We're going back to basics today with a Math Journey covering the three broad categories of symbols. I've found this concept very handy when introducing Algebra to middle school students. So let's go!
Math is a language, and I find it often helps to think of it as such right from the beginning. Just as there are different parts of speech in a language, so there are different 'parts of speech' in math. Where a spoken language includes parts of speech such as nouns, verbs, and adjectives, math has three major types of symbols: constants, operators, and variables. Let's go over each one in detail.
These would be the equivalent of your nouns. A Constant is a number – it has a single, discrete place on the number line. Even if the number itself is ugly – a non-terminating decimal, for example – it still does exist in a specific spot somewhere on the number line. In addition to the obvious constants, math frequently uses what I refer to as 'special constants'...
In the calculation below the mathematical symbols have been removed.
Using only +, -, x and / can you make it correct?
7 32 6 14 9 12 = 112
For the 8th consecutive year, all the students whom I tutored for the New York State Common Core examinations, have passed. All have been promoted to the next grade, and or graduated. Some of the students have received Academic Awards from their schools. Tutoring takes much diligence, patience and determination. There may be good and bad days, depending on how the students feel, but we did it. I could not have done it without the parents, who are committed to their children's success. I am very delighted.
1. No one was born to lose. The best of my students understand this principle like the backs of their hands. No, there is no inherent genetic formula or organic compound you can use to get an A in a class. We are all products of our hardwork and investments. Whoever decides to put in excellent work will definitely reap excellent results.
2. Always aim for gold. Have you heard that there is a pot of gold lying somewhere at the end of the rainbow? It's true! Okay, I'm just joking, but my best students always aim for the gold. The very best. As, not Bs, or Cs, or Ds. Just the very best. The one thing people don't think they are capable of achieving is the best. The top of the class. Or the valedictorian.
3. Never settle for less. My best students are innovative, inquisitive thinkers. They tend to think outside the box, never settling for "just what they got from class." They love to use real life examples and explore how theory comes alive in their personal experiences...
I recently sent this as advice to one of my clients having trouble with linear systems of inequalities. I thought I would share it here on my blog for students, parents, and tutors who have use for it.
EXPLANATION OF LINEAR SYSTEMS OF INEQUALITIES
A system with regular lines (the ones with equals signs in them that you have done before) shows the single point where the two lines cross each other on the graph. The X and Y at that point are the two numbers that make the equal sign true. For instance, with the equations 3 = 5X +Y and 10 = 2X -Y, the answer is x = 7/13 and y = 4/13 because if you plug those numbers into both equations you get true statements, 3=3 and 10=10. The point (7/13, 4/13) is the point where the two lines cross each other. Inequalities, where you have "less than" or "greater than" signs work the same way. But, instead of getting a point where the equations are true, you get a whole area on the graph where they are true. So, the answer...
As you may know, I am a big fan of the well-known author and brain specialist, Dr. Daniel Amen. He mentions in several of his books that Physical Exercise is good for the brain. I have read of research studies that showed a clear correlation between IMPROVEMENT in students' test scores in math and science, and their level of physical activity (for example, when math class followed PE class, the students had significantly higher scores). Maybe we should schedule PE before all math classes in our schools. What do you think about that idea?
This morning I read an online article on the myhealthnewsdaily site, entitled "6 Foods That Are Good for Your Brain," and another article about how Physical Exercise helps maintain healthy brain in older adults too. The second article, "For a Healthy Brain, Physical Exercise Trumps Mental Workout" was found under Yahoo News.
The remainder of this note is quoted from that article:
Regular physical exercise appears to...
Hello, if you are a student frantically searching for help with a math problem, take a second here and I will repost answers to any MATH related questions you may have.
I wanted to take a moment to share a recent "success story". Recently, a Student contacted me because he needed to pass a formal standardized exam, known as the "Praxis I". The Praxis tests are used by State Governments and Colleges of Education to ensure they bring only quality students into their programs to be trained as educators. My Student had unfortunately previously failed all 3 components of the Praxis test, and was now "under the gun", since a second failing score would have resulted in his expulsion from his School.
