April 27, 2015

Increased - that's a plus or multiply?

If you have a question: "Increased - is that a plus or multiply?", then definitely no answer to it. You need to read or listen to what's next. If "increased by as much as something something", then this addition and the plus sign. For example, the number 7 is increased by 2 units. What are we doing? To previous value we added increase and get new value:

7 + 2 = 9

Increased. This is a plus or multiply. Mathematics for blondes.
Increased

If "increased in so many times," it is multiplication. For example, the number 7 is increased in 2 times. It is necessary to multiply the previous value of the specified number of times, and we get a new meaning:

7 * 2 = 14

As you can see, the numbers in the above examples are the same, and the result is different. It all depends on how to formulate a thought.

April 21, 2015

Snail on a pole

Objective: Snail crawling on a pole height of 10 meters. During the day, she rises to 4 meters per night descends to 3 meters. For some time the snail reaches the top of the column?

Snail on a pole. Snail crawling on a pole problem. Mathematics for blondes.
Snail on a pole

To solve this problem about a snail on a post need to revise some of its condition. The day the snail takes off the pole to a height of 4 meters. The post has a height of 10 meters. Reformulate the question: at what height should be a snail to a running start to fly to the top of the column? The answer to this question is easy to calculate:

10 - 4 = 6 meters

Now the second question: how many days should snail fidgeting up and down the pole to reach a height of 6 meters. Again, take the hands of the math and think:

6 / (4 - 3) = 6 days

As a result, we have six days cochlea will need to enter the starting position for a breakthrough victory, and one day for the race victory. The result is that the triumph of a snail on a post we will see after 7 days.

The same mathematical problem can be formulated with a modern twist. Look how it will sound in the Russian mathematical textbooks soon: From the border to Russia at the borders of NATO distance of 10 thousand kilometers. Green men move per day from the Russian border to NATO at the border at a distance of 4000 kilometers. During the night, NATO troops cast of unknown green men on the 3,000 kilometers back to Russia. How many days disappear NATO? The answer to this problem is the same: after 7 days of the great light of the Russian world illuminate the whole of Europe. By the way, the light of the Christian faith had once lit up Europe funeral pyres of the Inquisition. And the Third Reich sprawling across Europe recently. We must not forget history.

April 20, 2015

9 multiply 7 equals

Immediately write a ready answer: 9 multiply 7 equals 63. So we have the multiplication table says that we all have been taught, but to the end, many of us have not learned. I'm not an exception. Natural laziness and chronic multiple sclerosis did not allow me to remember this vital piece of mathematics. But it is knowledge of mathematics allows me to easily cope with this column multiplication tables. I just calculate the result I needed.

We all know that when multiplied by a permutation of the factors does not change the result. If somewhere sometime mathematics will tell you otherwise, do not believe them - they are wrong. Simple math multiplication like to call it, that in fact is not a multiplication. For example, if we call a cow "Multiplication", it then becomes to turn milk and poop in the hay. After all, the cow to the mathematical action has nothing to do.

So, I always use the following equation:

9*7 = 7*9

Next is quite simple. That number, which must be multiplied by 9, I multiply by 10 and take away his once:

7*9 = 7*10-7 = 70-7 = 63


Multiplied by 10 is very simple - have added to the number of the toe and the result is ready. Subtract this number from a small number of possible even without a calculator. Now you can easily cope with any stitch multiplication table 9.

For example, if we need to multiply 9 by 9, proceed as follows:

9*9 = 90-9 = 81

If needed multiply 9 by 3:

9*3 = 3*9 = 30-3 = 27

In all these transformations have no miracles, we just use the normal math.

9 multiply. The multiplication table for 9. Mathematics for blondes.
9 multiply

My personal opinion is: better to use the thinking abilities of their brain than clog the brain remembering any nonsense. Even if this nonsense forced to learn mathematics.

Stupid triangle top view

Maybe it's someone not very pleasant, but stupid triangles does not happen, there are only stupid people. But wits on the Internet are often looking for a stupid triangle, top view which they would very much like to see. Show.

