Level D Lesson 13
Great Common Factor, Prime and Composite Numbers
In lesson 13 the objective is, the student will investigate greatest common factors and prime and composite numbers.
The skills students should have in order to help them in this lesson include multiplication and division facts.
We will have three essential questions that will be guiding our lesson. Number 1, how do factors relate to a product in multiplication? Number 2, how do I determine the greatest common factor of two or more numbers? And number 3, how can we use prime factorization to determine the greatest common factor for a set of two or more numbers?
The SOLVE problem for this lesson is, Mr. Morgan is making balloon decorations for a party. He has 24 red balloons, 18 blue balloons, and 12 green balloons. Mr. Morgan wants to tie the balloons in bunches with the same number of each color balloon in each bunch. What is the greatest number of decorations he can make?
We will start by Studying the Problem. First we will underline the question. What is the greatest number of decorations he can make? Now we want to take this question and put it in our own words in the form of a statement. This problem is asking me to find the greatest number of decorations he can make.
During this lesson we will learn how to find greatest common factors of 2 or more numbers. We will use this knowledge to complete this SOLVE problem at the end of the lesson.
Throughout this lesson students will be working together in cooperative pairs. All students should know their role as either Partner A or Partner B before beginning this lesson.
We are going to start this lesson by talking about factors. Remember that factors are numbers that when multiplied together give us a product. Each pair of students will need 30 centimeter cubes in order to complete this activity. Let’s name the factors of 12. Each factor of 12 that we name we will place a centimeter cube over that factor on our chart. The factors of 12 are, 1, and 12. Since I can multiply 1 times 12 to give me 12, 2 and 6, since I can multiply 2 times 6 to give me 12, and 3 and 4 as I can multiply 3 times 4 and this also equals 12. Now let’s name the factors of 15 following the same method. 1 and 15, 3 and 5.
Let’s talk about the word common. What does the word common mean? It means something that is shared or alike. For example, these two shapes have something in common. They are both blue, and they are both circles. Looking at your chart, you may have noticed that some of the factors of 12 were the same as the factors of 15. This happened every time you had to stack a centimeter cube! When two number share the same factor, that is called a COMMON FACTOR. Using your chart, name the common factors between 12 and 15. Remember that our common factors are those that have two centimeter cubes stacked on them. Our common factors are 1 and 3. We found two common factors for 12 and 15. These were 1 and 3. The largest, or greatest, factor is 3. We call this the GREATEST COMMON FACTOR or GCF. Let’s find the GCF of 4, 6, and 24.
First let’s name the factors of 4. 1 and 4, because 1 times 4 is 4, and 2 because 2 times 2 is 4. Now name the factors of 6;1, 6, 2 and 3. Finally, name the factors of 24; 1 and 24, 2 and 12, 3 and 8, 4 and 6. Our common factors are those factors that have 3 centimeter cubes stacked on them, 1 and 2 are our common factors. We found two common factors for 4, 6, and 24; 1 and 2. The GCF of 4, 6, and 24 is 2 because 2, is the greatest of these common factors.
Now let’s find the GCF or greatest common factor of 8 and 12. Instead of placing a centimeter cube on the chart, we’ll complete this pictorially. First we want to find the factors of 8, We will use a red colored pencil to help us to draw this pictorially on the chart. We will draw a diagonal line from the top left to the bottom right of the square. For example 2 is a factor of 8 so we will draw a diagonal line from the top left to the bottom right in red over the number 2. Let’s mark the factors of 8; 1 and 8, 2 and 4. Our second step is to find the factors of 12. This time we will use blue. We will draw a diagonal line from the top right to the bottom left of the square. For example 2 is a factor of 12. So we will draw a diagonal line from the top right to the bottom left of this square in blue. Let’s find the factors of 12. 1 and 12, 2 and 6, 3 and 4. Our common factors should have both a blue and a red diagonal line through them. Our common factors of 8 and 12 are 1, 2, and 4. So the common factors of 8 and 12 are 1, 2, and 4. We need to find the greatest common factor. So this will be the greatest of these three numbers. The greatest number is 4. So the GCF is 4.
