Math Tutor in San Diego

How to Solve Percentage Word Problems

2011-04-01T15:36:00.000-07:00

When they are short, English to Math simple translation works Percentage word problems are important because they have many real life applications, as in discount sales, interest rates, calculating tips, commissions, taxes, and so on. These problems also show up in a wide variety of school exams, and standardized tests. Percentage word problems constitute an exceptional category among word

Guest Post: Tutoring at a Community College

2011-02-24T22:35:00.000-08:00

By: Kathryn Lewallen I just received the following letter from a reader, who graciously accepted my invitation to publish it here as the first guest post in this blog: Greetings, Juan~ I am a math tutor at our local community college and want to thank you for your blog and website, which I found with a Google search. I am currently tutoring six students, all of whom are women over forty, back

Math is the art of stating the obvious

2011-02-19T23:20:00.000-08:00

Simple questions go a long way towards understanding Recently I had the following dialogue with a sixth-grader during a tutoring session: The little girl says: I have a question I wanted to ask you. We were working on this problem earlier today in class, and I am confused, I don’t know how to do it. It’s 159 – 26.6 because, what do you do? This number 159 has no decimal point. Is it O.K. if you

Geometry and Optical Illusions

2010-12-12T09:38:00.000-08:00

Optical illusions show the need for proofs in Geometry Yesterday I had two Geometry tutoring sessions. The first one in the day was only my second session with that student, while the second one was my first session with the other student [ a curious coincidence: (1,2) and (2,1); that is a symmetric pair in that relationship, with x = ordinal number of the session in the day; and y = ordinal

Random Comments From Students

2010-12-02T12:48:00.000-08:00

A math tutoring day with nine sessions After nine math tutor sessions in the same day I barely remember where I was during the day yesterday but I remember some random comments my students made during the tutoring sessions. The show must go on, so I have no time to go into detail about any thoughts or opinions I may have about the following topics. I just present here some of the comments I

The disconnect between expectations

2010-11-30T23:07:00.000-08:00

Individual student emotions are often neglected during math class Students have their own expectations about how much they should learn, and how fast. Usually these expectations differ from what their teacher expects them to do. Generally, math syllabuses are designed with an “ideal student” in mind. The material is covered and organized in a way that suites the teachers, and the small

One more tutoring day

2010-11-29T14:45:00.000-08:00

A new writing style (for me, in this blog) This is my first post here in four months. It has been a long time without writing anything. Thankfully, I have been very busy lately. I have decided that, in order to keep this blog active, I need to change my writing style to something lighter, more alike social media status updates, if I want to find the time to keep posting entries in this blog at

Finding better explanations

2010-07-28T02:55:00.000-07:00

Another advantage for individual students taking math tutoring sessionsRecently, one of my students told me:“I like your lecture style better than my professor’s. You have a much better way of explaining the subject. He just starts doing problems, and that’s it. Last time he was having a hard time explaining how he was using the absolute value to solve a problem. We were all confused about it,

Associative and distributive properties of multiplication

2010-06-02T14:50:00.000-07:00

Illustrating a not uncommon confusion Recently, working with a student who was preparing for the Algebra portion of a test, we came across this problem about simplifying the radical expression 3*sqrt(27).He factored 27 = 9*3, and he wrote 3*sqrt(27) = 3*sqrt(9*3) = 3*(sqrt(9)*sqrt(3)) = 3*(3*sqrt(3)) Then he paused, hesitated, stopped, and he silently looked at me, telling me with his look that

Why are there so many numbers?

2010-03-26T07:44:00.000-07:00

Where are they?In my math tutor practice I constantly answer questions. I usually get a lot of questions from my students. All kinds of math related questions. Some of them are very common, meaning, I get them all the time. For example, Calculus and Pre-Calculus students very often ask: “What is the domain?” “How do I find the domain of a function?” The vast majority of standardized test takers

Helping Students Find Their Own Motivation To Learn

2010-02-06T07:13:00.000-08:00

What is in it for the student, from the student's own perspective?It is important for students to understand teachers are helping them to figure out what they want to do in life, and are helping them achieve those goals. It is not enough for teachers to give students examples of what students do not want to do in life. Teachers want to inspire students to learn; to give them an appreciation for

