Math 7 (Period 4)

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