like x and y intercepts of graphs. The basic formula for slope is rise over run, which translates to m = (y2 — y1) / (x2 — x1)
it is a common mistake to put y as the numerator and x as the denominator, so to remember this think of the word run. If you were to run on the graph, you would be running straight across, or on the x axis. So x would go where run is, which is on the bottom. Here is a written example of me finding the slope using a given pair of coordinates,
When plugging in the y and x into the equation it does not matter which is which. However, if the y in one set of coordinates is y2, then the x must be x2 as well, not x1.
Unit 9: Equations of linear relations
An important skill learned in unit 9 is being able to convert general form into slope y-int form and vice versa. Most questions will either start off in general form, or start in slope y-int form and answer in general form. To solve an equation in general from, you must first change it to slope y-int form. Here is a video I found useful while doing this,
when converting to general form, it is important to note that the first number or Ax, must be positive. If it is negative you must divide it by -1, along with all the other numbers.