# Blogs

## Algebra Blogs

Sometimes the same procedure shows up in two different contexts. This is especially common in the fields of math and science, as science employs in real-world application many of the techniques we learn in their abstract form in math class. For some reason, the principle as shown in a high-school science class is often much harder for students to understand than it was in the math class. (My personal theory is that science teachers are applying the concept in a way that changes how they explain how it works, and they probably have not collaborated with the student's math teacher to ensure they're reinforcing the same terminology.) Last week one of my students ran into this phenomenon in her own work; a concept from last year's math class showed up in her physics class. To help her understand it, we went back to the original math concept and talked about proportions. The science homework she was struggling with was the old chestnut about unit conversions; rows and rows of fractions... read more

A few summers ago I wrote a blog post about finding math in unexpected places as a way to keep skills sharp through the summer break. One of the unexpected places I talked about was the world of tabletop Role-Playing Games (RPGs) such as Dungeons & Dragons. Such games are essentially communal storytelling exercises which use chance elements to help guide the story via a set of polyhedral dice. I've been running a D&D game for a group of friends for several months now, serving as “Game Master.” As Game Master I serve as lead storyteller for the group, while the others each create a character to experience the story firsthand. My job is to create the framework for the story. I devise and flesh out the world that the story takes place in, present challenges for the players to overcome, and rationalize the effect their actions have upon the world. Overall, my goal is to create circumstances that will allow the players to be heroes. Today I'd like to delve a little deeper... read more

We did it!  With hard work, determination, my high school students passed their regents exams.  I tutored US and Global History, Living Environment, Earth Science, Algebra Core, Algebra 2/Trigonometry, Geometry and Chemistry and the students passed.  One student passed with a 70%, another 75%, 76% and another 79%.  All the other students scored 80% and up.    I am so proud of my students.  Well done students and parents, we did it!

Math Student's Civil Rights   I have the right to learn Math (Math is learnable like other subjects) I have a right to make mistakes, erase then, and try again (Failure points to what I have not learned yet) I have the right to ask for help (asking for help is a great decision) I have the right to ask questions when I don't understand (understanding is the primary goal) I have the right to ask questions until I understand (perseverance is priceless) I have the right to receive help and not feel stupid for receiving it (asking for help is natural) I have the right to not like some math concepts or disciplines (i.e. trigonometry, statistics, differential equations, etc.) I have the right to define success as learning no matter how I feel about Math or supporters I have the right to reduce negative self-talk & feelings I have the right to be treated as a person capable of learning I have the right to assess a helper's ability to... read more

Math is all around us. We use math to calculate the speed of the earth rotating about its axis. We use math to calculate the radius and height of a water tank to store enough water for a town. We use math to calculate the amount of carpeting material to purchase for our houses, and we use math to calculate the amount of fabric material to purchase to sew a pillowcase for our pillows. This means that you cannot run away from math. Even the dosage of painkiller medicine that your body needs depends on your weight and the use of math. I have another example of the applications of math in our everyday lives. Movie theaters like any other for-profit business, have a budget with expenses and income columns. In order for the movie theater to break even, it needs to sell a minimum amount of tickets. This movie theater needs to sell a minimum of 100 tickets as the sum of the of tickets purchased. Also it needs to make a minimum of \$100 from the sum of the tickets purchased in order to break... read more

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. Constants 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'... read more

Growing up as a bright student who wanted to be cool meant that I stayed away from math subject like everyone else - then I got to Algebra.  The ability to find the unknown through remembering formulas simply intrigued me and I learned as much as possible.  As I went into High School and college, I found that the use of Algebraic equations could be a life saver if you simply needed to find the value of something upon demand.  I used them for sales to get the best prices, for Excel because the macros are not as easily remembered, for budgeting money as an adult to assure I stay on target and any other time I need to find the answer quick.  Once you learn the rules of Algebra, the rest is a piece of cake.  I will make you a believer.

All my grade 8 & 9 students (10 students) passed the Algebra Core Regents exam. Only one student had to retake it in August and she passed with an 83%. In June she scored 53%. My two Trigonometry students passed the Regents, but only 2 out 4 students passed the Geometry Regents exams.

**If you wrote down the wrong step once, you must write down the correct step twice.** Re-write the correct steps, start to finish, at least twice, at most 5 times start to finish. That way, you will have recorded the incorrect method once, but you will have recorded the correct method twice. That will reinforce the right method more, so you don't accidentally memorize your mistake. It's easy to teach yourself the wrong way to solve a problem if you don't correct your mistakes more often than you make them in the first place. Everybody learns visually (reading), verbally(hearing), and kinetically(writing), so if you wrote down the wrong ?? step once, you're one step closer to teaching yourself your mistake. Do not review any mistakes by only looking at the answer given to you on the sheet. It won't sink in (without writing it down.) Like I always say, nobody likes studying, least of all me. ;) Correcting yourself in the short run will allow you to study less in... read more

