1. Do this.

( 1)

Cut off the volume of a cuboid with square sides.

1cm 324cm3

2cm 512cm3

3 cm 588 cubic cm

4 cm 576 cubic cm

5 cm 500 cubic cm

6 cm 384 cubic cm

7 cm 252 cubic cm

8 cm 128 cubic cm

9 cm 36 cubic cm

10cm 0 cm3

(2)

I found that the cuboid has the smallest volume when the side length of the small square is 10 cm, and the cuboid has the largest volume when the side length of the small square is 3 cm.

(3)

When the side length of the small square is 3 cm, the volume of the uncovered cuboid is the largest, and the volume of the uncovered cuboid is 588 cubic centimeters.

Do it

( 1)

Cut off the volume of a cuboid with square sides.

0.5cm 180.5cm

1.0 cm 324 cm 3

1.5cm 433.5cm。

2.0cm 512cm3

2.5 cm 562.5 cubic cm

3.0 cm 588 cubic cm

3.5 cm 59 1.5 cm.

4.0 cm 576 cubic cm

4.5 cm 544.5 cubic cm

5.0 cm 500 cubic cm

5.5 cm 445.5 cubic cm

6.0 cm 384 cubic cm

…… ……

(2)

I found that the cuboid has the smallest volume when the side length of a small square is 0.5 cm, and the cuboid has the largest volume when the side length of a small square is 3.5 cm. Moreover, when the side length of the cut square is an integer, the volume of the cuboid is also an integer, and when the side length of the cut square is a decimal, the volume of the cuboid is also a decimal.

(3)

When the side length of the small square is 3.5 cm, the volume of the uncovered cuboid is the largest, and the volume of the uncovered cuboid is 59 1.5 cubic cm.

About "0"

0, can be said to be the earliest human contact number. Our ancestors only knew nothing and existence at first, and none of them was 0, so 0 isn't it? I remember the primary school teacher once said, "Any number minus itself is equal to 0, and 0 means there is no number." This statement is obviously incorrect. As we all know, 0 degrees Celsius on the thermometer indicates the freezing point of water (that is, the temperature of ice-water mixture at standard atmospheric pressure), where 0 is the distinguishing point between solid and liquid water. Moreover, in Chinese characters, 0 means more as zero, such as: 1) fragmentary; A small part. 2) The quantity is not enough for a certain unit ... At this point, we know that "no quantity is 0, but 0 not only means no quantity, but also means the difference between solid and liquid water, and so on."

"Any number divided by 0 is meaningless." This is a "conclusion" about 0 that teachers from primary school to middle school are still talking about. At that time, division (primary school) was to divide a copy into several parts and figure out how many there were in each part. A whole cannot be divided into 0 parts, which is "meaningless". Later, I learned that 0 in a/0 can represent a variable with zero as its limit (the absolute value of a variable is always smaller than an arbitrarily small positive number in the process of change) and should be equal to infinity (the absolute value of a variable is always larger than an arbitrarily large positive number in the process of change). From this, another theorem about 0 is obtained: "A variable whose limit is zero is called infinitesimal".

"Room 203 105 in 2003", although all of them are zeros, they are roughly similar in appearance; They have different meanings. 0 indicator vacancy of 105 and 2003 cannot be deleted. 0 in Room 203 separates "Building (2)" from "House Number". (3) "(that is, Room 8 on the second floor) can be deleted. 0 also means that ...

Einstein once said: "I always think it is absurd to explore the meaning and purpose of a person or all living things." I want to study all the numbers of "existence", so I'd better know the number of "non-existence" first, so as not to become what Einstein called "absurd". As a middle school student, my ability is limited after all, and my understanding of 0 is not thorough enough. In the future, I hope (including action) to find "my new continent" in the "ocean of knowledge".

Mathematics in life

There is a riddle: there is something invisible and intangible, but it is everywhere. What is this? The answer is: air. And mathematics, like air, is invisible and intangible, but it exists all the time around us.

Wonderful "golden numbers"

Take a line segment, find a point on the line segment, and let this point divide the line segment into two parts, one is long and the other is short, and the ratio of long segment to short segment is exactly equal to the ratio of whole segment to long segment. This point is the golden section of this line segment. This ratio is: 1: 0.6 18… and 0.6 18… This number is called "golden number".

