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Videos and Practice

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Learn how to read and use two-way frequency tables.

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Practicing Algebra 1 math: 'Scatter plots: line of best fit'

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Fun math practice! Improve your skills with free problems in 'Scatter plots: line of best fit' and thousands of other practice lessons.

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Get your students engaged with interpreting data from different sources through the use of Mangahigh's adaptive quiz 'Interpret data from different sources'. Students will start with 'Easy' questions to help introduce new ideas and then move onto 'Medium' questions with some scaffolded support.

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Common Core Math Alignment:

8.SP.A.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
8.SP.A.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

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XP Math - Math Games Arcade - Naruto: Line of Best Fit - FREE

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Common Core Math Alignment: 8.SP.A.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. 8.SP.A.2 Know that straight lines are widely used to model relationships between two quantitative variables.

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Let's find the volume of a few more solid figures and then if we have time, we might be able to do some surface area problems. So let me draw a cylinder over here. So that is the top of my cylinder. And then this is the height of my cylinder. This is the bottom right over here. If this was transparent, maybe you could see the back side of the cylinder. So you could imagine this kind of looks like a soda can. And let's say that the height of my cylinder, h, is equal to 8. I'll give it some units. 8 centimeters, that is my height. And then let's say that the radius of one of the top of my cylinder of my soda can, let's say that this radius over here is equal to 4 centimeters. So what is the volume here? What is the volume going to be? And the idea here is really the exact same thing that we saw in some of the previous problems. If you can find the surface area of one side and then figure out how deep it goes, you'll be able to figure out the volume. So what we're going to do here is figure out the surface area of the top of this cylinder, or the top of the soda can. And then we're going to multiply it by its height. And that'll give us a volume. This will tell us essentially, how many square centimeters fit in this top. And then if we multiply that by how many centimeters we go down, then that'll give us the number of cubic centimeters in this cylinder or soda can. So how do we figure out this area up here? Well the area at the top, this is just finding the area of a circle. You could imagine drawing it like this. If we were to just look at it straight on, that's a circle with a radius of 4 centimeters. The area of a circle with a radius 4 centimeters, area is equal to pi r squared. So it's going to be pi times the radius squared, times 4 centimeters squared, which is equal to 4 squared, is 16, times pi. And our units now are going to be centimeters squared. Or another way to think of these are square centimeters. So that's the area. The volume is going to be this area times the height. So the volume is going to be equal to 16 pi centimeters squared, times the height, times 8 centimeters. And so, when you do multiplication, the associative property, you can kind of rearrange these things, and the commutative property. It doesn't matter what order you do it if it's all multiplication. So this is the same thing as 16 times 8. Let's see. 8 times 8 is 64. 16 times 8 is twice that. So it's going to be 128 pi. Now you have centimeter squared times centimeters. So that gives us centimeters cubed. Or 128 pi cubic centimeters. Remember, pi is just a number. We write it as pi, because it's kind of a crazy, irrational number, that if you were to write it, you could never completely write pi. 3.14159 keeps going on, never a repeat. So we just leave it as pi. But if you wanted to figure it out, you can get a calculator. And this would be 3.14 roughly, times 128. So it would be close to 400 cubic centimeters. Now, how would we find the surface area? How would we find the surface area of this figure over here? Well, part of the surface area of the two surfaces, the top and the bottom. So that would be part of the surface area. And then the bottom over here would also be part of the surface area. So if we're trying to find the surface area, it's definitely going to have both of these areas here. So it's going to have the 16 pi centimeters squared twice. This is 16 pi. This is 16 pi square centimeters. So it's going to have 2 times 16 pi centimeters squared. I'll keep the units still. So that covers the top and the bottom of our soda can. And now we have to figure out the surface area of this thing that goes around. And the way I imagine it is, imagine if you're trying to wrap this thing with wrapping paper. So let me just draw a little dotted line here. So imagine if you were to cut it just like that. Cut the side of the soda can. And if you were to unwind this thing that goes around it, what would you have. Well, you would have something. You would end up with a sheet of paper where this length right over here is the same thing as this length over here. And then it would be completely unwound. And then these two ends-- let me do it in magenta-- these two ends used to touch each other. And-- I'm going to do it in a color that I haven't used yet, I'll do it in pink-- these two ends used to touch each other when it was all rolled together. And they used to touch each other right over there. So the length of this side and that side is going to be the same thing as the height of my cylinder. So this is going to be 8 centimeters. And then this over here is also going to be 8 centimeters. And so the question we need to ask ourselves is, what is going to be this dimension right over here. And remember, that dimension is essentially, how far did we go around the cylinder. Well, if you think about it, that's going to be the exact same thing as the circumference of either the top or the bottom of the cylinder. So what is the circumference? The circumference of this circle right over here, which is the same thing as the circumference of that circle over there, it is 2 times the radius times pi. Or 2 pi times the radius. 2 pi times 4 centimeters, which is equal to 8 pi centimeters. So this distance right over here is the circumference of either the top or the bottom of the cylinder. It's going to be 8 pi centimeters. So if you want to find the surface area of just the wrapping, just the part that goes around the cylinder, not the top or the bottom, when you unwind it, it's going to look like this rectangle. And so its area, the area of just that part, is going to be equal to 8 centimeters times 8 pi centimeters. So let me do it this way. It's going to be 8 centimeters times 8 pi centimeters. And that's equal to 64 pi. 8 times 8 is 64. You have your pi centimeters squared. So when you want the surface area of the whole thing, you have the top, you have the bottom, we already threw those there. And then you want to find the area of the thing around. We just figured that out. So it's going to be plus 64 pi centimeters squared. And now we just have to calculate it. So this gives us 2 times 16 pi, is going to be equal to 32. That is 32 pi centimeters squared, plus 64 pi. Let me scroll over to the right a little bit. Plus 64 pi centimeters squared. And then 32 plus 64 is 96 pi centimeters squared. So it's equal to 96 pi square centimeters, which is going to be a little bit over 300 square centimeters. And notice, when we did surface area, we got our answer in terms of square centimeters. That makes sense, because surface area, it's a two-dimensional measurement. Think about how many square centimeters can we fit on the surface of the cylinder. When we did the volume, we got centimeters cubed, or cubic centimeters. And that's because we're trying to calculate how many one by one by one centimeter cubes can we fit inside of this structure. And so that's why it's cubic centimeters. Anyway, hopefully that clarifies things up a little bit.