Solving systems of equations by substitution

2010-01-30T19:14:00.000-08:00

A topic that is hard to explain because it is so simpleThe substitution method oftentimes works as a powerful technique for solving systems of equations. This method is widely taught in middle and high schools, as part of the Algebra curriculum, along with the other standard methods for solving systems of linear equations in two variables: the elimination method, the graphing method, and the

When it comes to math homework, slower is often better

2010-01-23T22:15:00.000-08:00

Take your time to use the correct set of problem-solving questionsStudents often approach math homework problems with the same level of interest and excitement many adults show when preparing their tax returns. They want to get them done, and over with, as soon as possible so they can do something else. Many times, this is the only interest students have in their math problems, just getting them

Exploring Math with the TI-89

2009-11-21T18:11:00.000-08:00

Technology changes possibilitiesThe TI-89 is a graphing, programmable calculator with symbolic manipulation capabilities. I know that is a mouthful but we can break it down to three main features:1) This device graphs not only functions of one variable; it also graphs functions of two variables, parametric curves, functions given in polar coordinates, and data from number tables.2) Programmable

Math as a Freedom Source

2009-11-14T09:49:00.000-08:00

An ongoing dialogMath gives you freedom because it takes freedom to learn it. Learning math makes you ask questions all the time, like: "What do we have here?" "What could that possibly mean?" "Have we seen something like that before?" "How can I get that?" "How can I figure the answer to that?" "Am I supposed to do something here?" "What would happen if I do this operation?" "Am I off the mark?

My views about ADHD

2009-09-24T22:51:00.000-07:00

The ADHD label only makes sense within the framework of a massive, standardized educational system.Sometimes, parents tell me that their child has ADHD, and they ask me if I can help them raise their child’s math scores. I tell them it would take me more than a few sessions to figure out whether, or to what extent I can help them, because this ADHD situation works out differently with each child.

Is it possible to visualize a 4th spatial dimension?

2009-08-21T18:34:00.000-07:00

A classic question about higher dimensionsLast week I had a tutoring session with a college student I help with his Advanced Calculus class. We did mostly exercises on line integrals, integrating functions of two variables along paths in the XY plane. After that, near the end of the hour, my student asked me some questions on Linear Algebra. He wanted to clarify some details about the dimension

Is there a connection between Mathematical Writing and Fractal Geometry?

2009-07-31T18:56:00.000-07:00

When it comes to math formulas, and equations, is it possible to assign to them some sort of measure similar to a fractal dimension? For the above question to even start to make sense, let’s make one main assumption: let’s say that plain text has dimension one, whereas pictures of faces, landscapes, and other objects, have dimension two. Here is the case for assigning dimension one to plain text.

"Made-up" operations

2009-06-28T01:04:00.000-07:00

Sparing some test-takers the abstraction of modern algebraHere is a specific type of problem that usually confuses many students who are preparing for standardized tests like the GMAT, GRE, and SAT:Let the operation Δ be defined as aΔb = (a2 - b)/(a+b) for all real numbers a, b such that a does not equal -b. If a = 15 and aΔb = 5, what is the value of b?One source of confusion here is the symbol

Who is driving?

2009-06-20T21:03:00.000-07:00

Transferring control of the tutoring session to the studentAlong the lines of my previous post, about how I help students in our math tutoring sessions, here is another ingredient of my tutoring method: I transfer as much control as possible to the student, over the tutoring session. The keywords here are “as much as possible,” meaning, making sure the students still learn all they need to learn.

To like or dislike math is an individual choice

2009-05-20T08:49:00.000-07:00

Helping students regardless of whether they like mathOne basic way I help my students is by respecting their right to dislike math. I do not try to make them like math. I refrain from insinuating, or even thinking to myself, that they should like math, because I believe they are free to make that choice by themselves. I am no longer one of those teachers who are always telling students how

Negative exponents

2009-05-03T21:58:00.000-07:00

Using powers of two for explaining the concept of negative exponentsNegative numbers confuse many students. Usually students tend to struggle with almost everything related to negative numbers. From the very concept of using the (-) sign to refer to conventional spatial directions (left, down, back), to the different rules for adding, subtracting, or multiplying positive and negative numbers;

Another Testimonial

2009-03-26T22:46:00.000-07:00

This one from a GMAT student:"I did tell you that I ended up with 690 on the GMAT, right? It was a good score for my purposes. I am accepted at the PhD program at University of South Florida. :) Thanks for all your help."MAIA F. - March 16, 2009

