Solving Proportions
By definition, ratios must be the same in order for them to be proportionate. Using the process of cross-multiplication we are able to prove if any given set of fractions are proportionate. In solving proportions, you use the same process. In these problems, you are trying to find the value which makes the fractions proportionate.
Example 1:
3/n and 5/15
Step 1: Set-up cross multiplication
3*15 = 5*n
Step 2: Solve for the variable.
45 = 5*n
/5 /5
Divide both sides by 5
9 = n
Solution: value of n is 9
Example 2:
Find the value of y which makes the fractions proportionate.
y/4 and 4/3
Set-up cross multiplication:
y * 3 = 4 * 4
3y = 16
Divide each side by 3
/3 /3
y = 16/3 or 5.33
Example 3:
n/8 and 13/2
Set-up cross multiplication:
n * 2 = 8 * 13
2n...
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DEFINITIONS
When given two ratios (in the form x:y) or two relations (in the form of fractions), if the ratios of each element are the same they're said to be proportionate.
Example: 3/6 and 1/2 are proportionate because 3 out 6 is the same as 1 out of two (half).
PROVING PROPORTIONALITY
When given two fractions to prove as proportionate, such as
1
and
3
2
6
you solve through cross-multiplication.
Cross multiplication involves multiplying the numerator (number on top) by the denominator (number on bottom) of the other fraction, and then comparing the results. If the values are the same, the fractions are proportionate.
The set-up above will be set-up as such:
1 * 6
?
2 * 3
(6)
=
(6).
Because both values are the same, these fractions are proportionate.
Example 2:
3/2
and
18/8
The cross-multiplication...
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Algebra Word Problems, Part II: Real World Problems.
In this type of world situations, you will need to establish every variable in the situation as well as all fixed values. You generally will be given a relationship between the variable or variables.
Example 1:
“Richard wants to buy a shirt that is on sale for 20% off the regular price. Write the expression which represents the sale price of the shirt”.
In this situation, there are two variables: regular price and sales price. Accordingly, there is a fixed value which is a rate of change: the 20% off.
Start by writing the relationship between the variables (operation) using words:
Sales Price = Regular Price – 20% off the regular price.
The sales price is going to be the regular price minus the 20% off that regular price. Now you can substitute any symbol/variable in their place. In this case I will use s, in place of the Sales Price; and, p as the regular Price.
Substituting the variables,...
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I was a fairly typical young person and, like my peers, counted down the days until summer. My mother was a math professor, so I never stopped doing math during the summer, but felt like other parts of my brain became a little mushy in the summer. Come September, it was difficult to get back into the swing of writing papers and studying history and memorizing diagrams. I was out of practice and lost my routine. As an adult, I have almost continually taken classes, because I enjoy learning and find that from class to class, I need to maintain a routine, i.e. a study area and a time of day that I complete my assignments. I have also found that reviewing material a week or two before the course begins helps me to start the class with more confidence and competence. I am a big believer in confidence fueling success and I wonder if younger students practiced assignments in the week or two prior to return to school, if that confidence would help the transition to the school year routine...
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I was excited on Tuesday, July 16th, 2013. This was my third meeting with this student and I finally had a breakthrough with him. On the first meeting it was clear that he saw Algebra I almost as a foreign language. I began with one of the test packet, and had him do 10 questions and reviewed the questions he had done wrong. So this continued for a while, and of course sometimes he would say that he understood, but it was clear that he did not. Anyway, after reviewing the entire packet I began a teach and learn session, in which I picked a variety of topics and had him practice various equations. After which I gave him a quiz.
He failed the quiz miserably, so of course he still did not understand. Anyway, I gave him another packet for homework. When I saw the student again, I reviewed with him, but still not much improvement, but at least he tried. I did the teach and learn session again, of which some of the questions were from the previous session, and I gave him the same...
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The Summer session has just begun. The stress has already begun to set in, but this week I had a break through with a few of the students. So this is my second week with a student who I am tutoring for both Algebra I and Earth Science. So far he seems stronger in Earth Science but still needs much practice, before I can be very confident about his ability to pass the Regents exam in August. After the first session of Algebra, I walked away thinking about how am I going to get him ready by August 13th. I recommended an additional session to the parents, but so far they have said no. I did several practice examples, and made the second session mainly a teaching and learning session. Then I ended the session with a quiz, but he failed :(.
So when I had to meet him again for Earth Science, my mind was swirling as to how I can help him, and will I at least be successful with this subject. When I checked the homework, there was a slight improvement but not enough to celebrate. So I came...
