Translating Phrases
into Expressions 2.1
Algebraic
Expressions represent phrases that have an unknown or variable
Numeric expressions
have only numbers
The main strategy
to create an algebraic expression is called TRANSLATING. You translate each word by word to create the expression
That is when you
need to know the typical operation words like sum, difference, product and quotient,
You also need to know other operation words
such as ‘more than’, times,
minus, divided by… and so on….
For most
translations you keep the same order of the
way the phrase is stated…
The sum of a number and 12
n + 12
The difference of
12 and a numbers
12- n
The product of 12
and a number
12n
The quotient of 12
and a number
12/n
[PLEASE DO NOT USE THE DIVISION SYMBOL—USE THE
FRACTION BAR]
You NEVER use x
for times because x is a variable!!
Use the dot in the middle or use ( )
There are some “Algebra
Style Notes” for writing expressions:
1. You do not need any symbol if you are
multiplying either 2 or more variables or a number with a variable because just
putting them next to each other means multiplication.
3xy means triple
the product of a number and another number.
6z means the
product of 6 and a number
2. For addition and multiplication with a number
and a variable, generally the number is FIRST
You would not see (y)(3) You would see 3y. Generally you change 3 + n
to n + 3
3. You generally use a fraction bar for
division—especially when you have variables!
4. You keep variables in alphabetical order and
if there are exponents with the same variable, you ALWAYS list them with the
HIGHEST power first. If you have 3a+ 4a2 you would use the Commutative Property of Addition
and switch it to 4a2 + 3a
There are a couple
of translations where the ORDER IS SWITCHED—from the way it is written. They are both for
subtraction. When in doubt of the order, always substitute any number for the variable and you should be able to determine
the order:
For example:
5 less than a number… Not sure?
Think: Change a
number to let’s say 20
5 less than 20
is? 15
What did you do in
your hear? 20 -5
Notice that the 5 is
SECOND even though the word phrase
has it FIRST
So now change the
20 back into the variable
n - 5
Whenever you see LESS THAN… switch the order!!
The same is true
for
5 subtracted from a number
You again have a
number and then you subtract 5 from that number
n – 5
A more complicated
translation is
when you have a number times a SUM or a DIFFERENCE
For example:
5 times the SUM of
a number and 3
Thinkà change the number to any number of your
choice
5 times the SUM of
4 and 3 is ?
Wait… that’s 5
times 7 or 35
How did you get
35?
First you found
the sum (4 + 3) or 7
Then you found 5 time
7 or 35
Now substitute
back the variable for the number of your choice
5(n+3)
So you need a set
of ( ) to tell the reader to find the SUM first and then multiply.
The same is true
of DIFFERENCE times a number
Translate: Twice the DIFFERENCE of 5 and
a number.
2( 5 -n)
When you just can’t
translate word by word you will need
other strategies to help you
Stratgey # 2
DRAW a picture
When in doubt,
Draw it out!!
I have 5 times the
number of quarters as I have dimes.
I try translating
it to 5q = d
I check If I assume
that I have 20 quarters then 5(20) = 100 dimes.
Does that make
sense? That would mean I have a lot more dimes than I have quarters and that
isn’t what the words say that I have! The original problem says I have a lot
more quarters. So my Algebra is WRONG
I need to switch
the variables
5d = q
I check
If I assume that I
have 20 quarters then 5d = 20
Does this make
sense? YES
If I have 20
quarters and only 4 dimes
Sometimes it helps
to make a quick pictures.
Imagine 2 piles of
coins
The pile of quarters
is 5 times as high as the pile of dimes
You can clearly
see that you would need to multiple the number of dimes to make that pile the same height as the
number of quarters.
Strategy #3
Make a T-Chart
To translate known
relationships to Algebra, it often helps to make a T-Chart
You ALWAYS put the unknown variable on the LEFT side AND what you known on the right
Fill in the chart
with 3 lines of numbers and look for the relationship between the 2 columns.
Then you use that mathematical relationship with a variable.
Ask yourself... What
do I do to the left side to get the number on the right?
Example The number
of hours in d days
Your unknown is d days so that does on the left side
d days----# of
hours
1----------à 24
2---------à 48
3 --------à 72
Before you put the
variable… look at the relationship between the left column and the right column.
Ask yourself... What do I do to the left side to get the number on the right?
Specifically …
What do I do to 1
to get 24?
What do I do to 2 to get 48?
What do I do to 3 to get 72?
The answer to all
of those questions is… Multiply it by 24
You must multiply
the left column by 24 to get the right column.
So now back at
your T chart
d days----# of
hours
1----------à 24
2---------à 48
3 --------à 72
d---------à 24d
The last line of
the column will then use your variable d
Notice your answer is 24d
What if the questions
was to state the number of days in h hours ( the lip of the first example)
Your unknown is h hours this time so
that does on the left side
h hours ---à #
of days
24 --------à 1
48 --------à 2
72 --------à 3
Why did I start
with 24 not 1 hour this time?
Now look at the
relationship between the left column and the right column… Ask yourself... What do I do to the left side
to get the number on the right?
Specifically
What do I do to 24
to get 1?
What do I do to 48 to get 2?
What do I do to 72 to get 3?
The answer to all
those questions is… DIVIDE by 24
You must divide
the left column by 24 to get the right column.
The last line of
the T-Chart will then use your variable h
h hours ---à #
of days
24 --------à 1
48 --------à 2
72 --------à 3
h --------à h/24