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Cylinder volume & surface area (video) | Khan Academy

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Let's find the volume of a few more solid figures and then if we have time, we might be able to do some surface area problems. So let me draw a cylinder over here. So that is the top of my cylinder. And then this is the height of my cylinder.

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Pythagorean Theorem on a Coordinate Plane

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Pythagorean Theorem on a Coordinate Plane

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Pythagorean Theorem on a Coordinate Plane

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Uploaded by Mr. de la Garza on 2015-02-25.

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Practicing Eighth grade math: 'Pythagorean theorem: word problems'

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Fun math practice! Improve your skills with free problems in 'Pythagorean theorem: word problems' and thousands of other practice lessons.

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Videos and Practice

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Video and Practice

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Let's say we have two lines over here. Let's call this line right over here line AB. So A and B both sit on this line. And let's say we have this other line over here. We'll call this line CD. So it goes through point C and it goes through point D. And it just keeps on going forever. And let's say that these lines both sit on the same plane. And in this case, the plane is our screen, or this little piece of paper that we're looking at right over here. And they never intersect. So they're on the same plane, but they never intersect each other. If those two things are true, and when they're not the same line, they never intersect and they can be on the same plane, then we say that these lines are parallel. They're moving in the same general direction, in fact, the exact same general direction. If we were looking at it from an algebraic point of view, we would say that they have the same slope, but they have different y-intercepts. They involve different points. If we drew our coordinate axes here, they would intersect that at a different point, but they would have the same exact slope. And what I want to do is think about how angles relate to parallel lines. So right over here, we have these two parallel lines. We can say that line AB is parallel to line CD. Sometimes you'll see it specified on geometric drawings like this. They'll put a little arrow here to show that these two lines are parallel. And if you've already used the single arrow, they might put a double arrow to show that this line is parallel to that line right over there. Now with that out of the way, what I want to do is draw a line that intersects both of these parallel lines. So here's a line that intersects both of them. Let me draw a little bit neater than that. So let me draw that line right over there. Well, actually, I'll do some points over here. Well, I'll just call that line l. And this line that intersects both of these parallel lines, we call that a transversal. This is a transversal line. It is transversing both of these parallel lines. This is a transversal. And what I want to think about is the angles that are formed, and how they relate to each other. The angles that are formed at the intersection between this transversal line and the two parallel lines. So we could, first of all, start off with this angle right over here. And we could call that angle-- well, if we made some labels here, that would be D, this point, and then something else. But I'll just call it this angle right over here. We know that that's going to be equal to its vertical angles. So this angle is vertical with that one. So it's going to be equal to that angle right over there. We also know that this angle, right over here, is going to be equal to its vertical angle, or the angle that is opposite the intersection. So it's going to be equal to that. And sometimes you'll see it specified like this, where you'll see a double angle mark like that. Or sometimes you'll see someone write this to show that these two are equal and these two are equal right over here. Now the other thing we know is we could do the exact same exercise up here, that these two are going to be equal to each other and these two are going to be equal to each other. They're all vertical angles. What's interesting here is thinking about the relationship between that angle right over there, and this angle right up over here. And if you just look at it, it is actually obvious what that relationship is-- that they are going to be the same exact angle, that if you put a protractor here and measured it, you would get the exact same measure up here. And if I drew parallel lines-- maybe I'll draw it straight left and right, it might be a little bit more obvious. So if I assume that these two lines are parallel, and I have a transversal here, what I'm saying is that this angle is going to be the exact same measure as that angle there. And to visualize that, just imagine tilting this line. And as you take different-- so it looks like it's the case over there. If you take the line like this and you look at it over here, it's clear that this is equal to this. And there's actually no proof for this. This is one of those things that a mathematician would say is intuitively obvious, that if you look at it, as you tilt this line, you would say that these angles are the same. Or think about putting a protractor here to actually measure these angles. If you put a protractor here, you'd have one side of the angle at the zero degree, and the other side would specify that point. And if you put the protractor over here, the exact same thing would happen. One side would be on this parallel line, and the other side would point at the exact same point. So given that, we know that not only is this side equivalent to this side, it is also equivalent to this side over here. And that tells us that that's also equivalent to that side over there. So all of these things in green are equivalent. And by the same exact argument, this angle is going to have the same measure as this angle. And that's going to be the same as this angle, because they are opposite, or they're vertical angles. Now the important thing to realize is just what we've deduced here. The vertical angles are equal and the corresponding angles at the same points of intersection are also equal. And so that's a new word that I'm introducing right over here. This angle and this angle are corresponding. They represent kind of the top right corner, in this example, of where we intersected. Here they represent still, I guess, the top or the top right corner of the intersection. This would be the top left corner. They're always going to be equal, corresponding angles. And once again, really, it's, I guess, for lack of a better word, it is a bit obvious. Now on top of that, there are other words that people will see. We've essentially just proven that not only is this angle equivalent to this angle, but it's also equivalent to this angle right over here. And these two angles-- let me label them so that we can make some headway here. So I'm going to use lowercase letters for the angles themselves. So let's call this lowercase a, lowercase b, lowercase c. So lowercase c for the angle, lowercase d, and then let me call this e, f, g, h. So we know from vertical angles that b is equal to c. But we also know that b is equal to f because they are corresponding angles. And that f is equal to g. So vertical angles are equivalent, corresponding angles are equivalent, and so we also know, obviously, that b is equal to g. And so we say that alternate interior angles are equivalent. So you see that they're kind of on the interior of the intersection. They're between the two lines, but they're on all opposite sides of the transversal. Now you don't have to know that fancy word, alternate interior angles, you really just have to deduce what we just saw over here. Know that vertical angles are going to be equal and corresponding angles are going to be equal. And you see it with the other ones, too. We know that a is going to be equal to d, which is going to be equal to h, which is going to be equal to e.

