Key points of final knowledge of fourth grade mathematics (Volume II) published by People's Education Press.
Unit 1 Four Operations
1, the meaning of addition and the relationship between parts
(1) The operation of combining two numbers into one number is called addition.
(2) Adding two numbers together is called addend. The added number is called sum.
(3) the relationship between the parts of addition:
Sum = Appendix+Appendix
Appendix = Sum-Another Addendum
2. The significance of subtraction and the relationship between the parts.
(1) The operation of finding another addend by knowing the sum of two numbers and one of them is called subtraction.
(2) Subtract the relationship between the parts:
Difference = minuend-minuend
Subtraction = minuend-difference
Negative = negative+difference
3. Subtraction is the inverse of addition.
4, the meaning of multiplication and the relationship between the parts.
(1) The simple operation of finding the sum of several identical addends is called multiplication.
(2) The multiplication of two numbers is called a factor. Multiplied numbers are called products.
(3) the relationship between the parts of multiplication:
Product = factor? Factor/coefficient
Factor = product? Another factor
5, the significance of division and the relationship between the parts.
(1) The operation of finding the product of two factors and one of them is called division.
(6) the relationship between the parts:
Quotient = dividend? divisor
Dividend = dividend? business
Bonus = business? divisor
Division with remainder: dividend = quotient? Divider+remainder
6. Division is the inverse of multiplication.
7. Addition, subtraction, multiplication and division are called four operations.
8. The order of elementary arithmetic
(1) In the formula without brackets, if there are only addition and subtraction operations or only multiplication and division operations, they should be calculated in order (from left to right);
(2) In the formula without brackets, if there are both multiplication and division and addition and subtraction, it should be calculated first (multiplication and division) and then (addition and subtraction); (Multiply first, then divide, then add and subtract)
(3) For the formula with brackets, count the inside of brackets first and then the outside of brackets.
Calculation of 9.0
(1) a number and 0, the result is the original number:
a + 0 =a 0 + a = a
(2) A number minus 0 results in this number:
a - 0 = a
(3) a number minus itself, the result is zero:
a - a = 0
(4) a number multiplied by 0, the result is 0:
Answer? 0 = 0 ; 0 ? a = 0
(5) 0 divided by a non-zero number, the result is 0:
0 ? a = 0；
⑥ 0 inseparable:
Answer? 0 = (meaningless)
10, chartering
The solution to the chartering problem: first assume, then adjust.
Suppose we rent a cheap boat and calculate the result. If the ship is not full, we will make adjustments.
The second unit observes the object (2)
1, observing objects from different positions.
Identify the shapes of objects viewed from above, in front and on the left.
Count the faces you see first, and then look at their arrangement. Pay attention when drawing figures, only draw the upper and lower numbers.
2. When observing the same object from different positions, you may see the same or different graphics.
3. When observing different objects from the same position, the figures you see may be the same or different.
Only by observing from different positions can we know an object more comprehensively.
The third unit operation method
1, addition algorithm
① additive commutative law: Two numbers are added, the position of the addend is exchanged, and the sum is unchanged.
a+b=b+a
(2) law of addition and association: when three numbers are added, the first two numbers can be added first, and then the third number can be added; Or add the last two numbers first, and then add the first number, and the sum remains the same.
(a+b) +c=a+(b+c)
These two laws of addition are often used in combination.
2. The nature of continuous reduction
One number minus two numbers in a row equals the sum of this number minus those two numbers.
a-b-c=a-(b+c)
3. Law of multiplication operation
① Multiplication and exchange law: when two numbers are multiplied, the position of exchange factor remains unchanged.
Answer? b=b？ a
② Multiplication and association law: When three numbers are multiplied, you can multiply the first two numbers and then the third number, or you can multiply the last two numbers and then the first number, and the product remains unchanged.
(a？ b)？ c=a？ (b？ c)
These two multiplication laws are often used in combination.
③ Multiplication and distribution law: When the sum of two numbers is multiplied by a number, the two numbers can be multiplied by these two numbers respectively, and then the products can be added.
