The task of optimizing the solution through the discovery of a pattern or pattern (pattern).

The basics of motivation are interest and compulsion. In other words, desire and fear. Let’s try to play it. At all times, the optimal or rational solution is valued, which means that you can offer tasks to become the first or declare yourself the author of the idea. Or tasks that require completion in a limited time, use the fear of losing something as a motivator. Invite students to choose the situation in which they will solve the problem. For starters, invite them to feel themselves in the chair of the responder on the program “Who Wants to Be a Millionaire?”. Let them decide how much is at stake.

Multiplication in the mind

Let’s start with simple mathematical operations. Invite two students to compete one in the role of leader, you must not lose the amount, the other wants to win it. For example, who will quickly multiply a four-digit number by 5. Have four people from the class name four numbers. Write them down on the board. Who will answer the host or invited player faster? After several times, ask if it is possible to simplify the procedure? By asking such a question, you will interest and begin to motivate students to complete this task. Often, many do not even suspect that in order to get the result faster, it is necessary to perform two mathematical operations, namely, divide a four-digit number by two and multiply by ten. For example, 3456 x 5 = 3456/2 x 10 = 17280

You can also do it by multiplying by 4. The algorithm is also very simple. With the first action we multiply the original number by 2, with the second action we multiply the result of the first action by 2 again. For example: 79 x 4 = (79 x 2) x 2 = 158 x 2 = 316.

Multiply by 9. And again, we do it in two steps. We multiply the original number by ten and subtract the original number from the product of the first step. For example: 69 x 9 = 69 x 10 – 69 = 690 – 69 = 621

The Magic of Numbers

What are the numbers? Natural, real, complex, perfect, delicious, etc.

Consider the natural number 666 (six hundred sixty six). What can be said about it besides the mysticism associated with it, since it is shunned by the Christian church, which calls it the number of the beast?

This number is a palindrome, i.e. a number that reads the same in both directions. It is also called the Harshad number (that is, it is divided by the sum of its digits): 666 / (6 + 6 + 6) = 666 / 18 = 37. I also call it a triangular number, so 666 points can be arranged in the form of a regular triangle. On the game roulette in the casino there are 37 sectors from 0 to 36 inclusive and the sum of numbers from 1 to 36 is 666.

Amazing connection

Divide the class into groups of 2. Let everyone choose a four-digit number for themselves, which can have no more than one repetition of digits. For example, you can select 4345, 6565, or 1234. But you can’t use 8888 or 5535.

Now from the received 4 digits, form the smallest number and the largest number. For example, if you chose 4510, then you will get the highest number 5410 and the lowest 145.

Now subtract the smallest from the largest. 4510 -> 5410-145 = 5265, repeat the operation at least 6 times. The task of the group is to find repeated numbers, as the difference between the smallest and largest in all selected variants of the initial numbers. The group that finds the number writes it on a piece of paper but does not show it to others. The representative of the group goes to the blackboard and waits for the teacher’s signal to turn the sheet over at the command. For example, for the number 4510:

  1. 4510 -> 5410-145 = 5265
  2. 5265 -> 6552-2556 = 3996
  3. 3996 -> 9963-3699 = 6264
  4. 6264 -> 6642-2466 = 4176
  5. 4176 -> 7641-1467 = 6417

Let’s take the second number 2015 -> 5210-125 = 5085

  1. 5085 -> 8550-558 = 7992
  2. 7992 -> 9972-2799 = 7173
  3. 7173 -> 7731-1377 = 6354
  4. 6354 -> 6543-3456 = 3087
  5. 3087 -> 8730-378 = 8352
  6. 8352 -> 8532-2358 = 6174

Answer: In all examples, the number 6174 is repeated. This amazing connection between 6174 and four-digit numbers was discovered in 1950 by a mathematician named Kaprekar.

Looking for amazing numbers

The next problem is good for those who decide ahead and then wait for the class to pull up. To keep their minds busy, ask them to find a four-digit number that is divisible by 25, its sum of digits is divisible by 25, and its product of digits is divisible by 25.

