Pre-Algebra
Table of Contents
Unit 1 Overview: Whole Numbers
1.1 Introduction to Whole Numbers
1.2 Add Whole Numbers
1.3 Subtract Whole Numbers
1.4 Multiply Whole Numbers
1.5 Divide Whole Numbers

➕pre-algebra review

1.3 Subtract Whole Numbers

Citation:

Subtraction of whole numbers is a fundamental skill in math. It's all about finding the difference between two numbers or taking away one amount from another. We use it daily, from calculating change to figuring out how much time is left until an event.

Understanding subtraction symbols, visual models, and efficient techniques is key. These skills help solve word problems and real-world applications. The base-10 system plays a crucial role in subtraction, especially when regrouping is needed for larger numbers.

Subtraction of Whole Numbers

Correct subtraction symbols and notation

  • Subtraction symbol represented by a minus sign (-) indicates taking away or finding the difference between two numbers
  • Minuend is the first number in a subtraction problem from which another number is subtracted
    • Subtrahend is the second number in a subtraction problem being subtracted from the minuend
    • Difference is the result obtained by subtracting the subtrahend from the minuend
  • Subtraction is not commutative meaning changing the order of the minuend and subtrahend produces a different result
    • $8 - 3 \neq 3 - 8$ because $8 - 3 = 5$ while $3 - 8 = -5$

Visual models for subtraction

  • Represent minuend using physical objects (counters, blocks, or chips) and remove the number of objects equal to the subtrahend
    • Remaining objects after removal represent the difference (12 counters - 5 counters = 7 counters)
  • Illustrate subtraction on a number line by starting at the minuend and moving to the left by the value of the subtrahend
    • Endpoint of the movement on the number line represents the difference (start at 10, move 6 units left, end at 4)

Efficient subtraction of whole numbers

  • Subtract each place value column from right to left (ones, tens, hundreds) when subtracting without regrouping
    • $456 - 123 = 333$ by subtracting $6 - 3 = 3$, then $5 - 2 = 3$, and finally $4 - 1 = 3$
  • Borrow from the next place value column in the minuend when the top digit is smaller than the bottom digit
    • Decrease the next place value in the minuend by 1 and add 10 to the current place value ($52 - 17 = 35$ by borrowing 1 ten from 5 tens, making it 4 tens and 12 ones, then $12 - 7 = 5$ and $4 - 1 = 3$)
  • Use mental math strategies for quick calculations (e.g., rounding to nearest 10 or 100)

Word problems to subtraction expressions

  • Identify the minuend as the total amount or starting quantity given in the word problem
  • Recognize the subtrahend as the amount being removed, taken away, or subtracted from the minuend
  • Represent the word problem as a subtraction expression with the minuend and subtrahend separated by a minus sign
    • Jill has 15 apples. She gives 6 apples to her friend. How many apples does Jill have left? $15 - 6 = 9$ apples
  • Solve the subtraction expression to determine the difference or remaining amount

Real-world applications of subtraction

  • Calculate the difference in quantities (heights of two buildings: 100 meters - 85 meters = 15 meters)
  • Determine the remaining amount after removal from a total (25 students - 3 absent students = 22 present students)
  • Compute the decrease in a quantity over time (population of 10,000 in 2020 - population of 9,500 in 2021 = decrease of 500)
  • Measure the distance between two points on a number line (distance between cities at mile markers 120 and 95 is $120 - 95 = 25$ miles)

Understanding the base-10 system and its role in subtraction

  • Place value determines the value of each digit based on its position in a number
  • The base-10 system uses powers of 10 to represent larger numbers
  • Subtraction and addition are inverse operations in the base-10 system
  • Understanding place value is crucial for regrouping in subtraction problems

Key Terms to Review (17)

Number Line: A number line is a visual representation of the number system, where numbers are arranged in a linear fashion along a horizontal or vertical axis. It serves as a fundamental tool in understanding and working with various numerical concepts, including whole numbers, integers, fractions, and rational and irrational numbers.
Place Value: Place value is a fundamental concept in mathematics that describes the value of a digit based on its position within a number. It is the foundation for understanding and working with whole numbers, decimals, and other numerical representations.
Whole Numbers: Whole numbers, also known as natural numbers, are the set of positive integers that begin with 1 and continue indefinitely. They are the most fundamental and commonly used numbers in mathematics, with wide applications in various fields.
Mental Math: Mental math refers to the ability to perform mathematical calculations and operations solely in one's mind, without the use of any external tools or devices. It involves the manipulation of numbers and mathematical concepts using cognitive processes, allowing individuals to quickly and efficiently solve problems without relying on written work or technological aids.
Regrouping: Regrouping, also known as carrying or borrowing, is a fundamental mathematical process used in the addition, subtraction, and multiplication of whole numbers. It involves reorganizing the place value of digits to facilitate calculations when the value of a digit in one column is not sufficient to perform the operation in the next column.
Subtrahend: The subtrahend is the number that is being subtracted from another number in a subtraction operation. It represents the quantity that is being taken away or removed from the minuend, which is the number being subtracted from.
Minuend: The minuend is the number from which another number is to be subtracted. It is the larger of the two numbers involved in a subtraction operation, and it represents the original quantity or amount before the subtraction takes place.
Compensation Method: The compensation method is a technique used in the context of subtracting whole numbers, where a number is added to both the minuend and the subtrahend to make the subtraction easier to perform. This approach aims to simplify the calculation by transforming the original problem into a more manageable form.
Equal Additions Method: The equal additions method is a technique used in the subtraction of whole numbers. It involves adding the same amount to both the minuend and subtrahend, which does not change the difference between the two numbers, making the subtraction process easier to perform.
Difference: The difference between two numbers is the amount by which one number exceeds the other. It is the result of subtracting one number from another and represents the magnitude of the separation between the two values.
Borrowing: Borrowing is a mathematical operation used to facilitate subtraction when the minuend (the number being subtracted from) does not have a sufficient digit in a particular place value to subtract the corresponding digit in the subtrahend (the number being subtracted). It involves temporarily 'borrowing' from the next higher place value to make the subtraction possible.
Subtraction: Subtraction is a mathematical operation that involves removing or taking away a quantity from another quantity. It is one of the four basic arithmetic operations, along with addition, multiplication, and division. Subtraction is used to find the difference between two numbers or to remove a part from a whole.
Integers: Integers are a set of positive and negative whole numbers, including zero. They are the foundation for many mathematical operations and concepts, and are essential in understanding and working with various topics in pre-algebra.
Traditional Algorithm: A traditional algorithm is a step-by-step procedure used to solve a specific problem or perform a calculation. It is a well-defined, systematic approach that follows a predetermined sequence of operations to arrive at a solution. Traditional algorithms are often taught in mathematics and computer science courses as fundamental problem-solving techniques.
Partial Differences Method: The partial differences method is a technique used in the context of subtracting whole numbers. It involves breaking down the subtraction process into smaller, more manageable steps, focusing on the differences between the individual place values of the numbers being subtracted.
Inverse Operations: Inverse operations are mathematical operations that undo or reverse the effects of another operation. They are used to solve equations, simplify expressions, and perform calculations in various mathematical contexts.
Base-10 System: The base-10 system, also known as the decimal system, is a numeral system that uses ten as the base. It is the most commonly used number system in the world, where digits 0 through 9 are used to represent all numbers.