SOPHIA UNIT 1 CHALLENGE 1 ESSENTIAL
CONCEPTS NOTES
SOPHIA PATHWAYS College Algebra Notes
Unit 1 Challenge 1 – Essential concepts
Rea; Number Types
Differentiate between rational and irrational numbers.
1. Which of the following contains all of the rational numbers in the set?
A.) Correct.
Recall that rational numbers can be written as a ratio of two integers and have a terminating or
repeating decimal. is a ratio, equals , is equivalent to , is an
integer, and is a terminating decimal.
,2 — Adding and Subtracting Positive and Negative Numbers
Evaluate a sum or difference with positive and negative numbers.
Evaluate the expression.
8 – (-12) + 20 – 13 + (-15)
B.) = 12
First, subtract a negative 12 from 8, which is the same as adding a positive 12 to 8. Next, add 20 and 20
to get 40. Now subtract 13 to get 27. Finally, add negative 15 to get 27 + (-15) = 12.
8 – (-12) + 20 – 13 + (-15)
8 + 12 + 20 – 13 + (-15)
20 + 20 – 13 + (-15)
40 – 13 + (-15)
27 + (-15)
12
, 3 — Multiplying and Dividing Positive and Negative Numbers
Determine a product or quotient with positive and negative numbers
Which of the following will yield a positive number?
C.) -3 multiplied by -5 = 15 positive Correct.
Recall that if the signs match (both positive or both negative), then the answer is a positive number.
Incorrect. The product of -2 and 9
Incorrect. The quotient of 28 and -7
Incorrect. 24 divided by -4
4 — Introduction to Order of Operations
Evaluate an expression using Order of Operations
Evaluate the following expression:
[6 + 9 × (4 – 2)] ÷ 4 + 2
=8
When simplifying expressions, it is important that we simplify them in the correct order.
Consider the following problem done two different ways:
Add First Multiply
Multiply Add
Solution Solution
The previous example illustrates that if the same problem is done two different ways we will
arrive at two different solutions. However, only one method can be correct. It turns out the
second method, 17, is the correct method. The order of operations ends with the most basic of
operations, addition (or subtraction). Before addition is completed we must do repeated addition
or multiplication (or division). Before multiplication is completed we must do repeated
multiplication or exponents. When we want to do something out of order and make it come first
we will put it in parenthesis (or grouping symbols). This list then is our order of operations we
will use to simplify expressions.
CONCEPTS NOTES
SOPHIA PATHWAYS College Algebra Notes
Unit 1 Challenge 1 – Essential concepts
Rea; Number Types
Differentiate between rational and irrational numbers.
1. Which of the following contains all of the rational numbers in the set?
A.) Correct.
Recall that rational numbers can be written as a ratio of two integers and have a terminating or
repeating decimal. is a ratio, equals , is equivalent to , is an
integer, and is a terminating decimal.
,2 — Adding and Subtracting Positive and Negative Numbers
Evaluate a sum or difference with positive and negative numbers.
Evaluate the expression.
8 – (-12) + 20 – 13 + (-15)
B.) = 12
First, subtract a negative 12 from 8, which is the same as adding a positive 12 to 8. Next, add 20 and 20
to get 40. Now subtract 13 to get 27. Finally, add negative 15 to get 27 + (-15) = 12.
8 – (-12) + 20 – 13 + (-15)
8 + 12 + 20 – 13 + (-15)
20 + 20 – 13 + (-15)
40 – 13 + (-15)
27 + (-15)
12
, 3 — Multiplying and Dividing Positive and Negative Numbers
Determine a product or quotient with positive and negative numbers
Which of the following will yield a positive number?
C.) -3 multiplied by -5 = 15 positive Correct.
Recall that if the signs match (both positive or both negative), then the answer is a positive number.
Incorrect. The product of -2 and 9
Incorrect. The quotient of 28 and -7
Incorrect. 24 divided by -4
4 — Introduction to Order of Operations
Evaluate an expression using Order of Operations
Evaluate the following expression:
[6 + 9 × (4 – 2)] ÷ 4 + 2
=8
When simplifying expressions, it is important that we simplify them in the correct order.
Consider the following problem done two different ways:
Add First Multiply
Multiply Add
Solution Solution
The previous example illustrates that if the same problem is done two different ways we will
arrive at two different solutions. However, only one method can be correct. It turns out the
second method, 17, is the correct method. The order of operations ends with the most basic of
operations, addition (or subtraction). Before addition is completed we must do repeated addition
or multiplication (or division). Before multiplication is completed we must do repeated
multiplication or exponents. When we want to do something out of order and make it come first
we will put it in parenthesis (or grouping symbols). This list then is our order of operations we
will use to simplify expressions.
