The sixth grade mathematics knowledge points Book 1 1
Unit 1 Fractional Multiplication
Importance of (1) fractional multiplication:
1, the fractional multiplication of integers has the same meaning as integer multiplication, and it is a simple operation to find the sum of several identical addends.
"Fraction multiplied by integer" means that the second factor must be an integer, not a fraction.
2. Multiplying a number by a fraction means finding the fraction of a number.
"A number multiplied by a fraction" means that the second factor must be a fraction, not an integer. The first factor is anything. )
(2) Calculation rules of fractional multiplication:
1, the arithmetic of decimal times integer is: numerator times integer, denominator remains unchanged.
(1) In order to make the calculation simple, it can be reduced first and then calculated. (Integer and denominator divisor)
(2) The divisor is to subtract the greatest common factor from the following integer and denominator. (Integer cannot be multiplied by denominator, and the calculation result must be the simplest fraction).
2. The arithmetic of fractional multiplication is: use the product of molecular multiplication as numerator, and use the product of denominator multiplication as denominator. (numerator times numerator, denominator times denominator)
(1) If the fractional multiplication formula contains a band fraction, the band fraction should be converted into a false fraction before calculation.
(2) The method of fractional simplification is to divide the numerator and denominator by their greatest common factor at the same time.
(3) In the process of multiplication, the divisor is to cross out two divisible numbers in the numerator and denominator, and then write the divisor above and below respectively. (After reduction, the numerator and denominator can no longer contain common factors, so the calculated result is the simplest score).
(4) The basic nature of the fraction: the numerator and denominator are multiplied or divided by the same number (except 0) at the same time, and the size of the fraction remains unchanged.
(3) the relationship between products and elements:
A number (except 0) is multiplied by a number greater than 1, and the product is greater than this number. A× b = c, when b > 1, c > a.
A number (except 0) is multiplied by a number less than 1, and the product is less than this number. A× b = c, and when b < 1, c < a (b ≠ 0).
A number (except 0) is multiplied by a number equal to 1, and the product is equal to this number. A× b = c, and when b = 1, c = a.
When comparing the sizes of factors and products, we should pay attention to the special situation when the factor is 0.
(d) mixed operation of fractional multiplication
1, the mixed operation order of fractional multiplication is the same as that of integers. Multiply first, then divide, and then add and subtract. If there are parentheses, count them first and then count them outside.
2. The law of integer multiplication is also applicable to fractional multiplication; Algorithms can make some calculations simple.
Multiplicative commutative law: a× b = b× a
Law of multiplicative association: (a× b )× c = a× (b× c)
Multiplicative distribution law: a× (b c) = a× b a× c
(V) Meaning of reciprocal: Two numbers whose product is 1 are reciprocal.
1 and reciprocal are two numbers, which are interdependent and cannot exist alone. A number cannot be called reciprocal. (It must be clear who is the reciprocal of who)
2. The only criterion to judge whether two numbers are reciprocal is whether the product of the multiplication of two numbers is "1". For example: A× B = 1, then a and b are reciprocal.
3. Reciprocal method:
① Find the reciprocal of the fraction: exchange the positions of numerator and denominator.
② Find the reciprocal of an integer: 1 of an integer.
③ Find the reciprocal of the score: first turn it into a false score, and then find the reciprocal.
(4) Find the reciprocal of the decimal: first find the number of components, and then find the reciprocal.
4. The reciprocal of1is itself because 1× 1= 1.
0 has no reciprocal, because the product of any number multiplied by 0 is 0, and 0 cannot be used as the denominator.
5. The reciprocal of the true score is a false score, and the reciprocal of the true score is greater than 1 and itself.
The reciprocal of the error score is less than or equal to 1. The reciprocal of the score is less than 1.
(6) Fractional multiplication is applied to solve problems.
1, what is the score of a number? (by multiplication)
Given the unit "1", find the fraction of the unit "1" and multiply the fraction by the unit "1".
2. Find the quantity of the unit "1" skillfully: in a sentence with a score, the quantity before the score is the corresponding quantity of the unit "1", or the quantity after the words "Zhan", "Yes" and "Bi" is the unit "1".
3. What is speed?
Speed is the distance traveled per unit time.
Speed = distance/time
Time = distance/speed
Distance = speed × time
Unit time refers to 1 hour, 1 minute, 1 second and other time units with the size of 1, such as minutes, hours and seconds.
4. How much is A more (less) than B?
Duo: (A-B) B.
Minus: (b-a) B.
The sixth grade, the first volume, mathematics knowledge point two
Position and direction of the second unit (2)
1, what is a number pair?
Number pair: consists of two numbers, separated by commas and enclosed in parentheses. The numbers in brackets are the number of columns and rows from left to right, that is, "columns first, then rows".
