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Whole Numbers Module Overview

Acknowledgments This presentation is based on and includes content derived from the following OER resource: Psychology An OpenStax book used for this course may be downloaded for free at: https://openstax.org/https://openstax.org/details/books/psychology


Introduction In this module you will learn about: The Whole Number System. This number system includes the numbers used for counting along with zero. We will also cover the four arithmetic operations, important properties, and some relevant vocabulary.

Counting numbers and whole numbers Counting Numbers or Natural Numbers include 1, 2, 3, 4, … and so on. The notation “…” is called ellipses and indicates that the numbers continue endlessly. Whole Numbers include the counting numbers and zero, so they are: 0, 1, 2, 3, 4, 5, … Our number system is called a place value system since the value of a digit depends on its position or place in a number. We use a base-10 system. Number lines provide a helpful visual guide for our numbers

Model whole numbers Base-10 blocks provide a way to model place value, as shown in the figure below. The blocks can be used to represent hundreds, tens, and ones. Notice that the tens rod is made up of 10 ones, and the hundreds square is made of 10 tens, or 100 ones. (Image: Prealgebra, OpenStax Fig 1.4)

Round whole numbers At times, all one needs is an approximation of a number, depending upon the level of accuracy required. This process is called rounding, and the specific place value needs to be indicated. To round, take the following steps: Locate the given place value. All digits to the left of that place value do not change. Underline the digit to the right of the given place value. If the underlined digit is greater than or equal to 5, add 1 to the digit in the given place value. If the underlined digit is less than 5, do not change the digit in the given place value. Replace all digits to the right of the given place value with zeros.

(see the next slide for solutions!) Round whole numbers, sample problems To practice rounding whole numbers try the following: Round 175 to the nearest ten. Round 1543 to the nearest hundred. Challenge: round 1981 to the nearest hundred. (see the next slide for solutions!)

Round whole numbers, sample solutions Round 175 to the nearest 10. Since the number to the right is 5, we increase the 7 by 1 and replace the 5 with a 0 to get 180. Round 1543 to the nearest 100. Since the number to the right is 4, we leave the 5 alone and replace the 43 with 0’s to get 1500. Round 1981 to the nearest 100. Since the number to the right is an 8, we increase the 9 by 1. Note that this makes it a 10, so we need to put a 0 and carry the 1 into the thousands place to get a 2! We then replace the 81 with 0’s to get 2000.

Addition of whole numbers We use the notation “+” and some common words to indicate addition, including plus, sum, increased by, more than, added to, and total of. To add whole numbers: Write the numbers so each place value lines up vertically, starting with the ones digit and moving left. Add the digits in each place value. Work from right to left starting in the ones place. If a sum in a place value is more than 9, carry to the next place value. Continue.

(see the next slide for the solution!) Addition of whole numbers, sample problem Sam took five tests and her scores were: 100, 93, 75, 88 and 81. What is the total points she earned on the five tests?   (see the next slide for the solution!)

Addition of whole numbers, sample solution Sam took five tests and her scores were: 100, 93, 75, 88 and 81. What is the total points she earned on the five tests?   Answer: 437

Properties of whole number addition Identity Property of Addition The sum of any number “a” and 0 is the number. Specifically: a + 0 = a and 0 + a = a Commutative Property of Addition Changing the order of the numbers you are adding does not change the value of their sum. Specifically, for numbers “a” and “b”: a + b = b = a

Subtraction of whole numbers We use the notation “-” and some common words to indicate subtraction, including minus, difference, decreased by, less than, and subtracted from. To subtract whole numbers: Write the numbers so each place value lines up vertically. Subtract the digits in each place value, working from the right to the left, starting in the ones place. If the digit on top is less than the digit below, borrow as needed. Continue. Check by adding. Example of translating and subtracting: A pair of skis is on sale for $799. their regular price is $1050. What is the difference between the regular price and the same price? (try this, you should get $251).

Subtraction of whole numbers, sample problem A pair of skis is on sale for $799. Their regular price is $1050. What is the difference between the regular price and the sale price? (see the next slide for the solution!)

Subtraction of whole numbers, sample solution A pair of skis is on sale for $799. Their regular price is $1050. What is the difference between the regular price and the sale price? Answer: $251

Multiplication of whole numbers We use the notations “×”,  ”*” or “()” as well as some common words to indicate multiplication, including times, product, and twice. Please also note the terminology for multiplication as shown in the equation a × b = c. We say that “a” and “b” are factors and “c” is the product.

How to multiply whole numbers Write the numbers so each place value lines up vertically. Multiply the digits in each place value working from right to left, starting with the ones place in the bottom number. Multiply the bottom number by the ones digit in the top number, then by the test digit, and so on. If a product is more than 9, carry to the next place value. Write the partial products, lining up the digits in the place values with the numbers above. Repeat for the tens place in the bottom number, the hundreds place, and so on. Insert a zero as a placeholder with each additional partial product. Add the partial products.

Properties of whole number multiplication Multiplication Property of Zero The product of any number and 0 is 0. Specifically, a × 0 = 0 and 0 × a = 0. Identity Property of Multiplication  The product of any number and 1 is that number. Specifically, 1 × a = a and a × 1 = a. Commutative Property of Multiplication Changing the order of the factors does not change the value of the product. Specifically, a × b= b × a

Division of whole numbers  

How to divide whole numbers Divide the first digit of the dividend by the divisor. If the divisor is larger than the first digit of the dividend, divide the first two digits of the dividend by the divisor, and so on. Write the quotient above the dividend. Multiply the quotient by the divisor and write the product under the dividend. Subtract that product from the dividend. Bring down the next digit in the dividend. Repeat from Step 1 until there are no more digits in the dividend to bring down. Check by multiplying the quotient by the divisor.

Properties of whole number division Division Properties of One Any number divided by 1 is itself: a / 1 = a Any number (except 0) divided by itself is 1: a / a = 1 Division Properties of Zero Zero divided by any number is 0: 0 / a = 0 Dividing a number by zero is undefined: a / 0 = undefined

UNDERSTANDING is the goal. How to study this module “One learns by doing the thing, for though you think you know it, you have no certainty until you try.” -Sophocles Start by reading through the chapter in the ebook. Don’t try to read the entire chapter at once; take it slowly and read one section at a time. The ebook authors work through many examples step-by-step. You will want to have a pencil and paper and work them alongside the text. Access the available resources for additional lessons, explanations and examples. Don’t hesitate to take notes, reread, press pause, and rewind to review again. Upon completing the above, work all of the suggested exercises as you proceed through each section. Remember: reading or watching others working the math does not translate into you being able to work it on your own proficiently……the only way to learn math is to practice, practice and, then, practice some more! Keep in mind an old Chinese Proverb: “I listen and I forget; I see and I remember; I do and I understand.” UNDERSTANDING is the goal.

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