Welcome, friends, to the Fractions page of the Math 25 Project! This page was made to make you a local in the wonderful land of fractions, and with the help of sample problems, explanations, and a glossary at the bottom of the page, I will make sure that you get there. So hold on to your calculators, folks, and enjoy the mathematical ride.
Before we launch ourselves into that, however, I feel that it would be appropriate for you to find out a bit about who will be teaching you. I am Eva Herman, mathematician extraordinaire. Before I came to Cate School, I was a new eighth grader at Crane Country Day School in Montecito, and before that, I lived in Geneva, Switzerland for eight years. My math teacher there was, let's just say, not my favorite - we spent the better part of seventh grade learning how to use protractors, along with a small unit on the metric system to jazz things up a bit. I finished that year with no knowledge of pre-algebra and no word problem solving skills, two of the most important things to have going into eighth grade. I started Algebra I somewhat afraid that I would be behind my classmates. I ended up doing very well, though, because the material was new to all of us and did not require much algebra experience. Fast-forward a few months and I have a solid understanding of Algebra but no foundation, which brought me to Math 25. In this class I am being filled in on concepts that I missed and solving problems that involve those concepts in a more complicated way - concepts like fractions.
Before we launch ourselves into that, however, I feel that it would be appropriate for you to find out a bit about who will be teaching you. I am Eva Herman, mathematician extraordinaire. Before I came to Cate School, I was a new eighth grader at Crane Country Day School in Montecito, and before that, I lived in Geneva, Switzerland for eight years. My math teacher there was, let's just say, not my favorite - we spent the better part of seventh grade learning how to use protractors, along with a small unit on the metric system to jazz things up a bit. I finished that year with no knowledge of pre-algebra and no word problem solving skills, two of the most important things to have going into eighth grade. I started Algebra I somewhat afraid that I would be behind my classmates. I ended up doing very well, though, because the material was new to all of us and did not require much algebra experience. Fast-forward a few months and I have a solid understanding of Algebra but no foundation, which brought me to Math 25. In this class I am being filled in on concepts that I missed and solving problems that involve those concepts in a more complicated way - concepts like fractions.
Before we begin to solve these problems, we have to remember that fractions can only be added or subtracted if they have a common denominator. Problems in which the two fractions already have the same denominator can therefore be more easily solved than problems in which the fractions have different denominators. All of the problems that we are going to be doing today involve fractions with different denominators, but have no fear! You'll be solving these kinds of problems in no time. (Although if I am such a horrid teacher that you really need a di
Here is an example of a problem with fractions with common denominators:
Here is an example of a problem with fractions with common denominators:
Here is an example of a problem with fractions with different denominators:
Here is a little video to go over those concepts with the help of my mentee and good friend, Salman Kahn, owner of KahnAcademy:
Problem 1 - 13.2
I chose this problem because it takes the simple concept of combining fractions with unlike denominators and adds an interesting spin to it, which is using variables. I also chose it to go first because it is the first chronologically and one of the most simple problems.
This problem has given us more practice on adding fractions with unlike denominators. It has also reminded us of something very important that is not just included in fraction problems: one cannot combine unlike terms. For example, 2 and 4 are like terms, so one can combine them, but, say, 2x and 4 cannot be combined, because they are unlike terms. More examples of like terms: 5 and 347, y and 4y, 763x and 265x. More examples of unlike terms: 78 and 2m, x and y, 13h and -3z.
In this problem, the variable is in the numerator, but in this next problem, we'll find that there are variables in the denominators of the fractions. Again, though, the process will not change much.
In this problem, the variable is in the numerator, but in this next problem, we'll find that there are variables in the denominators of the fractions. Again, though, the process will not change much.
Problem 2 - 15.4
In this problem, every fraction has variables in the denominator. I chose it to show that variables being denominators instead of integers will not change the problem-solving process.
This problem has given us even more practice with adding fractions with like or unlike denominators and has taught us that even though variables may seem iffy, they are just like integers when solving. The next problem, 18.5, will also include combining fractions with unlike denominators, but with bigger numbers and more complex expressions.
Problem 3 - 18.5
Along with what was mentioned above, this problem also asks us to evaluate our answer to (c) with x = 4 and then with x = 10. It then asks to compare those answers to those of (a) and (b). Let's check it out, shall we?
The answer to the question about comparing the answers of c with those of (a) and (b) with 4 and 10 plugged in as x is that when you plug in 4 for x, you get the fractions used in problem (a), and when you plug in 10 for x, you get the fractions used in problem (b). So 18.5, along with teaching us about distribution, has also taught us about plugging numbers into expressions. The next problem, 24.3, does not involve that sort of thing, but it does involve - you guessed it- combining fractions with unlike denominators!
Problem 4 - 24.3
This is a fairly simple problem asking us to combine two fractions with variables in the denominator and an integer. With the practice we've had with those so far, you should nail this problem!
Wasn't bad at all, right? Good. This problem has given us more help with turning integers into fractions, too. Our next and final problem (don't be sad, though), 39.10, involves all of the concepts we have learned thus far, so it's a good one to end on.
Problem 5 - 39.10
So, now that we understand and have practiced all of the concepts involved in this problem, we should nail this one as well. Let's just use the steps that we have been using.
This problem hasn't taught us anything new, but it has given us great practice for the concepts we have learned from earlier problems, like distributing and making integers fractions.
When we were choosing our topics for this project, the only one that I was really interested in doing was Algebra Blocks, because, hey, it seemed as though it only involved building cubes out of smaller cubes, which looked pretty fun to me. But then I realized: doing this whole project on a topic that I was not as comfortable with in past years could be good for me. So Fractions it was. I had missed learning all about Fractions when I was in middle school and earlier, so I hadn't had much experience with them. But in doing this project - which was meant to teach others about fractions - ended up helping me as well. It also helped my website-building skills, which were less than developed. Though it was cumbersome to always have to take a picture of my work and then send it to myself and then post it and then crop it and all that jazz, I am pretty proud of my finished webpage. It was fun to play around with color and silly pictures, too. ( On a separate note: I was also going to have my reflexion in video format talking about my mathematic way of life as Nick Burns the Cate Historian, but that did not work out due to my lack of motivation and bad English accent. So sorry about that. I hope this is a good reflexion anyways.)
Glossary
fraction - a number which indicates that one number is being divided by another
common denominator- a number that can be divided by each of the denominators of a group of fractions
variable - a symbol for a number we do not know yet
like terms - terms whose variables (and their exponents) are the same
unlike terms - terms whose variables (and their exponents) are not the same
numerator - the expression written above the line in a common fraction showing how many of the parts indicated by the denominator are taken
denominator - the number below the line in a common fraction
integer - a whole number
common denominator- a number that can be divided by each of the denominators of a group of fractions
variable - a symbol for a number we do not know yet
like terms - terms whose variables (and their exponents) are the same
unlike terms - terms whose variables (and their exponents) are not the same
numerator - the expression written above the line in a common fraction showing how many of the parts indicated by the denominator are taken
denominator - the number below the line in a common fraction
integer - a whole number