Stupid triangle top view. Obtuse triangle. Mathematics for blondes.
Stupid triangle top view

First, let's talk about the name of this type of triangles. Call similar triangles stupid - a sign of illiteracy. As they say now, "politically correct" will call such triangles "obtuse". All the triangles which have one angle greater than 90 degrees, are obtuse triangle. It's not a shame, is not a defect, just that physique is obtuse triangles. However, each obtuse triangle always has a couple of sharp corners. So, just in case.

Now let's talk about the kinds. No triangles on the kinds and types of top side of the window. Obtuse triangle top looks exactly the same as the bottom. But the side view is not pleasing to the eye - is just a regular interval. From the window you are unlikely to consider anything, but in a notebook at a neighbor's party this triangle you can see from many angles. In this case the obtuse triangle will look quite different, what it looks like from above. Describe such a wonderful transformation is possible by means of projective geometry, descriptive geometry, trigonometry, or poems. Who like more.

April 8, 2015

How not to change the signs inside the brackets?

All of us are taught to change signs when opening or closing brackets of the expression in brackets, if before the brackets minus sign. Let's look at this tedious process for half-dead examples.

11-(2+5-4) = 11-3 = 8

Before the expression in brackets is a minus sign, it means that the disclosure of the brackets need to change all the signs to the contrary all the numbers that are inside the parentheses. Consider the same example, but without the brackets.

11-(2+5-4) = 11-2-5+4 = 9-5+4 = 4+4 = 8

Now let's try to take a part of the expression in parentheses.

1+2+3+4 = 10

Naturally, you ask: "Where is the minus sign?!" Do not worry, now appear.

1+2-(-3-4) = 3-(-7) = 3+7 = 10

I put in front of the brackets and the minus sign to change the sign before the numbers inside the parentheses. In disclosing the brackets again I changed the sign, because before my braces a minus sign. In the end, the result remained unchanged.

Now a more complex example.

17-6+9 = 20
17-(6-9) = 17-(-3) = 17+3 = 20

As you can see, continuous headache turns out, when suddenly appears before the brackets the minus sign. How not to change the signs inside the brackets? Very simply - no need to put a minus in front of the brackets. Look how it's done.

17-6+9 = 17+(-6+9) = 17+(3) = 17+3 = 20

Now consider the last two examples under the microscope. In the first case I put the first parenthesis after the minus sign. I like a knife cut a negative number into two parts - a minus sign and a positive number. The minus sign was in front of the bracket, and a positive number - within the brackets. Look.

17-(6........

In fact, we are in parentheses conclude a positive number, which until then was negative. Changing the sign of the first number in parentheses passed on complete autopilot without our intervention. A sort of a machine circumcision minus sign in numbers. But with the other numbers that fall into these brackets are already having problems. Signs they need to be changed manually.

In the second case I put the opening bracket before the minus sign. In fact, I put in brackets a negative number with a minus sign. Here's how it looks initially.

17(-6.........

Now, between numbers 17 and bracket there is no sign that mathematics implies multiplication. But I do not have anything to multiply. To answer in the solution remained the same example, I set the bracket additional "plus" sign.

17+(-6.........

Now everything is correctly written. Before the parentheses appears the plus sign and signs before the numbers inside the brackets intact. No mathematical crime I did not commit, just competently get rid of extra action on the replacement characters inside the brackets. Why mathematics always do? They had not taught this. If it is not in the curriculum, and then teach you that no one will. Math enough to know, you need to also know how to use her.

January 31, 2015

Two angles of a triangle

Consider a very basic problem about two angles of a triangle are known. This problem sounds like this:

Two angles of a triangle are 53 degrees and 57 degrees. Find it the third corner of the triangle.

In any triangle all three angles. That is why the triangle is called. The value of the two angles of the three we know. Now I ask you a couple questions that will help solve this problem.

The first question. What is the sum of the angles of a triangle? This sacred knowledge of mathematics tease "A theorem on the sum of the angles of a triangle." As if they did not call it a law of nature, its essence does not change. Incidentally, the sum of the angles of a triangle belongs to the category of the mathematical knowledge that is easily stored for a long time, but that you never use not awake in their daily lives. Useless knowledge? No, but people use this knowledge is very limited range of professions, such as surveyors.

