# Polygonal Numbers

The concept of polygonal numbers is closely related to regular polygons. Namely, polygonal numbers are numbers that are represented by dots arranged in a shape of regular polygons. In this investigation, students discover patterns for polygonal numbers (starting with triangular numbers and then square numbers, pentagonal numbers, and so on), derive explicit formulas for them, and combine their findings to verify and justify an explicit formula for one of the polygonal numbers.

Math Concepts: triangular numbers square numbers pentagonal numbers hexagonal numbers polygonal numbers quadratic sequence
MYP Related Concepts: patterns systems
MYP Key Concepts: relationships
MYP Global Context: Identities and Relationships

# The Leibniz Triangle

In this assessment, students investigate the terms in the Leibniz triangle in connection to a similar triangle they are already familiar with: Pascal’s triangle. In Part A, students observe the terms in the Leibniz triangle and state their predictions for the terms in its 5th row. Then, in Part B, students use guided questions to describe a pattern between the terms in Pascal’s triangle and in the Leibniz triangle, and verify the validity of their general rule. Finally, in Part C (extended version only), students select and apply their choice of problem-solving techniques to suggest an explicit formula for the general term of the Leibniz triangle, and verify and justify their formula.

Math Concepts: unit fraction Pascal's triangle Leibniz triangle binomial coefficient
MYP Related Concepts: patterns systems
MYP Key Concepts: relationships
MYP Global Context: Identities and Relationships

# 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