# The sum of ALL fractions

In this assessment, students work through a systematic approach to find the sum of all positive, proper fractions. Through a series of guided questions, students follow a step-by-step approach to recognize, describe, and later verify the pattern, the method, and/or the relationship that emerges.

Math Concepts: fractions proper fractions improper fractions number operations patterns
MYP Related Concepts: quantity validity
MYP Key Concepts: logic relationships
MYP Global Context: Identities and Relationships

In this assessment, students work with a simple concept: dividing squares (and later rectangles) into equally wide horizontal strips and cutting those trips with one of the diagonals. The resulting cut strips are shaded alternatingly (top strip shaded on the left, 2nd strip on the right, 3rd on the left, etc.). Then, after finding the total area of the shaded parts as a fraction of the total area, students are guided to find the answer to the question: is there a pattern for these areas for squares and rectangles of different size? Students explore squares and rectangles divided into an odd (at first) and into an even (later) number of horizontal strips.

Math Concepts: area square rectangle fraction ratio patterns
MYP Related Concepts: patterns
MYP Key Concepts: relationships
MYP Global Context: Identities and Relationships

# Homer's Theorem: Is It True?

This assessment is based on one of the episodes on The Simpsons, one in which Homer Simpson finds a pair of glasses that apparently makes him super smart the moment he puts them on – or does it? Homer states that “the sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side”, which sounds very similar to the Pythagorean theorem. In this assessment, students explore whether Homer’s statement could be true for isosceles or non-isosceles triangles. As they work through the mathematics, they not only become familiar with the triangle inequality, they also learn about and briefly explore degenerate triangles (at achievement level 7-8).

Math Concepts: square roots squares Pythagorean theorem triangles degenerate triangles
MYP Related Concepts: representation validity
MYP Key Concepts: relationships
MYP Global Context: Identities and Relationships

# Equable Shapes

In this assessment, students explore the concept or equable shapes, initially working with equable plane figures (the 5-12-13 triangle and different rectangles) and then with equable solid shapes (the sphere with r=3 and different cuboids). Throughout the investigation, students are guided towards exploring a general formula for both two- and three-dimensional equable shapes. At the end of the task (for Achievement Level 7-8), students are asked to algebraically prove their formulas.

Math Concepts: area perimeter surface area volume triangle rectangle sphere cuboid rectangular prism Pythagoras' theorem reciprocal
MYP Related Concepts: equivalence patterns
MYP Key Concepts: form relationships
MYP Global Context: Identities and Relationships

# Beautiful Roots

In this assessment, students investigate patterns of numbers in radical expressions that are claimed to be equivalent even though that claim seems to be fairly counter-intuitive. During the investigation, students use a calculator to find decimal values of such radical expressions, suggesting equivalence. By the end of the investigation, students verify that under certain conditions (i.e. the pattern established) radical expressions that follow a certain pattern are indeed equivalent and justify why having those conditions (i.e. the patterns) provide a reason for the equivalence.

Math Concepts: roots radicals patterns decimals equivalence algebraic expressions
MYP Related Concepts: equivalence quantity
MYP Key Concepts: relationships
MYP Global Context: Identities and Relationships