Interestingly, this number can be seen everywhere in life: the navel is the golden section of the total length of the human body; The angle between two adjacent leaves on the stems of some plants is exactly the angle between the two radii that divide the circumference into 1: 0.6 18 …. According to research, this angle has the best effect on ventilation and lighting of the factory building.

Architects especially prefer the number 0.6 18 ... Whether it is the pyramids of ancient Egypt, Notre Dame de Paris or the modern Eiffel Tower, the number 0.6 18 ... It is also found that most of the subjects of some famous paintings, sculptures and photography are at the 0.618 ... Musicians think that putting the bridge at 0.618 of the strings will make the sound softer and sweeter.

The number 0.6 18… also makes the optimization method possible. Optimization method is a method to solve the optimization problem. If it is necessary to add a chemical element to increase the strength of steel during steelmaking, it is assumed that the amount of a chemical element to be added per ton of steel is between 1000-2000g. In order to find the most suitable dose, we usually take the midpoint of the interval for experiments, and then compare it with the experimental results of 1000 g and 2000 g respectively, select two points with higher intensity as the new interval, and then take the midpoint of the new interval for experiments until the most ideal effect is obtained. However, this method is inefficient. If the measuring point is taken at the interval of 0.6 18, the efficiency will be greatly improved. This method is called "0.6 18 method". Practice has proved that the former method can be realized by using "0.6 18 method" to test 16.

There are so many examples and applications of "golden number" in life. Perhaps, there are more mysteries waiting for us to explore, so that it can serve us better and solve more problems for us.

Wonderful axial symmetry

If a straight line can be found on a graph, the graph can be aligned along this straight line, so that the two sides can completely overlap. Such a figure is called an axisymmetric figure, and this straight line is called an axis of symmetry.

Careful observation shows that the plane is a standard axisymmetric object. Looking down, its wings, fuselage and tail are all symmetrical. Axial symmetry makes its flight more stable. If the plane is not axisymmetric, it will fly stagger. At that time, who wanted to fly?

Looking closely, it is not difficult to find that many works of art are also axisymmetric. Take the simplest example: the bridge. It is the most common work of art in life. Take the bridges in Jinhua as examples: Tongji Bridge, Jinhongqiao, Shuanglong Bridge and Riverside Bridge. They are all axisymmetric. The ancient buildings in China are even more obvious. Ancient palaces were basically symmetrical. Another famous: the layout of Beijing. This is the most typical axisymmetric layout. It is symmetrical with the Forbidden City, Tiananmen Square, Monument to the People's Heroes and Qianmen as the central axis. The application of axial symmetry in art can make art look more beautiful.

Axisymmetry is also a biological phenomenon: human ears, eyes and limbs all grow symmetrically. The axial symmetry of the ear makes the sound we hear have a strong three-dimensional sense and can also determine the position of the sound source; The symmetry of the eyes allows us to see objects more accurately. It can be seen that our life is inseparable from axial symmetry.

Mathematics is very close to us, and it is reflected in all aspects of life. We can't live without mathematics, which is everywhere. These are just two very common examples, and there are countless such examples. I think mathematics in life can bring more discoveries to people.

But now I guess it's useless. There are so many words to write at this point.

Subject: knowledge of discount sales

research report

Knowledge of discount sales

First, the research content

Every holiday, shops and shopping malls will offer various discounts, such as "jumping price", "shocking price", "lowest price" and "big sale" ... For consumers, such activities are really overwhelming. Listening to these various advertisements, we have an idea: won't merchants lose money by offering too low a discount? How do they win more profits from discounted sales? How much knowledge is there in discount sales?

With these problems closely related to our lives, we embarked on a journey of exploring discount sales.

Second, research methods.

Consult parents, consult information, market research, calculation and reasoning.

Third, the research process

(A) the concept of discounted sales. First of all, let's learn what discount sales are:

Discount is one of the common means of business promotion. It can reduce the price of products in a short time, attract more consumers to buy, and thus increase sales in a short time. Discount is the original price multiplied by a few tenths or a few percent, which is the price discount. As a part of business activities, discount must serve the overall goal of the enterprise and improve its total profit. However, in order to pursue excess profits, some businesses gradually become easier.

