Mathematical Operations on the Number Line

Hi mathematicians, this is Mr. Almeida. In this video, I am going to give you the solution to many of your math questions. So, if you are ever asked to model a problem on the number line. I'm going to tell you what the meanings of the operations are on the number line, and you can use this to help you decide what am I actually being asked to model, and then depending on what you're being asked to model, you're either going to be using addition, subtraction, multiplication, or division when it comes to the four basic operations. Okay, so the first thing that I'm going to focus on is I'm going to focus on whether or not the parts are equal sized or not. If you take a look at this number line here, you notice that the numbers, these lengths are representing numbers. So say this was like the number five. This length would be represented here. Say this length was the number 15. This would be a bigger number than that. You notice that these lengths, these segments, these line segments... that are between these two points are not the same length. This one's longer or bigger than this number right here. So therefore, we know that these are not equal-sized parts. But if we look down here, we notice that the numbers are equal-sized parts. They're the same distance. If you took your left finger and put it right here, and your thumb and put it right here, and then you lifted it up, and you didn't move the distance from your pointing finger to your thumb, then... and then you moved your left finger here and your right finger here, or your right thumb here, they'd be the exact same length. These numbers are the exact same number. So if you're dealing with equal size parts, then you're going to be dealing with either multiplication or division. If you're dealing with numbers that are not the same size, they're not equal numbers, then you're going to be dealing with either addition or subtraction. Okay. So When you model this, I'm going to focus on just addition or subtraction now that we know how to distinguish if it's multiplication or division or addition or subtraction. All right. When it comes to addition, addition is what we call the concatenation or the putting together from end to end. So where this length ends, this one's going to start and then it's going to end over here. So it's the concatenation or putting together of two numbers. So. So as you can see, both of these numbers, this number A has been put together with this number B. So to find this number from this point all the way to this point, if you're asked to find this number, which is this question mark right here, then you're going to take the number A and you're going to add it to the number B. So in order to find this number, if the question is asking you in the word problem to find out what this number is, then you're going to have to add, put together these two numbers A and B, which is addition. If you're being asked to find this number, which is one of the unequal sized parts, then you are going to have to do subtraction. If you're asked to find this number here, this number, then you're going to start with the large segment from here to here, which is called C. And you're going to subtract from it the number B. So you're going to start with the number C and you're going to subtract or remove the number B. Imagine that I'm putting my hand on the segment that starts from that's this segment right here. here that starts here and here and I'm putting my my hand on it I'm grabbing it and I'm removing it from the number line the only thing that is left over is this right here so I'm starting with this number here which is C And I'm removing from this segment C a segment of length B. So that leaves us with just this segment right here. So if you're trying to find this segment right here, you're going to take C and you're going to subtract the number B. B from it. Okay? That's if the question was asking you to find this number. If the question is asking you to find this number, then you're going to start with the concatenation of all of the numbers put together, which is C. And then we're going to remove, if you want to find this number right here, we're going to remove this segment here. So imagine that I'm taking my hand and I'm pulling this segment A of length A away from this segment of length C. Okay? So to find B, I'm going to start with C and remove from that segment a length of A. And what I have left is B. So B is found by taking C, minusing A. So C minus A. Okay? So when you're dealing with addition, you are going to be putting two segments together from end to end, concatenating them. And if you're dealing with subtraction, you're going to have already the concatenation. segment, and you're going to remove from it a segment. So if you want to find this number, you're going to start with C and remove B. That's why A is C minus B. And if you want to find the number B, then you're going to start with the segment C, and you're going to remove from it the segment A. What do we call this? We call this C minus A. Okay? So we could write if we wanted to not write expressions and we wanted to say equations. Equations are just statements that both of these expressions are naming the same number. Well this number that we're trying to find can be found by doing C minus B and it can also be found by just telling me what the number is which is A. So we say that A, I'm just going to do this in like a light green. Okay, A equals C. Actually, let me just do this. Let me just copy so I don't have to delete. Copy this. A is equal to, I'll do the green equal sign. C minus B. If you have two names for the exact same number, or you have two expressions naming the number that you're trying to find, then we can say that both of those expressions are equal to each other. Okay, same thing here. B is going to be equal to, because this question mark is being named by both B and C minus A. So we say B. is equal to c minus a. So equations are found when you are trying to say two different expressions are the exact same number. Okay? All right, and then we can say that c down here is equal to a plus B because this question mark has two expressions it has C and it has a plus B therefore we can say that C has the same value as or is the same point on the number line as or has the same length as a plus B Okay, so next we're going to move to multiplication or division. Remember, if you're reading a word problem and you're thinking about multiplication or division, you are going to be thinking of equal size parts. If the size is... of the parts are not equal, then you cannot use multiplication or division. Cannot. But if they are equally sized, then you can use multiplication or division. All right. All right. Let's talk about what multiplication is. So let's start out with just an example. I'm going to do this in a different color. Let's do this in a light gray. All right, so say we have three times five. Well, three times five means, or is defined as, this number, which is five. Plus five plus dot dot dot means I'm going to keep on adding five and This first number tells me how many times I do it. I'm gonna do that three times So I'm actually just going to move this here because it might get in the way. All right. So multiplication is defined. So if you see three times five, it's defined as five plus five plus five. How many times are we going to do that? We're going to do that three times. OK. And so this first number actually tells us. This first number tells us how many times I am adding a number. This second number tells me what number is repeatedly being added. Okay, so this number tells me the number of times that I have something or the number of groups I have. And this second number in the ordered pair of multiplication tells us the size of each group. What is each group's value? So you see that this number here, the