Let’s talk about the steps we used to find the GCF. First we cross out the factors using colored pencils. Second, find the factors that each of the numbers share. And third, Circle the greatest common factor shared by the numbers. We were able to determine the GCF for 8 and 12 by crossing out the factors using colored pencils. Now let’s use a graphic organizer to help us with the same problem. Looking back at our chart we crossed out the factors of 8 and 12 and then circled the common factors. We’re going to use the graphic organizer below the chart to help us. First let’s list the factors of 8. We will do this in order from smallest to largest; 1, 2, 4, 8. Next let’s list the factors of 12. Again, we will do this from smallest to largest; 1, 2, 3, 4, 6, 12. Looking at our completed graphic organizers we can see that 4 is the greatest of our common factors. So we will shade the number 4 on each of our graphic organizers. The GCF is 4.
Let’s complete another example. This time we’re going to identify the GCF of 24, 44, and 48. We will use the number chart as well as the graphic organizer to help us to solve this problem. First let’s identify the factors of 24. The factors of 24 on our number chart are 1 and 24, 2 and 12, 3 and 8, and 4 and 6. Now we will list these numbers in order from smallest to largest in the graphic organizer; 1, 2, 3, 4, 6, 8, 12, 24. Using a different colored pencil let’s find the factors of 44 on the number chart, 1 and 44, 2 and 22, 4 and 11. Now let’s list these numbers in order from smallest to largest on the graphic organizer, 1, 2, 4, 11, 22, 44. And now let’s list the factors of 48, using the number chart first to help us we want to use a third colored pencil to mark the factors of 48. Let’s use a horizontal line through each of the numbers that are factors of 48; 1 and 48, 2 and 24, 3 and 16, 4 and 12, 6 and 8. And now we will list these factors of 48 in order from smallest to largest on the graphic organizer, 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. Looking at the factors of 24, 44 and 48, only one of these factors is in common this is the number 4. We will shade the number 4 on the graphic organizers as this is the greatest common factor. The greatest common factor of 24, 44 and 48 is 4.
Now let’s talk again about how we found the GCF? We crossed out the factors using colored pencils. We found the factors that each of the numbers shared. We used a graphic organizer to list the factors. We circled the greatest common factor shared by the numbers. And we shaded the GCF.
When we do not have a number chart or a graphic organizer to help us we can use a list to help us find the GCF. Let’s complete this example together. Find the GCF of 16 and 28. Use a list to help you solve. First we will list the factors of 16. The factors of 16 are, 1, 2, 4, 8, and 16. Now let’s list the factors of 28; 1, 2, 4, 7, 14, and 28. Now let’s look at the common factors. The common factors are 1, 2, and 4. The greatest of these common factors is 4. The GCF of 16 and 28 is 4.
Now we will add to the foldable helping us to organize the information we have learned so far in this lesson for future reference. The second flap will represent the Greatest Common Factor. On the inside of this flap, you will work with your teacher to include information about the GCF
Now we are going to explore Prime and Composite. Each pair of students will need their centimeter cubes in order to participate in this activity. Let’s start by using the grid and our centimeter cubes to create a rectangle using 8 cubes. An example of a rectangle using 8 cubes is shown. This rectangle has the dimensions of 2 and 4. 2 and 4 are factors of the area which is 8. We can turn the rectangle to show a 4 by 2 rectangle. The dimensions for this rectangle are the same factors 2 and 4. The area of the rectangle remains the same, it is still 8. But the rectangle has been turned or rotated. Let’s talk about this example. When we built a rectangle with 8 cubes, we started with a 2 by 4 or 2 times 4. When we rotated the rectangle the array displayed was 4 times 2. 2 and 4 are factors of the area of 8. Even if we rotate the rectangle, the factors will still remain the same. They are still 2 and 4.