(a+b)？ c=a？ c+b？ c
4. The nature of division
A number divided by two numbers in succession is equal to the product of these two numbers.
Answer? b？ c=a？ (b？ c)
Unit 4 Significance and Properties of Decimals
1, the meaning of decimal
When measuring and calculating, it is often impossible to get accurate integer results, so it is often expressed by (decimal).
Fractions with denominators of 10, 100 and 1000 can be expressed as decimals.
2. Composition of decimals
The number before the decimal point is called the integer part of the decimal point, and the number after the decimal point is called the decimal part of the decimal point.
3. Decimal Counting Unit
The first digit after the decimal point is a decimal, and the counting unit of decimal is one tenth, which can be written as 0.1;
The second digit after the decimal point is the percentile, and the counting unit of percentile is 1%, which can also be written as 0.01;
The third place after the decimal point is one thousandth, and the counting unit of one thousandth is one thousandth, which can also be written as 0.00 1.
4. The decimal series between every two adjacent counting units is 10.
5. How to read decimals
The integer part is read by integer method, the decimal point is read as "dot", and the decimal part should read each number in turn.
6. Decimal writing
The integer part is written as an integer, the decimal point is written in the lower right corner of each digit, and the decimal part should be written on each digit in turn.
7, the nature of the decimal
Add "0" or remove "0" at the end of the decimal, and the size of the decimal remains the same.
8. Comparison of decimal sizes
Compare the integer parts first. If the integer part is large, the decimal part is large. If the integer parts are the same, compare the decimal parts; If the decimals are the same, compare percentiles; If the percentile is the same, compare the thousandth.
9. Decimal size change law caused by decimal point movement
(1) Decimal point to the right: move one place by multiplying the original number by 10, and expand the decimal point to 10 times the original number; Moving two digits is equivalent to multiplying the original number by 100, and the decimal is expanded to 100 times of the original number; Moving three digits is equivalent to multiplying the original number by 1000, and the decimal number is expanded to 1000 times the original number.
(2) Move the decimal point to the left: moving one place is equivalent to dividing the original number by 10, and the decimal point is reduced to one tenth of the original; Moving two digits is equivalent to dividing the original number by 100, and the decimal number is reduced to 1% of the original number; Moving three digits is equivalent to dividing the original number by 1000, and reducing the decimal number to one thousandth of the original number.
10, rewritten between data of different quantity units.
The number of low-level units? Prepayment Rate = Advanced Unit
1 1, find the approximate number.
Keep an integer, that is, accurate to one place, and round it by looking at the number on the tenth place;
Keep one decimal place, that is, accurate to ten places, and round off the number in the percentile;
Keep two decimal places, that is, accurate to 1%, rounded to the nearest thousandth.
(When representing approximate value, 0 after decimal point cannot be deleted)
12, the number that is not an integer of 10,000 or an integer of 100 million is rewritten as a number in units of "10,000" or "100 million".
When rewriting, just put the decimal point to the right of 10000 or 10000000, and add the words "1000000" or "1000000" after the number.
Unit 5 Triangle
1, triangle
A figure surrounded by three line segments (the endpoints of every two adjacent line segments are connected) is called a triangle.
2. The base and height of a triangle
Draw a vertical line from the vertex of a triangle to its opposite side. The line segment between the vertex and the vertical foot is called the height of the triangle. This opposite side is called the base of a triangle.
? 3, the characteristics of triangle
Triangles are stable.
4, the relationship between the three sides of the triangle
The sum of any two sides of a triangle is greater than the third side, and the difference between any two sides is less than the third side.
5. Classification of triangles
(1) Triangle can be divided into acute triangle, right triangle and obtuse triangle.
(2) Triangle can be divided into isosceles triangle, equilateral triangle and equilateral triangle.
6. Sum of internal angles of triangle
Is the sum of the three internal angles of a triangle 180? .
7. Distance between two points
Among the connecting lines between two points, the line segment is the shortest. The length of this line segment is called the distance between two points.
8. Sum of interior angles of polygons
The sum of the inner angles of a polygon = (number of sides -2)? 180?
