# Areas of Lattice Polygons

In this assessment (which is based on Pick's Theorem), students investigate polygons drawn in coordinate planes whose vertices fall on lattice points. In Part A of the investigation, students observe lattice rectangles to discover a pattern between their areas and the number of their interior and boundary lattice points. Then, in Part B of the investigation, students use guided questions to verify and justify their findings using lattice triangles, both right-angled triangles and non-right-angled triangles. Finally, in Part C of the investigation, students select and apply their choice of mathematical problem-solving techniques to verify and justify their previous findings on convex or concave polygons.

Math Concepts: area polygon lattice lattice polygon Pick's formula
MYP Related Concepts: generalization
MYP Key Concepts: logic
MYP Global Context: Identities and Relationships

# 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 expansion
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