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Angles, parallel lines, & transversals (video) | Khan Academy

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Let's say we have two lines over here. Let's call this line right over here line AB. So A and B both sit on this line. And let's say we have this other line over here. We'll call this line CD. So it goes through point C and it goes through point D.

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Practicing Eighth grade math: 'Transversal of parallel lines'

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Fun math practice! Improve your skills with free problems in 'Transversal of parallel lines' and thousands of other practice lessons.

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Mangahigh turns our students into "maths addicts" who compete with each other for top scores and gold medals. And since the quizzes reward both accurate recall of knowledge and deep conceptual understanding, every hour they spent having fun makes them better mathematicians. Five stars.

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Mangahigh turns our students into "maths addicts" who compete with each other for top scores and gold medals. And since the quizzes reward both accurate recall of knowledge and deep conceptual understanding, every hour they spent having fun makes them better mathematicians. Five stars.

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IXL | Classify congruence transformations | Geometry math

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Improve your math knowledge with free questions in "Classify congruence transformations" and thousands of other math skills.

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Increasing, decreasing, positive or negative intervals

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Function values can be positive or negative, and they can increase or decrease as the input increases. Here we introduce these basic properties of functions.

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Linear functions word problem: fuel (video) | Khan Academy

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IXL | Identify linear and nonlinear functions: graphs and equations | 8th grade math

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Improve your math knowledge with free questions in "Identify linear and nonlinear functions: graphs and equations" and thousands of other math skills.

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Two functions, f and g, are described below. Which of these statements about f and g is true? So they defined function f as kind of a traditional linear equation right over here. And this right over here is g. So this right over here is g of x. And that also looks like a linear function. We see it's a kind of a downward sloping line. So let's look at our choices and see which of these are true. f and g are both increasing, and f is increasing faster than g. Well, when I look at g-- Well, first of all, g is definitely decreasing. So we already know that that's false. And f is also decreasing. We see here it has a negative slope. Every time we move forward 3 in the x direction, we're going to move down 7 in the vertical direction. So neither of these are increasing so that's definitely not right. f and g are both increasing. Well, that's definitely not right. So we know that both f and g are decreasing. So this first choice says they're both decreasing, and g is decreasing faster than f. So let's see what the slope on g is. So the slope on g is every time we move 1 in the x direction, positive 1 in the x direction, we move down 2 in the y direction. So for g of x, if we were to write our change in y over our change in x-- which is our slope-- our change in y over change in x, when we move one in the x direction, positive 1 in the x direction, we move down 2 in the y direction. So our change in y over change in x is negative 2. So g has a slope of negative 2. f has a slope of negative 7/3. Negative 7/3 is the same thing as negative 2 and 1/3. So f's slope is more negative. So it is decreasing faster. So g is not decreasing faster than f. f is decreasing faster than g. So this is not right. And then we have this choice-- f and g are both decreasing, and f is decreasing faster than g. This is right, right over here. We have this last choice-- g is increasing but f is decreasing. We know that's not true. g is actually decreasing.

