Solving Word Problems with Proportions and Relative Comparisons
These word problems are set-up where the dependent variable is not provided as is, but rather as a part of an operation. You will have to set-up each side of the equality with its own operations.
“Shelley finished x number of her math homework problems before dinner. Had she finished 3 more, she would have finished half her math homework. Write an equation which represents the relationship between y, total problems and x, number of problems Shelley
This isn’t set-up in the same way as problems presented in previous entries because there isn’t a defined rate of change right away. So, it will be set-up this way with one variable on each side of the equality. You're already given the variables to use in
Proportion of completed problems = proportion of total problems.
“3 more than completed problems” = “half her math homework” (half total problems)
Word Problems with Multiple Variables and Given Values
This type of problem will be presented such that you'll have to set-up the equation or relation between the variables. Additionally, you will be given the value of one or more variables. On all of these problems you are not asked to solve the problem, only
set-up the equation.
“A weather balloon is launched from a height of 100 meters above sea level. The balloon rises at a constant rate of 27 meters per minute. Write an equation that can be used to determine the time in minutes it will take the balloon to reach a height of 2889
meters above sea level.”
Start with the relationship of variables:
Dependent Variable = Fixed Value + Rate of Change * Independent Variable.
“Height of balloon” = “Initial height” + “27meters per minute” h = 100m + 27m
For the final step, substitute the given height of 2889:
2889 = 100 + 27m.
“The perimeter of a...
Writing Expressions Involving Rate of Change
These real-world problems can be best translated when broken down into their components (variables and operations). When you see the words “is” or “are”, this is the points where you set-up the equality. Whenever you see the word “per”, “each” the implication
is a multiplication. This indicates the rate of change between the variables.
The general format for these problems is:
Dependent Variable = Fixed Value + Rate of Change * Independent Variable.
The fixed value is generally a fixed value which does not change. Most commonly, it will be the initial value in a situation.
“Mark is purchasing a new computer. The cost of the computer is $2400 after tax. He will make monthly payments of $150. Write an equation which describes the balance on the account after any given number of months”
Variables present: balance and number of months.
The rate of change in this case is the $150 per...
The most obvious answer is cost. If a tutor charges the same rate for one or four students, it becomes cheaper per hour as you increase students and share the costs with other families. It is often believed a tutor is best when working 1:1 with a student.
In some instances it is well worth the time and money to have 1:1 tutoring and sometimes it is appropriate for students to study and do school work in small groups.
What is not obvious is the dynamics of small group tutoring. In a variety of circumstances it is invaluable for students to learn how to study “what needs to be studied”. The acts of independence and self regulating behavior have far reaching benefits.
Groups need to learn to share and take turns. This seems simple and yet there is the underlying tendency to allow the ‘smart one’ in the group to carry the burden of work. Assuming each student is in the class and has a different point of view/observation
about what is happening in class, they should share...
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.
3/n and 5/15
Step 1: Set-up cross multiplication
3*15 = 5*n
Step 2: Solve for the variable.
45 = 5*n
Divide both sides by 5
9 = n
Solution: value of n is 9
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
y = 16/3 or 5.33
n/8 and 13/2
Set-up cross multiplication:
n * 2 = 8 * 13
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).
When given two fractions to prove as proportionate, such as
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
Because both values are the same, these fractions are proportionate.
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.
“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,...
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...
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 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....
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...
Although learning is an awesome thing, it can be a difficult and frustrating journey for many students. This difficulty, however, is often times quite normal although most feel it means that a child may not be able to learn or that he/she is so frustrated
that learning is no longer taking place. This is where the experienced tutor steps in; for frustration in learning is a part of the learning itself.
I have taught and tutored many students and have seen first hand how this frustration can leave some students, and their parents, feeling helpless and hopeless. But there is ALWAYS Hope!!! What they have failed to realize is that as the brain learns difficult
concepts, it can only take in parts at a time, little parts at a time. So although it may seem no learning is taking place, it actually is, just in smaller segments. In fact, the most frustration comes right before a new concept is achieved. This is when most
give up. Had they stayed focused for perhaps one or two more...
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...
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...
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! :)
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.
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...
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
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...
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...