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When I was studying to be a teacher, one of the classes I had to take was Literacy in Secondary Education. Since the word
literacy is associated to reading and writing by most, it would strike many as a surprise that Math teachers have to take courses on literacy. However, literacy is the most practical and crucial aspect of ANY academic discipline, simply because it involves the ability to read and write in said subject. For mathematics, it could not be anymore important. If you cannot understand the words that I am using, then it is almost as if we were communicating to each other in different languages.
So whatever subject you are studying, I suggest you learn its vocabulary.
As the helpful tutor that I am, I will share a list of vocabulary terms that was distributed in my literacy class to all of you so that you can check your own vocabulary. Keep in mind that this is considered to be the Mathematics vocab that one should know by the time they finish high school...
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Hello Miss Gil, I received a 96% in Global History. I was so excited to hear these words from my student! At first she did not want to be tutored. Her father dropped her off at the Library. So I told her that if she did the practice test, and did well, she would never have to see me again. Well, she scored a 58%, and there were so many events and topics that she did not know.
We scheduled 3 additional three hour sessions. By the last session, her essays had improved and her overall score was an 83%. I told her that I believe that she can score as much as a 95% on the Regents Exam. She laughed and said "Yeah right". Well she scored a 96% and I am very proud of her.

Humans have a tremendous capacity to learn and adapt. However, we consistently build barriers that hinder our natural ability to change and grow. Many people, regardless of age, perceive themselves as not being talented enough to excel at math and science. They view math and science as the realms in which only scientists, engineers, mathematicians, and geniuses truly soar.
Nothing could be further than the truth. Sure, possessing a natural affinity towards these subjects helps. Yet, a supposed lack of talent does not prevent you from learning. The path may be more arduous. The journey may be longer. Nevertheless, you possess within you the fire to endure. Willpower, dedication, self belief, and an open mind can compensate for any lack of ability.
Bruce Lee was a legendary martial artist, actor, and philosopher who continues to inspire millions with the sheer intensity which he pursued his endeavors. Frail, sickly, and small as a child, Bruce Lee overcame many physical limitations...
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Each summer I have a few students who work on both math and reading to keep the 'flow' and/or prep for the upcoming year. These students and their parents are completely committed to the idea of
always learning as opposed to the idea of only learning in the classroom or merely learning during the school year... in essence, the parents are setting the foundation for lifelong learning.
I would never ask a student to do work which I would not be willing to do myself or work through with them in tutoring. To this end, I have the opportunity to do reading AND catch up on my practice. This summer I am reading 'The Joy of X-A Guided Tour of Math, from One to Infinity' by Steven Strogatz at Cornell University. I LOVE this book! It is almost as good as being in a lecture or small gathering and has helped me explore how I think about math and how to share these ideas with my students.
One of my students recommended 'Hoot' by Carl Hiassen and it is on my list for the library....
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Before I go run a marathon, play with my family at the pool, ride a roller coaster, head to the beach, or eat some serious amounts of ice cream, I will look back on the successful school year I have had.
I got to tutor over 15 students in Middle School, High School, SAT, and College Math...and even Chemistry! I watched GPAs rise for everybody-some were happy just to pass that College Math course to graduate, others enjoyed their hard earned As that were brought up from the D level. I must say, that things did get crazy with exams at the end of the year, but it all worked out amidst our busy-ness.
My tutoring schedule is light for the summer, and I am hoping nobody waits until December exams to contact me! I want things to be done the right way...after all of the swimming, adventure, and ice cream, of course! :)
-Becky

SUMMER OPPORTUNITIES
Now that students, teachers, parents and tutors have had a chance to catch their breath from final exams, it's time to make use of the weeks we have before school starts back. Consider all that could be accomplished in the next few weeks:
Areas of math that students NEVER REALLY GRASPED could be fully explained. This could be
elementary skills like adding fractions, middle school topics like systems of equations, or
high school areas like sequences and series.
Students could have a TREMENDOUS HEAD STARTon topics that will be covered in the first few weeks of school. Imagine your son or daughter being able to raise their hand to answer a question in the first week of school because they had worked several problems just like the ones that the teacher is demonstrating.
ENORMOUS PROGRESS could be made in the area of preparation for the standardized tests (PSAT, SAT, ACT and more) that are so important to getting into a great college.
STUDY SKILLS...
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A wise man once told me: "You can continue to beat your head against that rock, but you will not chip the rock, your head (on the other hand) will be deformed." I guess I should have seen it coming, my being summarily fired from a tutoring job - The parent (in this case the mother) demanding extra "busy-work" for her son between sessions, the lack of discipline, on the student's part (especially his inability to do homework or speak to his subject teacher) and his continual lack of attention during sessions. The call came, "You are not coming here anymore, Billy Ben (not his real name) ONLY got an 81 on his Geometry test. We want top performance, 95 or better, YOU failed." Did I tell you that this student, previous to my seeing him, was working on a solid average of 40? So, it was over. Had I failed? I'm not so sure. First, I didn't take HIS test, and second, knowing the student as I did, I actually thought that an 81 was pretty good and we might have...
