MindMap Gallery fifth grade mathematics
It is suitable for fifth-grade primary school textbooks published by the People's Education Press. The content is decimal multiplication, position, decimal division, possibility, simple equations, polygon area and tree planting problems. It is used for review and preview.
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This is a mind map about the reproductive development of animals, and its main contents include: insects, frogs, birds, sexual reproduction, and asexual reproduction. The summary is comprehensive and meticulous, suitable as review materials.
This is a mind map about bacteria, and its main contents include: overview, morphology, types, structure, reproduction, distribution, application, and expansion. The summary is comprehensive and meticulous, suitable as review materials.
This is a mind map about plant asexual reproduction, and its main contents include: concept, spore reproduction, vegetative reproduction, tissue culture, and buds. The summary is comprehensive and meticulous, suitable as review materials.
This is a mind map about the reproductive development of animals, and its main contents include: insects, frogs, birds, sexual reproduction, and asexual reproduction. The summary is comprehensive and meticulous, suitable as review materials.
Fifth grade
Chapter 1 Decimal Multiplication
1. Multiplying decimals by integers: meaning - a simple operation to find the sum of several identical addends.
For example: 1.5×3 means what is 3 times of 1.5 or what is 3 times of 1.5.
Calculation method: first expand the decimal into an integer; calculate the product according to the rules of integer multiplication; then look at how many decimals there are in the factor, and count the number of decimal points from the right side of the product.
2. Multiply decimals by decimals: meaning - to find out what fraction of this number is.
For example: 1.5×0.8 (the integer part is 0) is to find what eight tenths of 1.5 is.
1.5×1.8 (the integer part is not 0) is to find what is 1.8 times of 1.5.
Calculation method: first expand the decimal into an integer; calculate the product according to the rules of integer multiplication; then look at how many decimals there are in the factor, and count the number of decimal points from the right side of the product.
Note: In the calculation results, the 0 at the end of the decimal part should be removed to simplify the decimal; when there are not enough digits in the decimal part, 0 should be used as a placeholder.
3. Rule: When a number (except 0) is multiplied by a number greater than 1, the product is greater than the original number; when a number (except 0) is multiplied by a number less than 1, the product is smaller than the original number.
4. There are generally three methods for finding approximate numbers:
⑴Rounding method;
⑵ Further method
⑶Tail removal method
5. Calculate the amount of money and keep two decimal places to indicate that the calculation is to cents. Use one decimal place to calculate to the angle.
6. The order of four arithmetic operations for decimals is the same as for integers.
7. Operational laws and properties:
addition:
Commutative law of addition: a+b=b+a
Additive associative law: (a+b)+c=a+(b+c)
multiplication:
Commutative law of multiplication: a×b=b×a
Multiplicative associative law: (a×b)×c=a×(b×c)
Distributive law of multiplication: (a+b)×c=a×c+b×c or a×c+b×c=(a+b)×c
(When b=1, b is omitted)
Variation: (a-b)×c=a×c-b×c or a×c-b×c=(a-b)×c
Subtraction:
Subtraction property: a-b-c=a-(b+c)
division:
Division property: a÷b÷c=a÷(b×c)
Chapter 2 Location
To determine the position of objects, several pairs are used (first row: vertical, second row: horizontal). Using number pairs must be able to solve two problems: First, given a pair of numbers, it must be able to mark the point where the object is located in the coordinates. The second is to give a point in the coordinates, which must be represented by a number pair.
Chapter 3 Decimal Division
The meaning of decimal division: knowing the product of two factors and one of the factors, find the operation of the other factor. For example: 0.6÷0.3 means that the product of two factors is known to be 0.6, one factor is 0.3, and what is the other factor?
The calculation method of dividing a decimal by an integer is as follows: divide the decimal by the integer and divide it according to the method of integer division. The decimal point of the quotient must be aligned with the decimal point of the dividend. If the integer part is not enough to divide, the quotient is 0 and put the decimal point. If there is a remainder, add 0 and divide.