Now is 2 1 century, an era of consumption. Shopping has long been an indispensable activity in life, but it also makes businesses opportunistic. Don't the special pages of Suning and Gome in the newspaper make you feel excited? Aren't the coupons for Dicos and Baifu all drooling along the road? In fact, all discount activities are taking advantage of you! For example, in Kaiyuan, you can get 50 vouchers when the price is over 100. Before the discount, the merchants have raised the price a lot, and you still think it is cheap. Generally, the price of this kind of mobile goods makes you very helpless. Take sportswear as an example:

Brand object Li Ning jogging shoes Mizuno badminton shoes puma casual shoes

Price 289 399 492

These goods are close to the whole hundred, but in order to make up the 50 100 discount coupon, you have to buy a pair of socks or other accessories. This is how merchants make money bit by bit! Think about it, one person pays more than ten dollars, ten people pay more than one hundred dollars, and one hundred people pay more than one thousand dollars. After all, thousands of consumers! I dare not think about it any more.

There are many ways to discount. If you have enough energy, you can discount different products in different ways, which is more obvious.

You have to be flexible. There are usually the following discount methods:

Quantity discount Generally speaking, whether to give price discount and how much depends on the quantity purchased by customers. The greater the quantity, the greater the discount. One pen is one yuan, and ten pens are eighty cents each.

Seasonal discount depends on the time when customers buy it, and whether to give it and how much discount to give it.

Transaction discount Transaction discount occurs when consumers buy new products and sell their old products to merchants as part of the price offset of new products. When buying new bicycles, they sold the old ones to the merchants.

Differential price adjustment is based on different customer categories, product forms, sales locations, sales time, etc. The price of goods in special places such as amusement areas may be more expensive than that in general stores. The catering industry offers special discount meals to attract guests from 3 pm to 5 pm when business is slow.

Promotion discount is a temporary and short-term price reduction for the target goods in order to achieve a certain promotion purpose. 1From October 5th to 7th, the sweater went straight to 50 yuan.

It is particularly attractive to set the prices of a small number of products very low, but the prices of most products remain unchanged (some even rise). Its purpose is to attract consumers to visit the store and try it out actively with a small number of "special goods" as "bait" At the same time, it is hoped that consumers will buy some other products with normal prices while buying "special goods" to earn normal profits. The price of goods in special places such as entertainment areas may be more expensive than that in general stores in the catering industry at 3 pm.

Discount on special activities Use some special time and activities to promote goods in a big way, such as "clearance sale", "relocation reduction" and "debt repayment bleeding"

Psychological discount This is a method to reduce the price according to consumers' psychological perception and value of products, such as 1998 yuan, 499 yuan, etc. ), or prices with odd mantissa (such as 17.75 yuan, 45.25 yuan, etc. ) is more easily accepted by people's psychology. There are so-called lucky numbers to discount, such as changing 670 yuan into 666 yuan.

In order to further investigate, we conducted a survey in Xi 'an Anxi Street during the National Day holiday:

Commercial sports shoes women's table tennis racket watch

Pre-holiday price unit: RMB 31013052122.

Post-holiday price (dozen): RMB 356 172 56 137.

Post-holiday price (unpaid): RMB 396 196 70 208.

Discount 9 8.8 8 6.6

The survey data shows that these businesses have not brought benefits to customers through discounts, which is what we call false discounts. These behaviors of merchants seemed to be very effective at that time, and the sales of goods seemed to be very good. But after a long time, people gradually understand the "mystery" and are not so gullible. People's doubts are getting more and more serious, and they have reached the level of discount, but they still can't figure out: will it be a trap for merchants? Or is it really cheap? At this time, discount is no longer a means of small profits but quick turnover for merchants, nor will it arouse customers' shopping desire.

In fact, the discount is a trivial matter, but it reflects the business vision. Discount is a very effective marketing method, which can quickly increase product sales and widely promote commercial brands. If used properly, the effect will be very good. After investigation, we summed up some problems that businesses should consider before discounting.

