Algebra Students: Vocabulary Is Important In Math Class, Too! Ask If You Are Confused About Meanings

I recently read a blog from a math teacher whose main area of concern is elementary math and literacy. I had to laugh at the example he used--why do we use the word "borrow" in subtraction when we don't intend to give it back? This is a point well-taken. We tend, in mathematics, to use terms from everyday speech, but in mathematics, these terms often have very different meanings. This can be and often is a source of confusion for many Algebra students; and, unfortunately, students often don't even realize they have confused the meanings of a term, and the teacher doesn't catch it is until too late. By "too late" I mean that the mistake has been practiced and has become ingrained in your brain as a fact. These mistakes are difficult to fix. It is better to avoid these than try to fix them.

One term from your elementary days that still causes students difficulty is the term "value"--especially with respect to fractions. If I asked you, "Is 3/4 > 1/2" what you say? You and almost everyone else would say "Yes." In reality, the answer is "not necessarily." The problem here is that fraction symbols do not actually have a VALUE until you know the "of what." Is 3/4 of an inch greater than 1/2 of a foot? Of course not, you say? Why not? You just told me 3/4 > 1/2.

Now that you are thinking a little bit more about it, you realize that fractions can only be compared IF they are fractions of the SAME THING. So why have math books had homework sections with instructions to "compare the fractions?" Because someone assumed you knew that fractions can only be compared if they are of the same thing, and they left out an important part of the instructions. "Compare these fractions on a number line" or "Assume these fractions are of they same thing" would be appropriate directions. On a number line, 3/4 and 1/2 represent parts of the same size unit. We math teachers tend to assume that every student is picturing the same thing or understanding a definition the same way that we intend; but you and I both know this isn't always true.

In Algebra, there are two huge examples of extremely important concepts that students often get confused with their everyday meanings--or at the least, cannot really explain what the math meaning actually is. These two concepts are: "solve" as in solve an equation and "factor" as in factor this expression.

You know what it means to solve a puzzle or to solve a problem you are having making free throws in basketball; but what does it mean to solve an equation? To find the answer, you say? How do you know when you have an answer? It works? What does that mean? Very few Algebra students can actually say in words--with any real understanding--that to solve an equation means to find values for the variables that make the equation TRUE.

You know that washing hands is an important "factor" in slowing the spread of disease; but how does this apply to factoring an expression like a^2 - ab? The mathematics meaning of "factor" is totally different from the everyday meaning. In Algebra, to factor means to "re-write as multiplication." What? Well, a^2 - ab in factored form is a(a-b) since when you multiply a and (a - b) you get a^2 - ab.

All of mathematics--not just Algebra--is full of these terms with different meanings in the everyday world than in the mathematical world. For your own success, you must always memorize math definitions immediately, practice these definitions, and even discuss with yourself and with your teacher the differences in meanings. It is OK--in fact, important--to know that a term has several different meanings. It is equally important that you understand each of the meanings and know when to use which meaning.

If you get confused, or are ever in doubt, ASK YOUR TEACHER! It is the teacher's responsibility to teach you. We math teachers are not perfect human beings, even if we like to think we are. We often ASSUME more than we should. DO NOT BE AFRAID TO ASK QUESTIONS. That is your responsibility. Ours is to answer your questions.

Shirley Slick, "The Slick Tips Lady," is a retired high school math teacher and tutor with degrees in Mathematics and Psychology and additional training in brain-based learning/teaching. Her goals: (1) to help parents help their children with math, (2) to help eliminate the horrendous Algebra failure rate, and (3) to inform the general public about problematic issues related to the field of education. For your free copy of "10 Slick Tips for Improving Your Child's Study Habits," visit her website at http://myslicktips.com/

Algebra For Beginners - What Does It Means To Factor and Why Is It So Important In Algebra?

Algebra has become a "must take" course in high school. However, while technology keeps advancing and Algebra becomes a course in which everyone needs to be proficient, student success rates remain consistently low. There are many reasons for this, but one of those reasons is that students do not become proficient in the vocabulary part of mathematics. In fact, math students complain loudly if given vocabulary and spelling quizzes, claiming, "this isn't English class!" Students fail to understand that if they do not master the vocabulary, they will perform badly on tests because they won't understand what the directions are telling them to do. The instruction "Factor..." is a prime example of this problem.

What Is Factoring?

In simplest terms, the process of "factoring" or the verb "to factor" means to re-write an algebraic expression in terms of multiplication. Factoring is a process that is used throughout Algebra; but if you ask even the top students what the verb "to factor" means, very few will answer correctly.

Students understand factoring from arithmetic. Ask students to "factor 6," they will know to write 6 = 2 x 3. Asking students to "factor 12" may result in 12 = 2 x 6 or 12 = 3 x 4, and a few students might continue factoring to 12 = 2 x 6 = 2 x 2 x 3 = 2^2 x 3. These students know to factor to a product of prime numbers.

However, asking an Algebra class early in the year to factor the expression 2x + 2y will get you a room full of blank stares. They do not transfer their knowledge of the word "factor" from arithmetic to Algebra. For Algebra students to get proficient at the terminology, we need to give our students many examples of what we are doing; and we need to have our students saying and explaining definitions and properties and giving examples OUT LOUD.

