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Counting is a part of everyday life. It happens so often that it almost comes naturally. Think about deciding the amount of maple syrup poured on pancakes or keeping track of the number of snacks left in the pantry. Those are just a few examples for personal use. This lesson will explore situations where counting helps divide things into equal amounts.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Explore
Rectangular Arrangements
Take a look at the different arrangements of the squares in the applet. Switch between 5 and 6 squares to see the different arrangements for each option. Consider vertical and horizontal arrangements as different arrangements.
Now, think about the following questions.
- Consider when there are five squares. How many of the arrangements are rectangles?
- Think about each arrangement as a division. What do the orange squares mean?
- Consider the two previous questions for when there are six squares.
Discussion
Counting Is Everywhere
Numbers are used to counting things in everyday life. The most basic numbers for counting are natural and whole numbers.
Concept
Natural Numbers
Natural numbers, also known as counting numbers, are the base numbers for counting things and objects. A natural number contains no fraction, decimal part, or negative sign. A set of natural numbers is represented by the letter N and contains the numbers from 1 to infinity.
N={1,2,3,4,5,…}
Discussion
Whole Numbers
Whole numbers represented by the symbol W are a set consisting of all natural numbers and 0.
W={0,1,2,3…}
The dots indicate that the set continues to increase indefinitely one step at a time. These are some things to consider about whole numbers and natural numbers.
| Natural Numbers | Whole Numbers |
|---|---|
| N={1,2,3,4,5,…} | W={0,1,2,3…} |
| The smallest natural number is 1. | The smallest whole number is 0. |
| All natural numbers are whole numbers | Not all whole numbers are natural numbers. |
| Example: Marky's favorite toys are three car figurines, two puzzles, and a guitar. | Example: The school principal has one notebook and ten pencils, yet he has zero erasers. |
Discussion
Number Line
A number line is a horizontal straight line used to represent numbers visually. Numbers are placed along the line at equal intervals jointly with a vertical segment at each interval.
The arrows in the line indicate that the line extends infinitely on the left- and right-hand sides. A number on the left-hand side of the line is always less than a number on the right-hand side.
Natural Numbers on a Number Line
Natural numbers can be represented in a number line that starts from 1. Then 2, 3, 4, and so on, are marked at equal distances along the line to the right of 1.
Whole Numbers on a Number Line
A number line can also be used to represent whole numbers. The line begins at 0 and the rest of the whole numbers are drawn along the line to the right.
Pop Quiz
Whole Numbers in a Number Line
Find the whole number represented by the point on the number line.
Discussion
Relationship Between Division and Multiplication
The product of two or more numbers can be divided evenly by each number in the multiplication expression. This is true every time multiplication is performed. The concepts of factors and multiples are important characteristics to understand when performing such multiplication and division. Consider their definitions.
Concept
Factor
A number a is a factor of another number b if a divides b evenly and there is no remainder. This also means that the resulting number c is such that the product of a and c equals b.
ab=c⇔a⋅c=b
The numbers a and c are factors of b because of the Commutative Property of Multiplication. For this reason, each multiplicand in a multiplication expression is called a factor. A number can have many factors.
| Number | Factors |
|---|---|
| 48 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 |
| 21 | 1, 3, 7, 21 |
| 9 | 1, 3, 9 |
Discussion
Multiple
Consider when a number is multiplied by an integer. The result is a multiple of the first number.
One example shows that 12 is a multiple of 3. This also means that 3 is a factor of 12. Therefore, a number a is a multiple of a number b if and only if b is a factor of a.
Example
Giving Equal Amounts of Candy
Dominika loves Halloween. She plans to give candy to her friends at a Halloween party. Her parents give her a bag containing 20 pieces of candy. She wants to divide the candy into equal amounts.
Find the following information to help Dominika have a great Halloween party.
a Write the factors of 20 from least to greatest.
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b Dominika decides to give four pieces of candy to some of her friends. How many friends can she give the candy to?
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c After giving away all of the candy, Dominika asks her parents for more. She wants a total of 120 pieces of candy. How many bags of 20 pieces should Dominika's parents buy?
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Hint
a Make a list of pairs of whole numbers whose product is 20. Stop the list when the factors begin to repeat.
b Divide 20 by 4.
c List some multiples of 20 until the total is 120. What number must 20 be multiplied by to get 120?
Solution
| Factors of 20 | |
|---|---|
| 20=1⋅20 | |
| 20=2⋅10 | |
| 20=4⋅5 | |
| 20=5⋅4 | |
Factors of 20:1,2,4,5,10,20
b Dominika decides to give four pieces of candy to some of her friends. Divide 20 by 4 to find how many friends she can give the candy to.