first number in multiplication, is telling us the number of times or the number of groups. The second number is telling us the size of each group. Okay? The size of each group. So these numbers have very specific meanings in math. Okay? Alright, so now I'm going to... Okay. So now I'm going to talk about what each one is. All right, so with multiplication or division, here's going to be the number line that you use. Either you're going to be asked to find one of three numbers. Either you're going to... to be asked to find the size of each group because they're all equally sized that's the size of each group or you're gonna be asked to find the number of times that you have that number or the number of groups that you have that's the first number and multiplication and then what is all of this worth all of this is called 3 times 5 which is this right here okay so you are going to take a look so if we are asked to find the what F is we know how many times we are taking the number we're taking it e times And we want to find, and we know the size of each group, so that is 5, in this case D. So if we want to find what this number is, we're going to have to take the number E and multiply it by the number D. So that would be E times D. Okay? E. times D okay so if you want to find what your product is your you already know the number of times you're taking you know the size of each group and you're asked in the question find what how many total what Whatever these are, then you are going to have to use multiplication because, you know, these two numbers, you just don't know what their product is. When it comes to say you're being asked to find the. the size of each group. So say the question is asking you, what's the size of each group? So I don't know what this number is. But I do know that I have a certain number of groups, or a certain number of times. And I know what my product is. Okay. So I know what this number is, I trying to find out what this is. So I'm going to start with my, my product, which is the number F. And I'm going to divide by what's the other number that I know? I know the number of I know the number of times or groups that I have. So I'm going to divide that by E. Okay. So f divided by e is equal to d, okay? So if ever you're missing either the size of each group or the number of groups or the number of times, you are forced to do division, okay? So I'm going to write the division sign right here. But if you know that number of equal sized parts, and you know the size of each part, and you know the number of parts that you have, or the number of groups that you have, then you're going to have to do multiplication, okay? Because you know these two numbers, you just don't know what this number is. All right, and this last one is, anytime that you're doing division, you already know what all the numbers put together is. So you have F here. So we're going to divide F by, what's the, on the number? that I know I know the size of each group okay so in order to find what three times in order to find out what this number is I'm going to start with three times five I'm going to divide it by the size of each group which in this case is D And that is going to tell me the number of times that I have or the number of groups that I have. Okay? So, to summarize, if you are asked to find what all of these equal groups are put together, you know how many times there are, and you know the size of each group. You are going to have to do multiplication. If you are asked to find the size of each group, then you are going to do... division. You're going to take what all of those are and divide it by the number of groups or the number of times that you have, okay? This is called the number per something else. Like, for example, if this were the number of students and this were the number of buses, this would be the number of students per bus or the number of students in each bus, okay? And then if you were trying to find the number of times, say this is the number of students, say this is the number of students per bus, then you're trying to find the number of buses or the number of groups. So you're going to take the number of students and divide it by the number of students per bus, and that will give you the number of buses. Okay. So to summarize, if you only need to know these two pictures. in math when you're reading word problems. That's it. A lot of people think that you have to know keywords. Keywords don't actually help at all. More does not mean add. Less does not mean subtract. It all depends on what your question is asking you in the problem. All right. So all you have to do is think of these two number lines whenever you are trying to write algebraic expressions, solve word problems, and this will help you solve so many math problems. And you only have to know these two number lines. That's it. That's the secret to mathematics. Breaking it down to some very, very small, big ideas that you can apply across the board when you're doing math in word problems or writing expressions. All right. So, oh, I have to add one more thing here, which are the equations. Remember, if any time that you are naming the same number with two different expressions, you can set both those expressions equal because they're naming the exact same number. So we can say that D is equal to F divided by E. And this is just helping you see the connection between expressions and equations. If ever you're naming the exact same number with two different expressions, you can set those two expressions equal to each other. Therefore, you have made yourself an equation. So then F and E times D are naming the same number here. So I can set those equal to each other. I can say F is equal to E times D. Now sometimes when you are doing math, mathematicians like to write the letters in alphabetical order. So sometimes you might see this as like F equals D times E, but that's just a preference. thing, okay? This is still mathematically correct. All right, this last one is this number is being named by two numbers, two expressions. This number is being named by e, and it is being named by f divided by d. Therefore, we can say that e is equal to... F divided by D, because this number has two expressions that are... naming it so we can say they're both equal to each other. All right. So I hope that this helped you not only understand how to come up with the math that's involved, the operations involved when you're dealing with particular numbers, but it will also help you to see what the relationship between expressions and equations is all about. One last thing before we close this video out. Sometimes in math, math, we call division, division has two different interpretations or two different ways of thinking about it. If you're trying to find the size of each group or the size of each equal sized part, sometimes this division is called partitive division. And the way you can think about it is you are trying to find the size of each equal size part. So that's partitive division. And over here, if you are trying to find out how many groups or how many times you have a... a particular number that is repeatedly the same number, then this is what we call measurement division. You can think of it like you know what the ruler is, you know what you're counting by, you just need to know what how many times you have to count by that number and that's measurements you can think of it like a ruler because you know what the measurement of this equal sized part is you know you're measuring counting by this you just need to find out how many times you can count by this number to get to that number all right so I wish you all the best mathematicians this video is jam-packed packed with so much helpful information, and it only involves two number lines. Two number lines for almost many concepts that you can go over in math. All right, so take care.