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Comparing linear functions: equation vs. graph

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Two functions, f and g, are described below. Which of these statements about f and g is true? So they defined function f as kind of a traditional linear equation right over here. And this right over here is g. So this right over here is g of x.

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Each input has one output.

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IXL | Identify functions | Algebra 1 math

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Improve your math knowledge with free questions in "Identify functions" and thousands of other math skills.

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How do you solve systems of equations by substitution?

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Systems of equations with substitution: potato chips

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To solve a system of equations using substitution... 1. Isolate one of the variables in one of the equations, e.g. rewrite 2x+y=3 as y=3-2x. 2. You can now express the isolated variable using the other one. *Substitute* that expression into the second equation, e.g. rewrite x+2y=5 as x+2(3-2x)=5. 3.

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Solve each system by graphing the lines.

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IXL | Solve a system of equations by graphing | 8th grade math

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Improve your math knowledge with free questions in “Solve a system of equations by graphing” and thousands of other math skills.

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ALGEBRA - Solving Equations Jeopardy Style

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Solve multi-step equations using combining like terms, the distributive property and moving variables to one side of the equation.

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Play this two-step equation game alone or with another student. This game can also be played at school by dividing the classroom in two teams. Have fun solving equations!

You can play this game alone, with a friend, or in two teams. This game is a multi-player game that can be played on computers, Promethean boards, smart boards, iPads, and other tablets. You do not need to install an app to play this game on the iPad. Have fun evaluating algebraic expressions!

The game is based on the following Common Core Math Standards:

CCSS6.EE.2
Write, read, and evaluate expressions in which letters stand for
numbers.

CCSS6.EE.3
Apply the properties of operations to generate equivalent expressions.
For example, apply the distributive property to the expression 3 (2 + x) to
produce the equivalent expression 6 + 3x; apply the distributive property
to the expression 24x + 18y to produce the equivalent expression
6 (4x + 3y); apply properties of operations to y + y + y to produce the
equivalent expression 3y.

CCSS6.EE.4
Identify when two expressions are equivalent (i.e., when the two
expressions name the same number regardless of which value is
substituted into them). For example, the expressions y + y + y and 3y
are equivalent because they name the same number regardless of which
number y stands for.

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Play this two-step equation game alone or with another student. This game can also be played at school by dividing the classroom in two teams. Have fun solving equations! You can play this game alone, with a friend, or in two teams. This game is a multi-player game that can be played on computers, Promethean boards, smart boards, iPads, and other tablets.

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IXL | Solve one-step equations | 7th grade math

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Improve your math knowledge with free questions in "Solve one-step equations" and thousands of other math skills.

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Use rise over run to find the slope of each graph.

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IXL | Find the slope of a graph | 8th grade math

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Improve your math knowledge with free questions in "Find the slope of a graph" and thousands of other math skills.

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Rates & proportional relationships example (video) | Khan Academy

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Know how to add, subtract, multiply, and divide scientific notation.

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Learn how to simplify a multiplication and division expression using scientific notation. The expression in this problem is (7 * 10^5) / ((2 * 10^-2)(2.5 * 10^9)). Created by Sal Khan.

Video transcript

We have 7 times 10 to the fifth over 2 times 10 to the negative 2 times 2.5 times 10 to the ninth. So let's try to simplify this a little bit. And I'll start off by trying to simplify this denominator here. So the numerator's just 7 times 10 to the fifth. And the denominator, I just have a bunch of numbers that are being multiplied times each other. So I can do it in any order. So let me swap the order. So I'm going to do over 2 times 2.5 times 10 to the negative 2 times 10 to the ninth. And this is going to be equal to-- so the numerator I haven't changed yet-- 7 times 10 to the fifth over-- and here in the denominator, 2 times-- let me do this in a new color now. 2 times 2.5 is 5. And then 10 to the negative 2 times 10 to the ninth, when you multiply two numbers that are being raised to exponents and have the exact same base-- so it's 10 to the negative 2 times 10 to the negative 9-- we can add the exponents. So this is going to be 10 to the 9 minus 2, or 10 to the seventh. So times 10 to the seventh. And now we can view this as being equal to 7 over 5 times 10 to the fifth over 10 to the seventh. Let me do that in that orange color to keep track of the colors. 10 to the seventh. Now, what is 7 divided by 5? 7 divided by 5 is equal to-- let's see, it's 1 and 2/5, or 1.4. So I'll just write it as 1.4. And then 10 to the fifth divided by 10 to the seventh. So that's going to be the same thing as-- and there's two ways to view this. You could view this as 10 to the fifth times 10 to the negative 7. You add the exponents. You get 10 to the negative 2. Or you say, hey, look, I'm dividing this by this. We have the same base. We can subtract exponents. So it's going to be 10 to the 5 minus 7, which is 10 to the negative 2. So this part right over here is going to simplify to times 10 to the negative 2. Now, are we done? Have we written what we have here in scientific notation? It looks like we have. This value right over here is greater than or equal to 1, but it is less than or equal to 9. It's a digit between 1 and 9, including 1 and 9. And it's being multiplied by 10 to some power. So it looks like we're done. This simplified to 1.4 times 10 to the negative 2.