In my home State, students must achieve a combined Praxis I Score of at least 522 to be eligible for School. The passing score for the Reading test is 176, the Writing test 173, and the Math test 173. The minimum score on each test is 150, and the maximum score is 190. It should be noted that this is a fairly difficult exam series; the median scores (175-179) are barely above the minimum passing scores (173-176).
In mathematics, different functions has different rules and I can see a lot of students are struggling with the rules for integers. So I'll kindly discuss the rules for each operation: + - * /
(-) + (-) = (-)
(+) + (+) = (+)
(+) + (-) [Remember to always take the bigger number sign and use the opposing operation, which is subtraction to solve the equation.]
(-) + (+)
Ex: -9+8=-1 [Same rule follow as above]
(-) - (-)
Ex: -9-(-8) = -9+8
[When two negatives are next to each other you change to its opposing operation: addition and change the 8 into a positive integer.]
(+) - (+) = (+) [Unless the first integer is smaller than the second.
Ex: 5-8= 5+-8 [Then you follow the rule...
The Four Ones Problem
Use the digit "1" exactly 4 times, no other numbers, and any number of standard symbols from arithmetic or algebra to make a formula that equals 5.
(There may be more than one formula that works.)
0 = 1 – 1 + 1 - 1
1 = 1 * 1 * 1 * 1
2 = (1 + 1) / 1 / 1
Extra Credit: What is the SMALLEST whole number that CANNOT be calculated this way?
One of the reasons we get these migraines over integers is that, at least up to the point that we as students were actually introduced to operations with negative numbers, we had been taught (correctly) that addition is an operation that describes combination and subtraction describes extraction. We know, for instance, that adding values is like combining collections of objects, and subtracting values is like removing a collection of objects from another collection.
Then we get to integer math, at which point we are asked, judging by present-day treatments in textbooks, to understand the idea that we should be able to, for example, add a "negative collection" to another "negative collection." Or we must throw away and disregard as ridiculous all that "collection" talk.
Mathematics is always described as a beautifully and rigorously universal subject in every detail--when an idea is laid down and proven in mathematics, it applies everywhere and...
Today's post is about learning styles. One of the most important things that helps teachers provide better instruction is the knowledge of a student’s learning style. My belief is based upon the teachings of noted educational theorist, Dr. Howard Gardner. Dr. Gardner posits that there are “multiple intelligences,” that define our individual learning styles and complement each other (by working together) through our learning processes. His 1983 book, Frames of Mind, detailed his initial findings in this area.
In my educational practice, I attempt to identify my students' learning styles by doing extensive diagnostic testing in the very beginning. In my tutoring classes this may consist of having students to write a paragraph or two in the target language we are studying or work some basic math problems. Diagnostics also include inquiring about student preferences, because students generally do better in the areas that they like. After diagnostics, I set a plan that...
Should I get a tutor? Will it help my child? These are some of the most common questions posed to tutors by parents of students struggling in school. Tutoring can be expensive and difficult to schedule so parents must decide whether the time and money will be well spent. Instead of relying on a crystal ball, use these factors to help make the decision.
1. Does the student spend an appropriate amount of time on homework and studies?
While it can help with study skills, organization, and motivation, tutoring cannot be expected to keep the student on track unless you plan on having a session every night. If you can make sure the student puts in effort outside of tutoring, she will be more likely benefit from it.
2. Does the student have difficulty learning from the textbook?
If this is the case, the student will probably respond to one-on-one instruction that is more personalized. A tutor will help bring the subject to life and engage the student. A good tutor will explain...
How Was This Possible?
I hiked the Grand Canyon earlier this year, as I've done for many years. I started down the trail at exactly 7 am, and hiked at an irregular pace, slowing and stopping occasionally to enjoy the views. I’m not sure what time I got to the river, but I got there before dark and spent the night camped at the river. I started back up the same trail at exactly 7 am the next morning, hiking in the same leisurely and irregular way, and got to the top before dark.