1, scope: determine which goods are discounted. Here, make clear why these goods are discounted and see if they meet the purpose of discount.

2. Degree: determine the discount degree and profit rate, which can attract customers without losing profits.

3. Timing: Decide when the discount is most appropriate. For example, general operators regard holidays as the best time to discount.

4. Cycle: The time period during which the discount should last. The longer the discount, the better.

5. Frequency: the number of discounts in a year.

6. Mode: How to discount.

Below, we put forward some specific preferential schemes for your reference.

1, weekly discount

As we all know, Mondays to Thursdays are working hours, and fewer people shop. The general concept is weekend shopping, because there will be more time to choose carefully. The proposal of weekly discount now will inevitably change some people's consumption habits and concepts and attract more customers. Scheme: Monday 1 discount, Tuesday 2 discount, Wednesday 3 discount, Thursday 4 discount, Friday 50 discount and Saturday 60 discount.

2. Cash discount

Starting at six o'clock in the afternoon, there is a 60% discount from six to six, a 30% discount from seven to seven, and a 20% discount from eight to nine. In this way, the original customers' less time can be used in a short time, and the discount rate is at least 60%. This means will change the consumption concept of some people. We can foresee that the shopping center will be full immediately after 6: 00.

3. Countdown discount

Activity time 15 days: 20% off on the first and second days, 30% off on the third and fourth days, 60% off on the fourth and fifth days, 50% off on the sixth and seventh days, 40% off on the eighth and ninth days, 70% off on the tenth and eleventh days, 20% off on the twelfth and thirteenth days, and 14/fifteenth days 1 discount. Countdown discount is generally that the lower the discount rate, the better the product. Only classic brands will take the lead in time. I think if businesses use these methods reasonably, it will not only benefit themselves, but also change consumers' ideas and stimulate consumers' desire to buy.

(2) Calculation method. Having said that, we only examine the method of discount sales. We already know in advance that discounted sales will bring certain profits to businesses, but what is it? How to calculate the profit of discount sales? It is not enough to know how to discount, but also to thoroughly understand its principle!

Now, let's take a look at some specific profit calculation methods brought by discounted sales to merchants.

Let's clarify a few concepts first. 30% off the marked price means that the marked price is reduced by 70% when buying goods. Among them, the marked price refers to the original price of each commodity marked by the merchant, which is different from the selling price. The selling price refers to the price that the merchant buys for the customer. The marked price is different from the buying price, which refers to the wholesale price of the merchant from the wholesale department or the manufacturer.

Namely: purchase price = purchase price (cost price) of goods.

Sales price = actual sales price (sales price) of goods.

List price = original price

So, how do businesses make money from it? What is the money earned?

The shopping mall will increase the cost of a jacket with a price of 100 yuan by 50%, then price it as 150 yuan, and then sell it to a customer at a 20% discount. Please calculate, did the merchant make a profit in this transaction?

150× 80%- 100 = 20 (yuan), each jacket merchant earned 20 yuan.

Here, the cost price of the jacket is 100 yuan, the price is 150 yuan, and the price is 120 yuan, in which the earned 20 yuan is called profit, that is, the income in the process of selling goods. Parents also told us the concept of profit rate: the percentage of profit in the purchase price.

Formula: profit = selling price-buying price

Profit rate = profit/purchase price × 100%

I can't believe that the merchants are doing business at a loss on the surface, but they are actually doing such wishful thinking! After discounted sales, although a product earns less than the marked price, it can attract more customers and win more profits!

After mastering the basic formulas and concepts, let's start to discuss discount sales!

A jacket is marked with a price 50% higher than the cost price, and then sold at a 20% discount due to seasonal reasons. Every jacket is sold in 60 yuan. What is the cost price of each jacket?

How did the price in 60 yuan come from here?

(2) If the cost price of each of these jackets is not X yuan, how can the algebraic expression of X be used to express the marked price and actual selling price of each jacket?

60 yuan refers to 80% of the original price, that is, the original price × 80%.

If the cost price of each jacket is X Yuan, the listing price of each jacket is (1+50%) X Yuan, and the actual selling price of each jacket is (1+50%) X × 80% Yuan.

