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Dazzling DivisionGames and Activities That Make Math Easy and Fun
By Lynette Long
John Wiley & SonsISBN: 0-471-36983-7
Chapter OneAnatomy of a Division Problem
What is division? How do you write a division problem? How do you read one? What are you actually doing when you divide one number by another? Once you can answer these questions, you are well on your way to discovering the magic of division.
There are four basic operations in mathematics: addition, subtraction, multiplication, and division. Division is usually taught last since it is the hardest to master, but learning to divide is just as important as learning to add, subtract, or multiply.
Each of the four basic operations can be expressed as a symbol. The plus sign (+) tells you to add two numbers together. The minus sign (-) tells you to subtract one number from another. The multiplication sign (×) tells you to multiply one number by another. The division sign (÷) tells you to divide one number by another. The problems 6 + 4, 6 - 4, 6 × 4, and 6 ÷ 4 are different problems that have different answers.
There are other ways to indicate division besides using the division sign. Sometimes a division problem is written in the form 8[square root of (6,424)]. This problem is read as six thousand, four hundred twenty-four divided by eight. The problem 2[square root of (222)] is two hundred twenty-two divided by two. And 10[square root of (5)] is five divided by ten.
Youcan also write a division problem by using a slant line, /. The problem 12/3 is twelve divided by three. You can read 164/4 as one hundred sixty-four divided by four. Or you could write fifty-two divided by twenty-six as 52/26.
One more way to indicate division is with a bar. Twelve divided by four would be written as 12/4, and 14,955/5 would be read as fourteen thousand, nine hundred fifty-five divided by five.
You can write the same division problems five different ways.
Now that you know how to write division problems, it's time to find out the names of the three parts of a division problem. Look at the problem 24 ÷ 3. Read this problem as twenty-four divided by three. The number 24 is called the "dividend" (the number to be divided), and the number 3 is called the "divisor" (the number by which the dividend is divided). The answer to the problem (in this case, 8) is called the "quotient" (the number resulting from dividing one number by another).
What are the parts of 42 ÷ 6 = 7? 42 is the dividend 6 is the divisor 7 is the quotient
Try a different format. What are the parts of 10/2 = 5? 10 is the dividend 2 is the divisor 5 is the quotient
Identify the parts of this problem: 6[square root of (132)]. 132 is the dividend 6 is the divisor The quotient is not given.
Remember this problem is read as one hundred thirty-two divided by six. It is not six divided by one hundred thirty-two, even though the 6 is written first.
Here is a coded message. Match the answers to the division problems in the list to the number-letter codes here to spell a word and see how well you understand dividends, divisors, and quotients.
1. What is the divisor in the problem fifteen divided by five equals three?
2. What is the dividend in the problem 24 ÷ 6 = 4?
3. What is the divisor in the problem 12/3 = 4?
4. What is the quotient in the problem 30 ÷ 6 = 5?
5. What is the dividend in the problem 4[square root of (12)]?
6. What is the quotient in the problem sixty divided by five equals twelve?
7. What is the divisor in the problem 10/5 = 2?
8. What is the quotient in the problem 56/4 = 14?
9. What is the divisor in the problem 20[square root of (20)] = 1?
Did you spell a word? EXCELLENT!
Division is an essential mathematical skill. You will use division every day of your life, so start practicing and soon you'll become a division master. Then you can proudly display the division master certificate at the back of this book.
Chapter TwoDivision as Grouping
Try this tasting activity to learn what division really means. Division is the process of dividing items into equal groups. The dividend tells you how many things you are dividing into equal groups. The divisor tells you how many groups you are dividing these things into. The quotient tells you how many items are in each group.
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12 cookies (or crackers, pretzels, raisins, or any other snack)
large serving plate
12 small plates
1. Put the twelve cookies on the serving plate. Put two small plates on the table. You are going to divide the twelve cookies between two imaginary guests to solve the division problem 12 ÷ 2. The number of cookies (12) is the dividend. The number of plates (2) is the divisor. Divide the cookies equally between the two smaller plates. Remember, each guest has to get exactly the same number of cookies. How many cookies are on each plate? If you split the cookies equally, there should be six cookies on each plate. Twelve divided by two is six.
2. Place the twelve cookies back on the serving plate. (Don't eat any!) Now place three small plates on the table. You are going to solve the problem 12 ÷ 3. In this problem, 12 is the dividend and 3 is the divisor. Split the cookies equally among the three plates. There should be four cookies on each plate. Twelve divided by three is four.
3. Put the cookies back on the serving plate again and put four small plates on the table. Divide the cookies up. How many cookies does each guest get if each gets the same number? Each guest gets three cookies. Twelve divided by four is three.