The function of number pair: to determine the position of a point. Longitude and latitude are the principles.
2, the method of determining the position of the object:
(1) Find the observation point first; (2) Redirection (depending on the included angle of the direction); (3) finally determine the distance (see scale).
The key to drawing a road map is to select observation points, establish direction signs and determine the direction and distance.
Relativity of positional relationship: the positions of the two places are relative. When describing the positional relationship between the two places, the observation points are different, the narrative direction is just the opposite, and the degree and distance are just the same.
Relative position: east-west; North-south direction; East by southwest by north.
The first volume of the sixth grade mathematics knowledge points 3
Unit 3 Division of Fractions
First, the significance of fractional division: fractional division is the inverse operation of fractional multiplication. Knowing the product of two numbers and one of the factors, we can find the other factor.
Second, the calculation rules of fractional division: dividing by a number (except 0) is equal to multiplying the reciprocal of this number.
1, dividend/divisor = dividend × the reciprocal of divisor.
2. When division is converted into multiplication, the dividend must not be changed, and it becomes x, and the divisor becomes its reciprocal.
3. When there are decimals and fractions in the fractional division formula, the number of components and false fractions should be changed before calculation.
4. The change law of dividend and quotient:
① Divided by a number greater than 1, the quotient is less than the dividend: A ÷ B = C, and when B > 1, c < a. ..
② Divided by a number less than 1, the quotient is greater than the dividend: a÷b=c, and when B < 1, c > a ... (a ≠ 0, b≠0).
③ Divided by a number equal to 1, the quotient is equal to the dividend: A ÷ B = C, and when B = 1, c = a. ..
Third, the mixed operation of fractional division
1, the mixed operation is calculated by trapezoidal equation, and the equal sign is written in the lower left corner of the first number.
2. Operation sequence:
(1) Division: calculation at the same level, from left to right; Or convert all divisions into multiplication before calculation; Or follow the simple method of "dividing by several numbers is equal to multiplying the product of these numbers". Addition and subtraction are primary operations, and multiplication and division are secondary operations.
② Mixed operations: multiplication, division, addition and subtraction without brackets; Parentheses are counted in parentheses first, and then outside parentheses.
(a b)c = a \c b \c
The first volume of the sixth grade mathematics knowledge points 4
Fourth unit ratio
Ratio: The division of two numbers is also called the ratio of two numbers.
1. In the comparison formula, the number before the comparison symbol (:) is called the former item, the item after the comparison symbol is called the latter item, the comparison symbol is equivalent to the division symbol, and the quotient of the former item divided by the latter item is called the ratio.
Even the ratio, such as: 3:4:5 read: 3 to 4 to 5.
2. The ratio represents the relationship between two numbers, which can be expressed by a fraction, written in the form of a fraction, and read as several to several.
For example:12: 20 =12 ÷ 20 = 0.6?
12: 20 is pronounced as12: 20.
Distinguish ratio and ratio: ratio is a number, usually expressed as a fraction, and can also be an integer or a decimal.
A ratio is a formula that represents the relationship between two numbers. It can be written as a ratio or a fraction.
3. The basic nature of the ratio: the first term and the second term of the ratio are multiplied or divided by the same number at the same time (except 0), and the ratio remains unchanged.
4. Simplified proportion: the simplified result is still a proportion, not a number.
(1) Divide the first and last terms of the ratio by their greatest common divisor.
(2) The simplified method of the ratio of two fractions is to multiply the last item in the previous paragraph by the least common multiple of the denominator at the same time, and then simplify the integer ratio. You can also find the ratio and write it out in the form of ratio.
(3) the ratio of two decimal places, moving the decimal point to the right, is also converted into an integer ratio first.
5. Find the ratio: write the ratio symbol as a division symbol and then calculate it. The result is a number (or fraction), which is equivalent to quotient, not ratio.
6. The difference between ratio, division and fraction:
Division: Divider divisor symbol () Divider (not 0) Quotient invariant division is an operation.
Fraction: numerator fraction line (-) Denominator (cannot be 0) The basic property of a fraction is a number.
Ratio: the basic attribute ratio (not 0) of the item after the preceding comparison symbol (:) indicates the relationship between two numbers.
Quotient invariance: Divider and divisor are multiplied or divided by the same number (except 0) at the same time, and the quotient remains unchanged.
The basic nature of a fraction: the numerator and denominator are multiplied or divided by the same number at the same time (except 0), and the size of the fraction remains unchanged.
Application of Fractional Division and Fractional Ratio
1, the known unit "1".
2. The quantity of unknown unit "1" is divided.
3. The basic quantitative relationship of fractional application problems (by fraction)
(1)A is a fraction of B?
