Question

A group of students at Omega High School is using staples and popsicle sticks to build a scale model of the Great Wall of China as part of a project detailing China's military history. The number of staples the students will need is three times the number of popsicle sticks they will need. If the students determine they need 84 staples for this particular project, how many popsicle sticks will they need?

Short answer

The students will need 28 popsicle sticks to build their scale model of the Great Wall of China, as \(x = 28\).

Step-by-step solution

Write the equation that relates staples and popsicle sticks

Let x be the number of popsicle sticks required for the project. According to the problem, the number of staples required is three times the number of popsicle sticks. Therefore, the equation relating the number of staples (84) to the number of popsicle sticks (x) is: \[3x = 84\]

Solve the equation for x

To find the value of x, we just need to divide both sides of the equation by 3: \[\frac{3x}{3} = \frac{84}{3}\] \[x = 28\]

Interpret the result

The value of x represents the number of popsicle sticks required for the project. Since x = 28, the students will need 28 popsicle sticks to build their scale model of the Great Wall of China.

Key concepts

Algebraic Equations

When tackling SAT math problems like the one in our exercise, it is essential to have a solid understanding of algebraic equations. An algebraic equation is a mathematical statement that shows the equality of two expressions with unknown values, which we represent using variables, like 'x' or 'y'.

In our example, the given ratio was three staples to one popsicle stick, often interpreted as 'three times as many staples as popsicle sticks,' forming the basis for our equation. The equation represents a model of the relationship between two quantities and is a foundational skill for students not only in SAT math sections but also in higher-level math and real-world problem-solving. It is written as \(3x = 84\), with x being the unknown amount of popsicle sticks.

The beauty of algebraic equations is their utility in various contexts. Whether it's breaking down a scale model project or balancing chemical equations, mastering this concept allows you to express and solve problems systematically. By dividing both sides of the equation by 3, we isolated x, providing us with the answer, \(x = 28\). Such simplicity of methods in algebra is what demystifies math and opens the door to advanced concepts.

Scale Model Project

Scale model projects are a popular assignment in education for a reason. They require students to apply mathematical concepts, such as ratios and proportions, in a concrete and visual manner. In our example, students at Omega High School were tasked to build a scale model of the Great Wall of China. The project becomes a multi-disciplinary adventure–blending history, art, and mathematics.

A scale model is a three-dimensional representation of a physical object, maintaining accurate relationships between all important aspects of the model. For instance, if a wall of the scale model is three times as long as its height, the same should be true for the actual structure. The exercise provided uses algebra to determine the materials needed, revealing the practical application of math in real-life projects.

Approaching scale models with algebraic equations like \(3x = 84\) helps students visualize complex academic concepts while fostering creativity and technical skills. It's important to emphasize accuracy in scale modeling, as this precision determines the success and authenticity of the replica created.

Educational Math Activities

Educational math activities provide engaging ways for students to interact with mathematical concepts. The scale model project in the exercise is a perfect example of an educational activity that combines critical thinking with hands-on learning.

Activities such as these empower students to see relevance in their mathematical studies. Through constructing a scale model, they tackle real-world problems with a blend of arithmetic, geometry, and even engineering principles. It's not just about getting the right answer; it's about understanding the process, making decisions, and seeing the impact of mathematics beyond the classroom.

By adding a narrative, like the historical context of the Great Wall of China, math activities can become more memorable and engaging. Integrating stories or contextual problems makes learning dynamic and stimulating. The process of determining the number of popsicle sticks with the equation \(3x = 84\) encourages logical reasoning, enhances problem-solving skills, and makes math a genuinely interactive subject. These activities often yield the best results when they are collaborative, as group work and discussion further enhance understanding and retention of the concepts learned.