I'm just going to say it this is the best SAT Math video on YouTube because I'm going to go over literally every single concept process and formula that you will see on this test oh but Eric you can't go over all the SAT yes I can I've broken everything down into five levels from gooo gag slope intercept form to problems you didn't even know were on the SAT so don't let your Zoomer brain click off this video sit back relax and lock in until the end I promise it'll be worth your time level one you are in the absolute trenches of SAT Math but don't worry there are three things that I'll get you out in this order algebra Desmos linear functions when I say Algebra I mean basic algebra isolating variables adding fractions together order of operations which apparently in Britain they call bid Mass because they're pretentious I've put some basic algebra drills in the description but most of you are probably in good shape the second thing is Desmos I made a whole video on this so I'm not going to talk a lot about it in this video but the comment section agrees it's very important especially for single variable equations if you are simply asked to solve for x doesn't matter if there's a fraction or a square root or an absolute value just Chuck it into Desmos and the vertical lines are your Solutions and third linear functions a function in general is just a machine that takes an input and spits out an output so maybe you have a machine that takes a number and adds seven to it we would write that as F ofx equal x + 7 so if you input zero it outputs 7 if you input one the function outputs 8 we can think of each input as an x coordinate and each output as a y-coordinate and if we graph those points we get a straight line that's the linear in linear function and it's very important that you understand how this equation relates this table how the table relates the graph how the graph relates the equation it's this beautiful circle of life I'm sure a lot of you think this is piss easy here's a problem for you take these two points and find the line that connects them okay I'm I'm kidding this is a simple problem for most of you you find the slope and then you find the Y intercept to find the slope you look at the rate of change basically how much is the Y changing for a given change in X so our y goes up by six our X goes up by three therefore the slope is two the Y intercept you take your equation plug in the slope plug in one of the points in this case we're going to plug in 15 and then solve for b that's it the equation of this line is y = 2x + [Music] 3 level two is like the silver of SAT Math everybody is here and nobody wants to be variables and constants you're never going to be asked what these terms mean but you shouldn't be intimidated by them consider this linear equation if we increased X Y would increase but m and b would stay the same this is because X and Y are variable and M andb are constants the variables vary the constants St constant while we're at it a coefficient is a number that multiplies a variable so coefficient variable constant next you need to understand systems of equations really well conceptually a system is just two constraints on two things let's say I went to the store and I bought apples which cost $3 and bananas which cost $1 those are my two things then for the two constraints let's say first I have to spend $20 and second I have to buy 10 fruit each thing can be represented by a variable and each constraint can be represented by an equation getting as far as the hard part the rest is pretty simple a solution to a system of equations is just a combination of apples and bananas that satisfies these two constraints and so if we put both of these equations into Desmos the point of intersection is the solution in this case five apples and five bananas so since solutions to systems are points of intersection then two lines with no solution should be parallel and two lines with infinite Solutions should be the exact same line again you want to think about these questions graphically not algebraically you should also know that if two lines are parallel that means they have the same slope and if two lines are perpendicular that means they have negative reciprocal slopes so like if this slope is two then the slope would be ne- 1/2 here's a problem to practice systems of inequalities are solved very similarly and I talk about them in my Desmos video need to be comfortable translating English to math so if you get the sentence X is half of 50 you should be able to translate that into x = 12 * 50 notice that is means equals and of means multiplication this is just generally useful for word problems but the SAT will also give you problems like this next you need to memorize these six exponent rules if any of these are confusing like let's say the product Ru is confusing you can break it down into repeated multiplication so let's say you have x^2 * X 3r that breaks down into 2 X's * 3 X's which breaks down into 5 X's which is X 5th so you add 2 + 3 to get the final exponent 5 now it's time for [Music] geometry these two angles make up a right angle so they sum to 90° these two angles make up a semicircle so they sum to 180 these two angles make up a full circle so they sum to 360 any angles across from each other are the same angles on Parallel Lines from one another are the same all the angles in a triangle add up to 100 80° all the angles in a quadrilateral add up to 360° you can think of it as two triangles stapled together and generally speaking if you have a shape with n angles then it has 180 * nus 2 interior angles okay question for the audience if you're given a triangle with two side lengths how do you find a [Music] third that's right Pythagorean theorem you should know this anytime you have a missing side length in the right triangle use this formula it comes up a lot also very important is s cosine and tangent remember learning soaa in Geometry no uh well figure it out and then there are a couple of small things you need to know that density equals mass divided volume you should know that an isoceles triangle has two common sides and an equilateral triangle has three common sides you should know what perimeter area and volume are but don't worry about memorizing their formulas per is self-explanatory you add up all the side lengths