## Making Mathematical Arguments: Generalizing about Numbers

Students make and justify generalizations about signed number operations, squares, cubes, roots, factors, multiples, and primes. First, students use colored cubes to explore and develop rules for signed number operations. As they analyze different mathematical arguments, they learn about counterexamples to a conjecture. They then study squares, square roots, cubes and cube roots and analyze several mathematical arguments made about these kinds of numbers. Students then explore several number theory topics and develop their own mathematical argument.

### NCTM Standard: Number and Operations

• Understand numbers, ways of representing numbers, relationships among numbers, and number systems
• Understand meanings of operations and how they relate to one another
• Compute fluently and make reasonable estimates

### Primary Mathematical Goals

• Find counterexamples to define rules for adding, subtracting, multiplying, and dividing with integers
• Identify patterns in increasing perfect squares
• Write a mathematical argument based on an understanding of cubes and cube roots
• Identify prime numbers and prime factorization
• Create rectangles that model the different factor pairs of a number