Math adventure stories: Mystery of vanishing gemstone

Chapter 1: The coded scroll

Math adventure stories : One morning, disaster strikes in the magical village of Algebraville, where everything runs on the power of math.The Infinity Crystal—the heart of the town’s magic  is no longer there! Without it, the town’s floating bridges start to sway, the lights go down, and the fountains stop flowing.

The mayor, Ms. Equation, called an emergency  meeting .

“Citizens of Algebraville,” she said, “without the Infinity Crystal, our town’s magic will fade.” A trail of riddles left by the thief—the myterious Variable Master—could take us to the Crystal. “Who will solve the puzzles and save the town?”

To solve them, the kids must use their math skills. Only the cleverest can unlock the secrets of algebra.

A group of daring children stepped forward. They called themselves the Number Detectives and promised to return the crystal. Their first clue was a rolled-up scroll found on the pedestal. When unrolled, it presented a puzzle.

“To find where the map is hidden, compute the equation 3x + 5 = 14. “What is x?”

Solving the Riddles

The Number detectives huddled together to discuss the issue. “We need to isolate x,” said Mira, the group’s leader. “First, subtract 5 from both sides of the equation.”

• 3x+5=14
• 3x=14-5
• 3x=9

“Now divide both sides by 3,” added Leo.

• x=9/3
• x=3

As soon as they figure out the puzzle, a secret compartment in the library wall clicks open, and the numbers on the scroll glow brightly. They discover a golden map within, but a portion of it is missing. There is a message on the map’s reverse:

“Good job, Number Detectives! To finish this map, head to the Forest of Fractions and solve the next puzzle.”

The children gather their supplies and proceed on their journey, excited to discover what challenges await.

To be continued …

Stay tuned for chapter 2: The Forest of Fractions, where the number detectives encounter a  wise old turtle and tackle fraction puzzles to move forward in their adventure!

Two dimensional geometry formulas  sometimes known as plane geometry formulas , are the branch of geometry concerned with shapes, figures, and their properties in the two-dimensional plane. The plane has two dimensions: length and width, represented by the x-axis (horizontal) and y-axis (vertical) in a Cartesian coordinate system.

The basic ideas of two-dimensional geometry are points, lines, angles, forms (such as triangles, circles, and polygons), and their attributes. Let’s look at some essential aspects:

1. Cartesian Coordinates and Plane

The Cartesian coordinate system is the basis for two-dimensional geometry. The coordinate plane is a flat surface with points specified by a pair of numbers  (x,y).

•  x represents the abscissa (horizontal distance from the origin).
•  y represents the ordinate (vertical distance from the origin).

The origin (0,0) is where the x-axis and y-axis cross.

2. Points and coordinates

• A point is an location in the plane without size or dimension, expressed by coordinates  (x,y).
• The distance between two points (x_1, y_1​) and  (x_2, y_2) is given by the distance formula:

• The midpoint between two points (𝑥_1, 𝑦_ 1)  and (𝑥_ 2, 𝑦_ 2) is the point that divides the segment connecting them into two equal parts.

3. Lines and slopes

A line in the plane can be represented by the following linear equation:

• The slope (𝑚) of a line running through two points (𝑥_1, 𝑦_1)  and (𝑥_2, 𝑦_2)  can be calculated as:

• The equation of a line in slope-intercept form (when the slope $m$ and y-intercept c are known) is:

4. Angles

Angle between two lines: The  angle θ between two lines with slopes of m_1  and 𝑚_2  is as follows:

• The angle between two parallel lines is 0∘, and they both have the same slope.
• The slopes of perpendicular lines are negative reciprocals of one another, meaning that 𝑚_1 𝑚_2 = −1 , and the angle between them is 90∘ .