The calculation method of division when the divisor is a decimal: first expand the divisor and the dividend by the same multiple so that the divisor becomes an integer, and then calculate according to the rule of "division by decimals when the divisor is an integer".
Note: If there are not enough digits in the dividend, add 0 at the end of the dividend.
In practical applications, the quotient obtained by decimal division can also be used to retain a certain number of decimal places using the "rounding" method as needed to find the approximate number of the quotient.
Changes in division:
①Quotient invariance property: the dividend and divisor expand or contract at the same time by the same multiple (except 0), and the quotient remains unchanged.
②The divisor remains unchanged, the dividend expands (shrinks), and the quotient expands (shrinks) accordingly.
③If the dividend remains unchanged and the divisor shrinks, the quotient will expand; if the dividend remains unchanged and the divisor expands, the quotient will shrink.
Repeating decimals: The decimal part of a number. Starting from a certain digit, one number or several numbers appear repeatedly in sequence. Such decimals are called recurring decimals.
Repeating section: The decimal part of a recurring decimal, a number that repeats in sequence. For example, the cycle section of 6.3232... is 32. It is abbreviated as 6.32.
A decimal with a finite number of digits in the decimal part is called a finite decimal. A decimal that has infinite digits in the decimal part is called an infinite decimal. Decimals are divided into finite decimals and infinite decimals.
Chapter 4 Possibilities
There are three situations when an event occurs: it may happen, it may not happen, and it must happen.
Possible events, their likelihood. By adding the numbers of several possible situations as the denominator and a single possibility as the numerator, we can find out how likely the corresponding event is to occur.
Chapter 7 Tree Planting Issues
Problems with tree planting without closure:
(1) Planting trees on both sides of a road = road length ÷ interval 1; given the number of intervals and the number of trees, find the length of the road. Road length = number of intervals × (tree of trees - 1)
(2) Planting trees on both sides of a road = (road length ÷ interval 1) × 2
(3) No trees planted at both ends of a road = road length ÷ interval -1
(4) No trees should be planted on both sides of a road = (road length ÷ interval - 1) × 2
(5) The problem of wood sawing time: time to saw a section of wood = total time ÷ (number of sections - 1)
The problem of planting trees around a closed figure: planting trees = perimeter ÷ interval
Chicken and rabbit in the same cage problem: (turtle and crane problem, big boat and small boat problem)
(1) Arithmetic hypothesis method 1: Assume that several chickens are rabbits (they are all rabbits with many legs), first find the number of chickens.
Number of chickens: (Total number of heads × 4 - Total number of feet) ÷ (4-2 is the number of feet of a rabbit minus the number of feet of a chicken)
Number of rabbits: total number of rabbits - number of chickens
Arithmetic hypothesis method 2: Assume that all the chickens are chickens (all chickens with short legs), first find the number of rabbits.
Number of rabbits: (Total number of feet - Total number of heads × 2) ÷ (4-2 is the number of feet of a rabbit minus the number of feet of a chicken)
Number of chickens: total number of chickens - number of rabbits
(2) Equation method: Suppose there are x rabbits, then there are 2x rabbit feet. Then there are (total number of chickens - x) chickens.
According to the equation "rabbit feet chicken feet = total number of feet", first find the number of rabbits, and then calculate the number of chickens.
That is: 4x 2×(total number of heads-x) = total number of feet
When observing an object from different angles, the shapes seen may be different; when observing a cuboid or cube, up to three faces can be seen from a fixed position. (Usually we look at it from the left, front, and top, and these three views are collectively called three views)
Movement of figures: axially symmetrical figures.
(1) After folding in half along a straight line, a figure with both sides completely overlapping is called an axially symmetrical figure, and this straight line is called the axis of symmetry. A circle has countless axes of symmetry. A square has 4 axes of symmetry. An equilateral triangle has 3 axes of symmetry. A rectangle has 2 axes of symmetry. Isosceles triangles and isosceles trapezoids have 1 axis of symmetry.
