Number Sense and Nonsense
Building Math Creativity and Confidence Through Number Play
By Claudia Zaslavsky
Chicago Review Press Incorporated Copyright © 2001 Claudia Zaslavsky
All rights reserved.
Odds and Evens
What This Chapter Is About
All whole numbers are divided into two groups — odd numbers and even numbers. They follow one another in order — odd, even, odd, even, like the page numbers in this book. But sometimes an even number can be odd, and an odd number can be even. You will read about it in this chapter.
Knowing how numbers behave can give a big boost to your number sense. That knowledge can help you discover when numbers are used in ways that make no sense, or are nonsense. You will learn when to expect an even number and when to expect an odd number as an answer to a problem in arithmetic.
The ancient Egyptians, almost 4,000 years ago, had a way of multiplying numbers using mostly even numbers. You will learn it, too.
When Is on Even Number Odd?
"A sheep with two heads!" exclaimed Eddie, pointing to a picture in the book he was reading. "That's an odd number of heads for an animal."
"That's nonsense" replied his sister Molly as she came over to look. "No animal has two heads! And what's more nonsense, two is not an odd number. Two is an even number."
"See, there really was such a sheep," said Eddie. "But you're right, an animal should have only one head. An animal with two heads is unusual. When I said 'an odd number of heads,' I meant that the sheep had an unusual number of heads."
Numbers like 1, 3, 5, 7, and 9 are called odd numbers. The numbers 2, 4, 6, and 8 are called even numbers. Are the odd numbers unusual? No, there are just as many odd numbers as even numbers. So why are these numbers called "odd"?
Words that describe numbers, like "odd" and "even," may also have other meanings. The dictionary gives many meanings for the word odd. One meaning is "left over after others are paired." Another meaning is "peculiar" or "strange."
It makes number sense to call 7 an odd number. Think of seven eggs. Place them in a tworow egg carton so that you have three pairs. The one left over is the "odd" egg.
And it makes just plain common sense to call a sheep with two heads peculiar or strange or unusual.
How do you know whether a whole number is odd or even? We know that the one-digit numbers 1, 3, 5, 7, and 9 are odd, while 2, 4, 6, and 8 are even. How about larger numbers? Look at 825:
You know that 8 and 2 are even, while 5 is odd. But is 825 odd or even? How can you test the number?
You can put 825 eggs in two-row egg cartons to find out whether one egg is left over after all the other eggs have been paired. But there is a better way.
You can divide 825 by 2 and see whether there is a remainder. The answer is 412, with a remainder of 1. So 825 must be an odd number. But there is still a better way, a way that takes less time!
Look at the digit in the ones place, the digit on the right. You know that 5 is an odd number. It means that 825 is an odd number. It makes good number sense to look at the digit in the ones place to decide whether a number is odd or even. It doesn't matter whether the other digits in the number are odd or even.
1. Is zero odd or even? How about numbers ending in zero, like 10 or 50?
2. Decide whether each number is odd or even. To test your answer, divide each number by 2. Is there a remainder?
When Is an Odd Number Even?
Molly and her five friends were about to share a box of cookies. There were 18 cookies in the box. She wondered whether every child would have the same number of cookies, with none left over.
Molly's class had not yet learned how to divide. She first gave each person one cookie, and then a second cookie. Some cookies were left in the box, and she gave each person a third cookie. Now the box was empty.
"It came out even," Molly said to her friends. "Everybody has three cookies and no cookies are left in the box."
When Molly said, "It came out even," did she mean that each child got an even number of cookies? That's nonsense. She knows that 3 is an odd number. Why did she say "even"?
The word even has many meanings, just as the word odd has many meanings. One meaning of even is "exact." Each child got exactly the same number of cookies, even though that number is odd. Molly might have said, "It came out evenly." Then we would know that Molly was not talking about an even number. She was saying that each child had exactly the same number of cookies as every other child.
1. Look at the page numbers in this book. Which pages have odd numbers? Which pages have even numbers? Find the pattern.
2. Do you or people you know live on a street on which every house has a number? Notice which numbers are even and which are odd. Find the pattern.
