Math Super Powers
Want to know how Math Powers work? Read Math Super Powers.
We Need Math!
Math is very important in everyday life, more necessary than some may think. I often hear students complaining about how they do not think they will ever need to know any of the things they are learning in math class and how it is a waste of their time. It is more relevant to your life than you may think. Whether you are baking a cake with a recipe or deciding how long it will take for you to drive to your friend’s house, math is hidden in daily tasks and often goes unnoticed, but not unused.Trying to decide how much your lunch will cost? Math.
Setting an alarm in the morning to have enough time to drive to school? Math.
Many basics everyday activities depend on math and would be very difficult to figure out without it. Some people have the false impression that only complex jobs like engineering and accounting rely on math, however, a lot of different jobs include math. At least every employee has to know some math to understand their salary, wages, and hours, even when working at a job that seems completely unrelated to math. It is easy to overlook the necessity of math in our everyday lives because it becomes a habit, however in order to form these habits, everyone needs some extent of math knowledge to make these daily tasks involving math second nature.
Math is also important in other subjects and can be helpful to understanding things in science, English, and even the arts. There are many numbers and calculations involved in different sciences, like chemistry and physics. Studying the movement of objects, chemical reactions, or reading charts, graphs, and data all require one to know some math concepts. In English, poetry has a meter, or beat, and a certain number of lines, stanzas, or syllables that is easier to understand with knowledge of math.
Many students think that their high school math class is a waste of time and that since they are going into a career as a history teacher or a writer, whatever the case may be, that they do not need to pay attention or care about math. Of course, it is not necessary that everyone be as ecstatic about math as I am and wake up loving it every day, but everyone should be willing to learn it to an extent because of how much it is used in everyone’s life, regardless of careers. Knowing at least some basic math concepts is very helpful to have a simpler, easier life. Having to pull out a calculator every single time you go to the store and wonder how much change you will get can get annoying and time consuming and it is just easier and faster with a basic concept of math.
Math Manipulatives
Do Math Manipulatives Help Our Students Learn?
What are they?
A math manipulative is an object that is used in the teaching of mathematics that allows students to perceive the idea or concept they are learning through touching and moving the object. These manipulatives can range from anything like dice or money to pattern blocks, two-color counters, and even playing cards or dominoes.
What age groups?
All ages can benefit from the use of manipulatives while learning math. Math manipulatives are most commonly used in the early elementary ages or younger. Once students become more capable of abstracting concepts (older elementary, middle, and high school), teachers seem to have students spend more time doing math with paper and pencil, and less with hands on methods.
What are the benefits?
The use of manipulatives in the learning of mathematics allows students to represent math in multiple ways. More senses become engaged, including visual and tactile, which keeps a student more attentive. They are able to “see” math, which reinforces the conceptual understanding. This lays the groundwork for the mechanics that they will use later and allows the rules to be more meaningful and make sense, which in turn, will be less for them to “memorize”. Seeing math allows students to expand on ideas and uses of math in the world around them.
Why aren’t teachers using them?
Three reasons that math manipulatives are not used as often as they could, is time, money, and lack of knowledge. Developing the concept with a manipulative may require more time and so often, our teachers are burdened with getting through the material. While many math manipulatives on the market can be costly, not all manipulatives are expensive, but having enough for a class set could get pricey. Each math manipulative can be used to teach a variety of concepts. Often teachers may not know how to teach various concepts with these tools, and so they just do not get used. There are many companies out there that do trainings with their manipulative for teachers to learn.
This blog has an ultimate list of math manipulatives that can get you started!
Pinewood Derby
Pinewood Derby Physics
Have you ever seen a pinewood derby race, a race that is held every year in many Cub Scout dens? I have worked with my grandson to enter one a couple of years, and being the engineer that I am, I wanted to use my knowledge of math and physics to perhaps win the race.
We tried to add weights to the car, decreasing the friction of the wheels (it is illegal to use ball bearings), and experimented with the shape of the pinewood derby car to give it a more aerodynamic shape. All of these things help to one degree or another, and all are well known in the Cub Scout community as a simple search will prove (https://www.wired.com/2010/12/pinewood-derby-physics/).
Tautochrone Curve (or Brachistrochrone Curve)
There is something that can affect the speed of an object down a pinewood derby track and that is the shape of the track. You would think that the shortest distance would give you the fastest time (We all know that the shortest distance between two points is a straight line). But in the real world with gravity pushing an object down a slope, the distance takes a peculiar shape, called the Tautochrone curve (also brachistrochrone curve), shown in the graphic above. The next simulation shows this.
