#### Math - 2019-20

# 8.1 - Compare/Order Real Numbers

8.1The student willcompareandorderreal numbers.

*Adopted: 2016*

### 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

· 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 ´ 10^{5 }= 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**

·
8.1_{1}** 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.

·
8.1_{2} **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: Jul 30, 2019*