3. Write five telephone numbers, yours and those of your friends and relatives. Count the number of odd digits and the number of even digits. Are they about the same or is one group quite a bit larger than the other?
The Dating Problem
Some people write dates the short way, in numerals. Instead of March 27, 2001, they write 3-27-01, or 3/27/01. To save space, they write only the last two digits of the year: 01 instead of 2001.
When computers became popular, in the late 1900s, people saved space in the computer by writing a date like 1975 as just plain 75, using the last two digits. But that caused a big problem, called the Year 2000 problem, or Y2K problem. Many people were afraid that at the end of 1999, computers would record the next year as 1900 instead of 2000. That mistake might bring about terrible disasters, like shutting down water systems and electricity, and maybe setting off nuclear warfare! Billions of dollars were spent as the year 2000 approached to correct this error and avoid possible disaster.
Speaking of the year 2000 reminded Maria of a question she heard on the radio in the year 2000. She asked Eddie, "What is the earliest date starting in 2000 that has all even numbers when written in numerals?"
Eddie thought about it this way: "The year is 2000, and that's even. The first even-numbered month is February. So the date must be February 2, 2000, written 2-2-00."
"OK," said Maria. "Now tell me the last date before it that could be written with all even numbers."
"It can't be the year 1999, because that's an odd number. It must have been 1998. December is the twelfth month, and that's an even number. The date is December 30, 1998, written 12-30-98."
"Good thinking," replied Maria. "Try this one: What was the last date, before the year 2000, that could be written in numerals using only even digits?"
"Isn't it 12-30-98?" asked Eddie.
"Those are even numbers. I said all even digits.'
Figure out the answer to the last question. Then ask this question of a friend or grown-up.
Number Sense About Odds and Evens
Knowing about odd and even numbers and how they behave can be a great help when you are working with whole numbers.
Kwan added the different kinds of cards in his collection:
12 + 26 + 8 + 16 = 63
Maria looked at his addition and in two seconds told him that he had made a mistake. His answer didn't make sense — it was nonsense. She didn't add the numbers. All she did was notice that all the numbers are even. Her number sense told her that the sum of even numbers is always even. Kwan's answer was an odd number, so it had to be wrong.
Number Sense in Addition
Here is a way that you can develop your number sense. Complete each sentence with the word even or odd. Try many examples before you decide.
The sum of two even numbers is always _______.
The sum of two odd numbers is always _______.
The sum of one odd number and one even number is always _______.
Before you carry out each of the addition problems, decide whether the sum is odd or even. Then check your answer by doing the addition. You may want to use a calculator, but you will develop better number sense without the calculator. Even better, maybe you can do some of the addition in your head. That's called mental arithmetic.
Number Sense in Subtraction
Now let's see about subtracting, which is finding the difference between one number and another. Complete each sentence with the word odd or even. First try several examples of each type.
The difference between two even numbers is always _______.
The difference between two odd number is always _______.
The difference between an even number and an odd number is always _______.
Does it matter whether the even number or the odd number is larger?
Ask a friend to write some subtraction problems. Then tell your friend, without doing the subtracting, whether the difference is odd or even using the subtraction rules you learned above.
Number Sense in Multiplication
When two numbers are multiplied together, the answer is called the product. Now complete these sentences with "odd" or "even":
The product of two odd numbers is always _______.
The product of two even numbers is always _______.
The product of an odd number and an even number is always _______.
Knowing how odd and even numbers behave should help you to remember the multiplication facts.
1. Write the products for multiplying by 4: 4 × 1, 4 × 2, 4 × 3, and so on. That's the same as skip counting by fours. Then do the same for multiplication by 7: 7 × 1, 7 × 2, 7 × 3, and so on. Are the products odd or even? Do they follow a pattern?
2. Open a book and look at the page numbers of the two facing pages. Then open the book to another place and note the page numbers. Do this several times. Complete these sentences with the word odd or even:
The sum of the two page numbers is always _______.