Ratio Word Problems in Standardized Tests

2009-03-26T22:34:00.000-07:00

Look for all the numbers the problem does not show youStandardized tests like GMAT, GRE, CBEST, and ASVAB include ratio word problems. These may be, for example, problems about mixing water with alcohol, or about the ratio of girls to boys in a classroom, or any other type of situation where it makes sense to talk about ratios. There is a consistent pattern that shows in nearly every ratio word

Testimonial

2009-03-14T23:11:00.000-07:00

From a Calculus student:"The only class keeping me from a 4.0 GPA has always been my math class. Math has always felt impossible. However, Mr. Casteneda's tutoring has changed all of this! I just took my first Calculus exam, and scored 105 out of 100, so over 100%! His tutoring has helped me to not only get over my Math anxiety, but helped me to master the subject."COLETTE D. - March 10, 2009

A Number Divided By Another Number

2009-02-28T20:49:00.000-08:00

One more instance of Math not being EnglishLately I had a few students who had some trouble with fractions. Part of the problem was a very specific type of confusion at the time of setting up a division calculation. When asked “How much is 5 divided by 12?” for example, they would sometimes correctly calculate the result but most other times they would set up the division as 12 divided by 5. One

Ubiquitous Numbers

2009-01-28T21:05:00.000-08:00

One and Zero are always everywhereThe more you tutor math, the more skilled you become in finding good, clear ways to explain all types of math concepts to students. However, some concepts are more elusive than others. The difficulty of grasping a concept depends not only on the concept itself but also on the student who is assimilating it. One person, for example, can easily understand

Tutoring Vector Calculus

2008-12-25T23:17:00.000-08:00

Rich ProblemsI really like tutoring Vector Calculus because the problems are very rich. Very often you get to do a lot of stuff in a single problem, like: partial derivatives; determinants; dot-product; graphing three-dimensional shapes, parametrizing curves and/or surfaces; substitution; double or triple integrals; polar, cylindrical, or spherical coordinates; trigonometric substitution and/or

Math is not English

2008-12-10T21:52:00.000-08:00

The order of operations messes with our reading habits.The mathematical order of operations seems to be a source of confusion for some students, sometimes even frustration. For example, when presented with the expression3 + 4(x-1)some students ask: “Why can’t we just start by adding 3 + 4, and then multiplying 7 times (x-1)?”My short answer is: “Because math is not English.” Then I ask: “Is there

The Square of a Sum

2008-11-13T22:44:00.000-08:00

An arithmetic and geometric approach to AlgebraA very common mistake test takers make when they have been out of school for a while is that they automatically try to expand the square of a sum as if it was the same as the plain sum of the squares of the individual terms. For example, sometimes some students, when presented with say, (x + 5)2, they wrongly expand it as x2 + 25, leaving out the

Tutoring for the CSET

2008-10-29T06:15:00.000-07:00

Matrix Multiplication, Fields, and Other Abstract ConceptsLately I have been tutoring a few CSET takers. The CSET tests for math knowledge equivalent to what is expected from a math major. Consequently, the CSET covers a lot of topics, and some of them are quite abstract. One of the common CSET preparation guides out there starts with one of its most abstract questions right at the very beginning

Look for Solutions with Less Math and More Logic.

2008-09-22T02:22:00.000-07:00

One instance where often “less is more.”The following question is an excellent guideline for solving math word problems:“How can I solve this problem by doing the least possible amount of math?”Oftentimes there are several pathways from the setting of a problem to its final solution. Some routes are safer, while some are riskier, more error-prone. Some routes are faster, while some are

Missing Pieces of Information

2008-09-14T23:46:00.000-07:00

Some search for doors, sometimes some do not want to see themLast week I showed a student how to solve two linear equations in two unknowns. He knew perfectly well how to solve one equation with one variable but did not know how to combine two separate equations into one.Also last week another student made the remark: “I do not know how to start solving this problem. What does ‘isoceles’ mean?”