4. Return the cookies to the serving plate. Set the table for six guests. Now divide the cookies among these six guests. Each guest gets only two cookies. Oh, well. Twelve divided by six is two.
5. Finally, set the table for twelve guests. Divide the cookies fairly among the twelve guests. Each guest gets only one cookie. Twelve divided by twelve is one.
6. Write down the five different division problems you just solved:
12 ÷ 2 = 6
12 ÷ 3 = 4
12 ÷ 4 = 3
12 ÷ 6 = 2
12 ÷ 12 = 1
If you had enough cookies and enough plates, you could solve any division problem no matter how large the numbers.
7. Now you and your friends can eat all the cookies.
Chapter ThreeDivision as Repeated Subtraction
Another way to think of division is a method called "repeated subtraction." It is called repeated subtraction because you repeatedly subtract the divisor from the dividend until you get to 0 and have nothing left. The number of times you subtract the divisor from the dividend to get to 0 is the quotient. Try this activity to practice division as repeated subtraction.
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30 paper clips
1. The problem is 30 ÷ 10. Here's how you find the quotient with repeated subtraction. Start with thirty paper clips. Thirty is the dividend. Take away ten paper clips and put them in a pile. Ten is the divisor. Take away ten more paper clips and put them in a second pile. Take away the last ten paper clips and put them in a third pile. How many times did you subtract 10 from 30 to get to 0? Count the piles of paper clips you made. You should have three piles of ten paper clips, so 30 ÷ 10 = 3. Three is the quotient.
2. If you don't have paper clips handy, you can subtract 10's using pencil and paper instead.
30 - 10 = 20
20 - 10 = 10
10 - 10 = 0
You subtracted 10 from 30 three times to get to 0, so 30 ÷ 10 = 3.
3. Now solve 30 ÷ 6. How many times can you subtract six paper clips from a pile of thirty? Put your thirty paper clips in one big pile. Then make piles of six paper clips until you have no paper clips left. How many piles did you make? Count them. There should be five piles of six paper clips, so 30 ÷ 6 = 5. Five is the quotient. If you don't have paper clips, you can use pencil and paper to repeatedly subtract 6's.
4. What is 30 ÷ 2? Put your thirty paper clips in one big pile again, then make piles of two paper clips until none are left. How many times did you subtract 2 from 30 to get to 0? You should have fifteen piles of two paper clips, so 30 ÷ 2 = 15. Fifteen is the quotient.
Chapter FourDivision as the Opposite of Multiplication
There is a third way to understand and solve division problems. Just think of them as backward multiplication problems. In fact, division is often called the "inverse" (opposite) of multiplication.
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1. In multiplication, the two numbers that are multiplied are called "factors," and the answer is called the "product." In the multiplication problem 3 × 4 = 12, the numbers 3 and 4 are the factors and 12 is the product. To change this to a division problem, first read it backward starting with the product: 12 = 4 × 3. Then change the equals sign to a division sign and the multiplication sign to an equals sign: 12 ÷ 4 = 3. The two problems are the opposite or the inverse of each other: 3 × 4 = 12 and 12 ÷ 4 = 3. You can also swap the factors in the original multiplication problem to get 12 = 3 × 4 and 12 ÷ 3 = 4.
2. Look at the multiplication problem 7 × 7 = 49. Write it backward: 49 = 7 × 7. Now change it to a division problem, 49 ÷ 7 = 7. The two problems are the inverse of each other: 7 × 7 = 49 and 49 ÷ 7 = 7. Both factors are the same, 7, so you can only change 7 × 7 = 49 into one division problem.
3. You can change any multiplication problem with two different factors into two division problems. Take the problem 72 × 128 = 9,206. You can write it backward as 9,206 = 128 × 72 or as 9,206 = 72 × 128. The two division problems become 9,206 = 128 = 72 and 9,206 ÷ 72 = 128.
4. Now that you understand the relationship between division and multiplication, you can use this relationship to solve a division problem. What is 18 ÷ 2? Rewrite this as a multiplication problem. Oops! Something is missing. First write it as 18 ÷ 2 = what? Now rewrite this division problem as a multiplication problem. It becomes what × 2 = 18? Can you figure out what factor the word what stands for? The answer is 9, because 9 × 2 = 18. Now change this problem back to a division problem: 18 ÷ 2 = 9. Nine is the quotient. It's easy to understand and solve division problems when you know that they are the opposite of multiplication problems.
Excerpted from Dazzling Division by Lynette Long Excerpted by permission.
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