Solution progress. Let’s write the number in the form: abcd

A number is divisible by 25 without a remainder if it ends in two zeros, itself, fifty, or seventy-five. Then the desired number could look like this: ab00 / ab25 / ab50 / ab75. Call 4 students to the board to check each of these numbers against the criteria. It is important that each student at the blackboard commented on the course of the solution and there are no those left in the class who did not understand the logic of reasoning, otherwise motivating students to solve this problem will disappear. Therefore, it is possible to allow asking questions from the spot.

Solution:

Let’s check the numbers against the second condition, the sum of the digits must be divisible by 25 without a remainder, i.e. a + b + c + d = 25.

1) The number ab00 is not suitable, since the sum of the digits, if c=0 and d=0, even with the maximum values a=9; b=9; a + b + c + d = 9 + 9 + 0 + 0 = 18≠25

2) The number ab25 is suitable, since the sum of the digits a + b + 2 + 5 = a + b + 7, which is only possible if a=9; b=9, since 25 – 7 = 18 = a + b. Then the number 9925 remains to be checked for compliance with the third condition, that the product of digits must be divisible by 25. Let’s check the product: 9 9 2 5 = 810. But 810 is not divisible by 25. So the number ab25 is not suitable.

3) The number ab50 is also not good, since the sum should be 25 = a + b + 5 + 0. a + b=20, which cannot be.

4) Check ab75. There are options in which the sum will be equal to 25 = a + b + 7 + 5 = a + b + 12; i.e. a + b = 25 – 12 = 13. Consider these 6 options and their correspondence to the 3rd condition:

4.1) 9 + 4 = 13 or 4 + 9 = 13; and check for compliance with the 3rd condition: 9 4 7 5 = 1260. 1260 is not even divisible by 25. So neither the number 9475 nor 4975 is good.

4.2) 8 + 5 = 13 and 5 + 8 = 13; and check 5 8 7 5 = 1400. The number 1400 is evenly divisible by 25. A couple of great numbers found. These are 8575 and 5875.

4.3) Let’s check one more pair 7 + 6 = 13 and 6 + 7 = 13; 7 6 7 5 = 1260. 1260 is not even divisible by 25.

So, two four-digit delightful numbers: 5875 and 8575.

Sum of integers 1 to 100 per minute.

Ask students to calculate the sum of all integers from 1 to 100 in a minute. This way, you will be able to motivate students to solve this problem. Explain that the task is not to type the keys on the calculator, but to determine the pattern. The first time a 10-year-old boy coped with this task was back in 1787. After a minute, tell me that the bright boy’s name was Carl Friedrich Gauss.

How did he do it? He singled out 49 pairs of numbers: 1+99=100; 2+98=100; 3+97=100; …; 48+52=100; 49+51=100; In total, each pair of numbers was one hundred, and there were two unpaired numbers 50 and 100. Therefore, 49×100+50+100=5050.

The second way is to find a simplifying action algorithm. If you add the first and last in pairs, 1+100=101; 2+99=101; 3+98=101. Then, get 50 pairs of repetitions. That is, 101 must be multiplied by 50. The answer will also be 5050.

For those who didn’t know, Carl Friedrich Gauss is considered one of the greatest mathematicians of all time, and his portrait was featured on 10 German marks, Germany’s old currency before the introduction of the euro. For those who said they understood, let them add up from 1 to 200.

Graphic display of numbers

When children are tired of numbers, talk to them about what is used to display them quickly and visually. Most likely, tables, graphs and charts will be named by many. It will not be superfluous to recall that there are bar charts, dot charts, pie charts, secondary pie charts, tier charts, area charts, etc. And also for visual display, histograms, graphs and maps are used. Ask what a map will make from a picture of the outlines of an island? Remember what scale is.