Let’s try to create a different rectangle from 8 cubes. Another example of a rectangle using 8 cubes would be a rectangle that has the dimensions of 8 and 1. 8 and 1 are also factors of the area 8. We can turn the rectangle to show the dimensions of 1 and 8. Again, this rectangle has the same factors of 8 and 1. The area of the rectangle remains the same. It is still 8 but the rectangle has been turned or rotated. Let’s talk about that example. Our next array of cubes shows an 8 by 1 rectangle or 8 times 1. When we rotated the rectangle the array displayed 1 times 8. 1 and 8 are also factors of the area 8. Even if we rotate the rectangle, the factors will still remain the same they are still 1 and 8. How many different rectangles can be created with 8 centimeter cubes? 2, even though we can rotate the rectangles and read their dimensions differently, we only created two unique rectangles
Now let’s use 5 cubes to create a rectangle. The rectangle shown has 5 cubes. Its dimensions are 1 by 5 or 1 times 5. The area of this rectangle is 5. This same rectangle can be rotated and the dimensions are 5 by 1 or 5 times 1. This is still the same rectangle as it still has the same factors for 5. Those factors are 1 and 5. This is the only rectangle that can be built using 5 cubes. When we built a rectangle with 5 cubes, we started with 1 times 5. When we rotated the rectangle the array displayed 5 times 1. 1 and 5 are factors of the area 5. Even if we rotate the rectangle, the factors will still remain the same. The factors are still 5 and 1. The number 5 only has two factors: 1 and 5.
When a number has more than two factors the number is COMPOSITE. In the examples that we’ve completed 8 is a composite number because it has more than two factors. The factors of 8 are 1, 2, 4, and 8. This is what makes it a composite number.
When a number has only two factors the number is PRIME. In the examples that we’ve completed 5 is a prime number. Because the only two factors of the number 5 are 1 and 5.
In this next example instead of using the grid and the centimeter cubes we will use pictures to help us to determine if the number is Prime or Composite. We want to draw pictures of rectangles that have 6 squares for this example. There are two rectangles that we can draw. One rectangle has the dimensions of 6 and 1 and our second rectangle has the dimensions of 3 and 2. Even though we can rotate these rectangles to show two other rectangles, remember that the rectangle shows the same two factors when it’s rotated. So we have included all of our factors with the two pictures that we’ve drawn. Our factors are 1, 2, 3, and 6. Now that we know all of the factors let’s talk about what numbers 6 is divisible by. If a number is divisible by another number it means that you get a quotient that is a whole number with no remainder when you divide the numbers. What numbers can we divide 6 by that will give us a whole number answer? 6 divided by 1 equals 6. So 6 is divisible by 1. 6 divided by 6 equals 1. So 6 is divisible by 6. 6 divided by 2 equals 3. So 6 is divisible by 2. And 6 divided 3 equals 2. So 6 is divisible by 3. In our column Numbers Divisible By let’s list the facts and the numbers that 6 is divisible by. 6 divided by 1 equals 6, 6 divided by 2 equals 3, 6 divided by 3 equals 2, and we know that a number is always divisible by itself. So 6 divided by 6 equals 1. Our numbers divisible by are 1, 2, 3, and 6. Because 6 has more than two factors the number 6 is Composite.
Let’s take a look at another example together. This time we’ll talk about the number 15. We will start by talking about the pictures of rectangles that we can draw with 15 squares. We can draw two different rectangles with a total of 15 squares. The dimensions of our first rectangle are 15 and 1. And the dimensions of our second rectangle are 5 and 3. The factors of 15 are 1, 3, 5, and 15. Let’s talk about the numbers that 15 is divisible by. 15 divided by 1 equals 15, 15 divided by 3 equals 15 (should be 5), and 15 divided by 5 equals 3. Remember that a number is also always divisible by itself. So 15 divided by 15 equals 1. Our numbers that 15 is divisible by are 1, 3, 5, and 15. Because 15 has more than two factors, 15 is a composite number.
We will complete one more example together using the number 11. Let’s draw pictures of rectangles that we can make using 11 squares. There is only 1 rectangle that can be made using 11 squares. This rectangle has the dimensions of 11 and 1. So the factors of 11 are 1 and 11. Next let’s talk about the numbers that 11 is divisible by. 11 divided by 1 equals 11 and we also know that 11 is divisible by itself 11 divided by 11 equals 1. So the numbers that 11 is divisible by are 1 and 11. Remember that when a number has only two factors 1 and itself, the number is Prime. So 11 is a Prime number.
There is another way to determine Prime and Composite numbers using a number grid. It is called a Sieve of Eratosthenes.