There's a new addition at HomeschoolMath.net, free exponent worksheets.
You can create an unlimited supply of free printable exponents worksheets there. The worksheets concentrate on calculations with exponents, such as solving 33 or (1/2)4 or (-5)0 or 8-2. You can choose to include negative or zero exponent. You can choose fractions, decimals, or negative numbers as bases. You can also make worksheets that have one other operation besides exponentiation (several operations with powers).
Showing posts with label exponents. Show all posts
Showing posts with label exponents. Show all posts
Friday, February 26, 2010
Exponent worksheets
There's a new addition at HomeschoolMath.net, free exponent worksheets.
You can create an unlimited supply of free printable exponents worksheets there. The worksheets concentrate on calculations with exponents, such as solving 33 or (1/2)4 or (-5)0 or 8-2. You can choose to include negative or zero exponent. You can choose fractions, decimals, or negative numbers as bases. You can also make worksheets that have one other operation besides exponentiation (several operations with powers).
You can create an unlimited supply of free printable exponents worksheets there. The worksheets concentrate on calculations with exponents, such as solving 33 or (1/2)4 or (-5)0 or 8-2. You can choose to include negative or zero exponent. You can choose fractions, decimals, or negative numbers as bases. You can also make worksheets that have one other operation besides exponentiation (several operations with powers).
Monday, April 20, 2009
Teaching negative exponents
Continuing on with the exponents, I have another video about negative exponents. Check also the "prelude" to this, my video on zero exponent.
Negative Exponents: Learn Them with a Pattern!
In books we usually find the definition that x−n = 1/xn or perhaps x−n = (1/x)n (you take the reciprocal of x and use the opposite of the exponent).
But again, I want to highlight that we DON'T have to just "announce" to students this definition or how negative exponents are done. We can justify it. In the video, you can see this done first with the help of a pattern, where we go from the positive exponents to zero exponent to negative ones.
I also show another way of showing your students why negative exponents have been defined the way they are. If you look at the shortcut for dividing powers with the same base (I used 63/65 in the video), we can simplify as usual and arrive at 1/62. However, the shortcut says we can just subtract the exponents, so 63/65 also equals 6−2. The same reasoning applies for any non-zero x and any positive whole number exponent.
This way, we teach our students not only the "how" but also the "why" of mathematics, and help them become better thinkers. Mathematics is not just about memorizing things and then learning to "spit" them out at tests - it also has a LOT to do with proof and logical thinking. And surely our kids need that today, more than ever.
Negative Exponents: Learn Them with a Pattern!
In books we usually find the definition that x−n = 1/xn or perhaps x−n = (1/x)n (you take the reciprocal of x and use the opposite of the exponent).
But again, I want to highlight that we DON'T have to just "announce" to students this definition or how negative exponents are done. We can justify it. In the video, you can see this done first with the help of a pattern, where we go from the positive exponents to zero exponent to negative ones.
I also show another way of showing your students why negative exponents have been defined the way they are. If you look at the shortcut for dividing powers with the same base (I used 63/65 in the video), we can simplify as usual and arrive at 1/62. However, the shortcut says we can just subtract the exponents, so 63/65 also equals 6−2. The same reasoning applies for any non-zero x and any positive whole number exponent.
This way, we teach our students not only the "how" but also the "why" of mathematics, and help them become better thinkers. Mathematics is not just about memorizing things and then learning to "spit" them out at tests - it also has a LOT to do with proof and logical thinking. And surely our kids need that today, more than ever.
Teaching negative exponents
Continuing on with the exponents, I have another video about negative exponents. Check also the "prelude" to this, my video on zero exponent.
Negative Exponents: Learn Them with a Pattern!
In books we usually find the definition that x−n = 1/xn or perhaps x−n = (1/x)n (you take the reciprocal of x and use the opposite of the exponent).
But again, I want to highlight that we DON'T have to just "announce" to students this definition or how negative exponents are done. We can justify it. In the video, you can see this done first with the help of a pattern, where we go from the positive exponents to zero exponent to negative ones.
I also show another way of showing your students why negative exponents have been defined the way they are. If you look at the shortcut for dividing powers with the same base (I used 63/65 in the video), we can simplify as usual and arrive at 1/62. However, the shortcut says we can just subtract the exponents, so 63/65 also equals 6−2. The same reasoning applies for any non-zero x and any positive whole number exponent.
This way, we teach our students not only the "how" but also the "why" of mathematics, and help them become better thinkers. Mathematics is not just about memorizing things and then learning to "spit" them out at tests - it also has a LOT to do with proof and logical thinking. And surely our kids need that today, more than ever.
Negative Exponents: Learn Them with a Pattern!
In books we usually find the definition that x−n = 1/xn or perhaps x−n = (1/x)n (you take the reciprocal of x and use the opposite of the exponent).
But again, I want to highlight that we DON'T have to just "announce" to students this definition or how negative exponents are done. We can justify it. In the video, you can see this done first with the help of a pattern, where we go from the positive exponents to zero exponent to negative ones.
I also show another way of showing your students why negative exponents have been defined the way they are. If you look at the shortcut for dividing powers with the same base (I used 63/65 in the video), we can simplify as usual and arrive at 1/62. However, the shortcut says we can just subtract the exponents, so 63/65 also equals 6−2. The same reasoning applies for any non-zero x and any positive whole number exponent.
This way, we teach our students not only the "how" but also the "why" of mathematics, and help them become better thinkers. Mathematics is not just about memorizing things and then learning to "spit" them out at tests - it also has a LOT to do with proof and logical thinking. And surely our kids need that today, more than ever.
