Good day students.
Welcome to part 1 of the January 2015 Regents exam for Algebra I Common Core.
In this clip we're going to be going over problems 1 through 5.
Let's take a look at problem 1.
It reads: The owner of a small computer repair business has one employee, who is paid an
hourly rate of $22.
The owner estimates his weekly profit using the function P(x)= 8600 - 22x.
In this function, x represents the number of.
So if you take a look at this paragraph right here, you have 22x appearing here and you
also have 22 over here.
There clearly is a relationship between these two values.
So let's talk about the 22 related to the employee first and then we'll see if it can
help us determine what this 22, what the x and the 22 mean.
So hourly rate, what does hourly rate mean?
Hourly rate means how much this employee gets paid an hour.
So the owner pays the employee so he's basically taking money out of his wallet to pay the
employee.
So let's say the employee worked 1 hour how much would he get paid?
He would get paid 22 times 1 which is 22 dollars.
Ok 22 time 1 which is 22 dollars, that's how much he'll get paid.
Now what if the employee works 2 hours? he's going to be paid 22 times 2 which is 44 dollars.
What if the employee works, x amount of hours? he's going to be paid 22 times x, which is
22x dollars.
So you see this 22x basically represents how much is coming out of the owners pockets to
pay the employee.
so the 22x, the 22 represents what?
the 22 represents the hourly wage.
Since he's getting paid hourly, what does x represent?
x represents the number of hours the employee works per week.
Ok so if you look at these options right here, what is the meaning of x, what does x represent?
x represents the hours worked per week.
Option number 2.
Let's take a look at problem number 2.
It says: Peyton is a sprinter who can run the 40-yard dash in 4.5 seconds.He converts
his speed into miles per hour, as shown below.
Which ration is incorrectly written to convert his speed?
So we're looking for the incorrect form, so for problems like this you have to be really
careful to watch out for the words that are italicized, in this case we're looking for
what is wrong, which is incorrect?
So this might look like a complicated problem but it's actually very straight forward.
Setting up this conversion factor right here, at the end of the cancellation you want to
have miles per hour.
Miles per hour, how do you write this in fraction form?
you have miles in the numerator divided by hours in the denominator.So in this problem
we're changing two things, we're changing yards into miles, we're going from yards to
miles and then we're going from seconds to hours.
Because the initial measure was yards per second, and now we're going to miles per hour.
So there are 2 conversions happening here.
Now why did I say it was easy?
Well all you have to do is cancel out the units and see, which one results in a complication.
So let's start with the time, time conversion, we want to convert 4.5 seconds into hours.
so at the end of my cancellation I want to be left with only hours in the denominator.
So focusing on this I'm going to focus on the time conversion factors, seconds and minutes
is time and then minutes and hours this is time.
So these two numbers right here are used to convert from seconds to hours.
So watch this cancellation happen, it's pretty cool.
Seconds, seconds cancel out, so 60 over 4.5.Minutes, look at this the minutes and the minutes cancel
out and you're left with hour on the bottom.
Is that what we want?
Absolutely!
So the conversion factors for the time are perfectly fine.
So 60 seconds and minutes, 60 minutes and hour, that's fine so let's see, we can eliminate
these two, they're not causing any problems, so the problem should lie with the conversion
of the length or distance.
Ok so we're going from yards to miles, which ever one prevents us from ending up with miles
in the denominator, then that is the incorrect one.
So let's examine this, you see you have a yard here and a yard here, that's nice, this
helps us go from yard to feet and now we have a feet here and uh oh we don't have a feet
down here, feet is upstairs, this is written incorrectly, it was supposed to be written
as 1 mile over 5280 ft so that the feet can cancel out, so since we have a problem here
this number is what is causing the problem.
It was written in the wrong setup, ok? so this one is good, this ratio right here is
good, the problem is 5280 ft per 1 mile.
So we're going to eliminate 1, the incorrect answer is option 2.
Alright, now let's take a look at question number 3.
It says: which equation has the same solutions as 2x square plus x minus 3 equals 0?
So this is assessing your ability to factor quadratics, quadratic trinomials.
That's all we're doing here, that's all we're just factoring.
So we have 2x square plus x minus 3 equals 0.
There are different ways you can factor this expression, I'm going to be using the x game,
that's the method I like to show my students to factor quadratic expressions.
So I'm going to set up my x game right here.
ac goes on the top and b goes on the bottom.
Now A the coefficient of the first number, b is the coefficient of the second number
which is 1 and then c is the constant.
Remember ax square plus bx plus c, the standard form of the equation of a quadratic, so I'm
just extracting the A, B and the C from there to populate the spaces in my x.
Now what is the objective of this?
the objective of this whole factoring process is to express this in four terms, if you have
four terms then you can factor by grouping.