420=5 friends
This means that Dominika can give 4 pieces of candy to 5 of her friends before she runs out of candy.
c Dominika wants 120 additional pieces of candy. Consider that each bag of candy contains 20 pieces. List the multiples of 20 until 120 appears to find how many bags her parents should buy.
| Multiples of 20 |
|---|
| 20⋅1=20 |
| 20⋅2=40 |
| 20⋅3=60 |
| 20⋅4=80 |
| 20⋅5=100 |
| 20⋅6=120 |
The sixth multiple of 20 is 120. This means that Dominika's parents should buy 6 bags of candy. That is because 20⋅6=120. Great! The candy is ready and Dominika can now have fun at the Halloween party.
Example
Creating Halloween Decorations
Oh no! Dominika forgot to arrange balloons around her house for the party. She has 100 balloons. She wants to create arrangements of 6 balloons without having any balloons leftover. She also has 12 boxes of 5 spider decorations to decorate the walls.
Help Dominika complete her Halloween decorations!
a Dominika creates the 6-balloon arrangements with the 100 balloons she has. Which of the following statement describes this situation?
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b Dominika's little brother claims that there are 58 spider decorations in total. Is this claim true?
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Solution
a Dominika wants to create the 6-balloon arrangements without having any balloons leftover. The total number of balloons must be divided evenly by 6 to accomplish that. That is, 6 must be a factor of 100.
6⋅166⋅17=96=102××
There is not a whole number that can be multiplied by 6 to equal 100. This means that 6 is not a factor of 100. Dominika cannot create the 6-balloon arrangements without having any balloons leftover. She could go for 5-balloon arrangements instead, though, because 5⋅20=100.
b Dominika has 12 boxes of 5 spider decorations each. This means that the total number of spider decorations is a multiple of 5. List the multiples of 5 until a number that is greater than or equal to 58 appears.
| Multiples of 5 | 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 |
|---|
5⋅12=60
There are actually 60 spider decorations in total. Dominika's house looks really scary now. What an incredible Halloween party this will be! Discussion
Using Factors to Classify Whole Numbers
There are infinitely many whole numbers. Yet, they can be classified into two groups depending on the number of factors they have. Consider, for example, the factors of 11 and 9.
| Number | Factors | Number of Factors |
|---|---|---|
| 11 | 1 and 11 | 2 |
| 9 | 1, 3, and 9 | 3 |
The number 11 belongs to a set called prime numbers, while 9 belongs to a group called composite numbers.
Concept
Prime Numbers
A prime number is a whole number greater than 1 that is only divisible by 1 and the number itself. In other words, the only factors of a prime number are 1 and itself.
| Number | Factors | Prime? |
|---|---|---|
| 5 | 1 and 5 | ✓ |
| 6 | 1, 2, 3, and 6 | × |
| 7 | 1 and 7 | ✓ |
Discussion
Composite Numbers
A composite number is a whole number with more than two factors. This means that there is at least another number different than 1 and the number itself that divides this number evenly.
| Number | Factors | Composite? |
|---|---|---|
| 4 | 1, 2, and 4 | ✓ |
| 6 | 1, 2, 3, and 6 | ✓ |
| 7 | 1 and 7 | × |
The set of composite numbers contains infinitely many numbers. These are some facts about prime and composite numbers.
| Prime | Composite |
|---|---|
| A prime number has only two factors. | A composite number has more than two factors. |
| The only even prime number is 2. | Every even number greater than 2 is composite. |
| Any whole number greater than 1 is either prime or composite. | |
| The number 1 is neither prime nor composite. | |
Pop Quiz
Prime or Composite?
Closure
What About the Number 1?
The number 1 is a unique number because it is neither prime nor composite. Why is this?
1 Is Not Prime
The definition of a prime number states that a prime number is a whole number greater than 1 whose factors are 1 and the number itself. Suppose that the restriction about being greater than 1 is removed. List the factors of 1.Factors of 1:1
It seems that number 1 can be divided by 1 and itself, which is also 1. This fits the definition of a prime number, right? However, recall that every whole number greater than 1 has a unique prime factorization. This means that a whole number can be expressed as the product of prime numbers. Here are some examples. | Number | Prime Factorization |
|---|---|
| 2 | 2=2 |
| 3 | 3=3 |
| 4 | 4=2⋅2 |
| 5 | 5=5 |
| 6 | 6=2⋅3 |
| 12 | 6=2⋅2⋅3 |
4=4=4=2⋅22⋅2⋅12⋅2⋅1⋅1⋅1⋅1⋅1
The factor 1 could be included any number of times by this logic. Therefore, any number greater than 1 would have infinite prime factorizations. The restriction about prime numbers being greater than 1 is necessary so cases like this do not happen. 1 Is Not Composite
A composite number is defined as having more than two factors. This means that 1 cannot be composite because its only factor is itself.
Conclusions
Now that it is known that 1 is neither prime nor composite let's summarize these facts.
1 is not a prime number.
1 is not a composite number.
The only factor of 1 is itself.