The product of the two page numbers is always _______.
Number Sense in Division
This is a little more difficult to explain. When the answer comes out evenly, that is, when there is no remainder, you can figure out some rules. Use the multiplication facts to help you. You might want to work with a friend.
Using Number Sense and Just Plain Sense
Teacher: "Seven is an odd number. How can it be made even?"
Student: "Take away the S.'
Puzzles About Odd and Even Numbers
Here are some statements about the possible numbers on the two facing pages of an open book. The pages of the book are numbered in the usual way. Say whether each statement makes sense, or is nonsense.
1. The sum of the page numbers is 21.
2. The product of the page numbers is 380.
3. The product of the page numbers is 420.
4. The sum of the page numbers is 46.
5. The product of the page numbers is 99.
6. The sum of the page numbers is 99.
Make up some questions like these for your friend to answer. Try them out on a grown-up.
How Many Boys and How Many Girls?
Maria, Kwan, and Eddie were having a discussion about the number of boys and the number of girls in their class. Each one had a different opinion. Only one made number sense. The other two were talking nonsense. Decide which kid had a sensible answer, and why the other two didn't make any number sense.
1. Maria said there are 28 students in the class. There are five more boys than girls.
2. Kwan said there are 28 students in the class. There are four more boys than girls.
3. Eddie said there are 27 students in the class. There are four more boys than girls.
Multiplication by Doubling: The Ancient Egyptian Way
The ancient Egyptians must have liked to work with even numbers. They multiplied two numbers by doubling. We know that when any number is multiplied by two, the result is an even number. Then they kept right on doubling, as you will see in the example.
Suppose the problem is to multiply 14 by 13. This is how it was set up:
1. Set up two columns. Write the number 1 in the first column and 14 in the second column.
2. Multiply the numbers in both columns by two. Continue doubling until the next number in the first column would be greater than 13.
3. Check off the numbers in the left column that add up to 13.
4. Check off the numbers in the second column that are next to the checked numbers in the first column.
5. Add the checked numbers in the second column. This sum is 182, the product of 14 and 13.
1. For practice, multiply 13 by 14, using the Egyptian method of doubling. The first line of the two columns is:
2. Find several more products by the Egyptian method. Make up your own numbers.
3. Figure out why the method works. Note that 13 = 1 + 4 + 8, so 14 × 13 = 14 × (1 + 4 + 8) = 14 + 56 + 112 = 182.
How do we know about ancient Egyptian multiplication? The Egyptians wrote on papyrus, and soome of their original records remain.
Scientists learned to read the ancient Egyptian writing on papyrus, called hieratic writing.
An important source of this information is the papyrus of the scribe Ahmose (also called Ahmes), written more than 3,600 years ago. He began his papyrus with these words: "Accurate reckoning. The entrance into the knowledge of all existing things and all obscure secrets."
In a later chapter you will read how the ancient Egyptians and other people counted and wrote numbers long ago.
Prime and Not Piime
What's in This Chapter
Have you ever thought about a number as a rectangle of square tiles? Some numbers have several different rectangles. Some numbers have special rectangles called squares. By studying the rectangles you will learn about prime and composite numbers, about divisors and factors, about square numbers and multiples of numbers.
You will use a neat way to hunt for prime numbers, using a sieve to strain them out, while all the other numbers fall through the holes. You will test an idea about even numbers that people have been trying for over 250 years to prove either true or false. Maybe you will find the answer! You will learn to check your arithmetic by "casting out nines," and how to multiply by nine on your fingers.
Suppose you have a lot of small square tiles, each measuring one centimeter on the side. (If you don't use centimeter tiles, substitute the word unit for centimeter.) How many different rectangles can you make with three tiles? With four tiles? With a hundred tiles? (Continues...)
Excerpted from Number Sense and Nonsense by Claudia Zaslavsky. Copyright © 2001 Claudia Zaslavsky. Excerpted by permission of Chicago Review Press Incorporated.
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