Number Operations
Lessons in this chapter
About Number Operations
Number Operations closes out whole-number arithmetic and introduces exponents, the notation the rest of middle school algebra is written in.
Divide Multi-Digit Numbers is the last appearance of long division as something being taught rather than used. From here on it is a tool, applied inside fraction, decimal and algebra problems.
Understand Exponents and Squares makes the shorthand explicit. The exponent counts how many times the base is multiplied by itself, so 5 cubed is 5 × 5 × 5, which is 125. The error to head off early is reading 5 cubed as 5 × 3.
Squares get particular attention because they recur everywhere: in area, in the Pythagorean theorem, in quadratics. Knowing the squares up to 15 by sight saves real time later.
Solve Expressions with Exponents Using Order of Operations slots exponents into the sequence learned in 5th grade. Exponents come after parentheses and before multiplication, so 2 × 3 squared is 18, not 36, and getting that order wrong is the most common slip of the chapter.
All three lessons in Number Operations open as 6th grade math worksheets inside the page. Divide Multi-Digit Numbers, Understand Exponents and Squares and Solve Expressions with Exponents Using Order of Operations each give a run of math problems worked one at a time and marked as you go. They are free, and nothing has to be set up before you start.
Frequently asked questions
It starts with Divide Multi-Digit Numbers, which takes long division past the single-digit divisors of earlier grades. Understand Exponents and Squares then introduces exponent notation and what squaring a number actually does, and Solve Expressions with Exponents Using Order of Operations puts exponents into the order of operations, where they are evaluated before multiplication and division. The chapter sits in two Common Core domains: The Number System (6.NS) for the division, and Expressions and Equations (6.EE) for the exponent work.
A shorthand for repeated multiplication. In 5 cubed, the 5 is the base and the 3 is the exponent, meaning 5 × 5 × 5, which is 125.
Because the exponent counts multiplications, not a factor to multiply by. 5 cubed is 5 × 5 × 5 = 125, while 5 × 3 is 15.
After parentheses and before multiplication and division. So 2 × 3 squared is 2 × 9, which is 18, not 6 squared.
The squares up to 15 at least. They turn up constantly in area, in the Pythagorean theorem and in quadratics, so recognizing them saves time throughout middle school.
A power of ten, so 10 cubed is 1,000. This is the notation from 5th grade's Number System chapter, which is why it comes back here.