Saturday, April 18, 2009
Zero Exponent - with a Pattern!
Many school books just "announce" the rule or the fact that any number to zero exponent is one (excluding zero to zeroth power, to be exact). I like to call this "announced" mathematics - it's math without justifications, without explaining the "whys".
In the video below, I show you a better way, where we approach it through a simple pattern. This justifies the rule and makes it logical, instead of just a piece of "announced" mathematics without proof.
Zero Exponent video
I also show another way of justifying it, which has to do with the shortcut for multiplication of powers with the same base. For example:
Following the shortcut for multiplying powers with the same base,
x7 x0 = x7, because you can add the exponents 0 + 7 = 7.
This is an equation where x7 is multiplied by x0, and the result is x7. What must x0 therefore be? It must equal one.
See also my earlier article on this topic: Negative or zero exponent.
In the video below, I show you a better way, where we approach it through a simple pattern. This justifies the rule and makes it logical, instead of just a piece of "announced" mathematics without proof.
Zero Exponent video
I also show another way of justifying it, which has to do with the shortcut for multiplication of powers with the same base. For example:
Following the shortcut for multiplying powers with the same base,
x7 x0 = x7, because you can add the exponents 0 + 7 = 7.
This is an equation where x7 is multiplied by x0, and the result is x7. What must x0 therefore be? It must equal one.
See also my earlier article on this topic: Negative or zero exponent.
Zero Exponent - with a Pattern!
Many school books just "announce" the rule or the fact that any number to zero exponent is one (excluding zero to zeroth power, to be exact). I like to call this "announced" mathematics - it's math without justifications, without explaining the "whys".
In the video below, I show you a better way, where we approach it through a simple pattern. This justifies the rule and makes it logical, instead of just a piece of "announced" mathematics without proof.
Zero Exponent video
I also show another way of justifying it, which has to do with the shortcut for multiplication of powers with the same base. For example:
Following the shortcut for multiplying powers with the same base,
x7 x0 = x7, because you can add the exponents 0 + 7 = 7.
This is an equation where x7 is multiplied by x0, and the result is x7. What must x0 therefore be? It must equal one.
See also my earlier article on this topic: Negative or zero exponent.
In the video below, I show you a better way, where we approach it through a simple pattern. This justifies the rule and makes it logical, instead of just a piece of "announced" mathematics without proof.
Zero Exponent video
I also show another way of justifying it, which has to do with the shortcut for multiplication of powers with the same base. For example:
Following the shortcut for multiplying powers with the same base,
x7 x0 = x7, because you can add the exponents 0 + 7 = 7.
This is an equation where x7 is multiplied by x0, and the result is x7. What must x0 therefore be? It must equal one.
See also my earlier article on this topic: Negative or zero exponent.
Saturday, February 14, 2009
Exponents and negative numbers
Why do I get a negative answer even though the exponent is even and the other exponent is odd? Doesn't the exponent being odd or even determine if the answer is negative or positive?This is really a matter of notation.
Here are two examples:−(3/2)2 = −2 1/4.
−(3/2)3 = −3 3/8.
We have agreed that −(3/2)2 is not the same as (-3/2)2.
If you want to calculate -3/2 to the second power, it is written so that -3/2 is inside parenthesis, like this: (-3/2)2.
The answer to this is indeed positive: it is (-3/2) × (-3/2) = 9/4 = 2 1/4.
However, if the minus sign is outside the parenthesis, it means the opposite number of whatever it's in front of.
So, −(3/2)2 means: "First calculate (3/2)2, then take the opposite of that."
First you calculate (3/2)2 = 9/4, and its opposite is -9/4.
In other words, the location of the parenthesis makes a huge difference.
With exponent 3, we get:
−(3/2)3 = −(3/2 × 3/2 × 3/2) = −27/8.
(-3/2)3 = (-3/2) × (-3/2) × (-3/2) = -27/8 or -3 3/8.
So in this case, these two expressions have the same value.
An advanced student may notice:
If n is even, and x is any real number, then
(-x)n = xn ; in other words it won't matter if you calculate the power using x or using negative x - the result is the same.
And if n is odd, then
(-x)n = −xn ; in other words, it's like you can "pull the minus out" from inside those parenthesis and put it out front.
Exponents and negative numbers
Why do I get a negative answer even though the exponent is even and the other exponent is odd? Doesn't the exponent being odd or even determine if the answer is negative or positive?This is really a matter of notation.
Here are two examples:−(3/2)2 = −2 1/4.
−(3/2)3 = −3 3/8.
We have agreed that −(3/2)2 is not the same as (-3/2)2.
If you want to calculate -3/2 to the second power, it is written so that -3/2 is inside parenthesis, like this: (-3/2)2.
The answer to this is indeed positive: it is (-3/2) × (-3/2) = 9/4 = 2 1/4.
However, if the minus sign is outside the parenthesis, it means the opposite number of whatever it's in front of.
So, −(3/2)2 means: "First calculate (3/2)2, then take the opposite of that."
First you calculate (3/2)2 = 9/4, and its opposite is -9/4.
In other words, the location of the parenthesis makes a huge difference.
With exponent 3, we get:
−(3/2)3 = −(3/2 × 3/2 × 3/2) = −27/8.
(-3/2)3 = (-3/2) × (-3/2) × (-3/2) = -27/8 or -3 3/8.
So in this case, these two expressions have the same value.
An advanced student may notice:
If n is even, and x is any real number, then
(-x)n = xn ; in other words it won't matter if you calculate the power using x or using negative x - the result is the same.
And if n is odd, then
(-x)n = −xn ; in other words, it's like you can "pull the minus out" from inside those parenthesis and put it out front.
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