So that's the whole goal, you want to factor it in such a way that you have a term that
has a common factor with the first and another term with the common factor of the last one
in such a way that you can create a product of factors.
So ac is -6 and b is 1.
Now we're going to think of two that multiply to give -6 and add to give you 1.
So we can use 2 times 3, since the sum is positive that means the small number is negative,
since the product of the two numbers is negative.
so we're going to put these two numbers in the middle, so we're going to write this expression
as 2x square minus 2x plus 3x minus 3.
Now how many terms do we have here?
We have 4 terms, so with four terms you can break it down the center and factor by grouping.
So break it down the center in that orientation there, now from the first two terms I will
factor out the greatest common factor.
The greatest common factor is 2x, so then I factor out 2x I'm left with x minus 1.
And then from these two terms I'll factor out positive 3 and I'm left with x minus 1.
Now a quick check to make sure you're factorization process is accurate, these two quantities
in the parenthesis must always be the same, if they're different then you made a mistake.
Now what I'm going to do is I'm going to factor out the common factors.
The common factors are x plus 1, so if I pull out the common factors I'll have x minus 1
and I'm left behind with 2x plus 3.
So we have times 2x plus 3 equals 0.
Since multiplication commutes we can write this as 2x plus 3 times x minus 1 equals 0.
So let's see which equation matches this result that we got.
We can clearly see that the answer to number 3 is option number 4.
Alright let's take a look at problem number 4, it says: Krystal was given $3000 when she
turned 2 years old.
Her parents invested it at a 2% interest rate compounded annually.
No deposits or withdrawals were made. which expression can be used to determine how much
money Krystal had in the account when she turned 18?
So in this problem you need to know your exponential growth, discreet exponential growth to be
specific.
You want to know the continuous and discreet growth of course but this one since the interest
is being compounded at discreet periods, namely annually , we're going to be using the exponential
growth formula for discreet growth.
So the formula is the final amount equals the principal, the starting amount times one
plus r raised to the t.
So let's go over what these different variables mean and then we will apply it to this problem.
A represents the final amount.
P is the initial amount or principle, what you started with.
R is the interest rate in decimal form.
What do you think T is?
T is time.
So let's find out what these values are in this problem here.
Let's start with A the final amount after she turns 18 , what is A?
A is what we're looking for, we don't know what it is.
P how much did Krystal's parents start with?
They started with $3000, so that is the amount that she was given.
What is the interest rate?
It's 2% in decimal form 2% is 2 over 100, if you want to write this in decimal form,
you just put a decimal point here, a decimal point here, move this twice and move this
forward twice .02.
So the interest rate 2% is .02.
Then t, what is t?
well the money was given to here when she was 2 years old, and we want to find out the
amount when she was 18.
so this is a very tricky part here, you have to be really careful to make sure your time
is correct.
so from 2 to 18, how many years is that?
18-2 which is 16.
So the expression that can be used to express the amount of money that she will have when
she turns 18 is going to be, let's write down the formula again so we don't make mistakes
when inputting the values.
A is equal to P times 1 plus r to the t.
The A is going to be the initial amount which is 3000 times 1 plus the interest rate 0.02
raised to the t which is 16.
That's the expression for the final amount , so let's see what the correct answer is.
The answer for problem 4 is option number 1.
Let's take a look at problem number 5, it says: which table of values represents a linear
relationship?
So what is the general equation for a linear relationship?
A linear relationship can be written as y=mx+b, so what does this tell us?
it tells us that the growth is always constant, it's always a constant, so M is the slope
of the growth, so for linear relationships the growth is constant for linear relationships.
That is also known as the slope, the growth rate so rise over run is always the same for
linear relationships.
What we're going to do to find out the answer, we notice that all the x's are growing by
1's for all the options so we want to take a look at the output, the y's and see which
of them exhibits constant growth, which ever one has a constant growth that is your answer.
So let's take a look at option number 1, from 13 to 6, that is -7 because 13 minus 7 is
6, from 6 to 1 that is a growth of what?
-5, uh oh, these two are different, so since these 2 are different that means the growth
rate is not constant, hence this is not a linear relationship.
So 1 is not the answer.
Option 2 let's take a look at the growth rate from 3 to 2, from 3 to 2 that's negative 4,
8 minus 4 , from 4 to 2 how do you go from 4 to 2, you have to subtract 2, uh oh we have
a problem here, these two growth rates are different so this is not a linear relationship.
Remember linear relationships always have a constant or the same growth.
Let's shift to option number 3, and one thing you want to note is that you want to check
the entire thing to make sure.
so from 5 to 3 that's a growth of -2, because if you subtract 2 from 5 you get 3.
From 3 to 1 that's a growth of -2, from 1 to -1 that's another growth of -2 and then
from -1 to -3 that's another growth of -2 so this is excellent, why? because the growth
rate is constant, the growth rate is constantly -2.
This is a linear relationship.
So that's that.
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