Whole Numbers and Operations

Master of Arts in Child Study and Education

Math Fundamentals for Elementary Teachers

Introduction

We encourage teachers to complete the modules and practice problems at their own pace. While the modules are intended to build upon one other, they can also stand alone, depending on your individual needs. The order of the modules as laid out below provides a reasonably logical progression of skills and concepts.

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1. Whole Numbers and Operations
2. Rational Numbers (Fractions)
3. Algebra (Patterns and Equations)
4. Integers and Operations
5. Rational Numbers (Fractions, Decimals, Percents)
6. Algebra (Linear Relationships)
7. Geometry
8. Measurement

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Note that the corresponding problem set is linked at the end of each module.

Learning Objectives

• Place Value
• Connecting base-10 representations of written numerals with their value
• Multiplication
• Understanding whole number multiplication as repeated addition
• Multiplying multi-digit numbers using regrouping
• Division
• Understanding division as the inverse of multiplication
• Understanding long division and why it works
• Dividing multi-digit numbers using long division with remainders
• Factors and Prime Numbers
• Identifying factors of a whole number
• Identifying prime numbers

Place Value: Base-10 System

• All numbers consist of digits. For example, 1234 is a number with digits 1, 2, 3, and 4.
• In a base-10 (decimal) system¹, the position of each digit represents its value. For whole numbers, starting at n = 1 and moving to the left, the value of the nth digit is the digit multiplied by 10^n-1
• For example: 1234 = 1×10³ + 2×10² + 3×10¹ + 4×10⁰ = 1000 + 200 + 30 + 4
• Now try it yourself! Write 123456789 as in the form above.

¹ Of course, a base-10 system is not the only way to represent numbers. Base-2 is used in computer science, while base-60 was used in ancient Mesopotamia and continues to show up in our measurements of time.

Base-10 Representation: A Visual Model

What number does this represent?

The Base-10 Number System: Video

Multiplication as Repeated Addition

• Whole number multiplication involves repeatedly adding numbers.
• It is a mathematical operation where a number is added to itself a given number of times.
• For example, 5 multiplied by 3 = 5 + 5 + 5 = 15.
• We added the number 5 three times.
• We say “5 times 3” and write this as 5 × 3.
• The multiplicand is the number being multiplied (5).
• The multiplier is the number doing the multiplying (3).
• The answer is called the product (15).

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Exercise: Multiply 7 × 4 using repeated addition.

Multiplication as Grouping

• We can also represent multiplication as a method of totalling groups of equal quantities.
• (# of groups) × (# of objects in each group) = total # of objects
• Skip counting uses grouping to multiply. For example, if we have 9 groups of 3 (9 × 3), we can count by 3s 9 times to tally them.
• Example: Multiply 8 × 2 using grouping.

Repeated Addition and Grouping: Example

8 × 2 = 8 groups of 2 = 16

Properties of Multiplication

Multiplication is commutative¹, which means when we multiply two numbers, the order doesn’t matter. For numbers a and b, a × b = b × a.

Example: 3 × 4 = 4 × 3 = 12

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¹ You can think of the numbers as “commuting” or switching places, with the same result.

Properties of Multiplication

Commutative Property

You can multiply in any order.

a × b = b × a

3 × 4 = 4 × 3 = 12

Associative Property

You can group the numbers in any combination.

a × (b × c) = (a × b) × c

2 × (4 × 5) = (2 × 4) × 5

Zero Property�The product of 0 and any number is 0.

a × 0 = 0

9 × 0 = 0

Identity Property

The product of 1 and any number is the number.

a × 1 = a

6 × 1 = 6

Long Multiplication: Procedure

Long multiplication is a common strategy for multiplying multi-digit numbers. On slide 15, you can find a video that explains the process of multiplying multi-digit numbers.

First, write the problem and line up the numbers by place value.

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Then, multiply the top number by the ones digit of the bottom number.

Long Multiplication: Procedure

Multiply the top number by the tens digit of the bottom number.

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Finally, add the products. Remember to regroup if necessary!