A = B × what score?
What is the score of B =?
What score = A-B
(2) How much is A more (less) than B?
4. Proportional distribution: the method of distributing a quantity according to a certain proportion is called proportional distribution.
5. Draw a line graph:
(1) Find out the number of the unit "1", first draw the unit "1", and mark the known and unknown.
(2) Analyze the quantitative relationship.
(3) Find the equivalence relation.
(4) Column equation.
Draw two line graphs for the relationship between two quantities, and draw a line graph for the relationship between part and whole.
The first volume of the sixth grade mathematics knowledge points 5
Unit 5 circle
First, the characteristics of the circle
1. Circle is a plane figure surrounded by a closed curve in a plane.
2. Features of the circle: beautiful appearance and easy rolling.
3. center o: the point of the center is called the center. The center is generally represented by the letter o.
After the circle is folded in half for many times, the intersection of creases is at the center of the circle, that is, the center of the circle. The center of the circle determines the position of the circle.
Radius r: The line segment connecting the center of the circle and any point on the circle is called radius. In the same circle, there are countless radii, all of which are equal. The radius determines the size of the circle.
Diameter d: The line segment whose two ends pass through the center of the circle is called the diameter. The same circle has countless diameters, and all the diameters are equal. The diameter is the longest line segment in a circle.
The inner diameter of the same circle or equal circle is twice the radius: d=2r or r=d÷2.
4. Equal circles: circles with equal radii are called concentric circles, and equal circles can be completely overlapped by translation. Concentric circles: Two circles with coincident centers and unequal radii are called concentric circles.
5. The circle is an axisymmetric figure: if a figure is folded in half along a straight line, the figures on both sides can completely overlap, and this figure is an axisymmetric figure. The straight line where the crease lies is called the symmetry axis.
Figures with symmetry axis: semicircle, sector, isosceles trapezoid, isosceles triangle and angle.
A figure with two axes of symmetry: a rectangle.
A figure with three axes of symmetry: an equilateral triangle
A figure with four axes of symmetry: a square
Figures with or without symmetry axis: circles and rings.
Step 6 draw a circle
(1) The distance between two feet of a compass is the radius of a circle. (2) Draw a circle: fix the radius, center of the circle and make a circle.
Second, the circumference of the circle:
The length of a curve around a circle is called the circumference of the circle, and the circumference is represented by the letter C.
1, the circumference of a circle is always more than three times the diameter.
2. Pi: The ratio of the circumference to the diameter of a circle is a fixed value, which is called Pi and is expressed by the letter π.
Namely: pi = circumference ÷ diameter ≈3. 14.
Therefore, the circumference of a circle (c)= diameter (d)×π- circumference formula: c =πd, c = 2π r.
Pi π is an infinite acyclic decimal, and 3. 14 is an approximation.
3. Circumference change law: how many times does the radius expand, how many times does the diameter expand, and the circumference expansion multiple is the same as the radius and diameter expansion multiple.
4, semicircle circumference = half circumference+diameter = π r+d.
Third, the area of the circle.
1, derivation of the formula of circular area
As shown in the figure, divide a circle into several parts along the diameter and cut it into a rectangle. The more copies, the closer the image is to a rectangle.
Radius of circle = width of rectangle
Half of the circumference = the length of the rectangle.
Rectangular area = length × width
Therefore, the area of a circle = half of the circumference (πr)× the radius of the circle (r).
S circle =πr×r=πr2
2. For several figures, the circumference of a circle is the shortest and that of a rectangle is the longest when the areas are equal; On the contrary, in the case of equal perimeters, the area of a circle is the largest, while the area of a rectangle is the smallest.
At the same time, the circular area is the largest. Taking advantage of this feature, baskets and plates are made into circles.
3. Variation law of circular area: how many times the radius expands, how many times the diameter and circumference also expand, and the multiple of circular area expansion is the square of the radius and diameter expansion multiple.
4. Annular area = big circle-small circle =πR2-πr2.
Sector area =πr2×n÷360(n stands for the degree of the central angle of the sector)
Runways: The perimeter of each runway is equal to the perimeter of the circle formed by two semi-circular runways plus the sum of two straight runways. Because the lengths of two straight runways are the same, the starting lines of two adjacent runways are different, and the distance between them is 2×π× runway width.
When the radius of a circle increases by one centimeter, the circumference increases by 2π one centimeter.
The diameter of the circle increases by b cm, and the circumference increases by πb cm.
6. The diameter of the inscribed circle of any square, that is, the largest circle, is the side length of the square, and their area ratio is 4: π.
7. Public data
π=3. 14 2π=6.28 3π=9.42 4π= 12.56? 5π= 15.7
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