Solution: From the meaning of the question, x (1+50%) × 80% = 60,

Solve the equation: x = 50, then the cost price of each jacket is 50 yuan.

(2) If "every piece is bought in 60 yuan" is changed to "every piece is still profitable in 60 yuan", and the rest remains unchanged, what is the cost price of each of these coats?

If the cost price is X yuan, how to express the profit of each jacket with an algebraic expression containing X?

If the cost price of each jacket is X Yuan, the listing price of each jacket is (1+50%) X Yuan, and the actual selling price of each jacket is (1+50%) X × 80% Yuan.

"Profitable 60 yuan" means that the difference between selling price and buying price is 60 yuan, that is, selling price-cost price = 60.

Based on the above, we assume that the cost price of each jacket is X yuan.

X (1+50%) × 80%-x = 60, the solution is: x = 300,

So the cost price of these coats is 300 yuan.

If the above is changed to: A jacket is priced after a 20% increase on the cost price, and then sold at a 20% discount due to seasonal reasons, is it still possible for each jacket to make a profit in 60 yuan? Why?

If the cost price of each coat is X yuan, the equation is: X (1+20%) × 80%-X = 60.

Solution: x =- 1500.

The cost price is negative and unrealistic, so it is impossible for 60 yuan to make a profit, and in fact it will lose 40%).

Fourth, research results.

Through this research, I learned about the buying price, buying price, selling price,

The concept of profit and profit rate, and understand the enterprise.

Profit from discount promotion, their

The relationship is roughly as shown on the right:

At the same time, I also learned more equation solving skills.

That is, as shown in the following figure:

We can use the idea of equation to solve many discount sales problems in actual investigation.

After data collection and various practical explorations during the holidays, we have grown a lot and become calmer when we encounter things. At the same time, through many days of investigation and arrangement, we come to the conclusion that discount is actually a clever digital management. Discounting is not only as simple as reducing the price of goods, but also for the promotion and profit of enterprises. Discount contains a lot of profound truth and knowledge. In short, be kind and honest.

The Harvest and Reflection of verb (the abbreviation of verb)

Harvest: 1, learned some concepts of discount sales.

2. Know some equivalence relations related to discount sales.

3. In the process of solving, we should also consider the rationality of the answer.

I have mastered the knowledge of discount sales, so I won't suffer dumb losses in my future business.

Through the collection and arrangement of data, we all become more careful and considerate.

Mathematics comes from real life, and the purpose of learning mathematics is to solve some phenomena and problems in daily life with the mathematical knowledge we have learned! However, I also have some shortcomings in this survey. For example, when looking for information, it slightly disturbed the work of the clerk's aunt and uncle, and some data were extracted directly from the Internet. Of course, in my next inquiry, I will pay more attention to the combination with practice and strive to do better!

Remind everyone again: in fact, all discount promotions are just a trap for merchants to want you to buy more things, but don't fall into the trap of merchants and jump in ~

Six, parents' words

On the surface, the discount behavior of merchants is a very emotional behavior, which confuses customers to make perceptual choices. As a business, it is understandable. As consumers, most of them also understand the necessity of rational thinking, but they are often unable to start rationally. This kind of topic is close to the teaching content, and enables students to master how to think and solve problems rationally, and understand that incorrect thinking is more destructive. We deeply appreciate the teaching practice synchronized with the times!

Seven, teacher comments

Living in a bustling city, it is inevitable to encounter hawking scenes of "big sale" and "one fold up". So how should we treat discount promotion? The students of this research group, Yang, Zhao, He, used their spare time to study and summarize the reasons and means of discount promotion through interviews, data collection, statistical calculation and analysis. From this research report, we are glad to see that students can apply what they have learned in class to their lives. Isn't this one of the important purposes of our teaching? Their ability to investigate and analyze problems is gradually improving. They can use their own mathematical knowledge to compare the discounted price with the original price, reveal the essence of the problem, and guide everyone to consume rationally and understand consumption. Through this research and practice, students have not only exercised their comprehensive practical ability, but also formed a correct concept of financial management, which is conducive to the formation of a world outlook. It is also in line with the purpose of diversified evaluation of students to show their talents in the form of investigation.