Image courtesy of Curiosa Mathematica
Another strange property of the curve is that regardless of where an object is placed on the curve, the time required to reach the end is the same. The first illustration shows this, and is totally unintuitive.
The curve is a form of a cycloid (see https://en.wikipedia.org/wiki/Cycloid), that is the path a point on a wheel makes when it rolls on the ground. There is a lot of math involved in the description (but not too much for a student familiar with trigonometry).
The point to all this is that math is fascinating and has a lot of amazing applications for the student with an inquisitive mind. There is a lot of history behind the study of the cycloid, as noted from the wikipedia article cited above shows, so there is an endless supply of interesting applications of math and a lot of unsolved math mysteries for those who want to make a difference.
Factoring Polynomials
Factoring Polynomials: Learn It or Die!
Ask any adult if they know how to factor a polynomial. Most will say “What?”. Some will say “I learned that in high school, but I couldn’t tell you how to do it now”. If you happen to be asking an engineer or a scientist, they’ll probably say “Sure! It’s fun, but not really useful”. So why do we, as math educators insist that students must not only know how to factor polynomials but must master the subject as if their very lives depended upon it?
Most math educators would agree with the engineer’s first statement. It is fun. But why is it fun? The answer is that it is a puzzle, a mental exercise that yields satisfaction when solved. But let’s examine its utility. Factoring polynomials does lead to the Fundamental Theorem of Algebra (Wow! That sounds regal). It does lead to completing the square to solve a quadratic polynomial, which, in turn, leads to the quadratic formula. Are some adults required to solve a polynomial equation? The answer is yes. engineers, scientists, actuaries, programmers, and others are sometimes required to solve polynomial equations. But rarely is the solution even rational, much less an integer. Since the solution must be approximated, the clever maneuvers learned in high school are not applicable.
So why bother to teach factoring at all? The answer is that we, as educators, have been charged with teaching our students to think, to enjoy thinking, and to trust the results of their thought. We cannot possibly teach our students everything they will need to know to be productive citizens. But we can train their minds to empower them to figure it out. Factoring polynomials should be taught once, early in the math curriculum, not as a useful life skill, but as mental exercise to hone the mind. The trend in math education is to offer students various gimmicks or procedures (the X method, slide and glide, etc.) to allow students to factor a polynomial without really thinking about it. The mind is not honed. The procedures are robotic and forgotten as soon as they are regurgitated on a test.
That being said, here is a link to a great way to factor a polynomial.
How do You Eat an Elephant?
The old adage asks, “How do you eat an elephant? One bite at a time.” Here are some great ideas for making all those things you need to get done a lot more achievable.
Many thanks to author Maria Gracia for the basics of the list.
- GOALS– set realistic goals at the beginning of the school year (also semester, week, day, study period, etc). Write them down and keep them visible.
- AVOID THE RUSH– My dad always figures in “oops time” to every plan. If he thinks something will take 30 minutes, he’ll budget 45 minutes, in case something comes up unexpectedly. The same applies to time spent on homework and school projects. Plan for the unexpected.
- AVOID CLUTTER– At the beginning of the school year, there is no clutter yet. Don’t let it build up. Create separate folders for classes,for things that need to be signed,for tests and papers that have already been graded, etc. As things become outdated, move them to a separate file immediately. Avoid the paper build-up.
- MAKE TO-DO LISTS– Whether it’s a list of things to grab from your locker before you leave school, a list of what homework needs to be done that day, or a list of questions to ask your teacher about your math, lists help you to know exactly what you are doing and what’s still to do.
- EFFECTIVE STUDY AREA– Designate an area that is quiet and well-lit to do your schoolwork. If you can’t find a quiet place at home, go to the library. Study sitting at a table or desk, since getting too comfortable in a chair or on the bed may make you drowsy.
- AVOID OVERLOAD– While extra-curricular activities are a fun part of growing up, taking on too much may mean not having time to get everything done.
- USE A PLANNER OR CALENDAR– Write it down. Make sure that everything you need to do is on one calendar instead of separate social, school, and sports calendars to avoid the risk of scheduling conflicts and missed appointments. You may consider color-coding events such as marking all the tests in yellow, all the projects in blue and so forth.
- BREAK IT UP– Break up big assignments into smaller pieces that are more manageable. If you have 3 chapters to study, study one each day for 3 days instead of trying to do everything at once.
- EAT YOUR BROCCOLI FIRST– Eating your vegetables before dessert gives you something to look forward to. Likewise, doing the hard stuff first makes the rest of the work feel easier.