- Hello everyone and welcome
to Khan Academy live SAT class.
My name is Eric Li, I'm an SAT tutor
and one of the SAT experts
here at Khan Academy.
I'm so excited to be joining you today
and over the course of
the next three weeks
as we practice and prepare for the SAT.
Before we jump into SAT math,
which is the topic for today,
I wanna give you a sneak
peek at what's coming ahead
over the course of the next three weeks.
Let's take a look.
So, today, like I said,
we'll be covering SAT math,
but next Thursday, I'll walk us through
some SAT reading topics
and the following Thursday,
we'll cover SAT writing.
Now each of these classes
is going to take place
Thursday nights at 7:00 p.m.
Eastern, 4:00 p.m. Pacific,
and we're covering a
different topic each time
so make sure to tune in.
Now if you can't make
any of these classes,
don't worry because we'll
record all of the classes
and send them out to everyone afterwards
so you can review it whenever you want.
So I'm super excited to
be going through this.
And for today, we're gonna
focus on math like I mentioned.
So what's the plan for today?
Well, the first 25 minutes of today,
I'm gonna walk through how to avoid
careless mistakes in the SAT math section.
Next I'll show you how to approach
function notation problems,
which are a common concept
tested in the SAT math section.
Then in the next 25
minutes, you'll get a chance
to practice what you've
learned on Khan Academy.
And while you're practicing,
we'll have Khan Academy
and College Board staff on standby
who can chat with you one on one about,
and answer any question
you have about SAT math,
reading, writing, or anything SAT related.
You'll want to make sure to
practice during this time though
because these staff members
are only gonna be on
until the end of the hour.
Then in the last 10 minutes of class,
I'll answer a few
questions, discuss homework
and the plan for the next class,
and then you'll get your badge
for completing the class.
But remember you have to stay
to the very end of the
class to get that badge.
So, that's the plan for
the next three weeks
and what we're going to cover today.
And I'm so excited to
get started with SAT math
and to help you practice and
prepare for your upcoming test.
Let's jump in.
So why are we talking
about careless mistakes?
Well, let me know if
any of these questions
sound familiar to you.
So, have you ever solved
for the wrong variable
in an SAT math question?
Or maybe you missed a question because
your final answer was in the wrong units.
Or maybe you were
working through a problem
and you stopped one step
short of the right answer.
You were almost there, but not quite.
Or maybe you went one step too far.
Or maybe you made a simple
addition or subtraction error.
Or this one is probably
the most annoying to me,
maybe you even got the right answer,
but then you didn't transfer
the right answer to the answer sheet.
Well, all of these types of mistakes
are perfectly common human
mistakes that anyone can make.
But what I'm gonna show you today
is a process that's really helped me
and helped other students
reduce the number
of careless mistakes they make.
There are other ways to go about it,
but this is one process that's
worked really well for me.
So let's take a look.
So there are really four steps
that I'd recommend you take
to cut down on careless mistakes.
So the first one is just
to read and understand
the question and any charts
or tables that you have.
Now this is both the question
and any charts or tables,
so don't move on until you
really understand everything.
And one thing you can do
here is underline as you go
sentence by sentence to
break down the question
rather than trying to
take it all in at once.
The next thing I'd recommend
is that you think through
the steps you need to
take to solve the problem.
So this includes things like figuring out
which variable you're
solving for, so X or Y.
This is thinking about what units
your final answer needs to be in.
And any operations that you need to do
to get to the right answer.
Once you know what steps
you're going to follow,
you can then start
actually doing the math.
I would do this on paper
or in the calculator.
And I'd really discourage
you from doing the math
in your head, 'cause
no matter how confident
you might feel about the problem,
the moment you're doing mental,
you start doing more mental calculations,
you really up your chances
of making a careless mistake.
Now, the last and final step
that I'd suggest you take is
to mark down the answer
in your test booklet
before bubbling it in.
So say our answer was B 64.
Then I'd recommend you
circle it or you write it
and you box it so that when
you're transferring answers
for this problem to the answer sheet,
you have no doubt in the
world that your answer is B
and the right answer makes it over there.
So, to recap, you're gonna
wanna read and understand
the question and any charts or tables.
Think through the steps you need
to take to solve the problem.
Do the math on paper
or in your calculator.
And then mark down the
answer in your test booklet
before bubbling it in.
So this is a process that
we're going to now apply
to real SAT math questions.
And let's take a look
at what this looks like.