Hi mathematicians, this is Mr. Almeida. In this video, I am going to give you the solution to many of your math questions. So, if you are ever asked to model a problem on the number line.

I'm going to tell you what the meanings of the operations are on the number line, and you can use this to help you decide what am I actually being asked to model, and then depending on what you're being asked to model, you're either going to be using addition, subtraction, multiplication, or division when it comes to the four basic operations. Okay, so the first thing that I'm going to focus on is I'm going to focus on whether or not the parts are equal sized or not. If you take a look at this number line here, you notice that the numbers, these lengths are representing numbers. So say this was like the number five. This length would be represented here.

Say this length was the number 15. This would be a bigger number than that. You notice that these lengths, these segments, these line segments... that are between these two points are not the same length. This one's longer or bigger than this number right here.

So therefore, we know that these are not equal-sized parts. But if we look down here, we notice that the numbers are equal-sized parts. They're the same distance. If you took your left finger and put it right here, and your thumb and put it right here, and then you lifted it up, and you didn't move the distance from your pointing finger to your thumb, then... and then you moved your left finger here and your right finger here, or your right thumb here, they'd be the exact same length.

These numbers are the exact same number. So if you're dealing with equal size parts, then you're going to be dealing with either multiplication or division. If you're dealing with numbers that are not the same size, they're not equal numbers, then you're going to be dealing with either addition or subtraction. Okay. So When you model this, I'm going to focus on just addition or subtraction now that we know how to distinguish if it's multiplication or division or addition or subtraction.

All right. When it comes to addition, addition is what we call the concatenation or the putting together from end to end. So where this length ends, this one's going to start and then it's going to end over here. So it's the concatenation or putting together of two numbers.

So. So as you can see, both of these numbers, this number A has been put together with this number B. So to find this number from this point all the way to this point, if you're asked to find this number, which is this question mark right here, then you're going to take the number A and you're going to add it to the number B.

So in order to find this number, if the question is asking you in the word problem to find out what this number is, then you're going to have to add, put together these two numbers A and B, which is addition. If you're being asked to find this number, which is one of the unequal sized parts, then you are going to have to do subtraction. If you're asked to find this number here, this number, then you're going to start with the large segment from here to here, which is called C. And you're going to subtract from it the number B. So you're going to start with the number C and you're going to subtract or remove the number B.

