Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 1: Properties of Arithmetic

Lesson 1: Why Start with Arithmetic?

Grade 4 students using The Art of Problem Solving: Prealgebra explore why arithmetic serves as the foundation for algebra, learning key vocabulary such as integers, positive and negative numbers, and the number line. The lesson explains the distinction between arithmetic and algebra, introducing the idea that algebraic rules like the distributive property generalize arithmetic facts to work for any numbers. Students are challenged to understand not just how calculations work, but why, building the mathematical reasoning needed for more advanced problem solving.

Section 1

Counting and Whole Numbers

Property

Counting Numbers: The counting numbers start with 1 and continue. They are also called natural numbers. The notation "..." is called an ellipsis, which is another way to show "and so on," or that the pattern continues endlessly.

1,2,3,4,5...1, 2, 3, 4, 5...

Whole Numbers: The whole numbers are the counting numbers and zero.

0,1,2,3,4,5...0, 1, 2, 3, 4, 5...

Counting numbers and whole numbers can be visualized on a number line. The point labeled 0 is called the origin. When a number is paired with a point, it is called the coordinate of the point.

Examples

  • In the set {0,34,6,9.5,23,1100, \frac{3}{4}, 6, 9.5, 23, 110}, the counting numbers are 6,23,1106, 23, 110. The whole numbers are 0,6,23,1100, 6, 23, 110.
  • From the list {12,1,4,0,7.8,99\frac{1}{2}, 1, 4, 0, 7.8, 99}, the counting numbers are 1,4,991, 4, 99. The whole numbers are 0,1,4,990, 1, 4, 99.
  • Given the numbers {1.5,200,0,15,581.5, 200, 0, 15, \frac{5}{8}}, the counting numbers are 200,15200, 15. The whole numbers are 0,200,150, 200, 15.

Explanation

Counting numbers are the numbers you use to count things, starting from 1. Whole numbers are almost the same, but they also include zero. Think of it as starting your count from nothing (zero) instead of one.

Section 2

What is an Integer?

Property

Integers are counting numbers, their opposites, and zero.

…−3,−2,−1,0,1,2,3… \ldots -3, -2, -1, 0, 1, 2, 3 \ldots

Examples

  • To plot the number 4 on a number line, we move 4 units to the right of 0.
  • To plot the number -5, we start at 0 and move 5 units to the left into the negative side.

Section 3

Positive Numbers Definition

Property

Positive numbers are all numbers greater than zero: x>0x > 0

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Counting and Whole Numbers

Property

Counting Numbers: The counting numbers start with 1 and continue. They are also called natural numbers. The notation "..." is called an ellipsis, which is another way to show "and so on," or that the pattern continues endlessly.

1,2,3,4,5...1, 2, 3, 4, 5...

Whole Numbers: The whole numbers are the counting numbers and zero.

0,1,2,3,4,5...0, 1, 2, 3, 4, 5...

Counting numbers and whole numbers can be visualized on a number line. The point labeled 0 is called the origin. When a number is paired with a point, it is called the coordinate of the point.

Examples

  • In the set {0,34,6,9.5,23,1100, \frac{3}{4}, 6, 9.5, 23, 110}, the counting numbers are 6,23,1106, 23, 110. The whole numbers are 0,6,23,1100, 6, 23, 110.
  • From the list {12,1,4,0,7.8,99\frac{1}{2}, 1, 4, 0, 7.8, 99}, the counting numbers are 1,4,991, 4, 99. The whole numbers are 0,1,4,990, 1, 4, 99.
  • Given the numbers {1.5,200,0,15,581.5, 200, 0, 15, \frac{5}{8}}, the counting numbers are 200,15200, 15. The whole numbers are 0,200,150, 200, 15.

Explanation

Counting numbers are the numbers you use to count things, starting from 1. Whole numbers are almost the same, but they also include zero. Think of it as starting your count from nothing (zero) instead of one.

Section 2

What is an Integer?

Property

Integers are counting numbers, their opposites, and zero.

…−3,−2,−1,0,1,2,3… \ldots -3, -2, -1, 0, 1, 2, 3 \ldots

Examples

  • To plot the number 4 on a number line, we move 4 units to the right of 0.
  • To plot the number -5, we start at 0 and move 5 units to the left into the negative side.

Section 3

Positive Numbers Definition

Property

Positive numbers are all numbers greater than zero: x>0x > 0

Examples