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Multiplying & dividing in scientific notation (video) | Khan Academy

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Game

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This quiz is designed to test your reading comprehension and understanding of the Scientific Notation section of the Online Study Guide to Chemistry.

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Practice

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IXL | Scientific notation | 7th grade math

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Improve your math knowledge with free questions in "Scientific notation" and thousands of other math skills.

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Video and Practice

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I don't think it's any secret that if one were to do any kind of science, they're going to be dealing with a lot of numbers. It doesn't matter whether you do biology, or chemistry, or physics, numbers are involved. And in many cases, the numbers are very large. They are very, very large numbers. Very large numbers. Or, they're very, small, very small numbers. Very small numbers. You could imagine some very large numbers. If I were to ask you, how many atoms are there in the human body? Or how cells are in the human body? Or the mass of the Earth, in kilograms, those are very large numbers. If I were to ask you the mass of an electron, that would be a very, very small number. So any kind of science, you're going to be dealing with these. And just as an example, let me show you one of the most common numbers you're going to see, in especially chemistry. It's called Avogadro's number. Avogadro's number. And if I were write it in just the standard way of writing a number, it would literally be written as-- do it in a new color. It would be 6022-- and then another 20 zeroes. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. And even I were to throw some commas in here, it's not going to really help the situation to make it more readable. Let me throw some commas in here. This is still a huge number. If I have to write this on a piece of paper or if I were to publish some paper on using Avogadro's number, it would take me forever to write this. And even more, it's hard to tell if I forgot to write a zero or if I maybe wrote too many zeroes. So there's a problem here. Is there a better way to write this? So is there a better way to write this than to write it all out like this? To write literally the 6 followed by the 23 digits, or the 6022 followed by the 20 zeroes there? And to answer that question-- and in case you're curious, Avogadro's number, if you had 12 grams of carbon, especially 12 grams of carbon-12, this is how many atoms you would have in that. And just so you know, 12 grams is like a 50th of a pound. So that just gives you an idea of how many atoms are laying around at any point in time. This is a huge number. The point of here isn't to teach you some chemistry. The point of here is to talk about an easier way to write this. And the easier way to write this we call scientific notation. Scientific notation. And take my word for it, although it might be a little unnatural for you at this video. It really is an easier way to write things like things like that. Before I show you how to do it, let me show you the underlying theory behind scientific notation. If I were to tell you, what is 10 to the 0 power? We know that's equal to 1. What is 10 to the 1 power? That's equal to 10. What's 10 squared? That's 10 times 10. That's 100. What is 10 to the third? 10 to the third is 10 times 10 times 10, which is equal to 1,000. I think you see a general pattern here. 10 to the 0 has no 0's. No 0's in it. 10 to the 1 has one 0. 10 to the second power-- I was going to say the two-th power. 10 to the second power has two 0's. Finally, 10 to the third has three 0's. Don't want to beat a dead horse here, but I think you get the idea. Three 0's. If I were to do 10 to the 100th power, what would that look like? I don't feel like writing it all out here, but it would be 1 followed by-- you could guess it-- a hundred 0's. So it would just be a bunch of 0's. And if we were to count up all of those 0's, you would have one hundred 0's right there. And actually, this might be interesting, just as an aside. You may or may not know what this number is called. This is called a googol. A googol. In the early '90s if someone said, hey, that's a googol, you wouldn't have thought of a search engine. You would have thought of the number 10 to the 100th power, which is a huge number. It's more than the number of atoms, or the estimated number of atoms, in the known universe. In the known universe. It raises the question of what else is there out there. But I was reading up on this not too long ago. And if I remember correctly, the known universe has the order of 10 to the 79th to 10 to the 81 atoms. And this is, of course, rough. No one can really count this. People are just kind of estimating it. Or even better, guesstimating this. But this is a huge number. What may be even more interesting to you is this number was the motivation behind the naming a very popular search engine-- Google. Google is essentially just a misspelling of the word "googol" with the O-L. And I don't know why they called it Google. Maybe they got the domain name. Maybe they want to hold this much information. Maybe that many bytes of information. Or, it's just a cool word. Whatever it is-- maybe it was the founder's favorite number. But it's a cool thing to know. But anyway, I'm digressing. This is a googol. It's just 1 with a hundred 0's. But I could equivalently have just written that as 10 to 100, which is clearly an easier way. This is an easier way to write this. This is easier. In fact, this is so hard to write that I didn't even take the trouble to write it. It would have taken me forever. This was just twenty 0's right here. A hundred 0's I would have filled up this screen and you have found it boring. So I didn't even write it. So clearly, this is easier to write. This is just good for powers of 10. But how can we write something that isn't a direct power of 10? How can we use the power of this simplicity? How can we use the power of the simplicity somehow? And to do that, you just need to make the realization. This number, we can write it as-- so this has how many digits in it? It has 1, 2, 3, and then twenty 0's. So it has 23 digits after the 6. 