Sum of the angles of a triangle. The sum of three angles of a triangle is 180 degrees. Mathematics for blondes.
Sum of the angles of a triangle


The second question. If you know that the sum of all the angles of a triangle is 180 degrees, with the arithmetic yourself cope? Here everything is simple. From the sum of the angles of a triangle 180 degrees subtract two prominent corner and get the value of the third angle of the triangle.

180 - 53 - 57 = 70 degrees

I do not want to show here ready-made solution, but ... First, the calculator have a lot of different buttons and accidentally be confused. In such cases, the scientists disappear satellites of Mars. So a complete solution for monitoring, can not hurt. Just check yourself.

Secondly, this is a very good opportunity to do what we do mathematics is strongly not recommended. We are taught to perform tasks with minimal downtime, and possibly without saving intermediate results. Actually, I did. On the one hand, it is correct. On the other hand, it does not give us the opportunity to understand, but what do we actually do?

Personally, I like to consider solving mathematical problems under the microscope in slow motion. Sometimes the impression is that we observe the focus by illusionist and all the secrets of the focus immediately crawl out. Let's look at the detailed solution of this problem on two well-known corners of the triangle and one unknown. Here's how it looks.

Two angles of a triangle. Solution of the problem. Mathematics for blondes.
Two angles of a triangle

And so. Someone measured the angles in a triangle is real. The measurements were performed only for the two corners. Man in high school and knows that the third angle can be simply calculated. This is the condition of the problem. Now, a detailed description of the meaning of the decision and the action carried out by us.

1. Write a law that establishes a relationship between the angles of a triangle, in algebraic form. I have already said that in mathematics it is called "A theorem on the sum of the angles of a triangle." The geometric shape of this law is shown in the first picture.

2. Transform the algebraic form of the law on the corners of a triangle to solve our specific problem.

3. Enter this formula in the data from the task ahead of us. Pass from the algebraic form to the physical.

4.Analiziruem physical model for solving the problem. Mathematical apparatus introduced the decimal system of numbers, other notations are absent. The physical device is represented by a measure of the degree angles, other angle units available. Only under these conditions we can perform addition and subtraction.

5. Go to the mathematical model of the physical problem and perform mathematical operations with numbers using a calculator, a sheet of paper or in your mind.

6. Get ready solution to the problem in physical form.

Here's a novel in verse about I turned to a very simple task. The accuracy of the description of this literary opus does not claim because the school did not teach me this, had to invent on the fly. All the described actions we perform automatically, without going into detailed explanations. I agree with mathematicians that stupid every solution of the problem in as much detail paint. But even more stupid stupid to perform the actions that you teach. In this case, the formation is converted into a conventional animal training.

January 20, 2015

Right triangle trigonometry

I will not tell you the rules and definitions of trigonometric functions on a right triangle. Mathematics is happy to do without me. I'll just show you a picture, which shows a right triangle trigonometry.

Right triangle trigonometry. Mathematics For Blondes.
Right triangle trigonometry

The top row shows who is who in the trigonometric zoo. Sine and cosine of the angle alpha - a relationship of legs to the hypotenuse. Tangent and cotangent - a relationship of legs. With hypotenuse usually no problem, and it is located opposite one of the right angle. But the legs as much as two and they are different. One is located in front of the angle alpha is called opposed (the picture side a). Another gently pressed against the corner called alpha and an adjacent (on the picture side b). Now, looking at the picture, you can easily formulate the definition of trigonometric functions on a right triangle.

The bottom row of images shows how to find the sides of a right triangle, if we know one side and the angle alpha. Known side is green. Using this side and trigonometric functions, you can easily find the other two sides of a right triangle.

Spin this picture as you like, turn face down and look up to the light - trigonometric dependence in a right triangle from it are not changed.

Right triangle trigonometry. Spin this picture. Mathematics For Blondes.
Spin this picture

This picture you can come in handy in the future, the study of physics, theoretical mechanics, in engineering calculations. By the time you've forgotten how to define and use trigonometric functions in a right triangle.

And the saddest thing in the end. If you teach mathematics to use mathematics, these pictures you drew to themselves for a few minutes, without any textbooks. After all this is done simply elementary.