Writing upside-down, and sideways

2008-09-11T23:35:00.000-07:00

An indirect measure of tutoring experienceLast week, a student made this comment to me:“Wow! You write not only upside-down but also sideways!”Then I realized I have gradually acquired this ability over the years as a direct result of my continued math tutoring practice.As a tutor, many times you have to correct a result, or an equation the student has just written. You are sitting across the

Finding Problems That Motivate Students

2008-09-03T02:29:00.000-07:00

Time units may help with multiplication practiceAccording to my experience each child is a unique learner. Generally speaking, students are more motivated to solve a particular problem when the problem relates to something they find real, meaningful, or important. However, different children usually assign different degrees of importance to the same thing. So it is always helpful to find a topic

Calculus is harder than Algebra

2008-08-13T23:29:00.000-07:00

A few sources of confusion to be aware of.Sometimes I see students who used to get good grades in their Algebra classes but who are now struggling with Calculus.They tell me they are confused by Calculus; they do not know when to apply the Chain Rule; they get uncomfortable around dx; they have no clue how to start the problems, and so on and so forth.What most amazes them is they know they had

Dealing with the difficulty of memorizing products of digits higher than five

2008-07-17T10:14:00.000-07:00

Advantages of using Mayan and Roman numeralsWhen it comes to memorizing the multiplication tables, each child has his or her own pace. Some children find it easier than others to memorize the lists of numerical facts that make up the multiplication tables. Others need longer practice periods, maybe with the help of flash cards. Rote memorization is not everybody’s best act. Some children find

Lowest Common Multiple

2008-06-13T18:00:00.000-07:00

A Method and an Analogy to Clarify this TopicDuring the last few weeks I came up with a way to explain how to calculate the lowest common multiple (LCM) of two or more integers or two or more polynomials.Many students get confused by this LCM topic. One reason is the simplicity of the fact that, for any two expressions, their product is a common multiple, so, “Why look any further?” many students

Quadratic equations in rotated form

2008-04-24T22:47:00.000-07:00

Some long, time-consuming problemsThe last three weeks have been very busy for me. I have been tutoring all math subjects, from fractions to Statistics and multivariate calculus.Looking back over these past weeks it all seems kind of blurred but one topic stands out from the rest because, by coincidence, I had two sessions on the same topic with two different students, both during last week.The

Making an Icosahedron

2008-04-06T20:22:00.000-07:00

Geometry is fun!Yesterday I helped one student with his Geometry project.He had to build a 3-D model of a regular solid, so I showed him how to draw a net of equilateral triangles. We used the triangular net to cut out a template for icosahedrons.The icosahedron is one of the five Platonic solids (the other four are the tetrahedron, the cube, the octahedron, and the dodecahedron). The icosahedron

Solving the Rubik's Cube Puzzle

2008-03-30T21:43:00.000-07:00

Step-by-step solution in a couple of YouTube videos by Dan Brown.I just signed-up to YouTube yesterday, and this post is mostly meant as practice for myself posting videos into my blog.While exploring YouTube’s archives I found a few videos about solving the Rubik’s Cube puzzle. Rubik’s cube is one of my favorites puzzles because it is closely related to both Group Theory and Graph Theory,

Marathon GMAT Tutoring Session

2008-03-26T21:39:00.000-07:00

Very few students go for a three-hour-long tutoring sessionRecently I had a marathon (three hours in a row) GMAT tutoring session, with a student who had only a short time to prepare for the test. The coffee shop we were sitting in closed at 8:30 pm so we had to move to another coffee place nearby to finish the session.We went over exponents, roots, ratios, percentages, percent increases,

Welcome to Number City!

2008-03-16T22:55:00.001-07:00

Find your way around. Don’t get lost.Sometimes students ask me: “How long is going to take for me to pass this test?”To which I reply: “It all depends on how fast you get to the performance level you need for the score you want.”The key phrase here is “performance level,” which the tests are supposed to measure.Sometimes I have to be almost brutally honest by saying: “Look, realistically, as long

Tutoring Physics

2008-02-27T10:18:00.000-08:00

Just fine as long as it is not advancedCuriously enough, these days I find myself tutoring Physics sometimes.Three students I previously tutored in Calculus, as well as one I am currently tutoring for the GRE, all of them have recently asked me to help them with Physics.The first time it was a surprise for me because my student called me over the phone and all she said was she wanted to schedule

Integration by Parts in Calculus 2

2008-02-10T15:02:00.000-08:00

It is really neat when students make their own discoveries.This morning I had a Calculus 2 tutoring session about Integration by Parts.My student was having some trouble with those integrals that are calculated by applying the integration-by-parts formula multiple times.In this type of problem you get somewhat complicated expressions with parenthesis nested inside parenthesis, often with

Multiplication Tables

2008-01-31T11:42:00.000-08:00

A very important foundation for understanding mathSometimes I see students who are really intelligent but who are having problems at school understanding new material. Some of these students only need to go over a few specific examples to grasp the concepts and move on to the next topic. So it is sad and almost unbelievable to discover that, in a few cases, the real obstacle standing on their way

Absolute value expressions in 5th grade?

