# 6.5 and *6.4 - Exponents and Squares

6.5    The student will investigate and describe concepts of positive exponents and perfect squares.

6.4  The student will recognize and represent patterns with whole number exponents and perfect squares.

Bloom's Level:  Analyze, Understand

### BIG IDEAS

• I can find diagonal distance between opposite street corners and I can determine what size TV to purchase if I know length and width of the space where it goes.

• I will realize that an exponent is a short way to write repeated multiplication and understand the relationship between a perfect square and a geometric square.

### UNDERSTANDING THE STANDARD

• In exponential notation, the base is the number that is multiplied, and the exponent represents the number of times the base is used as a factor. In 83, 8 is the base and 3 is the exponent.
• A power of a number represents repeated multiplication of the number by itself  (e.g., 83 = 8 x 8 x 8 and is read “8 to the third power”).
• Any real number other than zero raised to the zero power is 1. Zero to the zero power (0) is undefined.
• Perfect squares are the numbers that result from multiplying any whole number by itself (e.g., 36 =  6 x 6 = 62).
• Perfect squares can be represented geometrically as the areas of squares the length of whose sides are whole numbers (e.g., 1 x 1, 2 x 2, or 3 x 3). This can be modeled with grid paper, tiles, geoboards and virtual manipulatives.

·  The symbol  can be used in grade six in place of “x” to indicate multiplication.

·  In exponential notation, the base is the number that is multiplied, and the exponent represents the number of times the base is used as a factor. In , 8 is the base and 3 is the exponent (e.g., ).

·  Any real number other than zero raised to the zero power is 1. Zero to the zero power () is undefined.

·  A perfect square is a whole number whose square root is an integer (e.g., ). Zero (a whole number) is a perfect square.

·  Perfect squares may be represented geometrically as the areas of squares the length of whose sides are whole numbers (e.g., , etc.). This can be modeled with grid paper, tiles, geoboards and virtual manipulatives.

·  The examination of patterns in place value of the powers of 10 in grade six leads to the development of scientific notation in grade seven.

### ESSENTIALS

• What does exponential form represent? Exponential form is a short way to write repeated multiplication of a common factor such as 5 x 5 x 5 x 5 = 54.
• What is the relationship between perfect squares and a geometric square? A perfect square is the area of a geometric square whose side length is a whole number.

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

6.51  Recognize and describe patterns with exponents that are natural numbers by using a calculator.

·  6.41  Recognize and represent patterns with bases and exponents that are whole numbers.

6.52  Recognize and describe patterns of perfect squares not to exceed 202, by using grid paper, square tiles, tables, and calculators.

·  6.42  Recognize and represent patterns of perfect squares not to exceed, by using grid paper, square tiles, tables, and calculators.

6.53  Recognize power of 10 by examining patterns in a place value chart:  104 = 10,000, 103 = 1,000, 102 = 100, 101 = 10, 100 = 1.

·  6.43  Recognize powers of 10 with whole number exponents by examining patterns in place value.

### KEY VOCABULARY

base, power, exponent, squared, cubed, square root, perfect square, squaring a number, exponential form, standard form, area of squares

Updated: Nov 27, 2017