Uploaded by PCCvideos on 08.01.2010

Transcript:

Instructor: Hi class.

Today we're going to go ahead

and take a look at how we would translate words

into basic math symbols.

And it's nice to have a basic set of words

that are often representing the different operations

for math before we start out.

And then I'm going to give you a process

that will help you to translate words into math symbols.

So I have a chart here

that gives you some of the basic words

that you see most often when you are translating words into math.

When you see the words 'more,' 'increase,' 'greater,' or 'sum,'

those will mean addition.

So if you see those in a sentence

you'll want to put down a plus symbol [+] for addition.

The most commonly seen subtraction words are

'less,' 'less than,' 'decrease,' and 'difference.'

Notice that I have an asterisk next to 'less than.'

There's a reason why.

Let's take a look at a sentence that has 'less than' in it.

Let's say that you have something that says

"5 less than a number."

If you have 5 less than a number,

that means that you take the number and you go 5 smaller than it,

which means you take the number and you subtract 5 off of it.

Now since we don't know what the number is,

we're going to give it a variable. Let's call it 'x.'

So this 'x' right here is representing the number.

The 'less than' right here is your subtraction.

And the value that we are subtracting is this 5.

Notice that in the sentence that we wrote

in mathematical terms that the x is first,

yet in the English sentence, the x, or the number,

or the words 'a number,' were second.

When you have 'less than,'

the order that you see the things in the sentence reverses.

And they're in the opposite order

that they're written in the sentences.

So that's why I had the asterisk there.

So, if you have two things

where something is 'less than' something else,

and like '5 less than a number,' or '8 less than a number,'

or 'a number less than 8,'

you reverse the order of the sentence,

and you go ahead and have them

in opposite order in the mathematical sentence.

Let's take a look at one more of those.

Let's say that we have '8 less than a number.'

Notice that 'a number' is second,

so it's actually going to go first

because we have 'less' attached to 'than' for our subtraction.

And then the 8, which was actually first,

will go last in our sentence.

So now if we want to represent something in multiplication

and we're writing it in, in English terms,

the two words that you see most often

are the word 'of' and the word 'product.'

'Product' is the answer after you multiply.

Just like over here in subtraction

'difference' is the answer after you subtract.

And 'sum' is actually the answer after you add.

So when you're multiplying,

or when something is telling you to multiply,

you'll often see the word 'of' or the word 'product.'

Other things that aren't written on here

is you might see something that says 'double' a number.

That means to take that number and multiply it by 2.

So that's telling you to actually multiply by a specific value.

Or 'triple' would be multiplying by 3.

'Quadruple' would be multiplying by 4.

And so on.

When you want to put a division symbol down,

you might see something like 'ratio' or 'quotient.'

'Quotient' is the answer after you divide.

And when we put division down,

often we write it as a fraction bar.

So you might see something that says,

'the quotient of a number and 6.'

When you see the words 'quotient' here,

that's telling you you're going to be dividing.

Now after that word 'quotient'

will be the things that you're going to divide.

So notice that we see 'a number and 6.'

So since number comes first,

that's going to be the first item that we have.

The quotient is actually a division symbol [躕,

which is going to be a fraction bar [/] in this case.

And then 6 is what we were dividing by.

So this sentence here

actually translates into this in math terms: [x/6]

Now a few other things that you may need to translate

We'll actually be translating them often

when you get to chapter 2 also, and 3

is if you see the word 'is' that means

that you're probably going to be using an equals symbol [=].

Sometimes there's things attached to it.

Like over here you'll notice 'is less than'

or 'is greater than' that might mean something else.

Like the word 'is' attached to 'greater than'

or 'is' attached to 'less than'

actually means the inequality symbol

of less than [<] or greater than [>].

So you want to watch and see if you have words

that are attached to 'is,'

because they might mean something different.

Also when you see 'same as' or 'equivalent',

those will be things that mean

that you want to put an equal sign [=]

in your particular sentence that you're translating.

Now let's take a look at some sentences

and how we would translate them into math symbols.

Let's say that we have

"6 more than double a number decreased by 5."

Now if we read that,

what I want to do is give you a strategy

for reading it and translating.

The first time you read through

you're going to just read it at a normal pace

to get an idea of what's going on.

After you read it through to get an idea,

you're going to go really slow, one word at a time,

and translate it from English into math symbols.

And as we do that I'm going to color code it

so that you can see which things match what

in our math sentence.