2008-01-22T21:08:00.000-08:00

Too abstract topics too earlyOne of my students is in 5th grade. Not long ago his homework consisted of writing down the full-blown English names of twelve-digit numbers, like 535,176,402,988. He kept busy writing line after line of tens of millions, and hundreds of billions. After a while, the assignment seemed pretty boring to me but my student was interested in the task all the way through. I

You Don't Have To Do Anything

2008-01-13T22:07:00.000-08:00

Think about what you can do, not what you "have to" doVery often students seem to freeze when they see some kind of problem. In this situations I ask: "What are you thinking? What is going through your mind right now? What thoughts, feelings or ideas do you have when you read this problem?" By asking this types of questions repeatedly, I have discovered that, in many cases, when students find a

Find and Replace

2007-12-22T21:19:00.000-08:00

The substitution methodMost students understand the concept of substitution when the task is to plug in a plain numerical value for a variable in a formula.A typical example would be to evaluate y = 3x^2 - 5x + 2when x = -1Things change dramatically when the task involves plugging in an algebraic expression to replace a variable in another expression, even when the expression we are plugging in

Different Learning Styles

2007-12-13T13:28:00.000-08:00

Everyone learns in his or her own individual waySome students like to go fast. They quickly pick up a new problem-solving method as soon as they see that it works. They take note of the new method and they are ready to move on to the next problem. Some other students want to stop and ask several questions and examine the new method from different angles before giving any credibility to it. They

Probability and the Binomial Distribution

2007-11-29T13:57:00.000-08:00

Some helpful key conceptsProbability problems describe a random experiment and some specific event its probability you are asked to calculate.The probability of the event is a fraction, a positive fraction between 0 and 1. This fraction is a smaller number divided by a bigger number.The big number at the bottom of the fraction is the total number of all possible outcomes of the experiment,

Repetition Is Key

2007-11-19T13:32:00.000-08:00

Practice makes perfectMost students need several exposures to a procedure before being able to perform it correctly. Successful performance of mathematical procedures has several ingredients: understanding the concept is fundamental but is not enough. Memory plays an important part as well, you have to put in the effort required to commit the formulas to memory.Then there are the concrete

How to Make the Most of Private Tutoring

2007-11-11T20:04:00.000-08:00

Useful tips for students looking for tutoringFind a math tutor who is right for you.Plan ahead, work out a budget, and find a tutor within your price range.Make sure you feel comfortable communicating with your tutor by e-mail, phone, and in person.Make sure you feel comfortable about the tutoring schedule, and the meeting place you use for the tutoring sessions, whether it is your own house, a

Some Funny Comments and Questions.

2007-10-25T19:20:00.000-07:00

From some of my studentsFollowing below are a few questions and comments made by some of my students in the last year or so. They are all from different students, and they are unrelated to each other. The only thing they have in common is that I found them very funny at the time I heard them, maybe in part because of the student voices or facial expressions when they made them. Anyway, I quote

Negative Numbers

2007-10-19T13:56:00.000-07:00

Building up concepts a little bit at a time.Beyond a certain age, most students can handle negative numbers. Some make mistakes sometimes, like forgetting writing the negative sign in front of the result, or subtracting the numbers when they should add them but in general they have the concept of negative numbers and their operations.This is different for little children who have not yet been

Factored Integers

2007-10-05T09:32:00.000-07:00

A memorization exerciseLast week I added a new page to my tutoring web site. The new page’s title is “Factored Integers.” It is a reference page listing about 200 positive integers, completely factored out as products of smaller numbers, including their prime factorization.The purpose of such a list is for some standardized test takers to do a memorization exercise. The idea is for the student to

Recommended Calculus Books

2007-09-26T21:27:00.000-07:00

Updating my "Recommended Books" web pageLast week I reorganized my “Recommended Books” web page, now including seven calculus titles. Three of these books I have come across through my students: Stewart’s Calculus; Larson/Hostetler/Edwards Calculus: Early Transcendental Functions; and Frank Blume’s Applied Calculus for Scientists and Engineers.Stewart’s Calculus book is popular among college

