# Investigating a Rule of Differentiation

In this assessment, students investigate one of the rules for derivative functions, namely for f(x)=x^n. In Part A of the investigation, students investigate the pattern for the gradient of the tangent drawn to f(x)=x^2 at x=2. In Part B of the investigation, students go one step further and investigate whether the rule for the derivative of f(x)=x^n is valid for negative and non-integer values of n. NOTE: The unit for which this assessment is created is Introduction to Calculus, which goes beyond the requirements of the MYP. It is intended as an assessment after the eAssessments, towards the end of the school year, as a preparation for Mathematics in the Diploma Programme.

Math Concepts: differentiation calculus derivatives tangent limit
MYP Related Concepts: generalization patterns
MYP Key Concepts: logic
MYP Global Context: Scientific and Technical Innovation

# Binomial Distribution Rules

In this assessment students investigate the rules of binomial distributions. In Part A, they are guided towards identifying the rules of expanding the product of two different binomials, both in terms of x. In Part B, they look closely at how perfect squares are expanded. Finally, in Part C, they follow steps similar to what they were guided through earlier, to discover further rules of expansions, this time for the difference of squares.

Math Concepts: binomials perfect squares difference of squares exponents FOIL
MYP Related Concepts: generalization simplification
MYP Key Concepts: relationships
MYP Global Context: Scientific and Technical Innovation

# Angles in Polygons

In this assessment, students use their knowledge of triangles, quadrilaterals, and other polygons to discover patterns in the number of diagonals, number of triangles, and the sum of interior angles in polygons with 3, 4, 5, ..., n number of sides/vertices. The students then verify their findings by comparing the results of a three-step process and the results of the formulas discovered.

Math Concepts: angles polygons interior angles angle sum diagonals
MYP Related Concepts: equivalence patterns
MYP Key Concepts: relationships
MYP Global Context: Scientific and Technical Innovation

# Triangular Numbers

In mathematics, a polygonal number is a number represented by dots arranged in the shape of a regular polygon, such as an equilateral triangle, a square, a regular pentagon, and so forth. These numbers are one type of two-dimensional figurate numbers, which were studied as early as in the times of Pythagoras. In this investigation, we find patterns for triangular numbers and discover the recursive and the explicit formulas to calculate the nth triangular number.

Math Concepts: patterns sequences triangular triangular numbers
MYP Related Concepts: patterns simplification
MYP Key Concepts: relationships
MYP Global Context: Scientific and Technical Innovation

# Carpets and Sponges

The construction of the Sierpinski carpet begins with a square, which is cut into 9 congruent sub-squares, from which the central sub-square is removed. The procedure is then repeated to the remaining sub-squares, ad infinitum. Similarly, the Menger sponge begins with a cube, which is cut into 27 identical sub-cubes, from which each central cube is removed. The procedure is then repeated to the remaining sub-cubes, ad infinitum. In this assessment students investigate the area of a Sierpinski carpet and the volume of the Menger sponge at each stage, and ultimately finding a formula for the area of the Sierpinski carpet and for the volume of the Menger sponge at infinity.

Math Concepts: patterns sequences algebraic expressions
MYP Related Concepts: patterns simplification
MYP Key Concepts: relationships
MYP Global Context: Scientific and Technical Innovation