Sure, we have all heard our math teachers say "Study for your test tomorrow." While we can all agree the importance of studying and getting prepared for an exam, not many math teachers actually tell you HOW to study. I am sure we have all spent time making flash cards, staring at our notes, or watching last minute videos on youtube, only to realize the test results often don't correlate to our effort. Before long, these upsetting experiences and test results created a scar in our minds, that statement we have all heard before: I am just not good at math.   The truth of the matter is, many people who have expressed their inability to understand and perform well on mathematics simply don't know how to study for a math exam. After all, those negative signs and multiple choice questions are often so tricky, even though you calculated every step correctly until the very end, all it took was one single mindless error that can well ruin the entire result. If we closely... read more

Physics students make up the lion's share of my current teaching efforts here at Wyzant. I've stuck mostly to AP and the first-year undergraduate level of physics, specifically in the non-calculus-based version of physics.   For non-calc physics, the mathematical skills required are surprisingly low. Since students at that level are rarely (if ever) asked to derive or define equations, the only math they need to succeed is the most basic form of algebra - we're talking about adding and subtracting variables from both sides of the equation! Without exception, every physics student I teach knows how to do at least that much. So why do they need our help?   My theory is that as a tutor, physics is best taught as a puzzle game. My students' classroom teachers provide the rules of the game at the beginning of every unit, and those rules are nothing more than the various equations and constants relating to whatever topic the students are learning at the time.   I... read more

There is some confusion among students about the distinction between "undefined" and "indeterminate". This article will give a couple examples of both, along with some explanation.   First, here's a generic example from algebra, where we solve an equation by finding the value of the variable.    2x + 4 = 12   The solution to this equation is the value of x that will make the equation true. We can simplify by subtracting 4 from both sides:   2x = 8   Then, divide both sides by 2:   x = 4   Placing 4 back into the original equation in place of x:   2(4) + 4 = 12 8 + 4 = 12 12 = 12   With that illustration in mind, let's look at an expression that is said to be undefined:   5/0   Rather than just saying "we can't divide by 0", let's see what happens when we turn this into an equation and try... read more

Settle in, folks, today's a long one. In The Function Machine, we learned that functions can be depicted as curves graphed on a coordinate plane. In What Does the Function Look Like?, we learned how to tell the general shape of a function's graph based on characteristics of its equation, and vice versa. Today, we'll be focusing on linear equations (meaning any equation that graphs into a straight line). The defining characteristic of a linear equation is that the highest power of x in the equation is x to the first. This denotes that for every y value, there is exactly one corresponding x value. Of course, there is always exactly one corresponding y value for every x, but this is one of those “square is a rectangle; rectangle is not necessarily a square” moments. We know there's exactly one y for every x because we choose our x's independently and the y's are dependent on them. There can't be more than one y for any given x; you've only got one output slot... read more

Here are 48 of my favorite math words in 12 groups of 4. Each group has words in it that can be thought of at the same time or are a tool for doing math.   between on over in   each multiply of many   ratio divisions distribution compartments   limit neighborhood proximity boundary   infinite infitesmal mark differentiation   graph width height depth   circle sphere point interval   hyper extra spacetime dimensional   geometry proportion sketch spatial   four table cross squared   target rearrange outcome result   area volume space place   What are your favorite math words? If you aren't sure, search for "mathematical words" and pick a few.

I'm going to list what I believe are the key concepts that you need to master across different math subjects. These are the tools that I have to use most often in order to solve problems, so you should get very familiar with the theory behind them and very comfortable with applying them.   Algebra 1: order of operation (PEMDAS) solving equations slope-intercept form of linear equations point-slope form of linear equations systems of linear equations (elimination and substitution methods) inequalities domain and range undefined and imaginary expressions asymptotes (horizontal and vertical) discontinuities (removable and non-removable) rational expressions factoring quadratic formula radical properties exponent properties transformations and translations of functions   Algebra 2:   1. recognizing and factoring the three most common polynomial forms: quadratic equations common factor expressions... read more

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... read more

Reading Formulas can make or break how a student comprehends the formula when alone - outside the presence of the teacher, instructor, tutor, or parent. Formula For Perimeter of Rectangle:  P = 2l + 2w How To Read:  The Perimeter of a Rectangle is equal to two (2) times the Length of the longer side of the rectangle (L) plus two (2) times the Width of the shorter side of the rectangle (W). When is reading formulas like this necessary?  At three particular moments, reading this formula in this manner can be effective.  When students are initially learning what the formula means When student are learning what it means when they should already know (remediation).  When students want to remind themselves (basics learning study skill habit) Remember, Formulas at their introduction are complete statements or thoughts.  Students cannot and will not recall complete thoughts or statements... read more

One of the first things you notice in algebraic expressions (besides the sometimes haphazard mix of operations) are numbers that appear with a smaller number above them (like this 54). These smaller numbers are called exponents and, in this post, I'll give a basic rundown of what they represent and a few basic rules that you will need to follow when dealing with them.   So, you're probably thinking, what do exponents represent anyway. In short, it's a special way of writing a special form of multiplication. I know it sounds hard to grasp, so I'll give you an example:   - Let's look a 3*3. Of course we know it as 9, but in dealing with the order of operations writing a number multiplied by itself may be combersome if you already have several parentheses in the expression. so the way that 3*3 would be written is 32 as your multiplying 3 by a second 3.   But what if you want to represent 4*4*4 or need to multiply 10 5's? Simply count up the... read more