3. Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Representative IMP Year 3 Lesson:

Proving Triples (Orchard Hideout), 127

After learning about the Pythagorean Theorem, students find that there are unique sets of numbers that one can find as the measures of the sides of right triangles: Pythagorean triples. Students are asked to examine two sets of measurements and use them to determine if a triangle is a right triangle or not. Students are then asked to write a proof about the multiplication of each member of a Pythagorean triple by a constant and whether or not the results will be the side measurements of a right triangle.