Math 7: Exercise 1, Page 78 Solved!

Hey guys! Let's dive into a math problem from the "Mathematics with a Key" textbook for 7th grade. We're tackling exercise 1 on page 78. If you're struggling with this one, you've come to the right place. We're going to break it down step by step so you can understand exactly how to solve it. This isn't just about getting the answer; it’s about understanding the process and building your math skills. So, grab your textbook, a pencil, and let's get started!

Understanding the Problem

Before we even think about solving anything, we need to really understand the problem. What's it asking us to do? What information are we given? These are crucial questions! Often, math problems look intimidating because they're packed with words and numbers, but the core concept is usually pretty straightforward.

Let's break this down like detectives. Think of each number and piece of information as a clue. What clues do we have? How do they connect? What's the ultimate question we need to answer? Once you can articulate the problem in your own words, you're already halfway to the solution. Don't underestimate the power of reading the problem carefully – maybe even a couple of times! Highlighting key information can also be super helpful. Seriously, give it a try! You'll be surprised how much clearer things become when you actively engage with the text.

Now, let's assume the problem involves some algebraic equations. Many 7th-grade math problems focus on algebraic thinking, even if they don't look like classic equations at first glance. This means we might need to use variables (like x or y) to represent unknown quantities. Don’t let that intimidate you! Variables are just placeholders, and we have the tools to figure out their values. We might be dealing with linear equations, where we need to isolate the variable to find its value. Or perhaps it’s a word problem that requires us to translate the given information into an equation. Whatever it is, understanding the type of problem is the first step to finding the right approach.

Another key aspect of understanding the problem is identifying the mathematical concepts involved. Is it a problem about fractions, decimals, percentages, or geometry? Recognizing the underlying concepts will guide you in choosing the appropriate formulas and techniques. For example, if the problem involves calculating the area of a rectangle, you know you'll need to use the formula: Area = length × width. Similarly, if it's about percentages, you'll need to recall how to convert percentages to decimals and use them in calculations. So, before you start crunching numbers, take a moment to identify the core mathematical ideas at play. This will make the solution process much smoother and more efficient.

Step-by-Step Solution

Alright, now that we've deciphered the problem, let’s get down to the nitty-gritty – the step-by-step solution. This is where we transform our understanding into action. The key here is to be organized and methodical. Each step should logically follow the previous one, leading us closer to the answer. Think of it like building a staircase; each step supports the next.

First, let's assume the exercise involves solving a linear equation. A typical linear equation looks something like this: 2x + 5 = 11. Our goal is to isolate the variable x on one side of the equation. To do this, we need to perform inverse operations. Remember, whatever we do to one side of the equation, we must do to the other to maintain balance. So, in this case, the first step would be to subtract 5 from both sides: 2x + 5 - 5 = 11 - 5. This simplifies to 2x = 6. Now, we need to get rid of the coefficient 2, which is multiplying x. To do this, we divide both sides by 2: 2x / 2 = 6 / 2. This gives us our solution: x = 3. Ta-da! We've solved for x. The most important part of solving equations like this is to keep the equation balanced, and undo operations in the correct order, like reverse PEMDAS.

But what if the problem is a word problem? Word problems often require a bit more translation. We need to convert the words into mathematical expressions. Let's say the problem states: "John has twice as many apples as Mary. Together, they have 15 apples. How many apples does Mary have?" The first step is to assign variables. Let's say Mary has x apples. Since John has twice as many, he has 2x apples. Together, they have 15, so we can write the equation: x + 2x = 15. Now, we can combine like terms: 3x = 15. Finally, divide both sides by 3: x = 5. So, Mary has 5 apples. The key with word problems is to carefully identify the relationships between the quantities and translate them into mathematical equations.