So once we get done translating it we're going to read it again,

and we're going to see if what we wrote

actually matches what was written in English,

and then keep revising it until we know what we have matches.

It's very similar to the revision process

you would do in writing a paper.

You read it. You revise it.

Then you read it again, and keep revising it

until you think that it's actually the way it's supposed to be.

So we're going to read our sentence.

First thing I come to is '6' here.

And that's a number. I know what it is.

I'm just going to write that down as 6.

Then if I continue reading I have 'more than.'

Well, 'more than' is something for addition.

So let's translate this 'more than' into a plus [+] symbol.

Then let's read on and see what else we have.

It says 'double a number.' 'Double' means you multiply by 2,

and we need to see what we're going to multiply by 2,

which comes right after it, and it says 'a number.'

Now since I don't know what that number is,

we're going to give it a variable.

Let's just call it 'x'

You can call it any letter you want.

So we're going to have to do 2 times that number.

No keep in mind when you have a number times a variable

all you need to do is place those right next to each other.

You don't actually need your multiplication symbol in there.

So if I translate 'double a number' it's just going to be a [2x]

That means 2 times that,

whatever that number x is going to end up being.

Now if I continue reading on it says, 'Decreased.'

'Decreased' means we're going to subtract.

So, this right here is going to translate

into a subtraction symbol. [-]

Now let's continue reading and see what we're going to subtract.

And it's 'decreased by 5.'

So that means we're going to go ahead and subtract 5.

Now we have the end of our sentence;

we've hit the period in the sentence.

So we need to stop and reread the sentence

and see if what we have matches what's written.

And it says '6' here, and we wrote down a 6.

Then it says 'more than'

and we wrote down [+] addition for that, which matches.

'Double a number' is 2 times some variable

and we wrote down a 2x, so that works.

'Decreased' is subtraction. And we're subtracting 5.

So what we have right here [ 6 + 2x - 5 ]

matches what was written in the English sentence.

If it didn't, we'd revise it

and then go back and read the sentence again

until it matched nicely.

Now let's do one more and see what it looks like.

So how about if we have

'triple the sum of a number and 4 is 12.

Now if we read this, the first thing that I'm coming to

says 'triple,' and 'triple' means to multiply by 3.

So what we need to do is see

what we're going to multiply by 3.

And if I continue on, it says 'the sum.'

Well, 'sum' is the answer after we add,

so we need to see what we're going to add,

and it says that 'the sum of a number and 4.'

So we're going to add a number and 4.

So translating this, the triple is a 3 times something,

and remember parentheses mean times,

means multiply, or times.

And then to figure out what I'm going to multiply by,

I'm going to multiply by the sum of a number and 4.

Well, we don't know what the variable is

so let's call it x, and that's x plus 4.

Because the sum is the answer after you add,

and this is stating right here ( x + 4 )

that we're going to add a number and 4.

So that gave us that part of the sentence.

Now if we continue on we have the word 'is.'

If you'll remember from our grid we had earlier,

'is' means equals [=] So we'll put an equals down.

And then after that we have 12 here, so we'll write that down.

Now we have a nice sentence here in math terms,

and we need to see if it matches what we have up above.

So we have 'triple' here, which was the 3.

The 'sum of a number and 4,' which is the x plus 4.

And triple meant that we would multiply this here by the 3.

'Is' was the equals. And our 12 was on the end.

So, this here does match our sentence that was in English.

Now, the other type of thing you might have,

rather than just having straight words,

is you might have a situation,

and you might have to think about some common sense

on what that situation is meaning.

For instance, you might have something that says,

'The cost of renting a car is 25 dollars a day

plus 50 cents a mile.

Find the cost for 10 miles.'

Which wouldn't be very far,

and you might not want to actually rent the car for that.

So let's read this, and what we're going to do is translate

this first sentence here into an equation.

And then we're going to use that equation

to do what the second sentence right here,

that starts right here, is asking us to do.

So if we read the first sentence it says 'the cost.'

Now we don't know what the cost is,

so how about we give it a letter, and 'Cost' starts with C,

so we're going to use a C. So the Cost is going to be a C.

And it's the cost of renting a car,

so the cost of renting the car is the C that we have down here.

Now often when you have variables for a story problem situation

what you'll do is you'll say 'let'

and then you'll define what it is.

So C is the cost of renting the car.

Now, let's read on.

We've got our C for our equation right here,

and we've defined it right here.

Reading on, it says 'is' which we know 'is' is an equals sign,

so we'll put an equals sign in our equation here.