9, the characteristics of isosceles triangle
Two waists are equal, and two bottom angles are equal. Two equal sides are called waist, and two equal inner corners are called bottom corners.
10 and the characteristics of equilateral triangles
The three sides are equal in length and the three internal angles are equal in size (all 60? )。
Unit 6 Addition and Subtraction of Decimals
1, the method of decimal addition and subtraction with a pen
(1) decimal point alignment, that is, the same digit alignment;
(2) Starting from the last digit, when calculating the addition, which digit adds up to ten, and the previous digit must be advanced1; When calculating subtraction, which bit is not reduced enough, it will be 1 from the previous bit.
(3) There is a 0 at the end of the number, which should generally be removed.
(4) Don't forget the decimal point.
2, decimal addition and subtraction mixed operation order
(1) has no brackets and is calculated from left to right;
(2) If there are brackets, first count the brackets.
3. Simple operation of decimal addition and subtraction
The law of integer operation is also applicable to decimal operation. Therefore, in the four decimal operations, it will be easier to apply the operation properties of additive commutative law, associative law and continuous subtraction appropriately.
4. When the number is decimal, the (ending) 0 should generally be removed.
Unit 7 Graphic Movement (2)
1, Properties of Axisymmetric Graphs
The distance between the corresponding point and the symmetry axis is equal.
2. The axis of symmetry of an axisymmetric figure
The symmetry axis is a straight line, so when you draw the symmetry axis, you should draw it outside the figure and use a dotted line.
Step 3 draw a symmetry axis
First, find the corresponding point with the same distance in the opposite direction of the symmetry axis, and finally connect the lines.
4. Drawing method of graphic translation
When translating, find the graphic points first, and then connect the points after translation.
5. Using translation, the area of irregular figure can be found.
Unit 8 Average and Bar Chart
1, the meaning of average.
Divide the sum of a set of data by the number of this set of data, and its quotient is called the average value of this set of data. The average value can not only describe the overall situation of a group of data itself, but also be used as a standard for comparing different groups of data.
2, the average method
(1) method of shifting more and supplementing less
(2) Formula method: total? Number of copies = average
3. Composite bar chart
After merging two simple bar charts, you get a composite bar chart.
(1) The composite bar chart should have a legend.
(2) There are two kinds of composite bar charts: horizontal and vertical.
(3) The double bar graph represents the number of one with two unit lengths, and is drawn into straight bar graphs with different lengths according to the number.
4. Drawing of horizontal composite bar graph
? (1) Prepare drawing tools such as rulers, pencils and erasers.
(2) Pay attention to the writing unit, the coordinates and abscissa in the painting, the date and name, and the "0" on the abscissa.
(3) If the positions are limited, such as 0 to 10 and 20, if 200 is written, the positions are absolutely limited. You can draw a wavy line on 0 and then write 100 (of course, other numbers are ok, but the most standard is to draw a lightning line).
(4) For example, the picture above should have different colors. If there are no markers, the first one can draw diagonal lines, and the second one can draw very tightly.
(5) Write a title at the bottom of each picture.
5, double bar chart
(1) The quantity is expressed by the length of the straight bar.
(2) We can clearly see the quantity, which is convenient for comparing the two sets of data.
Unit 9 Mathematics Wide Angle-Chicken and Rabbit are in the same cage
1, chicken and rabbit in the same cage is a hypothetical problem, which is contrary to the final result.
2. The solution to the problem of "chickens and rabbits in the same cage"
(1) hypothesis method
(1) If they are all rabbits.
2 if they are all chickens.
(2) the ancient "foot lifting method"
If every chicken and rabbit raises half a foot, then every chicken will become a "chicken with one leg" and every rabbit will become a "rabbit with two legs". In this way, the total number of feet of chickens and rabbits is reduced by half. This way of thinking is called reduction.
3. Formula:
What is the total number of feet of chickens and rabbits? 2- Total number of chickens and rabbits = number of rabbits;
Total number of chickens and rabbits-number of rabbits = number of chickens.