Imagine that I'm putting my hand on the segment that starts from that's this segment right here. here that starts here and here and I'm putting my my hand on it I'm grabbing it and I'm removing it from the number line the only thing that is left over is this right here so I'm starting with this number here which is C And I'm removing from this segment C a segment of length B. So that leaves us with just this segment right here.

So if you're trying to find this segment right here, you're going to take C and you're going to subtract the number B. B from it. Okay?

That's if the question was asking you to find this number. If the question is asking you to find this number, then you're going to start with the concatenation of all of the numbers put together, which is C. And then we're going to remove, if you want to find this number right here, we're going to remove this segment here.

So imagine that I'm taking my hand and I'm pulling this segment A of length A away from this segment of length C. Okay? So to find B, I'm going to start with C and remove from that segment a length of A.

And what I have left is B. So B is found by taking C, minusing A. So C minus A. Okay?

So when you're dealing with addition, you are going to be putting two segments together from end to end, concatenating them. And if you're dealing with subtraction, you're going to have already the concatenation. segment, and you're going to remove from it a segment.

So if you want to find this number, you're going to start with C and remove B. That's why A is C minus B. And if you want to find the number B, then you're going to start with the segment C, and you're going to remove from it the segment A. What do we call this? We call this C minus A.

Okay? So we could write if we wanted to not write expressions and we wanted to say equations. Equations are just statements that both of these expressions are naming the same number. Well this number that we're trying to find can be found by doing C minus B and it can also be found by just telling me what the number is which is A. So we say that A, I'm just going to do this in like a light green.

Okay, A equals C. Actually, let me just do this. Let me just copy so I don't have to delete.

Copy this. A is equal to, I'll do the green equal sign. C minus B.

If you have two names for the exact same number, or you have two expressions naming the number that you're trying to find, then we can say that both of those expressions are equal to each other. Okay, same thing here. B is going to be equal to, because this question mark is being named by both B and C minus A. So we say B. is equal to c minus a.

So equations are found when you are trying to say two different expressions are the exact same number. Okay? All right, and then we can say that c down here is equal to a plus B because this question mark has two expressions it has C and it has a plus B therefore we can say that C has the same value as or is the same point on the number line as or has the same length as a plus B Okay, so next we're going to move to multiplication or division.

Remember, if you're reading a word problem and you're thinking about multiplication or division, you are going to be thinking of equal size parts. If the size is... of the parts are not equal, then you cannot use multiplication or division.

Cannot. But if they are equally sized, then you can use multiplication or division. All right.

All right. Let's talk about what multiplication is. So let's start out with just an example. I'm going to do this in a different color. Let's do this in a light gray.

All right, so say we have three times five. Well, three times five means, or is defined as, this number, which is five. Plus five plus dot dot dot means I'm going to keep on adding five and This first number tells me how many times I do it. I'm gonna do that three times So I'm actually just going to move this here because it might get in the way.

All right. So multiplication is defined. So if you see three times five, it's defined as five plus five plus five. How many times are we going to do that?

We're going to do that three times. OK. And so this first number actually tells us.

This first number tells us how many times I am adding a number. This second number tells me what number is repeatedly being added. Okay, so this number tells me the number of times that I have something or the number of groups I have.

And this second number in the ordered pair of multiplication tells us the size of each group. What is each group's value? So you see that this number here, the first number in multiplication, is telling us the number of times or the number of groups.

The second number is telling us the size of each group. Okay? The size of each group.

So these numbers have very specific meanings in math. Okay? Alright, so now I'm going to...

Okay. So now I'm going to talk about what each one is. All right, so with multiplication or division, here's going to be the number line that you use.

Either you're going to be asked to find one of three numbers. Either you're going to... to be asked to find the size of each group because they're all equally sized that's the size of each group or you're gonna be asked to find the number of times that you have that number or the number of groups that you have that's the first number and multiplication and then what is all of this worth all of this is called 3 times 5 which is this right here okay so you are going to take a look so if we are asked to find the what F is we know how many times we are taking the number we're taking it e times And we want to find, and we know the size of each group, so that is 5, in this case D.

So if we want to find what this number is, we're going to have to take the number E and multiply it by the number D. So that would be E times D. Okay? E. times D okay so if you want to find what your product is your you already know the number of times you're taking you know the size of each group and you're asked in the question find what how many total what Whatever these are, then you are going to have to use multiplication because, you know, these two numbers, you just don't know what their product is.

