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| 14 Theory slides |
| 12 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Numbers are used to counting things in everyday life. The most basic numbers for counting are natural and whole 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,…}
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. |
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 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.
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.
Find the whole number represented by the point on the number line.
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.
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.
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 |
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.
Factors of 20 | |
---|---|
20=1⋅20 | |
20=2⋅10 | |
20=4⋅5 | |
20=5⋅4 |
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.
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!
Multiples of 5 | 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 |
---|
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.
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 | ✓ |
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. |
The number 1 is a unique number because it is neither prime nor composite. Why is this?
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 |
A composite number is defined as having more than two factors. This means that 1 cannot be composite because its only factor is itself.
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.
The number line has four points on it that represent whole numbers.
We want to determine which whole numbers the given points represent. We can find this information by using the numbers shown on the number line. Consider 0, 5, 10, or 15 as reference points.
Here are a few tips to help understand how to use reference points. First, consider when the reference point is to the left of the unknown point. Then consider when it is to the right of the unknown point.
Now let's try this method to find the value of each of the given points A, B, C, and D.
We are identifying what the value of point A is on the given number line.
We see that 0 is the point closer to A. Let's use this as our reference number and count how many ticks until we reach the value of A.
We move two units to the right of the number 0 to get to point A. That means we add 2 to 0 to get our answer. We found that A represents 2.
We will follow a similar process to find the number represented by point B.
Our reference number here is 5. We move 1 unit to the right of the number 5. That means we add 1 to 5 to get our answer. We found that B represents 6.
Let's now look at point C.
Our reference number here is 10. We move 1 unit to the left of the number 10. That means we subtract 1 from 10 to get our answer. We found that C represents 9.
Lastly, let's look at point D.
We have treated 15 as our reference point. Point D is two units to the left of 15. That means we subtract 2 from 15 to get our answer. We found that D represents 13.
Find the factors for each number. Write the factors from least to greatest.
We want to determine the factors of the number 27. This means all whole numbers that divide 27 evenly. This is equivalent to finding pairs of numbers whose product is 27. Each number in these pairs of numbers will be a factor of 27. Let's make a list to find these factors.
Factors of 27 |
---|
27= 1* 27 |
27= 3* 9 |
27= 9* 3 |
We can see that the factors in the third row are the same as the ones in the second but in a different order. Then, we stop the list. We can now list the factors of 27 from the least to the greatest. Factors of27: 1,3,9,27
We will follow a similar process to find the factors of 92.
Factors of 92 |
---|
92= 1* 92 |
92= 2* 46 |
92= 4* 23 |
92= 23* 4 |
The factors begin to repeat in the fourth row, so we stop the list. Now that we have the factors of 92, let's list them from the least to the greatest. Factors of92: 1,2,4,23,46,92
Find the first six multiples of each number. List them from the least to the greatest.
We can find a multiple of a given number by multiplying it by a whole number. In this case, we want the first six multiples of 7. This means we need to multiply 7 by 1, 2, 3, 4, 5, and 6 one by one. Let's do it!
First Six Multiples of 7 |
---|
7* 1= 7 |
7* 2= 14 |
7* 3= 21 |
7* 4= 28 |
7* 5= 35 |
7* 6= 42 |
Let's list them from the least to the greatest once we have the first six multiples of 7. Multiples of7 7,14,21,28,35,42
We can find the multiples of 71 in a similar fashion. We will multiply 71 by each number from 1 to 6 one by one.
First Six Multiples of 71 |
---|
71* 1= 71 |
71* 2= 142 |
71* 3= 213 |
71* 4= 284 |
71* 5= 355 |
71* 6= 426 |
These products are the first six multiples of 71. We can now list them from the least to the greatest. Multiples of71 71,142,213,284, 355,426
We want to find which numbers in Zain's list are not factors of 75. Zain's List 1, 3, 4, 7, 25, 75 We will test each of the numbers in the list one at a time. We will then see if there is a whole number such that the number in the list multiplied by that whole number is 75. The number in the list is not a factor of 75 if there is not such whole number. Let's start with the number 25. 25* 1=25 * 25* 2=50 * 25* 3= 75 ✓ We found that 25 times 3 is 75. This means that 25 and 3 are factors of 75. Let's cross these two numbers off the list because we only want the numbers that are not factors of 75. Zain's List 1, 3, 4, 7, 25, 75 We will follow a similar process to test the remaining numbers. Now, let's test the number 4. 4* 17=68 * 4* 18=72 * 4* 19=76 * 4* 20=80 * The numbers 72 and 76 are close to 75, but not quite. The product goes up if we continue multiplying by the following whole numbers. What does this mean? Any whole number in which 4 times that whole number is 75 should be between 18 and 19. However, no whole number between two consecutive whole numbers exists.
This means that there is not whole number that 4 times that whole number is 75. So, 4 is not a factor of 75. Now, let's see what happens with the next number, 1. 1* 75= 75 ✓ These two numbers are factors of 75. We will discard them from the list. Zain's List 1, 3, 4, 7, 25, 75 Finally, let's see what happens with the number 7. 7* 9=63 * 7* 10=70 * 7* 11=77 * 7* 12=84 * We found that 70 and 77 are close to 75. However, 7 is not a factor of 75 because there is no whole number between 10 and 11. It is safe to say that 7 is not a factor of 75. Zain's List 1, 3, 4, 7, 25, 75 This means that 4 and 7 are not factors of 75.
We are asked to find the numbers in the list that are not multiples of 14. Emily's List 30, 98, 142, 196, 224 There are methods to identify if a number is multiple of a given number. One way is to find a whole number that times the given number results in the number we are testing. The number is not a multiple of the given number if that whole number does not exist. We will test the numbers in Emily's list to see if they meet this standard.
Let's begin with the number 30 from the list. Then multiply some whole numbers like 1, 2, and 3 by our given multiple of 14 to try and get 30. 1* 14=14 * 2* 14=28 * 3* 14=42 * 4* 14=56 * The numbers 28 and 42 are multiples of 14. They are close to 30, but not quite. The product goes up if we continue multiplying by the following whole numbers. If there is any whole number that times 14 is 30, it should be between 2 and 3. However, consider that no whole number between two consecutive whole numbers exists.
This means that we cannot find a whole number that times 14 is 30, regardless of how hard we try. We can safely say that 30 is not a multiple of 14. Note it on the list. Emily's List 30, 98, 142, 196, 224 Now, let's analyze the number 98. 5* 14=70 * 6* 14=84 * 7* 14= 98 ✓ If we multiply 7 by 14, we get 98. This means that 98 is a multiple of 14. We can discard it from the list because we are looking for numbers that are not multiples of 14. Emily's List 30, 98, 142, 196, 224 The next number on the list is 142. This is close to 140. We know that 10 times 14 equals 140. Next, try 11. The product of 11 and 14 is 154. Well, 142 is between the two results. 10* 14=140 * 11* 14=154 * There is no whole number between 10 and 11 that meets our standard. The same concept was found earlier in this exercise. It is safe to say that 142 is not a multiple of 14. Emily's List 30, 98, 142, 196, 224 Next, we will test the last two numbers in the list, 196 and 224. We can get those numbers by multiplying 14 by 14 and 16 by 14. 14* 14= 196 ✓ 16* 14= 224 ✓ These two numbers are multiples of 14. Let's cross them off the list. Emily's List 30, 98, 142, 196, 224 The two remaining numbers are 30 and 142. They are our champions. They are not multiples of 14.