# 8.1 - Compare/Order Real Numbers

8.1  The student will compare and order real numbers.

### BIG IDEAS

• I can determine sale costs while shopping, create a budget, dilute mixtures, understand scales on maps, interpret probabilities and odds, and convert among metric units.
• I will be able to interchange fraction, decimal, and percent values to interpret real world situations.

### UNDERSTANDING THE STANDARD

2016 VDOE Curriculum Framework - 8.1 Understanding

·  Real numbers can be represented as integers, fractions (proper or improper), decimals, percents, numbers written in scientific notation, radicals, and p. It is often useful to convert numbers to be compared and/or ordered to one representation (e.g., fractions, decimals or percents).

·  Proper fractions, improper fractions, and mixed numbers are terms often used to describe fractions.  A proper fraction is a fraction whose numerator is less than the denominator.  An improper fraction is a fraction whose numerator is equal to or greater than the denominator.  An improper fraction may be expressed as a mixed number. A mixed number is written with two parts:  a whole number and a proper fraction (e.g., 3 ). Fractions can have a positive or negative value.

·  The density property states that between any two real numbers lies another real number.  For example, between 3 and 5 we can find 4; between 4.0 and 4.2 we can find 4.16; between 4.16 and 4.17 we can find 4.165; between 4.165 and 4.166 we can find 4.1655, etc. Thus, we can always find another number between two numbers. Students are not expected to know the term density property but the concept allows for a deeper understanding of the set of real numbers.

·  Scientific notation is used to represent very large or very small numbers.

·  A number written in scientific notation is the product of two factors: a decimal greater than or equal to one but less than 10 multiplied by a power of 10 (e.g., 3.1 ´ 105 = 310,000 and 3.1 ´ 10–5 = 0.000031).

·  Any real number raised to the zero power is 1. The only exception to this rule is zero itself. Zero raised to the zero power is undefined.

### ESSENTIALS

• How does the different ways rational numbers can be represented help us compare and order rational  numbers?
Numbers can be represented as decimals, fractions, percents, and in scientific notation. It is often useful to convert numbers to be compared and/or ordered to one representation (e.g., fractions, decimals or percents).
• When are numbers written in scientific notation?  Scientific notation is used to represent very large and very small numbers.

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

·  8.11  Compare and order no more than five real numbers expressed as integers, fractions (proper or improper), decimals, mixed numbers, percents, numbers written in scientific notation, radicals, and π. Radicals may include both positive and negative square roots of values from 0 to 400. Ordering may be in ascending or descending order.

·  8.12  Use rational approximations (to the nearest hundredth) of irrational numbers to compare and order, locating values on a number line. Radicals may include both positive and negative square roots of values from 0 to 400 yielding an irrational number.

### KEY VOCABULARY

simplify, numerical expression, exponent, variable, rational number, positive, negative, whole number, fraction, decimal, integer, square root, perfect square, order of operations, real numbers, compare and order, percents, scientific notation, ascending order, descending order

Updated: Nov 20, 2018