Fractions are a common source of problems for many students. There are difficulties that arise when we work with fractions. In particular, finding common denominators when we add or subtract fractions can be a messy process. Because of these difficulties, practical arithmetic is usually done with a limited set of fractions where finding common denominators is much easier than it is in the general case.
When we discuss length in inches, the fractions we use are halves, quarters, eighths, etc. When working with these fractions, the lowest common denominator will always be just the largest denominator among the group of fractions.
When we use the metric system, our fractions will be tenths, hundredths, thousandths etc. Again the lowest common denominator will just be the largest denominator among the group--although in this case our fractions will usually be written as decimal fractions rather than common fractions.
Common denominators are important because we can only add or subtract like things. If we want to add a half gallon and a quart we need to re-express the half gallon as 2 quarts--and then we can add 2 quarts and 1 quart to get 3 quarts.
Add 1 fifth and 2 fifths and we obviously get 3 fifths. Write the same problem as 1/5 + 2/5 and it is easy to see how some people might mke a mistake and think the answer is 3/10. When we multiple fractions we do multiply both the tops and the bottoms of the fractions--but applying a similar rule to additon will give us the wrong answer. Writing the denominator in words is probably a good way to get used to the fact that the denominator does not change when we add like fractions. Indeed, if we could simply add numerators and denominators there would be no need to find common denomionators when we add. The only reason we have to find common denominators is because the denominators have to be the same--and they stay the same when we add fractions. All of this is also true when we subtract like fractions.
The best way to start adding or subtracting unlike fractions is to keep it simple. Work with halves, quarters, and eighths. You don't even have to learn general rules for converting fractions. It is enough to know that 1 half is 2 fourths is 4 eighths, 1 fourth is 2 eighths, and 3 fourths is 6 eighths. Just convert everything to thelargest denominator you see, then add or subtract the numerators.
From this beginning we can proceed to consider more general problems where the largest denominator will work as a common denominator, or where we are given a common denominator that will work. We can convert fractions to other equivalent fractions by multiplying top and bottom by the same number, Start with 5 sixths of a pizza, divide each piece in half and we wind up with 10 pieces each half as large. Instead of six of these pieces to make up a whole pizza it now takes twelve--so we see that 5 sixths is equivalent to 10 twelfths.
So how do we find a common denominator if the largest denominator doesn't work and we aren't given a denominator we can use? Consider a dish of brownies. We cut the dish into individual brownies by making 5 rows and 6 columns for a total of 30 individual brownies. A row is 1/5 and the 6 columns divide the row into 6 brownies or 6/30. A column is 1/6 and the 5 rows divide the column into 5 brownies or 5/30. We see that 30, which is 5 x 6 will work as a commmon denominator for fifths and sixths--and this same idea will work whenever we want to find a common denominator for two fractions.
We often tell students they have to find the lowest common denomminator when they add or subtract fractions. We teach students complicated rules using factor trees for finding prime factorizations, and then teach them other complicated rules for so they can use these prime factorizations to find the lowest common denominator. But this is really overkill if all we want to do is is add or subtract unlike fractions. Multiplying the denominators will always give us a common denominator. Even if this isn't the lowest common denominator, it will work.
As a practical matter there is no need to worry if our common denominator is really the lowest. But it usually makes things easier if we can keep our common denominators as small as we can. There are practical methods that can help keep our common denominators small, and will often let us find the lowest common denominator.
One approach simply relies on experience. If you encounter problems with fourths and sixths often enough you will remember that 12 is the LCD or lowest common denominator. A more general method we can use is the following: Just multiply the two denominators and then divide by the largest number you can see that will divide into both denominators. In the case of fourths and sixths, we would multiply 4 x 6 and then divide the product by 2. With eighths and twelfths we would multiply 8 x 12 to get 96. Dividing by 4 we would get 24. These methods can help us keep down the size of our common denominator--but if we happen to use a denominator that is not the lowest possible, we can still solve the problem.
Saturday, February 23, 2008
Thursday, February 21, 2008
EZ Arithmetic: Beyond the Basics
Arithmetic doesn't get a lot of respect these days. On the one hand, calculators have turned computation into something approaching a forgotten art. Learning traditional methods of computation can provide a foundation on which we can build more advanced mathematical skills and understanding--but such methods no longer have great importance for their own sake.
