In this assessment, students investigate non-zero integer triples (called Addition Triples), in which (1) all three numbers are different non-zero integers, and (2) the first two numbers add up to the third number. For example, the number triple (2, 3, 5) form an Addition Triple as none of the three numbers are equal (i.e. they are all different), none of the three numbers are zeroes, and 2 + 3 = 5. Some other Addition Triples are (5, 6, 11) or (2, 5, 7). In this investigation, we are looking at the Total Number of Addition Triples that we can make from a list of n integers, considering whether n is an odd or even number.

MYP Related Concepts: patterns systems
MYP Key Concepts: relationships
MYP Global Context: Identities and Relationships

# Sum and Product of Quadratic Roots

In this assessment, students investigate the sum and the product of the roots of quadratic equations. While Parts A and B of the investigation provide guidance to finding the general formula for the sum of the roots, in Part C students are asked to conduct an independent investigation into the product of the roots of quadratic equations.

Math Concepts: quadratic equations quadratic formula roots sum of roots product of roots
MYP Related Concepts: patterns systems
MYP Key Concepts: relationships
MYP Global Context: Identities and Relationships

# 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, derive explicit formulas for them, and combine their findings to verify and justify an explicit formula for 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