5. Distance between a point and a line.

The distance between a point P(x_0​,y_0​) and a line Ax+By+C=0 is determined by the formula:

6. Types of 2D Triangle Geometric Figures

A triangle is a three-sided polygon. Triangles can be classified according to angles as follows:

• Acute triangle: All angles less than 90∘
• Right triangle: One angle is 90∘
• Obtuse triangle: One angle is greater than 90∘

A triangle’s area is:

The area of a triangle with vertices A(x_1, y_1), B(x_2, y_2), C(x_3, y_3), and  is as follows:

Quadrilaterals
A four-sided polygon is called a quadrilateral. Among the varieties are:

Square , Rectangular, Parallelogram, trapezium, rhombus
A rectangle’s area is:

Circles
The set of all points in the plane that are equally spaced from a fixed point (the center) is called a circle. The equation for a circle with radius r and center (h,k) is as follows:

The area and circumference of a circle are given by:

Area:

Circumference:

Polygons
A closed figure with straight sides is called a polygon. Among the examples are:

Pentagon (5 sides)
Hexagon (6 sides)
Octagon (8 sides)

The sum of the interior angles of a polygon with $n$ sides is given by:

Sum of interior angles=(n−2)×180∘

7. Conic Sections

Conic sections are curves that can be obtained by intersecting a plane with a cone. Each has a specific general equation.

• Circle:

• Ellipse:

• Parabola:

• Hyperbola:

8. Coordinated Geometry

In Two dimensional geometry formulas , coordinate geometry (also known as analytic geometry) employs the Cartesian coordinate system to algebraically represent and investigate geometric shapes.

For example:

• The distance formula finds the distance between two places.
• The slope of a line indicates its steepness and direction.
• A circle’s equation can be calculated using the distance between the center and a point on the circle.

Polynomial synthetic division

Synthetic division is a simplified and fast method for dividing a polynomial by a linear divisor of the type  x−c. It eliminates the requirement for lengthy division, making the operation faster and simpler, particularly for higher-degree polynomials. Here’s an explanation of how Polynomial synthetic division works, followed by an illustration.

Steps for Synthetic Division

Given a polynomial

​

Set up the coefficients: Record the coefficients of the polynomial P(x) in a row. If any powers of  x are absent (e.g., no 𝑥 2  phrase), include a zero for that term.

Write the root of the divisor: The divisor $x-c$ means the root is $c$. This is the value you’ll use for the division.

Perform synthetic division:

• Bring down the first coefficient as it is.
• Multiply the result by 𝑐 and add it to the following coefficient.
• Repeat the method for each coefficient.

Interpret the results:

• The last number in the row is the remainder.
• The other numbers are the quotient’s coefficients.

Polynomial synthetic division examples: Divide

Step-by-step process

1. Write the coefficients: The polynomial 2x^3 – 3x^2 + 4x – 5 has the coefficients $2,-3,4,-5$.

2. Set up the synthetic division table:To divide by x−1, use 𝑐 = 1 .

3. Begin the synthetic division:

• Bring down the first coefficient (2).
• Multiply 2 by 1 (the root of the divisor x – 1) to obtain 2.
• Add this to the following coefficient:− 3 + 2 = − 1
• Multiplying -1 by 1 yields -1.
• Add this to the following coefficient: 4 + ( − 1 ) = 3 .
• Multiply 3 by 1 to get three.
• Add this to the final coefficient: −5 + 3 = −2

The synthetic division table now looks like this:

4. Interpret the results:

The quotient is  2x^ 2 -x+3.
The remaining is -2.

Thus, the division gives:

Important Points:
Quotient: The quotient’s coefficients are  2x^2 −x+3.
remaining:  -2 is the remaining.
Interpretation: This indicates that P(x)=(x−1)(2x^ 2 −x+3)−2, indicating that the division was correct with the exception of −2.

Why It Works:

Synthetic division uses the remainder theorem and polynomial properties to divide by a linear term ( x – c). Rather than going through the long division procedure step by step, synthetic division integrates all of the processes into a more compact form.