The Tower of Hanoi classic puzzle

2007-09-16T11:23:00.000-07:00

The latest addition to my site, a page with an interactive puzzle.Last Friday I added a new page to my site.The new page is about the Towers of Hanoi puzzle.Solving this puzzle is a fun way to develop stamina for keeping your mental focus sharp for longer and longer, when you solve it with more and more discs each time.The new page is interactive, featuring a DHTML javascript that renders a

The Psychological Order of Operations

2007-09-05T06:17:00.000-07:00

The risks of seeking safety in the avoidance of difficult tasksSome students, when evaluating an expression like 8 / 3 - 1, they go:8 / 3 - 1 = 8 / 2 = 4When I stop them by asking, "What is going on there?" they sometimes look at me like, "What? I am just doing the operations here." Then I ask, "What about PEMDAS? Isn't there some sort of order we are supposed to follow for the calculations? Are

Buy the right math prep book

2007-08-17T18:52:00.000-07:00

Pay attention and read the book’s preface, not only the title.One of my students was going to take a standardized test specific to some job he wanted to apply for. The test includes a math section, and this is the only part of the test where this student felt he needed tutoring. My student has a friend who already took the same test, successfully. This friend told my student that, if he prepared

Reasoning, Creativity and Mental Flexibility

2007-08-15T21:35:00.000-07:00

The mental equivalent of a boot camp obstacle courseSometimes after solving a relatively hard problem, GMAT students ask: "Can you give me a few problems similar to this one? Can you make them up? Or, is there a book with several problems just like this one? I want more practice with this kind of problem."Seemingly, there is nothing wrong with wanting more practice but when GMAT or GRE students

Signs of Numbers vs. Signs of Variables

2007-08-02T22:17:00.000-07:00

The inner value vs. the outer look. A variable does not have a sign of its own. Numbers have signs of their own. Variables represent numbers. Variables may come with different signs in front of them in any particular expression. Still the number represented by the variable has a sign of its own. That is why sometimes I tell to some of my students: "Variables have an inner sign," which we cannot

Getting To Your Destination

2007-07-21T21:43:00.000-07:00

You can be the best driver in the world and still you can be driving for hours on end without ever reaching your destination. Some students have an interesting reaction when they first overcome some difficulties. For example, some people are used to making mistakes at solving equations, because they don't know the rules or they don't know how to apply them. Meaning, they cancel identical

Solving Equations For a Particular Variable

2007-07-08T19:56:00.000-07:00

A very basic principle An equation has one equal sign. The equal sign divides the equation into left hand side and right hand side. The two sides may look totally different from each other as expressions but the equal sign says their numerical value has to be the same. The fundamental principle of equations says that, when two expressions have the same

Word Problems Are Not That Much of a Mystery

2007-07-02T09:18:00.000-07:00

Most of it is just common senseMany times, my students are surprised by how simple some problems seem when I explain them. They go "Is that all there is to it? Can it be that simple? You didn't use any formula!"A big part of the difficulty students often have with word problems is they think there is or there should be a special type of formula suited for each particular problem. But, for many

Percentages of Percentages

2007-06-28T21:51:00.000-07:00

How to apply the percentage concept to itself Example problem: In a class, 40% of the students have blue eyes and 20% of those students with blue eyes have brown hair. What percentage of the whole class has blue eyes and brown hair? This problem does not say how many students are there in the class. So, we can pick 100 to simplify the first step. If there are 100 students in the class, then 40

The General Quadratic Equation Formula

2007-06-03T23:12:00.000-07:00

Benefits of showing where the formula comes fromI often show my student how the formula for solving a general quadratic equation is derived. This, understandably, seems quite complex to them. The benefits of showing them the development of the formula in full detail are:1) After that, memorizing the formula seems an easier task in comparison.2) Plugging in the correct values in the formula and

Self-Talk Is a Performance Factor

2007-05-20T06:51:00.000-07:00

A very important factor to achieve good resultsMany students, when they first start working with me, show the following behavior, it's really common: they are working on a problem and all of a sudden they start saying things like: "Well, I don't know what to do, I have never been good at math," "I really suck at math," "I always get these problems wrong," "I do not understand percentages," "When

