# 8.3 - Square Roots

8.3  The student will

a)  estimate and determine the two consecutive integers between which a square root lies;

b)  determine both the positive and negative square roots of a given perfect square.

### BIG IDEAS

• I can calculate area or volume, determine growth or decay, and figure out the impact of force.  I can determine national debt and world population, advertise with viral marketing, program a computer game, figure compound interest, and track the spread of viruses.
• I will understand the relationship between a perfect square and a geometric square and be able to use the area of a square as a means for estimating the square root of any number.

### UNDERSTANDING THE STANDARD

2016 VDOE Curriculum Framework - 8.3 Understanding

·  A perfect square is a whole number whose square root is an integer.

·  The square root of a given number is any number which, when multiplied times itself, equals the given number.

·  Both the positive and negative roots of whole numbers, except zero, can be determined. The square root of zero is zero. The value is neither positive nor negative.  Zero (a whole number) is a perfect square.

·  The positive and negative square root of any whole number other than a perfect square lies between two consecutive integers (e.g., lies between 7 and 8 since 72 = 49 and 82 = 64;  lies between -4 and -3 since (-4)2 = 16 and (-3)2 = 9).

·  The symbol  may be used to represent a positive (principal) root and - may be used to represent a negative root.

·  The square root of a whole number that is not a perfect square is an irrational number (e.g., is an irrational number). An irrational number cannot be expressed exactly as a fraction  where b does not equal 0.

·  Square root symbols may be used to represent solutions to equations of the form x2 = p. Examples may include:

-  If x2 = 36, then x is  = 6 or  =-6.

-  If x2 = 5, then x is  or −.

·  Students can use grid paper and estimation to determine what is needed to build a perfect square. The square root of a positive number is usually defined as the side length of a square with the area equal to the given number.  If it is not a perfect square, the area provides a means for estimation.

### ESSENTIALS

• How does the area of a square relate to the square of a number?
The area determines the perfect square number. If it is not a perfect square, the area provides a means for estimation.
• Why do numbers have both positive and negative roots?
The square root of a number is any number which when multiplied by itself equals the number. A product, when multiplying two positive factors, is always the same as the product when multiplying their opposites (e.g., 7 ∙ 7 = 49 and  -7 ∙ -7 = 49).

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

·  8.3b1  Determine the positive or negative square root of a given perfect square from 1 to 400.

·  8.3a1  Estimate and identify the two consecutive integers between which the positive or negative square root of a given number lies. Numbers are limited to natural numbers from 1 to 400.