Ask who came up with the first chart? With the ability to google the answers on your smartphone. Perhaps you will be told about the Sumerians, Egyptian or Greek scientists. There is no correct answer, but discussion of versions enriches the horizon. Here we should mention the contribution of the French scientists Rene Descartes and Francois Vieta, who laid the foundations for the concept of functions, they also developed a single alphabetic mathematical symbolism, which is still used today. Recall that the Cartesian coordinate system is directly related to René Descartes, and a graph cannot be plotted without a coordinate system. Explain why it is so important to set the origin of coordinates in the system? What is a wind rose and how is it related to movement? Remember the sundial and the clock face. It is noteworthy that from the German dial (German: Zifferblatt), literally – “a sheet of numbers”.

A good question for erudition:

What graphic image, invented by the ancient Greeks, is used to this day? Correct Answer: Zodiac signs. After all, in fact, the signs of the zodiac are relatively constant ratios between the linear distances of bright stars visible from the Earth in the celestial sphere. The celestial circle was divided into 360 angular fractions (this was done because this corresponds to the approximate number of days in a year). Now in geometry we call fractions degrees. The celestial sphere was divided into 12 equal sectors of 30 degrees. The vernal equinox point was chosen as the beginning. Direction along the course of the annual movement of the Sun. Points from bright stars that fell into the sectors were connected by segments. The resulting figures were given names reminiscent of real or mythological animals. The zodiac is translated from Greek as animal. But there is an exception to Libra – the only constellation of the zodiac that represents an inanimate object. So mathematics borders on astronomy.

Explain that the emergence of diagrams is connected with the economy. The first histogram was published by Scottish engineer and political economist William Playfair. His Commercial and Political Atlas was published in 1786. He also owns the first published image of a pie chart, or as it is sometimes called a pie chart in a circle. It happened in London in 1801. It reflects the performance of Turkey’s exports to Asia, Europe and Africa. His work served as an impetus for the development of graphic display methods in other sciences.

Physics plus mathematics

It is often difficult to explain what engineering thinking is without practice. To add variety to the lesson, break the class into groups and use the motion sensor ( Wireless Motion Sensor PS-3219 ) along with free software from PASCO SCIENTIFIC called Match Graph! software. The meaning of the tasks is to check how much the student can read the graph of the displayed function. To do this, you need to repeat the graph by changing your body position relative to the motion sensor.

Technically it looks like this:

  1. MatchGraph is installed on the teacher’s computer! software.
  2. wireless motion sensor connects via Bluetooth to MatchGraph! the image is displayed on the screen through the projector so that the class can see what is happening
  3. the motion sensor itself is installed on the edge of the table, so that there is about 3 meters of free space from the edge of the table in a straight line.
  4. The name of the participant is entered in the program and after pressing the start button, the student must repeat the graph as accurately as possible with his body movement along one line.
  5. Upon completion, points are given, which are calculated from how much the graph describing the movement of the student coincided with the given one.

At first, most students try to figure it out using the so-called “poke” method, that is, by trial and error. But if you explain how to read the graph, you can achieve a better result and thereby motivate students. How exactly to read the graph:

  1. Starting point. Where is she?
  2. How quickly the graph of the function changes on the first segment. Calculate speed knowing time and distance. When performing, count to yourself, and not just look at the chart. By the way, racing car drivers also mentally run the track before the race to get the best results.
  3. What happens at the bend points?

Since the use of technical means increases the likelihood of malfunctions. Don’t be afraid to bring in students to eliminate them. Discuss what noise on the graph means.

What is the ultrasonic sensor medium? The correct answer is air. Why can bats stray during strong winds? What other sensors for determining the position of the body in space do you know? (Infrared, video camera, satellite, lidar, radio tag, etc.). What are the limitations of using the sensors?

Conclusion

Using the connections of mathematics with history, physics, robotics, allowing you to cope with the solution in a playful way, helping classmates and the teacher in technological terms, you can motivate students to engage in mathematics with interest, explaining and proving in practice the statement of the famous German mathematician Karl Gauss: “Mathematics is the queen of sciences, and arithmetic is the queen of mathematics.”

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