When it comes to say you're being asked to find the. the size of each group. So say the question is asking you, what's the size of each group?

So I don't know what this number is. But I do know that I have a certain number of groups, or a certain number of times. And I know what my product is. Okay. So I know what this number is, I trying to find out what this is.

So I'm going to start with my, my product, which is the number F. And I'm going to divide by what's the other number that I know? I know the number of I know the number of times or groups that I have.

So I'm going to divide that by E. Okay. So f divided by e is equal to d, okay? So if ever you're missing either the size of each group or the number of groups or the number of times, you are forced to do division, okay?

So I'm going to write the division sign right here. But if you know that number of equal sized parts, and you know the size of each part, and you know the number of parts that you have, or the number of groups that you have, then you're going to have to do multiplication, okay? Because you know these two numbers, you just don't know what this number is.

All right, and this last one is, anytime that you're doing division, you already know what all the numbers put together is. So you have F here. So we're going to divide F by, what's the, on the number? that I know I know the size of each group okay so in order to find what three times in order to find out what this number is I'm going to start with three times five I'm going to divide it by the size of each group which in this case is D And that is going to tell me the number of times that I have or the number of groups that I have.

Okay? So, to summarize, if you are asked to find what all of these equal groups are put together, you know how many times there are, and you know the size of each group. You are going to have to do multiplication.

If you are asked to find the size of each group, then you are going to do... division. You're going to take what all of those are and divide it by the number of groups or the number of times that you have, okay?

This is called the number per something else. Like, for example, if this were the number of students and this were the number of buses, this would be the number of students per bus or the number of students in each bus, okay? And then if you were trying to find the number of times, say this is the number of students, say this is the number of students per bus, then you're trying to find the number of buses or the number of groups.

So you're going to take the number of students and divide it by the number of students per bus, and that will give you the number of buses. Okay. So to summarize, if you only need to know these two pictures. in math when you're reading word problems. That's it.

A lot of people think that you have to know keywords. Keywords don't actually help at all. More does not mean add.

Less does not mean subtract. It all depends on what your question is asking you in the problem. All right.

So all you have to do is think of these two number lines whenever you are trying to write algebraic expressions, solve word problems, and this will help you solve so many math problems. And you only have to know these two number lines. That's it.

That's the secret to mathematics. Breaking it down to some very, very small, big ideas that you can apply across the board when you're doing math in word problems or writing expressions. All right. So, oh, I have to add one more thing here, which are the equations.

Remember, if any time that you are naming the same number with two different expressions, you can set both those expressions equal because they're naming the exact same number. So we can say that D is equal to F divided by E. And this is just helping you see the connection between expressions and equations. If ever you're naming the exact same number with two different expressions, you can set those two expressions equal to each other. Therefore, you have made yourself an equation.

So then F and E times D are naming the same number here. So I can set those equal to each other. I can say F is equal to E times D.

Now sometimes when you are doing math, mathematicians like to write the letters in alphabetical order. So sometimes you might see this as like F equals D times E, but that's just a preference. thing, okay? This is still mathematically correct. All right, this last one is this number is being named by two numbers, two expressions.

This number is being named by e, and it is being named by f divided by d. Therefore, we can say that e is equal to... F divided by D, because this number has two expressions that are... naming it so we can say they're both equal to each other. All right.

So I hope that this helped you not only understand how to come up with the math that's involved, the operations involved when you're dealing with particular numbers, but it will also help you to see what the relationship between expressions and equations is all about. One last thing before we close this video out. Sometimes in math, math, we call division, division has two different interpretations or two different ways of thinking about it.

If you're trying to find the size of each group or the size of each equal sized part, sometimes this division is called partitive division. And the way you can think about it is you are trying to find the size of each equal size part. So that's partitive division. And over here, if you are trying to find out how many groups or how many times you have a...

a particular number that is repeatedly the same number, then this is what we call measurement division. You can think of it like you know what the ruler is, you know what you're counting by, you just need to know what how many times you have to count by that number and that's measurements you can think of it like a ruler because you know what the measurement of this equal sized part is you know you're measuring counting by this you just need to find out how many times you can count by this number to get to that number all right so I wish you all the best mathematicians this video is jam-packed packed with so much helpful information, and it only involves two number lines. Two number lines for almost many concepts that you can go over in math.

All right, so take care.

Transcript for:Mathematical Operations on the Number Line

Transcript for:
Mathematical Operations on the Number Line