Sample Problems: Writing and Solving Algebraic Expressions (Grade 5)

📚 Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases containing numbers, variables (letters that represent unknown numbers), and operation symbols (+, -, ×, ÷). Writing and solving these expressions is a key skill in mathematics, laying the foundation for more advanced algebra later on. Let's break down the process.

📜 A Brief History

The use of symbols to represent unknown quantities dates back to ancient civilizations. Early forms of algebra were developed in Mesopotamia and Egypt. The systematic use of symbols, like the variables we use today (e.g., $x$, $y$), became more widespread during the Renaissance. This allowed mathematicians to express general relationships and solve equations in a more efficient way.

• 🌍 Ancient Civilizations: Early forms of algebra existed in Mesopotamia and Egypt.
• 🕰️ Renaissance Period: The systematic use of symbols became more prevalent.
• 💡 Modern Notation: The notation we use today evolved over centuries.

🔑 Key Principles

When dealing with algebraic expressions, keep these principles in mind:

• 🔢 Variables: A variable is a symbol (usually a letter) that represents an unknown number. For example, in the expression $3 + x$, $x$ is the variable.
• ➕ Operations: The basic operations are addition (+), subtraction (-), multiplication (× or *), and division (÷ or /).
• ⚖️ Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• 📝 Writing Expressions: Translate word problems carefully. Look for keywords like "sum," "difference," "product," and "quotient."
• ➗ Solving Expressions: To solve an algebraic expression, substitute a given value for the variable and then perform the operations.

💡 Real-World Examples

Let's look at some examples.

Example 1: Writing an Expression

Problem: Write an algebraic expression for "five more than a number."

Solution: Let the number be represented by $x$. The expression is $x + 5$.

Example 2: Solving an Expression

Problem: Evaluate the expression $2y - 3$ when $y = 7$.

Solution: Substitute $7$ for $y$: $2(7) - 3 = 14 - 3 = 11$.

Example 3: A More Complex Problem

Problem: John has $x$ apples. Mary has twice as many apples as John. Together, they have 15 apples. How many apples does John have?

Solution: Mary has $2x$ apples. Together they have $x + 2x = 3x$ apples. Since they have 15 apples in total, $3x = 15$. Dividing both sides by 3, we get $x = 5$. John has 5 apples.

✍️ Practice Quiz

Here are some practice problems to test your understanding:

1. ➕ Write an algebraic expression for "the sum of a number and 8."
2. ➖ Write an algebraic expression for "the difference between 12 and a number."
3. ✖️ Write an algebraic expression for "the product of 6 and a number."
4. ➗ Write an algebraic expression for "the quotient of a number and 4."
5. 🔢 Evaluate the expression $3a + 2$ when $a = 4$.
6. 💡 Evaluate the expression $10 - 2b$ when $b = 3$.
7. 🍎 Sarah has $y$ candies. Tom has 3 times as many candies as Sarah. Together, they have 20 candies. How many candies does Sarah have?

✅ Solutions

1. $x + 8$
2. $12 - x$
3. $6x$
4. $x / 4$
5. $3(4) + 2 = 14$
6. $10 - 2(3) = 4$
7. $y + 3y = 4y = 20$, so $y = 5$. Sarah has 5 candies.

🎯 Conclusion

Writing and solving algebraic expressions is a fundamental skill in algebra. By understanding variables, operations, and the order of operations, students can confidently tackle a wide range of problems. Keep practicing, and you'll master these concepts in no time!