One Way To Get Familiar With More Square Numbers

2007-05-17T22:54:00.000-07:00

Using the algebraic formula for the square of a sumAt the high school level, most students know all square numbers between one and one hundred: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.That is, the squares of all numbers from 1 to 10. Many students also know 121 is the square of 11 and 144 is the square of 12.That is about how far most people go memorizing square numbers. Relatively few people know

Have a Calculator Handy When Doing Long Division

2007-05-14T01:36:00.000-07:00

Not to skip the work, but to check your resultsLong division problems are among the most boring, detail oriented, time-consuming problems math has to offer.I agree that 7th graders should know how to perform long division by hand, without a calculator. In my opinion, even 4th graders could be required to perform long division accurately.However, some teachers go into overkill mode when assigning

Some Questions In Test Prep Books Make No Sense

2007-05-06T00:15:00.000-07:00

But you can find the right answer anywayTest prep books are very well written and edited. They generally present the right solution for almost every single problem included but every once in a while (seldom though) you can find a mistake. Sometimes the given answer for a problem may be wrong or, even more rare, some questions don't make sense from a strict, formal, rigorous, mathematical point of

Successive Changes Given As Percentages

2007-05-03T22:09:00.000-07:00

Beware of addition in this caseA car increases its speed by 25%.After that, it increases it again, this time by 20%.By what percentage is the car's final speed greater than its original speed?I solve this problem as follows:(1.25)(1.2) = 1.5, so the answer is 50%.Most students solve it the following way:Assuming the initial speed is 100, the next speed is 125.Now, 20% of 125 is (20/100)(125)

Sessions on Pre-Calc and GRE prep

2007-03-05T08:58:00.000-08:00

A good tutoring dayYesterday I had two sessions. The first one was to prepare an entrance placement test where the student wants to qualify for a particular Calculus course, so he is being tested on Pre-calculus. The second session was with two students who are preparing for the GRE. Both tutoring sessions were early in the afternoon at the same Starbucks.In the pre-calculus session we covered a

Sessions in Trigonometry, GMAT and Pre-Calc

2007-03-05T08:46:00.000-08:00

A busy tutoring dayLast Saturday I had three students. The first session was about graphing sine and cosine functions, high school trig. The second one was a GMAT prep session, and the third one was about circles and parabolas, pre-calculus.During the first session I was partially awake, not very alert, so I had to do the problems myself, kind of slowly, organizing the data in a table format,

Moving over to Blogger

2007-02-02T07:12:00.000-08:00

My previous blog host went off line. I think Blogger.com will offer a much longer lasting service. Anyway, I am re-posting here the backup copies of my old posts, so I apologize for the inconvenience if you already saw this material. I expect to catch up pretty soon.

Negative Zero

2007-02-02T07:08:00.000-08:00

Yes, it is a real number.Every once in a while, I find students who show surprise or disbelief when they first encounter the concept of -0 (negative zero).Let's say they are solving some equation, and close to the end it reads like x = - (a - b), where a and b represent two numbers known to be equal by virtue of the conditions set at the beginning of the problem.So, in this example, the

Another Important Key to Problem-Solving

2007-02-02T07:05:00.000-08:00

Write down ~everything~Very often I see students struggling with confusion when they try to solve a word problem all in their head.It is so common, it's amazing. First of all, nobody says you have to produce the answer by just looking at the problem.This is what happens, they read the problem, they understand the first sentence, or the first few sentences, and they are already asking themselves:

One Key to Problem-Solving

2007-02-02T06:56:00.000-08:00

Not "How To?" but "What do we have here?"I always emphasize this to my students. When first facing a math problem, especially a word problem, do not try to get the answer right away. This is an unrealistic expectation. The answer will come as the result of properly developing all the relevant, detailed information contained in the problem. Pay attention to the wording. Create a clear mental image

Moving the Decimal Point

2006-10-06T02:21:00.000-07:00

When you have a fraction, meaning you are dividing one number by another, you can always move the decimal point in both numbers, provided you move it the same number of places in each number (both numerator, and denominator), and in the same direction.Now, when you are multiplying two numbers, you can always move the decimal point in both of them, provided you move it the same number of places in

Chords in a Circle

2006-09-29T01:08:00.000-07:00

The diameter is the chord of maximum length. If the circle has a chord of length L, then the diameter cannot be less than L. The circle can be as large as we want but it cannot be too small. It has to be large enough so the chord of length L fits inside.