and area and volume formulas are all on your formula sheet so they're just plug and chug here's an example problem which I solved using the sponsor of today's video Acy so the first thing I did was tell Acy but I have no idea what a right circular cylinder is I've never heard those words in my life then Acy broke it down for me said it's like a can of soup and here's the formula I need so I asked Acy if that formula is on my formula sheet and it said yes because it is built for people taking the SAT so I plug in my numbers solve the equation and get H equal 3 which Acy says is correct now if I want more practice I can come down here and get a similar question from acy's Team or this is what really shocks me I can just say hey create a problem like this but for another shape and boom a problem that is 100% plausible to see on the SAT is instantly generated listen I reject almost every company that tries to work with me this is seriously impressive it's essentially a private tutor but for a fraction of the cost if you want to try AC for free go to the link in my description and pin comment below and use code SAT Math 10 for $10 off a monthly subscription finally a bit of Statistics mean is the sum of all the values in the data set divided by the number of values in that data set and median is just the middle number you can use Desmos to find mean and median but typically these problems are more conceptual for example let's say we have data on the wealth of every single American which number would be larger the mean wealth or the median wealth the mean wealth is larger in fact by quite a bit it's five times larger because when you calculate the median Elon Musk counts just as much as the poorest person in the country right you cancel out the left you cancel out the right but when you're calculating mean you have to sum everything and so you're effectively waiting the mean by people's wealth Elon Musk single edly adds $3500 to the average wealth in the country which is crazy the mean is still useful however if we rearrange the original formula that we had we can use the mean to figure out the sum so if we take the average wealth per household and multiply it by the number of households you get the total household wealth in the entire country which is $139 [Music] trillion oh also the range is the max minus the Min it's it's very [Music] forgettable level three you're starting to know what you're doing you should know that an integer is a positive or negative whole number or or zero I guess cuz Zer is not positive or negative you should understand how to translate a function horizontally or vertically which by the way can often be solved with Desmos pretty quickly you need to understand percentages which might just be the most practical thing on the SAT 20% of X is 0.2x that's like your tip at a restaurant increasing X by 20% is 1.2x that's like your total after the tip and decreasing X by 20% would be 0.8x the analogy kind of falls apart here you please don't tip - 20% so what is 150% more than [Music] 10 if you said 15 you should give up on your hopes and dreams because the answer is 25 we want a 150% increase not a 50% increase and there is a difference here's a problem to practice let's talk about circles for a bit the general equation of a circle is very very important and very much worth memorizing but you can actually derive it yourself pretty easily we have a circle with a center and a point the line that connects these two points is the radius then we can break down this radius into a horizontal distance of x - 8 H since that's how far apart the x coordinates are and a vertical distance of y- K since that's how far apart the vertical points are now here's the magical part if we write out the Pythagorean theorem for this right triangle we get x - h^2 + y - k^2 = R2 that is the circle formula the circle formula is literally just a Pythagorean theorem that's all you need to know here's a problem using it here's another you might also be asked to find the slope of a line tangent to a circle that means this line and just know that this is perpendicular to your radius so think back to what we said about perpendicular slopes here's a problem to practice arc length and sector area are also important but don't try to solve them formulaically just think about what fraction of the circle the arc makes up for example if your circle has a circumference of 12 and you want to get the length of this Arc look at the central angle in this case it's 60° which is 1 6 of the entire circle and so we would take 1 16 multiply it by the circumference and that is your Ark length x same thing with sector area like this area in blue you would just replace circumference with area and find a fraction of that number then there's the inscribed angle theorem I have absolutely no idea why this is tested but basically this is an inscribed angle and it is 1/2 the measurement of this central angle so if x were 40 then the central angle would be 80 finally you need to know how to convert between degrees and radians I recommend memorizing that 180° equals Pi radians and then just use the appropriate fraction depending on which way you're converting triangle time congruent triangles are copies of one another notice they both start with a c this means they're literally the same triangle whereas similar triangles are scaled up or down versions of one another so like think of dragging the corners of the triangle to make it larger or smaller without actually stretching or compressing it since the two triangles are the same triangle just scaled up or down they have to have the same angles they have to have proportional sides and their trig ratios have to be the same so question for you guys are these two triangles similar they definitely look similar right these sides are proportional these sides are proportional and these angles are the same but because this is side side angle similarity they're actually not necessarily similar consider these two triangles I found online they share the same exact side length the same exact other side length and the same angle and yet they're completely different triangles and so in order to prove similarity you can use angle angle side side side side angle side but not side side angle and yes the SAT tests this also tested by the SAT is conditional probability