Book cover for Elementary Algebra

Elementary Algebra

Lynn Marecek

ISBN #9780998625713

1st Edition

6,645 Questions

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32,192 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section builds a strong foundation for algebra by reviewing essential arithmetic with whole numbers, including place value, naming and rounding numbers, and understanding multiples, factors, and prime factorization. The transition into algebraic language introduces variables and constants, setting the stage for expressing relationships mathematically. Mastery of these topics not only solidifies computational skills but also enhances problem-solving abilities in algebra.

Learning Objectives

1

Understand and apply place value concepts with whole numbers, including reading and writing numbers in both digit and word form.

2

Develop strategies for rounding numbers to a specified place value and using divisibility tests to identify factors.

3

Find prime factorizations of composite numbers and determine least common multiples using both listing and prime factors methods.

4

Introduce algebraic notation by using variables, constants, and the language of algebra to represent real-world relationships.

5

Translate between arithmetic operations and algebraic expressions to build a firm foundation for elementary algebra.

Key Concepts

CONCEPT

DEFINITION

Counting Numbers

The set of numbers used for counting: 1, 2, 3, ... (also known as natural numbers).

Whole Numbers

The set of counting numbers along with zero: 0, 1, 2, 3, ... .

Place Value

A system where the value of a digit depends on its position within a number, divided into periods such as ones, thousands, millions, etc.

Ellipsis

The notation '…' indicating that a pattern continues indefinitely.

Divisibility Test

Rules used to determine if one number is divisible by another without performing full division (e.g., a number is divisible by 2 if its last digit is even).

Prime Number

A counting number greater than 1 that has no factors other than 1 and itself.

Composite Number

A counting number that has factors other than 1 and itself.

Prime Factorization

Expressing a composite number as a product of prime numbers.

Least Common Multiple (LCM)

The smallest number that is a multiple of two or more numbers.

Variable

A letter or symbol used in algebra to represent a number that may change.

Constant

A fixed value that does not change, such as the difference between two always-related quantities.

Example Problems

Example 1

Find the place value of each digit in the given numbers. 51,493 (a) 1 (b) 4 (c) 9 (d) 5 (e) 3

Example 2

Find the place value of each digit in the given numbers. 87,210 (a) 2 (b) 8 (c) 0 (d) 7 (e) 1

Example 3

Find the place value of each digit in the given numbers. 164,285 (a) 5 (b) 6 (c) 1 (d) 8 (e) 2

Example 4

Find the place value of each digit in the given numbers. 395,076 (a) 5 (b) 3 (c) 7 (d) 0 (e) 9

Example 5

Find the place value of each digit in the given numbers. 93,285,170 (a) 9 (b) 8 (c) 7 (d) 5 (e) 3
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Step-by-Step Explanations

QUESTION

In the number 63,407,218, what is the place value of the digit 7?

STEP-BY-STEP ANSWER:

Step 1: Write the number in a place value chart, aligning digits with their periods (e.g., tens, hundreds, thousands, etc.).
Step 2: Identify the position of the digit 7; observing that in 63,407,218, 7 falls in the thousands place.
Step 3: Confirm by matching with the place value chart provided in class.
Final Answer: The digit 7 is in the thousands place.

Place Value Determination

QUESTION

Round 103,978 to the nearest hundred.

STEP-BY-STEP ANSWER:

Step 1: Locate the hundreds digit in 103,978 (which is the digit 9 in the hundreds place).
Step 2: Look at the digit immediately to the right (the tens digit). Since the tens digit is 7 (which is ≥ 5), increase the hundreds digit by 1.
Step 3: Replace all digits to the right of the hundreds place with zeros.
Final Answer: 103,978 rounded to the nearest hundred is 104,000.

Rounding Numbers

QUESTION

Find the prime factorization of 48.

STEP-BY-STEP ANSWER:

Step 1: Begin by dividing 48 by the smallest prime number, 2: 48 ÷ 2 = 24.
Step 2: Divide 24 by 2: 24 ÷ 2 = 12.
Step 3: Divide 12 by 2: 12 ÷ 2 = 6.
Step 4: Divide 6 by 2: 6 ÷ 2 = 3, noting that 3 is a prime.
Step 5: Write the factorization as 2 x 2 x 2 x 2 x 3.
Final Answer: The prime factorization of 48 is 2⁴ · 3.

Prime Factorization

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Common Mistakes

  • Misidentifying digit position due to misunderstanding high place values (e.g., confusing millions with hundred-thousands).
  • Failing to replace all digits to the right of the rounding place with zeros.
  • Overlooking the proper use of commas when writing large numbers in words.
  • Confusing prime numbers with composite numbers by not recognizing that 2 is the only even prime.
  • Incorrectly matching factors when finding LCM, leading to errors in using the prime factors method.