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Hi math students :)
When preparing for a mathematics tutoring session, try to have the following things at hand...
Textbook (online or e-text)
Syllabus, assignment, tips/hints/suggestions, answer sheet/key
Class notes
Pencils, pens, erasers, paper (graph paper, ruler, protractor)
All necessary formulas, laws, tables, constants, etc.
Calculator that you will use on tests
Do I really need my calculator? I can do most of my work in my head.
Having your calculator is just as important as paper and a pencil in most cases. You'll be using it on your test and if you don't know how to input what you want, you won't do very well. Have your tutor teach you about your calculator's functions beforehand. Learn how to check your simple math and how to input exponents, logarithms, or trigonometric functions before your test.
Why do I need my book, notes, or answer key? Isn't the tutor supposed to know everything?
Yes :), but even the most experienced tutor...
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Over the time that I have tutored, both at Austin Community College and as an independent tutor, the best thing that I have learned is that any student that is having fun while they learn make them actually want to come back and learn more. Although people came in to the tutoring lab at Austin Community College, they were coming in to the tutoring lab reluctantly because they just didn't understand what was going on and weren't really enjoying the subject because of it. My personal goal for any person that I tutor is to not only help someone understand the concepts while they are sitting with me, but also make it more fun. Not only does time fly when you're having fun but, as noted in "Brain & Behavior, An Introduction to Biological Psychology", a person is able to learn better and more efficiently as they have positive memories with those concepts rather than a neutral emotion of repetition. I love learning math and, as I look back, I did well in all my math classes,...
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I have been working with a few students who are ready to learn math much, MUCH faster than allowed by the traditional classroom model in which math is taught over 6 to 8 years. Based on this experience I believe that many students as young as 4th grade and as old as 8th grade (when starting in the program) can master math in 2 years from simple addition through the first semester of Calculus, with Arithmetic, Algebra 1, Geometry, Algebra 2, Precalculus, Probability, Statistics, and Trigonometry in between.
This is significantly faster than the traditional approach and is enabled by a combination of one-on-one teaching and coaching and a variety of media that I assign to students to complete in between our sessions. This is a "leveraged blended learning" approach that makes use of online software, selected games, and selected videos with guided notes that I have created that ensure that students pick up the key points of the videos, and which we discuss later. The result...
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When someone is interested in a topic, there is a heavy intuitive knowledge associated with that interest. A good musician intuitively knows what would be enjoyable to their target audience; a good fashion designer intuitively knows what would be fashionable for the next season; a good personal trainer knows intuitively how to work a particular person with a particular body and a particular mentality to make that person more active and more healthy. In all of these fields and any field you can think of, there is a certain amount of memorization required, a certain set of rules to follow, but is mainly following personal intuition within that field.
Science and math works the same way. Sure, there is a certain amount of memorization involved – terms, history, phrases – but there is a certain amount of intuition involved. There is a logic behind every concept in all of science and math, regardless of terms and phrases. This logic has its beginnings in the appropriate intuition.
Think...
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As the school year begins to wind down, I have noticed that many of the students I help have begun the journey of signing up for next years classes or, better yet, deciding where they will start the next chapter of their life in college. I began to reminisce on my Senior year of high school and how stressful that year was for me. It was so easy to become overwhelmed by all of the choices that (seemed to be) abruptly placed in front of me: what college should I go to? What should I major in? Should I choose a college close to home? Should I rush? Should I go to a college with all of my friends? Will I absolutely hate it?
I ended up choosing the wrong college and transferred twice until I finally ended up a college that I love! I say all of this to jump into the idea of NOT stressing about this time of year. Yes, I did say not to stress. College is a time of change. That change, no matter how terrifying it may seem, will take you on a wonderful journey that no one can plan for. If...
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Like any puzzle, it is difficult to find the solution without an order of operation, a guide that helps you take one step at a time.

I would like to encourage all of the 6th, 7th and 8th grade students with Interest in Mathematics to participate in Mathcounts competition. Mathcounts is the best Math competition available in the Nation to Middle school students. It is very helpful to build a Very strong foundation in Mathematics for the kids. Most students who have done well in this competition have done very well in their life. Many of them go to study in Top Engineering and Medical schools and many represent our nation USA in International Mathematics Olympiad. I love to help students prepare for this competition. More details about this can be found on
www.mathcounts.org. The kids who get selected from their state get to go to compete in National competition which is fully paid by Mathcounts organization for all flight, stay and food expenses and lots of freebies offered at the competition.