(2) Characteristics of axially symmetrical figures: Fold in half along the axis of symmetry, with both sides completely overlapping. 『The distance between each set of corresponding points and the axis of symmetry is equal. The lines connecting corresponding points are perpendicular to the axis of symmetry.
(3) You must be able to draw the other half of the symmetrical figure based on the axis of symmetry.
Numeric encoding:
(1) Numbers can not only be used to express quantity and order, but can also be used to encode.
(2) The postal code consists of 6 digits, the first 2 digits indicate the province; the first 3 digits indicate the postal area, the first 4 digits indicate the county or city, and the last 2 digits indicate the delivery office (Dadi Jixiang Delivery Office).
(3) 18-digit ID card: The 7th to 14th digits indicate the date of birth; the second-to-last digit indicates gender, odd numbers - male, even numbers - female.
(4) Fill in the coding rules based on the card number information, athlete number information, and house number information.
Chapter 6 Area of a Polygon
formula:
Derivation of the formula for the area of a parallelogram: shearing, translation
A parallelogram can be transformed into a rectangle;
The length of the rectangle is equal to the base of the parallelogram;
The width of the rectangle is equal to the height of the parallelogram;
The area of the rectangle is equal to the area of the parallelogram, because the area of the rectangle = length × width, so the area of the parallelogram = base × height.
Derivation of triangle area formula: rotation
Two identical triangles can be combined to form a parallelogram, and the base of the parallelogram is equivalent to the base of the triangle;
The height of the parallelogram is equal to the height of the triangle;
The area of a parallelogram is equal to twice the area of a triangle. Because the area of a parallelogram = base × height, the area of a triangle = base × height ÷ 2.
Derivation of trapezoid area formula: rotation
Two identical trapezoids can be combined to form a parallelogram. The base of the parallelogram is equal to the sum of the upper and lower bases of the trapezoid; the height of the parallelogram is equal to the height of the trapezoid; the area of the parallelogram is equal to twice the area of the trapezoid, because the area of the parallelogram = base × height, so the area of the trapezoid = (upper base and lower base )×height÷2.
Parallelograms with equal bases and equal heights have equal areas; triangles with equal bases and equal heights have equal areas.
The area of a parallelogram with equal bases and equal heights is twice the area of a triangle.
The rectangular frame is stretched into a parallelogram, the perimeter remains unchanged, and the area becomes smaller.
Calculating the area of combined graphics: it must be converted into simple graphics that have been learned.
When the combined figure is convex, use dotted lines to divide it into several simple figures, and add the areas of the simple figures to calculate.
When the combined figure is concave, use dotted lines to fill it up to form the largest simple figure, and calculate it by subtracting the area of several smaller simple figures from the area of the largest simple figure.
Chapter 5 Simple Equations
In expressions containing letters, the multiplication sign between the letters can be written as "·", or it can be omitted. The plus sign, minus sign, division sign, and multiplication sign between numbers cannot be omitted.
a×a can be written as a·a or a, a is read as the square of a, and 2a means a a.
In particular, 1a=a, we do not write the "1" here.
Equation: An equation containing unknown numbers is called an equation (★Conditions that an equation must meet: it must be an equation and it must have unknown numbers, both are indispensable). The value of the unknown that makes the left and right sides of the equation equal is called the solution of the equation. The process of finding the solution to an equation is called solving the equation.
Principles of solving equations: balancing the scale. If the same numbers (except 0) are added, subtracted, multiplied, and divided on both sides of the equation at the same time, the equation will still hold.
10 quantitative relations:
addition:
sum = addend + addend
One addend = sum - another addend
Subtraction:
Difference = Minuend – Minuend
Minuend = difference + subtrahend
Minuend = Minuend – Difference
multiplication:
Product = factor × factor
One factor = product ÷ another factor
division:
Quotient = dividend ÷ divisor
Dividend = Quotient × Divisor
Divisor = dividend ÷ quotient
All equations are equations, but not all equations are equal.
The solution of an equation is a number; solving an equation is a computational process.