Beginning Mathematics: Division (2nd ed, 1987)

[Music] [Applause] Got Ready to learn about another mathematics operation? How about division? How might we use division during gym class? Well, suppose you're in a gym class and there are 12 students altogether. Your gym teacher has two soccer balls and she wants you to practice passing the ball. What would you do? Well, you divide the class into two groups. One half of the class would use one soccer ball and one half the other. But how should we go about dividing the class? No, we can't split a student in half. Perhaps we'd better count to be sure we have the same number in each group. Maria in one group, Betty in the other. Chris in the first group, Bob in the second, then Lesie and Ramon and Angela and George and Fernando and Bonnie and Ellen and Renee. Now, you could count the number of students in each group and you'd find that there are six. But we could also say 12 / 2 equals 6. Now, we're using division. We can write the same operation in several different ways. 2 / 12 is 6. 12 / 2 = 6. They all describe the same operation. In division, we call the result the quotient. What if we now wanted to divide the class into four teams for a beanag race? If there are 12 children and four teams, how many children will there be on each team? How many fours are there in 12? Well, there's Maria, Betty, Chris, and Bob. And there's Lesie, Ramon, Angela, and George. And there's Fernando, Bonnie, Ellen, and Renee. Now, how many times did we subtract four people from the whole group of 12? 12 - 4 is 8. 8 - 4 is 4. 4 - 4 is 0. 1 2 3 times. So if you think of division as just repeated subtraction of the same number, we know that the answer to our equation is three. But this looks like the same kind of array we sometimes use in multiplication. Yes, this is a 4x3 array. Now we can think of our division problem in terms of multiplication. What number * 4 equ= 12? And we know that that number is 3. We can describe the division process in terms of multiplication because division is the reverse of multiplication. Now let's look at the multiplication table. If we want to know how many twos there are in 12, we can find it in this multiplication table. Very quickly, how many twos in 12? 6. We can prove this because we know that 6 * 2 is 12. We can also see how many sixes there are in 12. Two. Now, let's think about the properties of division. Let's suppose that you decided you wanted to plant some flower seeds and raise your own garden. You walk it off and find that each row is about 18 steps [Music] long. Now suppose there are six packages of [Music] seeds. Carrots. Who would want to plant vegetables in a flower garden? [Music] Now we have six different kinds of flowers. How can we give each kind of flower an equal amount of space in the row? Well, it looks as if we can solve this by division. But which way should we divide? Should we divide 18 by 6 or 6 by 18? Does it make any difference which way we divide? Well, let's try. 6 / 18 equals what number? Well, if you have six cookies and you want to divide them among 18 friends, how many cookies would each friend receive? Each would receive a very small part of one cookie. But if you have 18 cookies divided among six friends, how many cookies would each receive? Three. We can see that the order in which two numbers are divided does make a difference. So if we have a row 18 steps long and we want to divide it into six equal parts, we must divide 18 by six. We'll find that each part should be about three steps long. If there were three of you working and each wanted to take a turn at planting the seed in the section that was three steps long, how long a row would you each plant? How many threes are there in three? One. Any number divided by itself is always one. What about a number divided by one? How many ones are there in three? Three. And while we're talking about low numbers, what about 0? What is the quotient of 0 / 3? Let's use our cookies again for this problem. If you have three cookies and you divide them among three people, each one will receive one cookie. If you have no cookies and you divide them among three people, how many cookies will each person receive? None. The quotient of zero divided by any number is always zero. What about the other way around? What's the quotient of any number divided by 0? Well, just think about how many nothings there are in three cookies. Or think about how many times zero can be subtracted from three. You'll find that there is no number that can be an answer to this problem. Well, the garden is coming along fine. You can almost see the flowers growing right now. [Music] [Music] Well, we're just about ready to plant the seeds. These are the netoriium seeds. The instructions on the package say they should be planted in holes with three seeds in each hole. If we have 16 seeds in this package, how many holes should we dig? Well, how many threes are there in 16? Three in this hole, 13 left. Three in this hole, 10 left. Three in this hole, seven left. Three in this hole, four left. Three in this hole, one left. We've subtracted three five times. So our quotient is five, but there's a remainder of one left over. Five holes planted, one seed left over. Some division problems do not come out even and sometimes you'll have a remainder left over. These four mathematics symbols tell us about operations with numbers. Actually, they tell us what operation we're going to perform between two numbers. They tell us if we're going to add two numbers together, subtract one number from another, multiply one number by another, or divide one number by another. When we see these symbols together like this, it also tells us that these operations are related. We know that subtraction is the reverse of addition and that multiplication of whole numbers can be thought of as repeated addition of the same number. Today we found that division is the reverse of multiplication and that division of whole numbers can also be thought of as repeated subtraction of the same number. Well, that's enough for one day. But before we go, let's see how the flower garden is growing. It seems to be coming along nicely. These are the pansies. These will probably be nesters when they grow. Now, let's see. What will these be? It's a carrot. Now, who do you suppose wanted to plant carrots in a flower garden? [Music]