 Greetings and welcome to MathHelp for Science Courses. In this lecture, we are going to talk about showing work and what that means in terms of a science course and what your professor will likely expect from you when the professor asks for you to show your work when you're doing that. And there's some very good important reasons as to why we do ask you to show work in all of your problems. So let's talk about that a little bit first here. And first of all, why do I have to show my work? So that's often a big question from students and we will often tell you to show your work. And some of the reasons for this are because it does help you to show that you understand the concepts. Just writing down an answer doesn't show any understanding. And it can also help you find out where you might have gone wrong in a calculation. So when your professor is reviewing it, if all they see is a wrong answer, all they know is that you did something wrong but they don't know what and have no way to give you any kind of feedback other than that it's wrong. Sometimes it's a very simple math error. You typed the wrong number into your calculator and didn't realize it and got the wrong answer. Or is it a deeper problem with understanding the concepts? So it helps the professor to be able to see what you might have done wrong. And it's also helpful for the student because it does require you to think through the process that you are using. And it will minimize errors. It'll actually help you catch errors as you go by looking at each of the steps. And then you can also go back and review and correct any issues much more easily when all you have is an answer to review. You will not likely remember what you were doing or how you got that answer. When you have it looked at step by step, you have a much better idea and are better able to review that and to apply it to other situations. So how do we go about showing work? Well, we can do that in a number of ways, but primarily the best thing is to do is to write out every single step that you go through. So for some simple problems that we'll look at, you might only take a couple of steps. It might be a relatively quick process. Other more detailed problems may have many, many steps that you have to go through. So it might be a lot more work for you to be able to do that. What you should generally do is start off with any equations that you are given and any values that you are given. You might be given some numbers. You want to always put those down and you want to put down any equations that you are using. And then explain how you go through the process. So let's take a look at a couple of examples here as to how you might go about doing this. So starting off with an arithmetic problem here and the problem says 3.5 minus 1.4. That quantity times 1.6 minus 0.9 and you want to find out that equals something and we want to find that answer. Generally it is a good idea to write down what you know and in this case you would just write down the problem that you know minus 1.4 times 1.6 minus 0.9. Now I'm sure at some point you've gone through order of operations so perhaps that's something you know and that we always do what's in parentheses. In this problem it would be parentheses first then the multiplication and then finally the subtraction. So whatever is in parentheses is done first so we would be doing this first. 3.5 minus 1.4 would be 2.1. That is multiplied by 1.6 and then we're going to subtract 0.9. Now the next step we do the multiplication. So we want to multiply 2.1 times 1.6 and we would get 3.36. However remember that we also want to look at our significant figures when we do this. So we would actually write this in only two significant figures. Remember when you're multiplying or dividing numbers you can only use as many significant figures as the least and each of these has two significant figures so your answer would be 3.4 and you would then take that and subtract 0.9 and your final answer would then be 2.5. So that would be your final answer. It's also a good idea to always circle or clearly mark your final answer so that your professor knows that he is not searching through your work to find out what you intend your final answer to be. So in this case if we do this problem you would get 2.5. Now we can look at a couple of other examples as well so let's look at another example and this one is going to ask us to calculate an average speed. Now in this case you were given some numbers you were given a distance 14 kilometers and that's a distance and you're given a time of 2.5 hours. So generally a good idea is even though those are in the problem it shows that what you're doing to be able to put those up here. So we could abbreviate distance by D time by T and write the distance equals 14 kilometers and time is equal to 2.5 hours. So that writes down what we know. Now if we want to find a velocity we're going to need an equation to be able to do that and one equation that is often gone over in many science classes is that velocity equals distance divided by time and you would want to put that equation down tell the professor what equation you are using. Then you would substitute in the numbers that you know. So you would take for example the distance here and put it in for the D you would take the time that you know and put that in for T and you would then find that the velocity is equal to 14 kilometers divided by 2.5 hours. So what is your velocity? Well now we can divide those two numbers and if we divide 14 by 2.5 you would find 5.6 kilometers per hour for our velocity and again make sure you make very clear what your final answer is. So the steps that you're looking at there again write down what do you know, write down your equation, substitute