23 digits after the 6. So what happens if I use this-- if I try to get close to it with a power of 10? So what if I were to say 10 to the 23? Do it in this magenta. 10 to the 23rd power. That's equal to what? That equals 1 with 23 0's. So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23. You get the idea, that's 10 to the 23rd. Now, can we somehow write this guy as some multiple of this guy? Well, we can. Because if we multiplied this guy by 6-- if we multiply 6 times 10 to the 23rd, what do we get? Well, we're just going to have a 6 with twenty three 0's. We're going to have a 6, and then you're going to have twenty three 0's. Let me write that. You're going to have twenty three 0's like that. Because all I did, if you take 6 times this. You know how to multiply. You'd have the 6 times this 1. You'd get a 6. And then all the 6 times the 0's will all be 0. So you'll have 6 followed by twenty three 0's. So that's pretty useful. But still, we're not getting quite to this number. I mean, this had some 2's in there. So how could we do it a little bit better? Well, what if we wrote it as a decimal? This number right here is identical to this number if these 2's were 0's. But if we want to put those 2's there, what can we do? We could put some decimals here. We could say that this is the same thing as 6.022 times 10 to the 23rd. And now, this number is identical to this number, but it's a much easier way to write it. And you could verify it, if you like. It will take you a long time. Maybe we should do it with a smaller number first. But if you multiply 6.022 times 10 to the 23rd, and you write it all out, you will get this number right there. You will get Avogadro's number. Avogadro's number. And although this is complicated or it looks a little bit unintuitive to you at first, this was just a number written out. This has a multiplication and then a 10 to a power. You might say, hey, that's not so simple. But it really is. Because you immediately know how many 0's there are. And it's obviously a much shorter way to write this number. Let's do a couple of more. I started with Avogadro's number because it really shows you the need for a scientific notation. So you don't have to write things like that over and over again. So let's do a couple of other numbers. And we'll just write them in scientific notation. So let's say I have the number 7,345. And I wanted to write it in scientific notation. So I guess the best way to think about it is, it's 7,345. So how can I represent a thousand? Well, I wrote it over here, 10 to the third is 1,000. So we know that 10 to the third is equal to 1,000. So that's essentially the largest power of 10 that I can fit into this. This is seven 1,000's. So if this is seven 1,000's, and then it's 0.3 1,00's, then it's 0.4 1,000's-- I don't know if that helps you, we can write this as 7.345 times 10 to the third because it's going to be seven 1,000's plus 0.3 1,000's. What's 0.3 times 1,000? 0.3 times 1,000 is 300. What's 0.04 times 1,000? That's 40. What's 0.05 times 1,000? That's a 5. So 7.345 times 1,000 is equal to 7,345. Let me multiply it out just to make it clear. So if I took 7.345 times 1,000. The way I do it is I ignore the 0's. I essentially multiply 1 times that guy up there. So I get 7, 3, 4, 5. Then I had three 0's here, so I put those on the end. And then I have three decimal places. 1, 2, 3. So 1, 2, 3. Put the decimal right there. And there you have it, 7.345 times 1,000 is indeed 7,345. Let's do a couple of them. Let's say we wanted to write the number 6 in scientific notation. Obviously, there's no need to write in scientific notation. But how would you do it? Well, what's the largest power of 10 that fits into 6? Well, the largest power of 10 that fits into 6 is just 1. So we could write it as something times 10 to the 0. This is just 1, right? That's just 1. So 6 is what times 1? Well, it's just 6. So 6 is equal to 6 times 10 to the 0. You wouldn't actually have to write it this way. This is much simpler, but it shows you that you really can express any number in scientific notation. Now, what if we wanted to represent something like this? I had started off the video saying in science you deal with very large and very small numbers. So let's say you had the number-- do it in this color. And you had 1, 2, 3, 4. And then, let's say five 0's. And then you have followed by a 7. Well, once again, this is not an easy number to deal with. But how can we deal with it as a power of 10? As a power of 10? So what's the largest power of 10 that fits into this number, that this number is divisible by? So let's think about it. All the powers of 10 we did before were going to positive or going to-- well, yeah, positive powers of 10. We could also do negative powers of 10. We know that 10 to the 0 is 1. Let's start there. 10 to the minus 1 is equal to 1/10, which is equal to 0.1. Let me switch colors. I'll do pink. 10 to the minus 2 is equal to 1 over 10 squared, which is equal to 1/100, which is equal to 0.01. And you I think you get the idea that the--, well, let me just do one more so that you can get the idea. 10 to the minus 3. 10 to the minus 3 is equal to 1 over 10 to the third, which is equal to 1/1,000, which is equal to the 0.001. So the general pattern here is 10 to the whatever negative power is however many places you're going to have behind the decimal point. So here, it's not the number of 0's. In here, 10 to the minus 3, you only have two 0's but you have three places behind the decimal point. So what is the largest power of 10 that goes into this? Well, how many places behind the decimal point do I have? I have 1, 2, 3, 4, 5, 6. So 10 to the minus 6 is going to be equal to 0.-- and we're going to have six places behind the decimal point. And the last place is going to be a 1. So you're going to have Five 0's and a 1. That's 10 to the minus 6. Now, this number right here is 7 times this number. If we multiply this times 7, we get 7 times 1. And then we have 1, 2, 3, 4, 5, 6 numbers behind the decimal point. So 1, 2, 3, 4, 5, 6. So this number times 7 is clearly equal to the number that we started off with. So we can rewrite this number. Instead of writing this number every time, we can write it as being equal to this number. Or, we could write it as 7. This is equal to 7 times this number. But this number is no better than that number. But this number is the same thing as 10 to the minus 6. 