The first question will
be more straightforward
and then we'll see how this applies
to more complex questions as we go.
So, let's take a look
at this first question.
So, first step is to read
and understand the problem.
So during a timed test,
Alexander typed 742 words
in 14 minutes, okay,
742 words in 14 minutes,
that's seems worth underlining.
Assuming Alexander works at
this rate for the next hour,
which of the following best approximates
the number of words he
would type in that hour?
Okay, so I have some information
and for me it's helpful to write down
the information I have
since it just helps me
better understand it.
We have 742 words in 14 minutes
and then the question is, how many words
would Alexander type in one hour,
so in this case it's 60 minutes, okay.
So I understand the question
and the information
that's been given to me.
And the next thing I wanna do
is think through the process
I wanna follow to solve this question.
So I think the first thing I could do
is figure out how many
words per one minute,
per one minute, and then
because the question
is asking me how many words
would he type in one hour,
the second thing I can
do is multiply by 60.
So that seems like a good process
and will get me all the
way to the final answer.
So now we can actually
start doing the math.
So to calculate words per one minute,
I can just do 742 words
divided by 14 minutes
and I'll pull up my calculator.
In the real thing, you won't have this
digital calculator, you'll have your own.
But just bear with me here.
Here I have 53 words.
That's 53 words per one minute.
Words per one minute.
But I'm not done yet, right?
So the second step here is
to multiply by 60 minutes.
And the interesting
thing you'll notice here
is once I multiply by 60 minutes,
I've actually got a minutes
factor in the numerator
and a minutes in the denominator,
any unit divided by itself
is just gonna be one,
so that would go away.
And then if I do this math of
53 times 60, let's do that.
Multiply this by 60.
We get the final answer of 3,180.
And then last step is to
box that answer or write C
so that there's no chance
of me missing, misbubbling.
So one thing, the last thing
I wanna show you here is
what the process does
in questions like this.
So because we knew that
we needed to solve first
for the words per minute
and then multiply by 60,
it meant that some of these other answers
were less tempting.
So while we got 53 words here
and while it was an answer choice,
we knew that that wasn't
the right final answer
because we hadn't taken
the second step here
of multiplying by 60
minutes to get to the hour.
Hopefully this shows you
how to apply this process
with a question that has a few steps in it
and you're dealing with some units.
But let's see how this process
stacks up when we apply it to
a different kind of
question, different scenario.
So in this next question,
let's give it a read.
Geometry students
participated in an activity
to classify the shapes in the room
by number of sides and color, okay.
The table above displays the results,
and then if a triangle
is chosen at random,
what is the probability that it is blue?
Okay, so, we're classifying shapes
and we have, we wanna
understand the probability
that if a triangle is chosen at random,
what's the probability that it is blue?
So before I move on from here,
just 'cause I understand the question,
doesn't mean I'm ready to
actually start solving it yet.
Because in this case,
we have this table here
that we want to tackle first.
So let's take a look and make sure
we understand the table
before we go any further.
So we've got shapes on the left hand side,
we've got these colors
here of red and blue.
And then these total columns.
So total, this total column
on the right hand side
seems to be the number of shapes.
And then this one on the bottom,
this total column is number
of shapes that are by color,
and this one is shapes by type, okay?
Okay, so now I understand the question,
I understand the table,
and now I can start
to think about how I would solve this.
So, let's read that question again.
If a triangle is chosen at
random from this activity,
what is the probability that it is blue?
So that to me says that
I'm gonna be focusing here
on just the triangles because it says,
if a triangle is chosen at random,
so I'm not worrying about
these quadrilaterals
or the pentagons or the others.
And then it's asking me,
what is the probability that it is blue?
Well, I've got the blue triangles here.
There are three blue triangles
and probability is always how many chances
of getting something out of
the total number of chances.
That total number of
triangles then is here.
And you've got three out of four.
Now what I want to point out here is
in this problem, understanding
the table was crucial
because you'll notice if you look at some
of the other answer choices,
the totals that you have
in the rest of the table
are also present here
and if you didn't take your
time to understand the table,
it might've been tempting to use
some of these other variables
that you see in the table
like the 16, the eight
or even this one as well.
But if you follow an organized process
then that's how you can cut
down on these kinds of mistakes.
So, we've done now a question with units
and a few steps in the process.
We've done a question now with a chart
that you've been presented.
Now let's take a look at what happens
with a problem that has a few steps in it
and you don't have a diagram drawn for you
or a chart presented to you this time.