Then if we can continue on it says '25 dollars a day'

and we are going to just have one day in this case,

which I should have put in the second sentence.

So that's going to be a 25 right here.

And then we read the word 'plus', which is an addition symbol,

so this will translate into the plus.

And then we come to something that's a little tricky.

It says '50 cents a mile.'

50 cents a mile is going to be 50 cents

times how many miles you drive,

and we don't have any variables defined

for what the miles are going to be.

And miles starts with m, so how about if we define m

as being the number of miles driven.

So, right here we've defined

what the variables in our equation are going to be.

Now, for the 50 cents a mile,

that's going to be 50 cents, times the miles.

So we're going to have .5 times m,

which is the 50 cents a mile.

So now I have this sentence here in math terms

with the variables defined

that tells us what this first sentence here

should be representing.

So if we read through we have the cost of renting a car.

We defined C as being the cost,

and I have that first right here. 'Is' is my equals.

25 dollars a day... we're just going to do one 25 dollars,

which is right here. [ C = 25 ]

'Plus' is the addition symbol. [C = 25 +]

And 50 cents a mile, we define the number of miles as being m,

so that's .5 times m. [ C = 25 + .5m ]

Now in the second sentence it says "find the cost for 10 miles."

So what we're going to do then is we're going to let m be 10,

and plug this in right here [ C = 25 + .5(10) ]

in order to find our actual cost.

Now I'm going to go ahead and erase this up here

so we can finish off the problem.

So plugging our 10 in for that m

we find that the cost is 25 dollars plus .5 times 10.

So 50 cents times our 10 dollars ends up being 5.

And the cost for that 10-mile trip in that car we rented

[ 30 = 25 +5 ] was 30 dollars.

Now, most of you are probably saying

I could figure that out without writing the equation.

However, it is nice to know how to write the equation

so if you don't know how many miles you're driving,

you could have the equation

and every time you find out the number of miles

you could just plug it in and calculate.

So having that equation so you don't have to do too much

every time you rent this particular car can really help.

Or you can program a computer program

to have this equation in it

and then just enter the number of miles,

of miles and it will tell you what the cost is going to be.

So those are some basics

for translating words into math symbols.

During the chapter that we discuss all our story problems

we'll do quite a bit more on it.

This just gets you a head start

as to what we're going to be doing.

Today we're going to go ahead

and take a look at how we would translate words

into basic math symbols.

And it's nice to have a basic set of words

that are often representing the different operations

for math before we start out.

And then I'm going to give you a process

that will help you to translate words into math symbols.

So I have a chart here

that gives you some of the basic words

that you see most often when you are translating words into math.

When you see the words 'more,' 'increase,' 'greater,' or 'sum,'

those will mean addition.

So if you see those in a sentence

you'll want to put down a plus symbol [+] for addition.

The most commonly seen subtraction words are

'less,' 'less than,' 'decrease,' and 'difference.'

Notice that I have an asterisk next to 'less than.'

There's a reason why.

Let's take a look at a sentence that has 'less than' in it.

Let's say that you have something that says

"5 less than a number."

If you have 5 less than a number,

that means that you take the number and you go 5 smaller than it,

which means you take the number and you subtract 5 off of it.

Now since we don't know what the number is,

we're going to give it a variable. Let's call it 'x.'

So this 'x' right here is representing the number.

The 'less than' right here is your subtraction.

And the value that we are subtracting is this 5.

Notice that in the sentence that we wrote

in mathematical terms that the x is first,

yet in the English sentence, the x, or the number,

or the words 'a number,' were second.

When you have 'less than,'

the order that you see the things in the sentence reverses.

And they're in the opposite order

that they're written in the sentences.

So that's why I had the asterisk there.

So, if you have two things

where something is 'less than' something else,

and like '5 less than a number,' or '8 less than a number,'

or 'a number less than 8,'

you reverse the order of the sentence,

and you go ahead and have them

in opposite order in the mathematical sentence.

Let's take a look at one more of those.

Let's say that we have '8 less than a number.'

Notice that 'a number' is second,

so it's actually going to go first

because we have 'less' attached to 'than' for our subtraction.

And then the 8, which was actually first,

will go last in our sentence.

So now if we want to represent something in multiplication

and we're writing it in, in English terms,

the two words that you see most often

are the word 'of' and the word 'product.'

'Product' is the answer after you multiply.

Just like over here in subtraction

'difference' is the answer after you subtract.

And 'sum' is actually the answer after you add.