Initially, factoring in Algebra relies heavily on the Distributive Property. Surprisingly, students seem to quickly understand and effectively use the distributive property for factoring two uncomplicated terms. When looking at 2x + 2y, students generally recognize the common multiplier of 2. Then they learn to use the Distributive Property to re-write the expression using multiplication. 2x + 2y = 2(x + y).

Students have a little more difficulty as the terms get more complicated and/or the number of terms increases; but with several examples and practice, students can factor expressions like: 3a + 9ab - 15ac. Each term shares both 3 and a. Again, using the Distributive Property, the expression 3a + 9ab - 15ac becomes 3a(1 + 3b - 5c) when re-written as multiplication.

Note: One of the best things about factoring is that it can be easily checked by performing the multiplication to verify that the result is the original expression.

Now that we know WHAT factoring is, we need to understand the reasons WHY to factor.

Why Do We Need To Factor?

There are two main uses for factoring: (1) reducing fractions, and (2) solving equations.

Reducing Fractions:

As with factoring, students learn to reduce fractions in arithmetic. 12/14 = (2 x 6)/(2 x 7) = (2/2)(6/7) = 1(6/7) = 6/7.

Making the transfer of that knowledge to Algebra often causes trouble; but just as in arithmetic, reducing algebraic fractions uses factoring and the fact that x/x = 1.

Reduce: (3a + 9ab) / (3a^2 + 15ab).

This fraction becomes 3a(1 + 3b) / 3a(a + 5b) = (3a/3a) ((1 + 3b)/(a + 5b)) = 1 ((1 + 3b)/(a + 5b)) = (1 + 3b)/(a + 5b).

Solving Equations:

Some algebraic equations can be solved (which means to find the values that make the equation true) by first moving all the terms to one side of the equal sign. This leaves zero on the other side. Then, if possible, the algebraic expression is factored. Then the fact that if ab = 0, then either a = 0 or b = 0 allows us to find solutions.

Solve: a^2 = 3a.

This equation becomes a^2 - 3a = 0. Then factoring gives us a(a - 3) = 0. So either a = 0 or (a - 3) = 0. This tells that we have two values that make the original equation true: 0 and 3.

For every example in this article, the factoring method used has been the Distributive Property. There are, however, other methods for factoring different types of expressions. Those will be discussed in other articles. For now, it is important that you remember:

(1) To factor means to re-write as multiplication, and

(2) Factoring is important because it is used to reduce algebraic fractions and to solve equations.

Shirley Slick, "The Slick Tips Lady," is a retired high school math teacher and tutor with degrees in Mathematics and Psychology and additional training in brain-based learning/teaching. Her goals: (1) to help parents help their children with math, (2) to help eliminate the horrendous Algebra failure rate, and (3) to inform the general public about problematic issues related to the field of education. For your free copy of "10 Slick Tips for Improving Your Child's Study Habits," visit her website at http://myslicktips.com/

Algebra Students: Vocabulary Is Important In Math Class, Too! Ask If You Are Confused About Meanings

I recently read a blog from a math teacher whose main area of concern is elementary math and literacy. I had to laugh at the example he used--why do we use the word "borrow" in subtraction when we don't intend to give it back? This is a point well-taken. We tend, in mathematics, to use terms from everyday speech, but in mathematics, these terms often have very different meanings. This can be and often is a source of confusion for many Algebra students; and, unfortunately, students often don't even realize they have confused the meanings of a term, and the teacher doesn't catch it is until too late. By "too late" I mean that the mistake has been practiced and has become ingrained in your brain as a fact. These mistakes are difficult to fix. It is better to avoid these than try to fix them.

One term from your elementary days that still causes students difficulty is the term "value"--especially with respect to fractions. If I asked you, "Is 3/4 > 1/2" what you say? You and almost everyone else would say "Yes." In reality, the answer is "not necessarily." The problem here is that fraction symbols do not actually have a VALUE until you know the "of what." Is 3/4 of an inch greater than 1/2 of a foot? Of course not, you say? Why not? You just told me 3/4 > 1/2.

Now that you are thinking a little bit more about it, you realize that fractions can only be compared IF they are fractions of the SAME THING. So why have math books had homework sections with instructions to "compare the fractions?" Because someone assumed you knew that fractions can only be compared if they are of the same thing, and they left out an important part of the instructions. "Compare these fractions on a number line" or "Assume these fractions are of they same thing" would be appropriate directions. On a number line, 3/4 and 1/2 represent parts of the same size unit. We math teachers tend to assume that every student is picturing the same thing or understanding a definition the same way that we intend; but you and I both know this isn't always true.

In Algebra, there are two huge examples of extremely important concepts that students often get confused with their everyday meanings--or at the least, cannot really explain what the math meaning actually is. These two concepts are: "solve" as in solve an equation and "factor" as in factor this expression.

You know what it means to solve a puzzle or to solve a problem you are having making free throws in basketball; but what does it mean to solve an equation? To find the answer, you say? How do you know when you have an answer? It works? What does that mean? Very few Algebra students can actually say in words--with any real understanding--that to solve an equation means to find values for the variables that make the equation TRUE.