Let's take a look.
So let's give this problem a read.
Cara is hanging a poster
that is 91 centimeters wide
in her room, okay?
The center of the wall is 180 centimeters
from the right end of the wall, okay?
If Cara hangs the poster so
that the center of the poster
is located at the center of the wall,
how far will the left and
right edges of the poster be
from the right end of the wall?
Okay, so a little bit more information
to work with this time, right?
But this is where
breaking down the problem
piece by piece and drawing
a diagram as you go
can really be helpful.
So let's take a read.
Cara is hanging a poster,
it's 91 centimeters,
so I've got my beautiful poster,
excuse my inability to
draw straight lines.
So this is my poster.
And we've got 91 centimeters here.
And then I've got a wall that
I'm hanging this inside of.
So if I draw this wall, and
it says the center of the wall
is 180 centimeters from
the right end of the wall.
So then I'm just gonna estimate
the center's gotta be about there.
If this is the center, the center is
180 centimeters from the right.
So we've got 180 centimeters.
And then I have this diagram
and now I'm putting the poster
so that the center of the poster
is located at the center of the wall.
So then I'm gonna move
that poster inside here.
And then this again is 91 centimeters.
And now what is the question now?
Now I have this diagram that
can help me solve the problem.
The question is how far will
the left and right edges
of the poster be from the
right end of the wall.
Okay, so I've got this
right end of the wall.
And I wanna know this distance,
so the left edge of the poster
to the right end of the wall.
And then the right edge of the poster
to the right end of the wall.
And I can label this one l and then
I'll just change colors
here, this one I'll label r.
So now that I have the diagram,
it really helps to clarify
what I'm dealing with
and what I'm solving for.
So let's take these one by one.
So if want to solve for this distance of l
from the left edge of the
poster to the right wall.
It looks like this 180 centimeters here
is a pretty good starting point.
So if I, it's at least 180 centimeters
since I'm all the way to the center
and then I have this half of the poster
that is just to the left of the wall.
And because I know that this
poster is 91 centimeters wide,
I can add 91 divided by two.
Then this should get me the value of l.
If I do open up this calculator,
I've got 180 plus 91 divided by two.
And that'll get me 225.5.
225.5 centimeters.
And now I want to go to the right edge.
And I think I can use
this 180 centimeters again
as a reference point
since I know that poster
is right in the middle, right?
So we've got 180 centimeters.
And then this time,
this half of the poster
is actually closer to the
right edge of the wall
than this one, instead
of adding something,
it looks like we'll have to subtract.
And again since I know that
this is 91 centimeters total,
I can subtract 91 divided by two.
And then pull up that calculator again.
180 minus 91 divided by two
gets me 134.5 centimeters.
So then I want to look
at my answer choices now
and I've got the left edge at 225.5.
Okay, well that's one's
out and that one's out.
And then the right edge should be 134.5,
then my final answer again is C.
So in this problem, it was very important
to work through the
information given to you
step by step and to draw a diagram
to help you work through
it piece by piece.
Because if you had done the simpler,
tried to skip through and do it mentally,
you might've done 180 minus 91 or plus 91
and you would've missed
some of the nuances
in this problem, but once
you draw the diagram,
it becomes a lot easier to work through.
So we've now done a handful of problems
to show you how you can use this process
to really reduce your chances
of making careless mistakes.
And now I wanna move us on to
tackling function notation problems.
So let's keep going and let's take a look.
So, function notation.
Before I actually start
doing practice problems here,
I just want to walk you through
a quick review of function notations.
So what is a function first?
Well a function, a function
is a mathematical expression that takes in
one or more inputs,
inputs, and generates only
one output, one output.
And that's really key though,
having one or more inputs,
but all of those only ever
generating one unique output
is what really sets a function apart.
Now a few ways you might
see functions notated
in the SAT is something
like f of x or maybe h of x
or you might see a
composition of functions
where you have f of h of x,
and this is also equivalent
to this f o h of x,
these two expressions are the same thing
just a different kind of notation.
Now, let's say for example,
f of x equals x plus one.
Another way you might see function show up
is in a table format, so, in this case,
you have this nice table
here of x and f of x.
And for given values of x,
given this definition of f of x,
you'd have certain values of f of x.
So this is one way you might see it.
And then the last way you might see
a function show up is graphically.
So here we have a graph where
there's x and there's f of x.
And for example, there's one, two, three.
