Letters and Expressions

[Grade 7 Math] Welcome to the World of Algebraic Expressions!

Hello everyone! Let's start learning the unit called "Algebraic Expressions" together.
Some of you might feel that "seeing alphabets in math sounds difficult..." But don't worry!
In fact, using letters just makes the " \( \square \)" and " \( \triangle \)" symbols we used in elementary school math much more convenient and stylish to work with.
Once you master this unit, you'll be able to organize complex calculations neatly and clearly. Let's take it one step at a time!

1. Why do we use letters?

In elementary school, you used " \( \square \)" to represent an unknown number, like in " \( 100 + \square = 150 \)". In junior high school, we use alphabets (letters) like "\( x \)" or "\( a \)" instead of that " \( \square \)".

Why use letters?
By using letters, you can create "magic formulas" that apply to any number. For example, if an apple costs \( 1 \) yen, the cost for buying \( x \) apples can be written as " \( 100 \times x \)". Simply by plugging in any number for \( x \), you can calculate the total cost immediately. That is the beauty of using letters!

Fun Fact: Why do we often use " \( x \)"?

It is said that ancient mathematicians chose the alphabet closest to the first letter of the word for an unknown number. There is no deep meaning behind it, so feel free to think of it as "the most popular letter in math!"

2. Rules for Writing Algebraic Expressions (Very Important!)

There are universal rules for writing expressions that use letters. Learning these rules is the first major hurdle in this unit. But don't worry—once you get used to them, it's just like solving a puzzle!

Rule 1: Omit the multiplication sign " \( \times \)"

When multiplying a number and a letter, or a letter and a letter, you don't need to write the " \( \times \)" symbol; you just place them next to each other.
Example: \( a \times 5 \rightarrow 5a \)
Example: \( x \times y \rightarrow xy \)

Rule 2: Write numbers before letters

When writing a combination of numbers and letters, always put the number first.
Example: \( a \times (-3) \rightarrow -3a \)

Rule 3: Arrange letters in alphabetical order

When you have multiple letters in one term, it's easier to read if you put them in alphabetical order (the order they appear in the dictionary).
Example: \( b \times a \rightarrow ab \)

Rule 4: Omit the " \( 1 \)"

Instead of writing " \( 1a \)", just write " \( a \)". Similarly, " \( -1a \)" becomes " \( -a \)".
Caution! You cannot remove the " \( 1 \)" in " \( 0.1a \)". Leave it exactly as it is.

Rule 5: Write division as a fraction

In junior high school, we stop using the " \( \div \)" sign and switch to fractions. The number you are dividing by becomes the denominator (the bottom part).
Example: \( a \div 3 \rightarrow \frac{a}{3} \)

Key Takeaway:
"Remove the ' \( \times \)'!" "Turn ' \( \div \)' into a fraction!" Make sure you remember these two rules!

3. Values of Expressions and Substitution

Plugging a number into an expression in place of a letter is called "substitution." The result of the calculation is called the "value of the expression."

Example: Find the value of \( 5x - 2 \) when \( x = 3 \).

Step 1: Replace the letter with the "number." Remember to bring back the hidden " \( \times \)" sign.
\( 5 \times 3 - 2 \)
Step 2: Calculate as usual.
\( 15 - 2 = 13 \)
The answer is "\( 13 \)"!

Common mistake:
When substituting a negative number, always remember to use parentheses!
Example: When \( a = -2 \), then \( a^2 \) becomes \( (-2)^2 = 4 \). Forgetting the parentheses is a common cause of calculation errors!

4. Calculating with Algebraic Expressions (Addition and Subtraction)

You can perform calculations with expressions containing letters. However, there is a rule!

Combine terms with the same letter (Like Terms)

Terms that have the same letter part, such as " \( 3a \)" and " \( 2a \)", are called "like terms." Since they represent the same "type" of item, you can add or subtract them.

Example: \( 3a + 2a = (3+2)a = 5a \)
(Think of it like: 3 apples plus 2 apples equals 5 apples!)

Caution:
You cannot calculate things like " \( 3a + 5 \)" because one has a letter and the other doesn't!
Since you can't simplify it further, that is the final answer. Be careful not to force it into " \( 8a \)."

5. Representing Quantities (Application)

This is practice for turning word problems into algebraic expressions. It might feel difficult at first, but it becomes much easier if you think about it using concrete numbers.

① Total Cost
The cost of buying 3 pens, where each pen costs \( a \) yen:
\( a \times 3 \rightarrow 3a \) (yen)

② Change
The change received from 1000 yen after buying items that cost a total of \( x \) yen:
\( 1000 - x \) (yen)

③ Unit Conversion
What is \( x \text{ m} \) in \( \text{cm} \)?
Since \( 1 \text{ m} = 100 \text{ cm} \), we have \( 100 \times x \rightarrow 100x \) (\(\text{cm}\))

Advice:
If you're wondering, "Should I add or multiply?", try replacing the letter with a concrete number like \( 10 \) and ask yourself, "How would I calculate this?"

Summary: Key points to remember!

1. A letter is a "magic box that can hold any number"
2. Omit the multiplication sign " \( \times \)", and put numbers before letters!
3. Write division as a fraction!
4. You can only add/subtract terms that have the same letters!

"Algebraic Expressions" is a super important foundation for the "equations" and "functions" you will learn next. Start by mastering the writing rules and get comfortable with using letters. It's perfectly okay to take it slow at first!