So when you're multiplying,

or when something is telling you to multiply,

you'll often see the word 'of' or the word 'product.'

Other things that aren't written on here

is you might see something that says 'double' a number.

That means to take that number and multiply it by 2.

So that's telling you to actually multiply by a specific value.

Or 'triple' would be multiplying by 3.

'Quadruple' would be multiplying by 4.

And so on.

When you want to put a division symbol down,

you might see something like 'ratio' or 'quotient.'

'Quotient' is the answer after you divide.

And when we put division down,

often we write it as a fraction bar.

So you might see something that says,

'the quotient of a number and 6.'

When you see the words 'quotient' here,

that's telling you you're going to be dividing.

Now after that word 'quotient'

will be the things that you're going to divide.

So notice that we see 'a number and 6.'

So since number comes first,

that's going to be the first item that we have.

The quotient is actually a division symbol [躕,

which is going to be a fraction bar [/] in this case.

And then 6 is what we were dividing by.

So this sentence here

actually translates into this in math terms: [x/6]

Now a few other things that you may need to translate

We'll actually be translating them often

when you get to chapter 2 also, and 3

is if you see the word 'is' that means

that you're probably going to be using an equals symbol [=].

Sometimes there's things attached to it.

Like over here you'll notice 'is less than'

or 'is greater than' that might mean something else.

Like the word 'is' attached to 'greater than'

or 'is' attached to 'less than'

actually means the inequality symbol

of less than [<] or greater than [>].

So you want to watch and see if you have words

that are attached to 'is,'

because they might mean something different.

Also when you see 'same as' or 'equivalent',

those will be things that mean

that you want to put an equal sign [=]

in your particular sentence that you're translating.

Now let's take a look at some sentences

and how we would translate them into math symbols.

Let's say that we have

"6 more than double a number decreased by 5."

Now if we read that,

what I want to do is give you a strategy

for reading it and translating.

The first time you read through

you're going to just read it at a normal pace

to get an idea of what's going on.

After you read it through to get an idea,

you're going to go really slow, one word at a time,

and translate it from English into math symbols.

And as we do that I'm going to color code it

so that you can see which things match what

in our math sentence.

So once we get done translating it we're going to read it again,

and we're going to see if what we wrote

actually matches what was written in English,

and then keep revising it until we know what we have matches.

It's very similar to the revision process

you would do in writing a paper.

You read it. You revise it.

Then you read it again, and keep revising it

until you think that it's actually the way it's supposed to be.

So we're going to read our sentence.

First thing I come to is '6' here.

And that's a number. I know what it is.

I'm just going to write that down as 6.

Then if I continue reading I have 'more than.'

Well, 'more than' is something for addition.

So let's translate this 'more than' into a plus [+] symbol.

Then let's read on and see what else we have.

It says 'double a number.' 'Double' means you multiply by 2,

and we need to see what we're going to multiply by 2,

which comes right after it, and it says 'a number.'

Now since I don't know what that number is,

we're going to give it a variable.

Let's just call it 'x'

You can call it any letter you want.

So we're going to have to do 2 times that number.

No keep in mind when you have a number times a variable

all you need to do is place those right next to each other.

You don't actually need your multiplication symbol in there.

So if I translate 'double a number' it's just going to be a [2x]

That means 2 times that,

whatever that number x is going to end up being.

Now if I continue reading on it says, 'Decreased.'

'Decreased' means we're going to subtract.

So, this right here is going to translate

into a subtraction symbol. [-]

Now let's continue reading and see what we're going to subtract.

And it's 'decreased by 5.'

So that means we're going to go ahead and subtract 5.

Now we have the end of our sentence;

we've hit the period in the sentence.

So we need to stop and reread the sentence

and see if what we have matches what's written.

And it says '6' here, and we wrote down a 6.

Then it says 'more than'

and we wrote down [+] addition for that, which matches.

'Double a number' is 2 times some variable

and we wrote down a 2x, so that works.

'Decreased' is subtraction. And we're subtracting 5.

So what we have right here [ 6 + 2x - 5 ]

matches what was written in the English sentence.

If it didn't, we'd revise it

and then go back and read the sentence again

until it matched nicely.

Now let's do one more and see what it looks like.

So how about if we have

'triple the sum of a number and 4 is 12.

Now if we read this, the first thing that I'm coming to

says 'triple,' and 'triple' means to multiply by 3.

So what we need to do is see

what we're going to multiply by 3.

And if I continue on, it says 'the sum.'