On the other hand, the more advanced use of arithmetic in problem solving has been largely neglected because of our desire to teach students algebra as early as possible. I once heard an ad for a tutoring service. A boy was struggling with algebra--and they discovered that his real problem was with percents. At first this seemed ridiculous to me. There aren't a lot of algebra problems that deal with percents. How could that be the root of the boy's problems?
I then realized that experience with percents could be of great value. Percentage problems can give students a background using reasoning that is similar to what is used in algebra. The way we transform problems working with percentages and the way we transform problems in algebra are very much alike. But percent problems give students a chance to learn such reasoning in a less abstract setting, and most students learn better when things are less abstract. Once students become familar with using such methods in a less abstract setting, it becomes much easier to understand what we are doing in the more abstract setting of algebra. Students will at least have a sense that they have seen something like this before--and that can make things a lot easier for them.
But the methods we use with percentage problems are only the last remaining vestige of the arithmetic methods that were once used to solve "algebra" problems. False position was widely used.
A number and a seventh the number add up to 17. What is the number? If the number were 7, adding a seventh would give us 8. In fact, we want our answer to be 17 which is 17/8 times as large. This means our number should be 17/8 times our first guess of 7. Our answer is 7 times 17/8 = 119/8 or 14 and 7/8. This problem and its solution were found in the Rhind Papyrus dating from about 1650 BC. Our first guess of 7 is the false position--a preliminary test solution which we will later correct to get our answer. No effort is made to make a realistic guess our test solution, but it is nice if we can pick a number that is easy to work with. In this case picking 7 makes it easy to add a seventh, and thus simplifies our arithmetic.
False position is a technique we can use with mental math. The method we have illustrated only works with problems that turn out to be proportions. But a modified version, double false position, will work with other linear problems.
On the other hand, the more advanced use of arithmetic in problem solving has been largely neglected because of our desire to teach students algebra as early as possible. I once heard an ad for a tutoring service. A boy was struggling with algebra--and they discovered that his real problem was with percents. At first this seemed ridiculous to me. There aren't a lot of algebra problems that deal with percents. How could that be the root of the boy's problems?
I then realized that experience with percents could be of great value. Percentage problems can give students a background using reasoning that is similar to what is used in algebra. The way we transform problems working with percentages and the way we transform problems in algebra are very much alike. But percent problems give students a chance to learn such reasoning in a less abstract setting, and most students learn better when things are less abstract. Once students become familar with using such methods in a less abstract setting, it becomes much easier to understand what we are doing in the more abstract setting of algebra. Students will at least have a sense that they have seen something like this before--and that can make things a lot easier for them.
But the methods we use with percentage problems are only the last remaining vestige of the arithmetic methods that were once used to solve "algebra" problems. False position was widely used.
A number and a seventh the number add up to 17. What is the number? If the number were 7, adding a seventh would give us 8. In fact, we want our answer to be 17 which is 17/8 times as large. This means our number should be 17/8 times our first guess of 7. Our answer is 7 times 17/8 = 119/8 or 14 and 7/8. This problem and its solution were found in the Rhind Papyrus dating from about 1650 BC. Our first guess of 7 is the false position--a preliminary test solution which we will later correct to get our answer. No effort is made to make a realistic guess our test solution, but it is nice if we can pick a number that is easy to work with. In this case picking 7 makes it easy to add a seventh, and thus simplifies our arithmetic.
False position is a technique we can use with mental math. The method we have illustrated only works with problems that turn out to be proportions. But a modified version, double false position, will work with other linear problems.
Small cookies cost ten cents, and large ones cost thirty cents. If I paid 2 dollars for a dozen cookies, how many of each type did I buy? If we buy a dozen small cookies it costs $1.20 or 120 cents. In fact I spent 80 cents more.
Each large cookie costs 20 cents more than a small cookie. Since I paid an extra 80 cents, I must have bought 4 large cookies. 4 large cookies and 8 small cookies do cost 120 cents plus 80 cents or 200 cents as required.
Methods like false position extend what we can do with arithmetic. Flexible use of math at one level forms an important foundation for building math at the next level. Think in terms of gear shifts. When we shift gears, there is a range of speeds where we can make the shift. In a similar way there is a range of problem difficulty where we can make the shift from arithmetic to algebra. The best way to make the shift is for students to be able to see how a range of problems could be done using either approach. We start by learning the older methods using arithmetic.