7 times 10 to the minus 6. So now you can imagine numbers like-- imagine the number-- what if we had a 7-- let me think of it this way. Let's say we had a 7, 3 over there. So what would we do? Well, we'd want to go to the first digit right here because this is kind of the largest power of 10 that could go into this thing right here. So if we wanted to represent that thing, let me do another decimal that's like that one. So let's say I did 0.0000516 and I wanted to represent this in scientific notation. I'd go to the first non-digit 0-- the first non-zero digit, not non-digit 0, which is there. And I'm like, OK, what's the largest power of 10 that will fit into that? So I'll go 1, 2, 3, 4, 5. So it's going to be equal to 5.16. So I take 5 there, then everything else is going to be behind the decimal point. Times 10. So this is going to be the largest power of 10 that fits into this first non-zero number. So it's 1, 2, 3, 4, 5. So 10 to the minus 5 power. Let me do another example. So the point I wanted to make is you just go to the first-- if you're starting at the left, the first non-zero number. That's what you get your power from. That's where i got my 10 to the minus 5 because I counted 1, 2, 3, 4, 5. You got to count that number just like we did over here. And then, everything else will be behind the decimal. Let me do another example. Let's say I had 0.-- and my wife always point out that I have to write a 0 in front of my decimal points because she's a doctor. And if people don't see the decimal point, someone might overdose on some medication. So let's write it her way, 0.0000000008192. Clearly, this is a super cumbersome number to write. And you might forget about a 0 or add too many 0's, which could be costly if you're doing some important scientific research. Or, maybe doing-- well, you wouldn't prescribe medicine at this small a dose. Or maybe you would, I don't want to get into that. But how would I write this in scientific notation? So I start off with the first non-zero number, if I'm starting from the left. So it's going to be 8.192. I just put a decimal and write 0.192 times-- times 10 to what? Well, I just count. Times 10 to the 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. I have to include that number, 10 to the minus 10. And I think you'll find it reasonably satisfactory that this number is easier to write than that number over there. Now, and this is another powerful thing about scientific notation. Let's say I have these two numbers and I want to multiply them. Let's say I want to multiply the number 0.005 times the number 0.0008. This is actually a fairly straightforward one to do, but sometimes it can get quite cumbersome. And especially if you're dealing with twenty or thirty 0's on either sides of the decimal point. Put a 0 here to make my wife happy. But when you do it in scientific notation, it will actually simplify it. This guy can be rewritten as 5 times 10 to the what? I have 1, 2, 3 spaces behind the decimal. 10 to the third. And then this is 8, so this is times 8 times 10 to the-- sorry, this is 5 times 10 to the minus 3. That's very important. 5 times 10 to the 3 would have been 5,000. Be very careful about that. Now, what is this guy equal to? This is 1, 2, 3, 4 places behind the decimal. So it's 8 times 10 to the minus 4. If we're multiplying these two things, this is the same thing as 5 times 10 to the minus 3 times 8 times 10 to the minus 4. There's nothing special about the scientific notation. It literally means what it's saying. So for multiplying, you could write it out like this. And multiplication, order doesn't matter. So I could rewrite this as 5 times 8 times 10 to the minus 3 times 10 to the minus 4. And then, what is 5 times 8? 5 times 8 we know is 40. So it's 40 times 10 to the minus 3 times 10 to the minus 4. And if you know your exponent rules, you know that when you multiply two numbers that have the same base, you can just add their exponents. So you just add the minus 3 and the minus 4. So it's equal to 40 times 10 to the minus 7. Let's do another example. Let's say we were to multiply Avogadro's number. So we know that's 6.022 times 10 to the 23rd. Now, let's say we multiply that times some really small number. So times, say, 7.23 times 10 to the minus 22. So this is some really small number. You're going to have a decimal, and then you're going to have twenty one 0's. Then you're going ti have a 7 and a 2 and a 3. So this is a really small number. But the multiplication, when you do it in scientific notation, is actually fairly straightforward. This is going to be equal to 6.0-- let me write it properly. 6.022 times 10 to the 23rd times 7.23 times 10 to the minus 22. We can change the order, so it's equal to 6.022 times 7.23. That's that part. So you can view it as these first parts of our scientific notation times 10 to the 23rd times 10 to the minus 22. And now, this is-- you're going to do some little decimal multiplication right here. It's going to be-- some number-- 40 something, I think. I can't do this one in my head. But this part is pretty easy to calculate. I'll just leave this the way it is. But this part right here, this will be times. 10 to the 23rd times 10 to the minus 22. You just add the exponents. You get times 10 to the first power. And then this number, whatever it's going to equal, I'll just leave it out here since I don't have a calculator. 0.23. Let's see, it will be 7.2. Let's see, 0.2 times-- it's like a fifth. It'll be like 41-something. So this is approximately 41 times 10 to the 1. Or, another way is approximately-- it's going to be 410-something. And to get it right, you just have to actually perform this multiplication. So hopefully you see that scientific notation is, one, really useful for super large and super small numbers. And not only is it more useful to kind of understand the numbers and to write the numbers, but it also simplifies operating on the numbers.

