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...
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...
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...
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...
Purpose: This series shares tips on how to identify, manage, and overcome Mathematics Negative Self Talk (NST). We cannot avoid NST totally because the NST about Math skills in general is a widely accepted habit.
So what is Mathematics NST anyway? Mathematics NST is when we speak in our minds or to others about an inability to learn, do, and/or understand Mathematics in general. Focus here is what we cannot do or have never done in Mathematics. For example, "I hate Math." "I can't do Math!" "This is too complicated!" " I could never do Math!" "My parents aren't good at Math either." "What can we use Algebra for anyway?" "The teacher is confusing me." The NST phrases list is endless, but also popular in today’s culture.
Downside of NST: NST in Math is simply a bad habit of thinking and attitude. This habit limits learning Math...
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...
All too often, I hear students complain "I hate math!", or "Math is too hard (or boring, or pointless, or !)" Too many kids these days from the entitlement generation (uh, that's my generation's kids - sorry friends, we've spoiled our kids like we were told to!) think that math is just for engineers, computer geeks, math nerds, or smart folks who are decidedly NOT COOL. While it is all too often true that those with natural mathematical ability are introverted, and that they may lack social skills that make it difficult to have a lot of popular friends, why does our culture (the schools, the media, television programs, video games, even some parents and teachers, too) keep this myth, this lie, alive? Because of ego. Basically, we can reduce the kind of petty, bullying behavior towards our brilliant colleagues by first acknowledging the problem, then taking logical (what else) steps to curb it. Once we remove the taunting by their peers, we should execute a branding...
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...
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.
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...
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!!
IF I could go back in time and give my younger self some advice on how to be a better student, be more successful in school, life, etc, I would definitely tell myself that being involved in everything comes at a cost. It is better to find a few things that you like to do, do them well and often, than feeling stressed because there is so much on your plate at one time. Being a 'Jack of all Trades' it is natural for me to dip my toes in different waters- all at the same time, but that does not mean that I can give 100% to any of them at that time.
While I was able to get good grades (A- average) while in school, I was impressed by how much better I did- and felt about my work- the few times that I scaled back on my activities.
Another piece of advice that I wish that I could bestow upon my younger self would be to learn how to speak up in a group setting when someone is not fulfilling their part of an agreement. Now, this said, the best way to do this would be in a tactful manner-...
Math students often become nervous when approaching word problems, because mathematical symbols are buried in the sentences as words – I call these “math words” – and these math words have to be discovered in the text and arranged into equations before any calculation can be done! Never fear – there are some easy ways to translate math words into mathematical symbols so you can solve these tricky problems quickly.
Word problems generally use specific words and phrases that correspond to the four operations of addition, subtraction, multiplication, and division. When you see these words and phrases, simply replace them with the operator symbols and you will begin to see potential equations appear. Here are some of these clues:
Addition: increased by, more than, combined (with), together (with), totaling, total of, sum, added to
Subtraction: decreased by, minus, less, less than, fewer than, faster than, difference between, difference of
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...
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...
Probably the hardest thing about doing word problems is taking the words and translating them into a workable mathematical equation. For this reason many students fear and hate doing them. It can be confusing to know where to start and how to go about figuring out the answer. However, there are ways of breaking down a word problem that makes it clearer and easier to solve. The following is a list of helpful hints and strategies in tackling these challenging word problems.
1. Remember that when you are doing a word problem you are looking to convert the words into an equation, so read through the entire problem first. Don’t try to solve the problem when you’ve only read one sentence. It’s important to completely read the problem in order to get the whole picture and effectively translate and solve the problem.
2. Go back to the beginning. Reread the first sentence. Write down what you know and what you don’t know. Use variables to stand for the unknowns and clearly label what...
When interviewing a prospective tutor, parents should ask about the tutor's skills and experience, and find out if the tutor truly enjoys teaching. When the tutor feels enthusiastic about the subject, and communicates well, the student has an opportunity to learn to enjoy the subject too.
I recommend for parents to observe the first lesson to see the tutor's skills in action, and watch/listen carefully to future lessons when possible, to make sure the tutor has an encouraging, supportive attitude at all times. (Tutors should welcome and respond positively to the child's questions, and NEVER make the child feel "stupid," no matter what.) It is most important to have a safe and quiet place for studying, without distractions. I like to find a quiet table at a library, and work with students there. I welcome suggestions from parents, and I am always looking for ways to improve my teaching skills.
One of my dad's favorite sayings is, "If something seems too good to be true, it probably is." The website Coursera is an example of why that saying needs the word "probably". The idea of taking real college courses from top-notch instructors at prestigious schools for free sounds impossible, yet students around the world are doing just that.
When I first heard of Coursera, I was skeptical. To try it out, I enrolled in some basic undergraduate courses so that I could see how they stacked up against the classes I took at KU and Emporia State University. I am currently taking precalculus at UC Irvine, organic chemistry at Illinois, and calculus at The Ohio State University. All three classes are superlative. The video lectures give me new insights into familiar concepts, and the online quizzes motivate me to practice my skills and keep them sharp and up-to-date. Best of all, they haven't cost me a dime, and I can attend class from the...
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...