your known values into the equation and then get your answer. So just writing down for example if all you were to write down if you were given this problem is 14 divided by 2.5 and equals 5.6 while you're getting the correct answer you are not properly showing your work. You need to be showing exactly what steps you went through in order to get that and that really shows your understanding. Now let's look at one more example here and this example uses a small angle formula and the formula is given up here and it says that D, a linear size is equal to alpha which is an angular diameter so this is a linear diameter and an angular diameter multiplied by the distance. So that's the equation. Even though it's given here if you were using that you'd want to write that down in your solution. You would then want to write down what we know. D, the linear diameter, is 3,474 kilometers. We would then write down the distance, sorry the angular diameter, angular diameter equals half of a degree, so 0.5 degrees. Now if we want to put the values in here we need to use this equation but we need to actually convert the angular diameter here given from degrees into the appropriate set of units. So just dividing these two numbers or using these numbers would not give us the correct answer. So if you were just to put this in you would say that 3,474 kilometers divided by 0.5 degrees equals what? Well that would equal about 7,000 kilometers per degree but that doesn't make any sense. That's not giving us a distance. So there's an extra step that has to be done here. So let's clear off this portion that's incorrect. We don't want to look at that but what we're going to have to do is to instead change this into a different version of the formula which still really works out the same. What it says is that d equals alpha lowercase d divided by 206265. So this is the form that will often be used in astronomy which is often where we use this. In this case you need the angular size to be in arc seconds, not in degrees. As it's set up degrees does not work here it needs an even different set of units but let's look at the one that we would actually use. So in this case we would be using this version of the equation we would need to convert this alpha into arc seconds. So alpha is equal to 0.5 degrees and we could multiply that by the number of arc minutes in a degree which is 60 arc minutes per degree and we'd multiply that by 60 arc seconds per arc minute. So in terms of units we can look at what cancels here and the degrees will cancel, the arc minutes will cancel and it leaves us with just an answer in arc seconds which would be 1,800 arc seconds. So now we have a value that can actually be put in here for alpha. So now we want to put our numbers into our equation and we would say that d, the diameter actually let's go ahead and put that in there. So we know our diameter our diameter is 3,474 kilometers. So if we put that in 3,474 kilometers is equal to 1,800 arc seconds times d which is what we're looking for divided by 206265. So what do we have here? Well then we would say that this is, we can then solve for d. So d, lower case d, is equal to and as we bring these to the other side they get flipped over so it's equal to 206265 times 3,474 kilometers divided by the 1,800 arc seconds. So we would look for that and we put those numbers into our calculator at this point and we would find out that the moon is at a distance of about based on these numbers of about 400,000 kilometers and we would keep that with one significant figure because remember we only knew one of our numbers here to one significant figure. So we only knew that the moon was half a degree in angular diameter so that only had one significant figure so when we multiply everything together we need to keep this at one significant figure as well. So otherwise that will show you that the moon would then be about 400,000 kilometers away. So again make sure you're showing your work here and in terms of showing it that means so showing the equation that you used writing down the numbers that you were given if you have to do a conversion like this that work should be shown as well then putting those numbers into your equation and rearranging it as needed and then finally multiplying and dividing or whatever mathematical options you need to do to get your final answer and show each of those every step that you need to do along the way and we find in this case that the moon is about 400,000 kilometers away. So let's summarize here with what we found and what we've gone over and what we've talked about is that it is very important to show the work it's important to the student it's also important to the professor for the professor it shows that what you understand and the professor can help to tell you possibly find out where you might be going wrong or what you might not be understanding that they would not be able to do if you're just giving an answer so you need to be able to show your work the student it helps you in terms of following through and making sure you're doing each step and not skipping anything which can often give you an incorrect answer you should always show each step in any problem requiring a calculation and that does include giving the equations and any values that are given in your explanations so show whatever values you are given show what equations that you're going to be using plug your values in and work through those step by step and if you do it in multiple steps you'll generally find that you make far fewer errors so that concludes this lesson on showing work and we'll be back again next time for another lesson in math help for science courses so until then, have a great day everyone and I will see you in class