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Introduction to scientific notation (video) | Khan Academy

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I don't think it's any secret that if one were to do any kind of science, they're going to be dealing with a lot of numbers. It doesn't matter whether you do biology, or chemistry, or physics, numbers are involved. And in many cases, the numbers are very large.

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Practicing Eighth grade math: 'Cube roots of perfect cubes'

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Practice finding square roots.

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Exponent properties review

Exponent properties review

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Multiply & divide powers (integer exponents) (practice) | Khan Academy

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Videos and practice ordering rational numbers.

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Video transcript

I've got five numbers here, and the goal is to order them from least to greatest. And it might be obvious to you that all five of these numbers are negative numbers. So let's just think about which of these is the greatest number. And you might be tempted to say oh, well, you know, if these were all positive, 40 would be the greatest. So you might be tempted to say negative 40 is the greatest. But you have to realize what the negative is telling you, and to think about that. Think about if these were your dollar amounts in your bank account. Would you rather have negative $40 in your bank account or negative $7 in your bank account? Negative 40 means that you owe the bank $40. So it's 40 less than having nothing. Negative 7 would mean that you owe the bank only $7. So it's actually the case that negative 40 is less than negative 7. And out of all of these numbers, it is the least of all of them. So negative 40 is the very least that you could do. That's the least amount of money that you would want out of these comparisons in your bank account, or the smallest amount of money. You owe the bank $40. Not only do you have nothing, you owe the bank $40. Then the next smallest number would be negative 30. And then the next smallest after that would be negative 25. And then the next smallest after that would be negative 10. And then finally, the greatest of all of these numbers-- and I will do it in pink-- the greatest of all of these numbers is negative 7. And if it's still not obvious to you, you could also think of them in temperature terms. Which of these is the coldest temperature, whether you're talking about Celsius or Fahrenheit? Negative 40 would be the coldest temperature. Negative 7 would be the warmest temperature. There would be the most heat in the air at negative 7. Another way to think about it is we could draw a number line. So let's draw a number line right over here. If this is a zero, and we could put maybe positive 7 up here. Positive 7 isn't one of these numbers, but then we could plot all the other numbers. So this right here might be negative 7. If that's negative 7, then maybe this right over here would be negative 10. Notice we go further and further to the left of zero. And then we go a little bit further to the left. We get negative 25. A little bit further to the left than that is negative 30. And then you go even further to the left and you get to negative 40. So if you think of it this way, the least of these numbers is the one that's furthest to the left on the number line, and the greatest is the one that's furthest to the right.

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Ordering negative numbers (video) | Khan Academy

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How do you order real numbers on a number line?

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Ordering real numbers  from least to greatest

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Practice identifying rational and irrational numbers.

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Quia - ALGEBRA - Rational or Irrational?

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Identify whether a given number is rational or irrational.

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Practice identifying rational and irrational numbers.

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Practice identifying rational and irrational numbers.

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IXL | Identify rational and irrational numbers | 8th grade math

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Improve your math knowledge with free questions in "Identify rational and irrational numbers" and thousands of other math skills.

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Can you tell if a number is rational or irrational?

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Practice identifying whether numbers are rational or irrational.