Summary of knowledge points of junior high school Olympic mathematics
1. What is the problem with tap water? When a ship sails in the water, it will also be affected by the current. In this case, calculating the sailing speed, time and journey of the ship and studying the interaction between the current speed and the ship's own speed are called running water problems.
Second, what are the three basic quantities in the running water problem?
The running water problem is one of the travel problems, so the relationship between the three basic quantities of speed, time and distance in the travel problem is of course applicable here.
3. What is the relationship between the three basic quantities in the running water problem?
There are two basic formulas for the flowing water problem:
Downstream speed = ship speed+current speed, (1)
Current speed = ship speed-current speed. (2)
The ship speed here refers to the speed of the ship itself, that is, the distance traveled in still water per unit time. Water velocity refers to the distance that water flows in unit time. Downstream speed and countercurrent speed refer to the distance traveled per unit time when the ship sails downstream and countercurrent respectively.
According to the relationship between reciprocal operations of addition and subtraction, we can get from the formula (1):
Current speed = downstream speed-ship speed,
Ship speed = downstream speed-current speed.
From formula (2), we can get:
Water speed = ship speed-current speed,
Ship speed = current speed+current speed.
That is to say, as long as we know any two of the three quantities: the speed of the ship in still water, the actual speed of the ship and the current speed, we can find the third quantity.
In addition, given the current ship speed and current ship speed, according to formula (1) and formula (2), we can add and subtract them to obtain:
Ship speed = (downstream speed+upstream speed) ÷2,
Water velocity = (downstream velocity-upstream velocity) ÷2.
Summary of knowledge points of Olympic mathematics in the second day of junior high school
1. Features of simple encounter problems: (1) Generally, two moving objects start at the same time and do opposite movements at different times.
(2) Within a certain period of time, two moving objects meet.
(3) The key points to solve the encounter problem: the time required for meeting = total distance ÷ speed and.
To solve the problem of meeting, we must firmly grasp the key condition of "speed and harmony" The main quantitative relations are:
Second, * * * Similarities between simple meeting problems and chasing problems:
(1) Do you want to leave at the same time?
(2) whether to start at the same place
(3) Direction: same direction, opposite direction and opposite direction.
(4) Method: Drawing.
Third, the starting point of simple encounter problem solving and the places to pay attention to
The problems encountered are related to speed sum, distance sum.
(1) Do you want to leave at the same time?
(2) Are there any return conditions?
(3) Whether it is related to the midpoint: judge the location of the intersection.
(4) Whether it is multiple return: according to the multiple relationship.
(5) Under normal conditions, we should start with "harmony", but when the conditions are related to "difference", we should start with the difference and then analyze the time, so as to get the desired results.
Summary of knowledge points of junior high school Olympic mathematics
1. What is the clock travel problem? The stroke problem of the clock face is to study the relationship between the hour hand and the minute hand on the clock face. There are two common types:
(1) Study the angle between the hour hand and the minute hand, including overlapping, being in a straight line, being at right angles or being at a certain angle.
(2) Study the problem of time error.
On the clock face, every hand rotates clockwise, but because of the different speeds, the minute hand always catches up with the hour hand, or the minute hand exceeds the hour hand, so the common clock face problem is often transformed into the problem of catching up.
Second, what are the types of clock face problems?
The first category is the problem of traceability (there are often two situations when paying attention to the relationship between the hour hand and the minute hand); The second is the problem of meeting (the hour hand and the minute hand will never meet, but when the angle between the hand and the minute hand is equal to a certain scale, the distance sum can be found); The third is the problem of inaccuracy, the most critical point of this kind of problem: finding the proportional relationship between the watch and the real time.
3. What is the key problem of the clock face?
① Determine the initial positions of the minute hand and the hour hand;
② Determine the distance difference between the minute hand and the hour hand;
4. What are the basic methods to solve the clock face problem?
(1) framework method:
The circumference of the clock face is divided into 60 cells on average, and each cell is called 1 cell. The minute hand walks for 60 minutes every hour, which is a week; The hour hand only moves for 5 minutes, so the minute hand moves 1 minute, and the hour hand moves112 minutes.
② Degree method:
Viewed from the angle, the circumference of the clock face is 360, the minute hand turns 360/60 degrees per minute, which is 6 degrees, and the hour hand turns 360/ 12*60 degrees per minute, which is 1/2 degrees.
Five, examples of clock strokes.
Example 1: How long did it take for the hour hand and the minute hand to become a straight line for the first time from 5 o'clock sharp?
At 5 o'clock sharp, the minute hand points directly above and the hour hand points to the lower right. At this time, the interval between them is 25 cells (on the surface, there are 5 cells between each number). If you want to be in a straight line, the minute hand will walk 30 cells more than the hour hand, so during this time period, the minute hand will walk 55 cells more than the hour hand. From the minute hand moving1112, we can know that this time is 55/(11/2) = 60 minutes, that is, after 60 minutes, the hour hand and the minute hand come first.
Example 2: How many minutes have passed since 6 o'clock sharp when the hour hand and the minute hand first coincided?
At 6 o'clock sharp, the minute hand points directly above, the hour hand points directly below, and the interval is 30 squares. If they overlap for the first time, that is, the interval between them becomes 0, then the minute hand will go 30 squares longer than the hour hand, and this time is 30/(1112) = 360/11minute.