Monday, February 11, 2008
EZ Arithmetic--An Introduction to EZ Math
Mathematics begins with the study of numbers. Beyond a few small numbers that we can recognize at a glance, the study of numbers is based on counting. But counting is not just a part of mathematics, it is a part of our everyday language. We can count forward or backwards, starting with any number. We can count by ones, twos, fives, or tens. We are really quite fluent in our use of counting.
Arithmetic is different. Even intelligent people can sometimes struggle with basic arithmetic. Carrying and borrowing create problems for a lot of people. As an introduction to EZ Math we will look at how to better introduce students to carrying and borrowing.
Addition with carrying requires us to access basic addition facts, know when to carry, and then do the carrying when needed. For students who are struggling to remember the basic facts, all this can be quite difficult.
The standard approach in teaching is to write down little carry numbers. As a teaching tool this illustrates what is happening when we carry. For children, carry numbers mean they do not have to retain as much information in their heads. Unfortunately once learned, writing carry numbers is a hard habit to break. And if we never do anything as difficult as remembering that we are carrying when we do addition, then we will never be able to get very far in the study of math.
Writing carry numbers is only one way to make carrying easier for beginners. In EZ Math we use another. We start with vending machine addition. Add a pair of two digit numbers ending in 0 or 5. If both end in 5 there is a carry, otherwise there is not. We also start with small digits in the tens place. By keeping the addition facts very simple, students are able to focus on learning the process of carrying in a setting where no carry numbers need to be written.
As students become familar with the carrying process, we can introduce problems with larger digits in the tens place. Later we can shift to evens, where we add a pair of two digit even numbers. While harder than vending machine problems, limiting the ones digits to 0, 2, 4, 6, 8 greatly reduces the number of facts students may need to access at the start of the problem, so they will be less tired when they need to remember whether they need to carry or not. Again, we can start doing problems with small tens digits, and only later introduce problems with larger digits in the tens place.
Subtraction and borrowing can be taught in a similar way. We may wish to work through the vending machine problems for both addition and subtraction before proceeding to the evens. Once students become proficent working with these restricted problem types we can proceed we can proceed to the more genral two digit addition and subtraction problems--and then on to working with larger numbers where we might have to carry or borrow more than once.
The challenges you may have learning or teaching math at a more advanced level are not that different from the challenges faced by children and their teachers who need to learn or teach carrying and borrowing. Trying to simplify everything else as much as possible when introducing new concepts makes sense at almost any level.
Arithmetic is different. Even intelligent people can sometimes struggle with basic arithmetic. Carrying and borrowing create problems for a lot of people. As an introduction to EZ Math we will look at how to better introduce students to carrying and borrowing.
Addition with carrying requires us to access basic addition facts, know when to carry, and then do the carrying when needed. For students who are struggling to remember the basic facts, all this can be quite difficult.
The standard approach in teaching is to write down little carry numbers. As a teaching tool this illustrates what is happening when we carry. For children, carry numbers mean they do not have to retain as much information in their heads. Unfortunately once learned, writing carry numbers is a hard habit to break. And if we never do anything as difficult as remembering that we are carrying when we do addition, then we will never be able to get very far in the study of math.
Writing carry numbers is only one way to make carrying easier for beginners. In EZ Math we use another. We start with vending machine addition. Add a pair of two digit numbers ending in 0 or 5. If both end in 5 there is a carry, otherwise there is not. We also start with small digits in the tens place. By keeping the addition facts very simple, students are able to focus on learning the process of carrying in a setting where no carry numbers need to be written.
As students become familar with the carrying process, we can introduce problems with larger digits in the tens place. Later we can shift to evens, where we add a pair of two digit even numbers. While harder than vending machine problems, limiting the ones digits to 0, 2, 4, 6, 8 greatly reduces the number of facts students may need to access at the start of the problem, so they will be less tired when they need to remember whether they need to carry or not. Again, we can start doing problems with small tens digits, and only later introduce problems with larger digits in the tens place.
Subtraction and borrowing can be taught in a similar way. We may wish to work through the vending machine problems for both addition and subtraction before proceeding to the evens. Once students become proficent working with these restricted problem types we can proceed we can proceed to the more genral two digit addition and subtraction problems--and then on to working with larger numbers where we might have to carry or borrow more than once.
The challenges you may have learning or teaching math at a more advanced level are not that different from the challenges faced by children and their teachers who need to learn or teach carrying and borrowing. Trying to simplify everything else as much as possible when introducing new concepts makes sense at almost any level.
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