This year, I just happened to notice that there was a point on the trail that I reached at exactly the same time as the day before!
How was this possible? Was that just a coincidence? Considering that I've hiked about 1200 miles in the Grand Canyon over 40 years, what are the chances that has happened before?
(Try to observe your own thought processes as you work on this. How did you go about solving the problem? Did you approach this mathematically, trying to write...
Whenever you complete a math problem, it is paramount to go back and double check your work. Remember, no one is perfect and mistakes will be made from time to time. The first step is to always ask yourself "Does this answer make sense"? For example, if you're working on a geometry problem and you're trying to calculate an angle of a polygon, and you determine the answer is 110°, look at the angle and ask "Does this answer makes sense, does this angle look like it's greater than a right angle or a 90° angle"? If not, you know you've made an error and can go back to find the mistake. You can do it!!
Yesterday was a great day to be a tutor. Two of my students who were struggling in Math (one more than the other), have proved to me that with hard work and determination you can achieve. One of the students, his mom called me to say that he scored an A on his test. That was such great news as his grades were usually 60 - 70 on his quizzes and tests.
I tutored both of them yesterday. I gave them a revision test just to see how much they have progressed and where they may still need more practice. I was amazed at how much better they were in comparison to six weeks ago, when I first met them. The other student, most times he would just look at the paper without knowing what to do. This time he completed the test. He still needs some more practice with fractions and probabilities, but he is doing much better now. So proud of both of them.
Hopefully, we will be able to get through this quickly so we can start working on Geometry.
In my experience, teachers often recoil at the saying, "Those who can't do, teach." Indeed, the underlying tone implies that teachers are nothing more than those people who lack the ability required for a certain field of expertise. However, I like to ignore the intended insult and interpret the expression as a compliment and as an excellent description of an effective tutor. Every time I have struggled in an academic subject, the experience has given me insight into all the wrong twists and turns you can take in the process of trying to unravel a maze of skills and concepts. As a tutor, this invaluable insight now lets me meet a student in the familiar depths of his confusion and travel with him out to the light of understanding. To those for whom a subject comes naturally, this territory of incomprehension exists only in theory. Sometimes, explaining the correct approach in detail does not penetrate a tutoree's confusion. By understanding "not understanding,"...
It is the mark of an educated mind to be able to entertain a thought without accepting it. (Aristotle)
This quote provokes me never to accept the status quo and always challenge assumptions. It is the thought that through education we never stop learning or seeking after truth and knowledge.
Mathematics is the only language shared by all human beings regardless of culture, religion, or gender.
Pi is still approximately 3.14159... regardless of what country you are in. Adding up the cost of a basket full of groceries involves the same math process regardless of whether the total is expressed in dollars, rubles, or yen. With this universal language, all of us, no matter what our unit of exchange, are likely to arrive at math results the same way. Very few people, if any, are literate in all the world's tongues—English, Chinese, Arabic, Bengali, and so on. But virtually all of us possess the ability to be "literate" in the shared language of math. This math literacy is called numeracy, and it is this shared language of numbers that connects us with people across continents and through time.
With this language we can explain the mysteries of the universe or the secrets of DNA. We can understand the forces of planetary motion, discover cures for catastrophic...
I would absolutely love to assign my students practice problems if I knew they would do it and follow through. College students tend to be cramming what they can just to get by or to keep their scholarships and are often overloaded each semester. But I do emphasize to every student that I tutor that practice makes perfect, especially in math. There are usually many variations in solving equations and doing extra problems is the only way to truly master it. When I was studying math, whenever the professors gave us odd home work problems, I would make sure to do the even ones as well. If I had a hard time with a certain type of problem, I would seek out other additional problems very similar to it and do those as well. So, I do promote additional homework. However, it is up to the student to take advantage of the advice. ;)