Mastering writing complex algebraic expressions from scenarios Grade 7

📚 Understanding Algebraic Expressions

An algebraic expression is a mathematical phrase that combines numbers, variables (letters that represent unknown numbers), and operation symbols (+, -, ×, ÷). It doesn't have an equals sign (=) like an equation.

📜 History of Algebraic Expressions

The use of symbols in algebra evolved over centuries. Early mathematicians used words to describe problems, making them lengthy and complex. Gradually, symbols were introduced for unknowns and operations, simplifying the process. The development of algebraic notation is attributed to mathematicians from various cultures, including those in ancient Babylon, Greece, and India.

🔑 Key Principles for Writing Algebraic Expressions

• 🔍 Identify the Unknown: Determine what the problem is asking you to find. This will be your variable (usually represented by letters like $x$, $y$, or $n$).
• ➕ Look for Keywords: Certain words indicate specific operations:
  • ➕ Addition: sum, plus, increased by, more than
  • ➖ Subtraction: difference, minus, decreased by, less than
  • ✖️ Multiplication: product, times, multiplied by, of
  • ➗ Division: quotient, divided by, ratio
• ✍️ Translate the Words: Break down the sentence into smaller parts and translate each part into a mathematical symbol or expression.
• 🧮 Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS) when simplifying or evaluating expressions.
• 🔗 Combine Like Terms: If possible, simplify the expression by combining terms with the same variable and exponent.

🌍 Real-World Examples

Let's translate some scenarios into algebraic expressions:

1. Scenario: Sarah has $x$ apples. John has twice as many apples as Sarah, plus 3 more. How many apples does John have?

   Expression: $2x + 3$
2. Scenario: A rectangle has a width of $w$ and a length that is 5 units greater than the width. What is the area of the rectangle?

   Expression: $w(w + 5)$ or $w^2 + 5w$

3. Scenario: Michael earns $h$ dollars per hour. He worked 10 hours this week, but had to pay $25 in taxes. How much money did Michael have left?

   Expression: $10h - 25$

4. Scenario: The cost of a ticket is $t$ dollars. A group of 4 friends went to see a movie and shared an extra cost of $8 of popcorn. How much did each friend spend?

   Expression: $\frac{4t + 8}{4}$ or $t + 2$
5. Scenario: A shop sells shirts for $s$ dollars and pants for $p$ dollars. Someone buys 3 shirts and 2 pants. How much money did they spend?

   Expression: $3s + 2p$

6. Scenario: Peter is $y$ years old. His sister is half his age, minus 2 years. How old is his sister?

   Expression: $\frac{y}{2} - 2$

7. Scenario: A number $n$ is increased by 7, and then the result is multiplied by 3. What is the final result?

   Expression: $3(n + 7)$ or $3n + 21$

💡 Tips and Tricks

• 🧐 Read Carefully: Understanding the problem is half the battle. Read the scenario multiple times.
• ✏️ Underline Key Information: Highlight the numbers and keywords that indicate operations.
• 💬 Think Step-by-Step: Break down the problem into smaller, manageable steps.
• ✅ Check Your Work: Make sure your expression accurately represents the scenario. Substitute a value for the variable to see if it makes sense.

📝 Practice Quiz

Translate each of the following scenarios into algebraic expressions:

1. Maria has $b$ books. David has 5 fewer books than Maria. How many books does David have?
2. A taxi charges a flat fee of $3, plus $2 per mile. If someone travels $m$ miles, how much will they pay?
3. The length of a garden is 3 times its width, $w$. What is the perimeter of the garden?
4. A store sells apples for $a$ dollars each and bananas for $b$ dollars each. Someone buys 5 apples and 3 bananas. How much will they spend?
5. A number $x$ is doubled, and then 6 is subtracted from the result.
6. John has $d$ dollars. He spends $\$10 and then shares the remaining money equally with his two friends. How much money does each of them receive?
7. A pizza is cut into 12 slices. If you eat $s$ slices, how many slices are left?

✅ Answers to the Practice Quiz

1. $b - 5$
2. $3 + 2m$
3. $2(w + 3w) = 8w$
4. $5a + 3b$
5. $2x - 6$
6. $\frac{d - 10}{3}$
7. $12 - s$

🎉 Conclusion

Mastering the art of writing algebraic expressions from scenarios is a crucial skill in mathematics. By understanding the key principles, identifying keywords, and practicing regularly, you can confidently translate real-world situations into mathematical language. Keep practicing, and you'll become an algebra ace in no time!