Long Multiplication: Example

Long Multiplication: Khan Academy

Division is the Inverse of Multiplication

Division “undoes” multiplication and multiplication “undoes” division

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You can verify your multiplication or division answer using the inverse operation.

What is Long Division?

• Long division is a method for dividing large numbers into equal parts.
• Long division breaks down the division problem into a sequence of smaller steps.
• The dividend (typically the larger number) is divided by the divisor.
• The divisor is written outside the long division bracket and the dividend is written inside the long division bracket.
• The answer (quotient) is written above the dividend with place values aligned.
• Procedure:
• Divide
• Multiply
• Subtract
• Bring down
• Repeat or remainder

Long Division: Procedure

Look at the first two digits. The first two digits are greater than 19, so we start with the first 2 digits. Divide the hundreds.

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Bring down the tens and divide the tens.

Long Division: Procedure

Bring down the ones and divide the ones.

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Write the final answer.

Long Division: Example

Use long division to divide 2370 by 16.

Long Division: Why it Works

• It is not easy to understand why the conventional algorithm for long division works.
• The key is that in each step one does not actually divide by the divisor but by the divisor multiplied by a power of ten.
• For instance, in the following example you divide by 300, then 30, and then 3.

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• To calculate the hundreds digit in the quotient, ask yourself how many times 300 goes into 789 (789 divided by 300). Since 300 is a whole hundred, the tens and ones digits in 789 do not matter when you are dealing with the “7”.
• This logic and process is continued for the tens and ones digits!

NOTE: The digits shown in grey are not always written out.

Long Division with Remainders: Khan Academy

What is a Factor?

• Every whole number is made up of two or more factors. These factors produce the original number when multiplied together.
• Each factor divides the original number evenly, leaving no remainder.
• Take the number 16. 1 is a factor of 16 since 1 × 16 = 16. Similarly, 16 is a factor of 16 since 16 × 1 = 16. The other factors are 2, 4, and 8. From this example, we see that finding one factor allows you find another.
• Exercise: List all the factor pairs of 60 (pairs of numbers that multiply to 60).
• What are the factors of 7? 9? 23?

What is a Prime Number?

• A prime number is a number with exactly two distinct factors, 1 and itself.
• A composite number is a number with more than two factors.
• 1 is not prime since it only has one distinct factor (1). 1 is also not composite because it does not have more than two distinct factors. The number 1 is special because it is the unit (the building block) of the positive integers!
• Consider the number 7. Its only factors are 1 and itself. Therefore, 7 is prime.
• Consider the number 9. Its factors are 1, 3, and 9 (since 1 × 9 = 9, and 3 × 3 = 9). Therefore, 9 is not prime. 9 has more than two distinct factors, so 9 is compositie.
• Exercise: List the numbers from 1 to 20, and classify them as either prime or composite.

Prime Numbers as Factors

• Prime numbers are the “building blocks” of whole numbers. Every whole number can be uniquely factored into two primes (its “prime factorization”).
• Take the number 10. Its prime factorization is 5 x 2.
• Now, take the number 100. Its prime factorization is 5 x 5 x 2 x 2. Observe that prime factors can repeat!
• There is no shortcut to finding these prime factors. The best method is trial and error, but it gets easier with practice. Mastering multiplication tables helps a lot!

Prime Numbers as Factors: Video

Whole Numbers and Operations (Problem Set)

Sources for Images

Slide 5: https://www.mathswithmum.com/mab-dienes-place-value-base-10-blocks/

Slides 12–13: https://ca.ixl.com/math/lessons/multiplying-by-2-digit-numbers

Slide 16: https://www.scholastic.com/parents/school-success/learning-toolkit-blog/multiplication-and-division-models-and-strategies.html

Slide 17: https://www.onlinemathlearning.com/long-division.html

Slides 18–19: https://ca.ixl.com/math/grade-5/divide-4-digit-numbers-by-2-digit-numbers

Slide 21: https://www.homeschoolmath.net/teaching/md/long_division_why.php

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