let's say we have data on men and women and whether they're above or under 6 ft me personally I'm 61 and that's completely relevant to to the problem let's say we're trying to find the probability of selecting a woman that would just be 50% right there are 200 people in our data set half of them are women so there's a one half probability of selecting a woman but if we're trying to find the probability of selecting a woman given that we're choosing someone over 6 foot we're only selecting from this row we're choosing people who are over 6 foot and only one in 16 of them are going to be women here's a problem to practice [Music] box plots look like this and they tell you a data set's minimum 25th percentile 50th percentile which is the median 75th percentile and maximum and you'll notice this box plot is showing you SAT scores in fact if you're at level three you're probably around here Scatter Plots you need to know how to interpret them and how to find the line of best fit and you also need to know when a sample is biased let's say I'm trying to gather data on the birth rate in America I can't just go to a playground and ask every adult hey how many kids do you have because my sample is obviously going to be biased in fact I'd be very concerned if it weren't and this exact scenario was an SAT question you want to be collecting a random sample that is representative of your population and finally you want to start looking at quadratics problems you should already be comfortable factoring by this point you should also know that to find solutions to a quadratic you can just plug it into Desmos Desmos also makes it easy to find the vertex you just click on it that's literally it and with that you're ready for level four [Music] Welcome to Hell this is the standard form of a quadratic a determines whether we're upward facing if it's positive or downward facing if it's negative and C is the Y intercept this is the vertex form of a quadratic the a is the same a as in standard form and the HK is the vertex you can think of this as like a horizontal shift H and the vertical shift K from the origin you need to memorize this formula we can also compare these two forms with the equation h = b / 2 a so basically if you know A and B then you know the x coordinate of the vertex this is a useful formula you should memorize it but we're still not done we know that the vertex is the midpoint of our two solutions so like if our Solutions are 2 and six the vertex should be at four you should memorize this as well but wait we're still not done if we compar this formula with this formula we can see that the sum of our Solutions has to be equal to b/ a you should use the shortcut anytime the SAT asks for the sum of solutions okay got everything there's even more there's also a shortcut for the product of your two solutions and you can use it for problems like these and you're probably thinking oh now he's going to say we're we're still not done and we you got to know the discriminant formula too jokes on you you actually don't need to know the discriminate at least not until level five if you get a problem asking about a quadratic intersecting line I recommend you solve it with Desmos and I show you how to do that in my Desmos vid so I'm not going to repeat myself you also need to understand polinomial functions kind of it's a bit unclear what college board expects you to know but they've given a lot of weird polinomial problems like this one where if you understand functions at a deep level you'll be fine but definitely worth practicing some of these one thing that you probably should memorize is that the function goes through the xais for every solution with an odd power and bounces off the x-axis for every solution with an even power exponential functions let's say you put $1,000 into a savings account and it grows by 4% each year so you start off with $1,000 then you have 1,000 * 1.04 then you have ,000 * 1.04 * 1.04 and so on since we're just doing repeated multiplication over and over again we can express the amount of money we have with an exponent so the exponential function would be 1,000 * 1.04 to the T you should be comfortable constructing exponential functions in the same way that I just did right now you should also think about what happens if we shift the exponent or if we scale the exponent how that changes the interpretation of the problem here's a problem to practice here's a second one and here's a third one you should also try out some hard algebra problems like this one and those stupid real number problems that I absolutely despise what I recommend for these is just plug in R equals 0 into all the answer choices and if that doesn't give you a concrete answer then try r equal 1 shifting gears to Geometry consider this right triangle the S of a is 3 over 5 the cosine of C is also 3 over5 the cosine of a is 4 over 5 and the S of C is also 4 over5 so notice a pattern the S of one angle is the cosine of its compl angle the SAT loves this formula so definitely do memorize it and they love to test it in weird ways like this problem right here now consider the square it's 1T by 1T so what is the area of the square in square [Music] feet if you answered one square foot you're correct that was an that was an easy problem but what is the area in square in it's 144 it's it's 144 square in because 1 foot is 12 in and the area is your side length squared um so the square unit conversion is the base unit conversion but squared and this is a nice shortcut to keep in mind this also applies to scaling up units if we double the side lengths of a square we 4X the area and I'm not drawing a visual for this cuz I don't want to but if you double the side lengths of a cube you 8X the volume because the formula for volume of a cube is your side length cubed and 2 cubed is 8 here's an example problem applying this idea to cylinders you should also be comfortable with surface area but do not memorize any formulas College Board loves to give these weird scenarios like two boxes glued together and there's no formula to memorize for that you just need to understand how to set up the equation yourself finally before you get to level five you want to be more comfortable with Statistics you want to be a master at reading frequency tables cuz College Board loves to use these and