You know that washing hands is an important "factor" in slowing the spread of disease; but how does this apply to factoring an expression like a^2 - ab? The mathematics meaning of "factor" is totally different from the everyday meaning. In Algebra, to factor means to "re-write as multiplication." What? Well, a^2 - ab in factored form is a(a-b) since when you multiply a and (a - b) you get a^2 - ab.

All of mathematics--not just Algebra--is full of these terms with different meanings in the everyday world than in the mathematical world. For your own success, you must always memorize math definitions immediately, practice these definitions, and even discuss with yourself and with your teacher the differences in meanings. It is OK--in fact, important--to know that a term has several different meanings. It is equally important that you understand each of the meanings and know when to use which meaning.

If you get confused, or are ever in doubt, ASK YOUR TEACHER! It is the teacher's responsibility to teach you. We math teachers are not perfect human beings, even if we like to think we are. We often ASSUME more than we should. DO NOT BE AFRAID TO ASK QUESTIONS. That is your responsibility. Ours is to answer your questions.

Algebra For Beginners - What Does It Means To Factor and Why Is It So Important In Algebra?

Algebra has become a "must take" course in high school. However, while technology keeps advancing and Algebra becomes a course in which everyone needs to be proficient, student success rates remain consistently low. There are many reasons for this, but one of those reasons is that students do not become proficient in the vocabulary part of mathematics. In fact, math students complain loudly if given vocabulary and spelling quizzes, claiming, "this isn't English class!" Students fail to understand that if they do not master the vocabulary, they will perform badly on tests because they won't understand what the directions are telling them to do. The instruction "Factor..." is a prime example of this problem.

What Is Factoring?

In simplest terms, the process of "factoring" or the verb "to factor" means to re-write an algebraic expression in terms of multiplication. Factoring is a process that is used throughout Algebra; but if you ask even the top students what the verb "to factor" means, very few will answer correctly.

Students understand factoring from arithmetic. Ask students to "factor 6," they will know to write 6 = 2 x 3. Asking students to "factor 12" may result in 12 = 2 x 6 or 12 = 3 x 4, and a few students might continue factoring to 12 = 2 x 6 = 2 x 2 x 3 = 2^2 x 3. These students know to factor to a product of prime numbers.

However, asking an Algebra class early in the year to factor the expression 2x + 2y will get you a room full of blank stares. They do not transfer their knowledge of the word "factor" from arithmetic to Algebra. For Algebra students to get proficient at the terminology, we need to give our students many examples of what we are doing; and we need to have our students saying and explaining definitions and properties and giving examples OUT LOUD.

Initially, factoring in Algebra relies heavily on the Distributive Property. Surprisingly, students seem to quickly understand and effectively use the distributive property for factoring two uncomplicated terms. When looking at 2x + 2y, students generally recognize the common multiplier of 2. Then they learn to use the Distributive Property to re-write the expression using multiplication. 2x + 2y = 2(x + y).

Students have a little more difficulty as the terms get more complicated and/or the number of terms increases; but with several examples and practice, students can factor expressions like: 3a + 9ab - 15ac. Each term shares both 3 and a. Again, using the Distributive Property, the expression 3a + 9ab - 15ac becomes 3a(1 + 3b - 5c) when re-written as multiplication.

Note: One of the best things about factoring is that it can be easily checked by performing the multiplication to verify that the result is the original expression.

Now that we know WHAT factoring is, we need to understand the reasons WHY to factor.

Why Do We Need To Factor?

There are two main uses for factoring: (1) reducing fractions, and (2) solving equations.

Reducing Fractions:

As with factoring, students learn to reduce fractions in arithmetic. 12/14 = (2 x 6)/(2 x 7) = (2/2)(6/7) = 1(6/7) = 6/7.

Making the transfer of that knowledge to Algebra often causes trouble; but just as in arithmetic, reducing algebraic fractions uses factoring and the fact that x/x = 1.

Reduce: (3a + 9ab) / (3a^2 + 15ab).

This fraction becomes 3a(1 + 3b) / 3a(a + 5b) = (3a/3a) ((1 + 3b)/(a + 5b)) = 1 ((1 + 3b)/(a + 5b)) = (1 + 3b)/(a + 5b).

Solving Equations:

Some algebraic equations can be solved (which means to find the values that make the equation true) by first moving all the terms to one side of the equal sign. This leaves zero on the other side. Then, if possible, the algebraic expression is factored. Then the fact that if ab = 0, then either a = 0 or b = 0 allows us to find solutions.

Solve: a^2 = 3a.

This equation becomes a^2 - 3a = 0. Then factoring gives us a(a - 3) = 0. So either a = 0 or (a - 3) = 0. This tells that we have two values that make the original equation true: 0 and 3.

For every example in this article, the factoring method used has been the Distributive Property. There are, however, other methods for factoring different types of expressions. Those will be discussed in other articles. For now, it is important that you remember:

(1) To factor means to re-write as multiplication, and

(2) Factoring is important because it is used to reduce algebraic fractions and to solve equations.