Another common type of problem in 7th grade involves geometry. This might include calculating areas, perimeters, or volumes of different shapes. For instance, if the problem asks for the area of a triangle, you need to remember the formula: Area = 1/2 × base × height. The problem will usually give you the base and height, or provide enough information for you to calculate them. Be sure to use the correct units (e.g., square centimeters for area). Always draw the shapes out if you can, it helps you to picture the problem better! Geometry problems are often visual, so a diagram can make a huge difference in understanding and solving the problem.

Common Mistakes to Avoid

Okay, we've walked through the solution, but let's talk about some common mistakes that students often make. Recognizing these pitfalls can save you a lot of grief and help you ace your math assignments. Trust me, we've all been there!

One of the biggest mistakes is careless arithmetic. It’s so easy to make a small error when adding, subtracting, multiplying, or dividing, especially when you're working quickly or under pressure. Always double-check your calculations, even if you think you've got it right. It’s often those tiny slips that lead to wrong answers. A good strategy is to write out each step clearly and neatly. This not only helps you spot potential errors but also makes it easier for your teacher to follow your work and give you partial credit, even if the final answer is incorrect. Seriously, neatness counts!

Another common mistake is misunderstanding the order of operations. Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? This is the golden rule of arithmetic. If you perform operations in the wrong order, you'll likely get the wrong answer. For example, if you have an expression like 3 + 2 × 4, you need to multiply 2 and 4 first, and then add 3. If you add 3 and 2 first, you’ll end up with the wrong result. So, always keep PEMDAS in mind, and break down the problem into smaller steps to ensure you're following the correct order.

For word problems, a frequent mistake is misinterpreting the wording. Math problems often use specific language that you need to translate into mathematical terms. For example, "twice as many" means multiplying by 2, and "less than" means subtracting. Pay close attention to these keywords and phrases, and take your time to understand what the problem is really asking. It can be helpful to rewrite the problem in your own words or to draw a diagram to visualize the situation. Also, make sure you're answering the question that was actually asked. Sometimes, you might solve for a variable, but that's not the final answer the problem is looking for. Always reread the question to make sure you're providing the correct information.

Tips for Success in Math

Alright, let's wrap things up with some tips for success in math. These aren’t just quick fixes, but rather strategies that will help you build a solid foundation and excel in your math journey. Math isn't just about memorizing formulas; it's about developing problem-solving skills and logical thinking. So, let’s make you a math whiz!

First and foremost, practice makes perfect. This is a cliché, but it’s absolutely true when it comes to math. The more problems you solve, the more comfortable you'll become with different concepts and techniques. Don’t just rely on the examples your teacher does in class; do extra practice problems on your own. Work through the exercises in your textbook, and look for additional resources online or in workbooks. The key is to challenge yourself with a variety of problems, so you're prepared for anything that comes your way. Math is like a sport; you need to train regularly to improve your skills.

Another crucial tip is to ask for help when you're struggling. Don't let confusion fester. If you're stuck on a problem or a concept, reach out to your teacher, classmates, or a tutor. There's no shame in asking for help; in fact, it's a sign of strength and a commitment to learning. Your teacher is there to support you, and often, a different explanation or perspective can make all the difference. Collaboration with classmates can also be incredibly beneficial. Explaining a concept to someone else helps you solidify your own understanding, and you might learn new strategies from your peers. So, don’t hesitate to seek assistance when you need it. We're all in this together!

Finally, develop a positive attitude towards math. Many students have math anxiety, which can hinder their performance. Believe in your ability to learn and improve. Approach math problems with curiosity and a willingness to try. Don’t get discouraged by mistakes; they're a natural part of the learning process. View them as opportunities to learn and grow. Celebrate your successes, no matter how small they may seem. Each problem you solve correctly builds your confidence and momentum. Remember, math is a skill that you can develop with effort and persistence. So, cultivate a positive mindset, and watch your math abilities soar.

So there you have it! We've tackled exercise 1 on page 78, broken down the solution step by step, and discussed common mistakes to avoid and tips for math success. Remember, math is a journey, not a destination. Keep practicing, stay curious, and you'll be amazed at what you can achieve!