Well, 'sum' is the answer after we add,

so we need to see what we're going to add,

and it says that 'the sum of a number and 4.'

So we're going to add a number and 4.

So translating this, the triple is a 3 times something,

and remember parentheses mean times,

means multiply, or times.

And then to figure out what I'm going to multiply by,

I'm going to multiply by the sum of a number and 4.

Well, we don't know what the variable is

so let's call it x, and that's x plus 4.

Because the sum is the answer after you add,

and this is stating right here ( x + 4 )

that we're going to add a number and 4.

So that gave us that part of the sentence.

Now if we continue on we have the word 'is.'

If you'll remember from our grid we had earlier,

'is' means equals [=] So we'll put an equals down.

And then after that we have 12 here, so we'll write that down.

Now we have a nice sentence here in math terms,

and we need to see if it matches what we have up above.

So we have 'triple' here, which was the 3.

The 'sum of a number and 4,' which is the x plus 4.

And triple meant that we would multiply this here by the 3.

'Is' was the equals. And our 12 was on the end.

So, this here does match our sentence that was in English.

Now, the other type of thing you might have,

rather than just having straight words,

is you might have a situation,

and you might have to think about some common sense

on what that situation is meaning.

For instance, you might have something that says,

'The cost of renting a car is 25 dollars a day

plus 50 cents a mile.

Find the cost for 10 miles.'

Which wouldn't be very far,

and you might not want to actually rent the car for that.

So let's read this, and what we're going to do is translate

this first sentence here into an equation.

And then we're going to use that equation

to do what the second sentence right here,

that starts right here, is asking us to do.

So if we read the first sentence it says 'the cost.'

Now we don't know what the cost is,

so how about we give it a letter, and 'Cost' starts with C,

so we're going to use a C. So the Cost is going to be a C.

And it's the cost of renting a car,

so the cost of renting the car is the C that we have down here.

Now often when you have variables for a story problem situation

what you'll do is you'll say 'let'

and then you'll define what it is.

So C is the cost of renting the car.

Now, let's read on.

We've got our C for our equation right here,

and we've defined it right here.

Reading on, it says 'is' which we know 'is' is an equals sign,

so we'll put an equals sign in our equation here.

Then if we can continue on it says '25 dollars a day'

and we are going to just have one day in this case,

which I should have put in the second sentence.

So that's going to be a 25 right here.

And then we read the word 'plus', which is an addition symbol,

so this will translate into the plus.

And then we come to something that's a little tricky.

It says '50 cents a mile.'

50 cents a mile is going to be 50 cents

times how many miles you drive,

and we don't have any variables defined

for what the miles are going to be.

And miles starts with m, so how about if we define m

as being the number of miles driven.

So, right here we've defined

what the variables in our equation are going to be.

Now, for the 50 cents a mile,

that's going to be 50 cents, times the miles.

So we're going to have .5 times m,

which is the 50 cents a mile.

So now I have this sentence here in math terms

with the variables defined

that tells us what this first sentence here

should be representing.

So if we read through we have the cost of renting a car.

We defined C as being the cost,

and I have that first right here. 'Is' is my equals.

25 dollars a day... we're just going to do one 25 dollars,

which is right here. [ C = 25 ]

'Plus' is the addition symbol. [C = 25 +]

And 50 cents a mile, we define the number of miles as being m,

so that's .5 times m. [ C = 25 + .5m ]

Now in the second sentence it says "find the cost for 10 miles."

So what we're going to do then is we're going to let m be 10,

and plug this in right here [ C = 25 + .5(10) ]

in order to find our actual cost.

Now I'm going to go ahead and erase this up here

so we can finish off the problem.

So plugging our 10 in for that m

we find that the cost is 25 dollars plus .5 times 10.

So 50 cents times our 10 dollars ends up being 5.

And the cost for that 10-mile trip in that car we rented

[ 30 = 25 +5 ] was 30 dollars.

Now, most of you are probably saying

I could figure that out without writing the equation.

However, it is nice to know how to write the equation

so if you don't know how many miles you're driving,

you could have the equation

and every time you find out the number of miles

you could just plug it in and calculate.

So having that equation so you don't have to do too much

every time you rent this particular car can really help.

Or you can program a computer program

to have this equation in it

and then just enter the number of miles,

of miles and it will tell you what the cost is going to be.

So those are some basics

for translating words into math symbols.

During the chapter that we discuss all our story problems

we'll do quite a bit more on it.

This just gets you a head start

as to what we're going to be doing.