notice that if a data set is symmetric that means the mean equals the median if it doesn't make sense why just take a minute and think about it you also need to know standard deviation you'll never need to solve for it but you need to know what it measures which is the spread of the data so if data is closer together like in data set a the spread is lower and so the standard deviation is lower versus in data set B the data is more spread out and so we would say it has a higher standard deviation last but not least margin of a it sounds simple but most people don't actually know how it works let's say your school has a th000 students you sample a 100 of them and ask them what their SAT Math score is the average score from your sample is a 600 with a margin of error of 30 points this means that we believe in the entire School the average SAT Math score is somewhere between a 570 and a 630 here's my question for you is it possible that the school's average SAT Math score is a 240 the answer is yeah it actually is possible imagine if we got really unlucky with our sample and happened to pick the 100 best students and everyone else is stupid and got a 200 in this case our average among all 1,000 students is a 240 so then why is our margin of only 30 points well margin of error just reflects some level of confidence so maybe we're 95% confident that the average score is between 570 and 630 or maybe we're 90% confident or 99% confident it's technically up to the statistician but this is too much to worry about for the SAT all you need to know is that a 240 and any other value outside the margin of air is possible but not plausible those are the terms that the SAT uses and one last thing if we want to reduce the margin of a we just have to take a larger sample and just like that we're at level [Music] five finally after all this time hours of studying sleepless nights hundreds of practice questions you have now become an average student in China but don't worry there's not a lot that you have to learn before you're at a perfect score the first thing is the altitude of a triangle which is this dotted line that goes from the right angle to the hypotenuse it is perpendicular to the hypotenuse it is not necessarily the midpoint but it is always perpendicular to the hypotenuse the altitude splits the big triangle into two smaller triangles this one and this one and the important thing is that all three of these triangles are similar which means these angles are the same these angles are the same and a bunch of sides are proportional to each other to show you how the sides relate to each other let's draw this top triangle such that it is proportional to ABC the big triangle first we want to put angle D in the bottom left because it's the right angle then we're going to put angle a on top because that is the same angle in the same position as in ABC and then the remaining angle is B and know that this is not the right angle B this is like the red angle B the acute angle so we would say triangle ABC is similar to ADB and once you set this up on paper then you can solve it like a normal similar triangles problem here's a problem to practice then there are these problems where you're given a quadratic and mystery integer constant factors and you have to figure out what answer Choice has to be an integer for these problems there's no easy way to do this you have to set up all the equations comparing each of your factors to the ABC terms then you kind of just have to think about them so KJ equal -45 which means J = 45 K which means negative J = 45 K and since J is an integer that means negative J is an integer which means 45 over K is an integer which means the answer is D here's a similar problem to practice and here's a different problem but sort of The Same Spirit College Board is actually trolling they put a pyramid surface area problem on the August sat the October sat and the November sat despite never mentioning it on any Blue Book practice test I I cannot tell you why they like pyramids now but they do for some reason now you might think this is simple the surface area of a square right pyramid is just the square base plus the four triangular sides which all have the same area and yes that part is easy what makes these problems difficult is that they often involve the height of the pyramid which is not the same as the height of each triangle that's called the slant height so what you can do is make a right triangle where the height is the pyramid height the base is the ground and the hypotenuse is the slant height and you can use this right triangle to so try out the August question the October question and the November question for yourself the other way college board can be evil is by putting a bunch of constants in the problem and this is evil because often times it prevents you from being able to use Desmos so if you're going for a perfect score you need to know how to solve every single Problem by hand this means memorize the quadratic formula memorize the discriminant of a quadratic and how to use it for problems like these and know how to complete the square for Circle problems there are also a few things that are nice to know like the slope of a line in standard form is A over B The Arc Length can be found by multiplying your radius by the central angle in radians 345 triangles and 51213 triangles are pretty common and the triangle inequality theorem which states that the sum of two sides of a triangle must be greater than the third side finally you should practice the hardest problems in every category but I recommend paying special attention to mean problems involving integers exponential functions with complicated fractional exponents and quadratic problems where you solve for some combination of a b and c and of course focus on your weaknesses go back to every problem that you've missed in the past and ask yourself how can College Board make this even harder and would I be able to solve that harder variant if it showed up on my SAT this is a very effective strategy you're basically predicting the SAT before you take it and if you want to get one-on-one tutoring from me personally head over to learnmath outcom I will get you results and finally this video took way too long to make so if you enjoyed it be sure to like and I don't know share with your grandma