And you've got one, two, three.
For certain values of x, you might have
corresponding values of f of x or y.
With a chart like this
you can really trace
on the x-axis and then go up and see what
the corresponding y
value or f of x value is
at that value of x.
So these are really,
this really covers a lot
of the different ways
that function notation
can show up in the SAT.
But you haven't seen a
practice problem yet,
let's now apply what we've
seen here to see it in action.
So let's take a look here.
Just 'cause we're no
longer focusing exclusively
on careless mistakes,
doesn't mean we're changing
our process though, so
same process as before.
Consider the table shown above.
What is the value of f of f of four?
Okay, so I've got that as the question.
And then I have a table here,
so for a given values of x,
then I have corresponding
values of f of x.
We've got f of f of four.
So when you have a function
notation question like this,
the key thing to remember
is that you want to work
from the inside out.
Now what do I mean by that?
I mean that the first
thing you wanna solve for
is this f of four and
then you can worry about
the f of f of, I may just rewrite that.
Then you can worry about the f,
f of f of four.
If I solve first for f of four,
it's really asking me,
what is the value of f of x
when x is four here?
So if I've got, use the
table, I've got four
and the corresponding
value of f of x is zero.
So then this zero I can
substitute into here
and then this expression
simplifies to just f of zero.
Then I can return to the table.
And for that value, when
x is zero, f of x is two.
So I've got two, I box the right answer.
Realizing now all of the answers
to the sample problems seem to be C.
(laughing)
Promise that was not on purpose.
But let's keep going and
see what this looks like
when the table's a bit more complicated
or there are a few more pieces
of information to work with.
Okay, consider the table shown at left.
What is the value of g
of f of negative one?
Okay, so for me I prefer to see this as
g of f of negative one,
that's just easier for me to understand.
It's always okay to rewrite a problem
so that it's a, or rephrase it,
so that it's easier for you to understand.
So again this is similar to the last one
and I want to work from inside out.
But, let me make sure I
understand the table first.
So this time I've got
given values of x here.
And then I've got f of x.
And then I've got g of x up here, okay.
So I want to solve then
inside and then outside.
f of negative one then, so
for, let's go to the table,
when x is negative one,
what is the value of f of x?
So it looks like we're right there.
We've got two.
And then we can plug this in
similar to the last problem
and simplify this to say g of two.
And in this case we return to the table.
We're looking at two.
And then this time I'm looking at g of x,
so then I'm looking over here at four.
So this is another example
of function notation
just with a different
way of expressing it.
And now we'll do one final
function notation problem,
this time with a graph.
So let's keep moving.
So what is the approximate
value of g of f of five?
So, that one makes sense to me.
And then on this left hand side,
I have the graphs of the function of f
and function g shown below.
So here's f, here's g, and then
for certain values of x here
the y values of f and g are shown.
And okay, so I think I understand that.
Now again, I want to solve inside out.
So f of five and then
I've got g of f of five.
And what we can do here is,
we'll now have to reference
the chart here, so.
When x is five, to solve for f of five,
what is the value of f?
So what I can do is trace up through here.
And it looks like I'm
intersecting f here at nine.
Then the value of f when x
is five is going to be nine.
And I can substitute that into here.
And this becomes g of nine.
And then I can look at the table here.
So when x is nine, I'm
here, I can trace up
and this time I'm looking at x.
And it looks like I'm
intersecting here at seven.
So then this value is seven.
I mark down seven.
And you're good to go.
So now you've seen a few ways to
reduce the chances of making
careless mistakes on the SAT,
you've seen a few
different flavors of that
with a few different problems.
And you've also seen a
few different examples
of function notation problems.
So now, it's probably
easier to follow along
as I was working through it,
but to see if you've understood it,
it's now your turn to
practice what you've learned.
So head to khanacademy.org/sat,
the link is also in the event description.
And head over there and join me
as I head over there and to
practice what you've learned.
Now for the next 30, for the
next about 25, 30 minutes,
Khan Academy and College Board experts
are available on standby to
chat with you one on one.
And there will just be a
little box in the bottom right
of Khan Academy where you
can click ask us a question,
and send us any SAT
related question you have.
Then at 6:50 Eastern, 4:50 Pacific,
come back here, join me here,
I'll wrap up, answer any
questions you might have.
And then I'll share a
badge, I'll share the badge
that you earned by completing this class.
So head on over, I'll head over there too.
And then I'll see you in just a bit.
