Observing Objects
1. The scope of observing objects varies depending on the position.
2. The shape of an object observed varies depending on the position.
【Key Knowledge Points】
Festival Gifts (The scope of observing objects varies depending on the position)
1. As the height and distance of the observation position change, one can judge the corresponding changes in the observed object's image.
2. Based on the observed image, determine the observer's position.
Tiananmen Square (The shape of an object observed varies depending on the position)
1. By observing and comparing some photos, one can identify and determine the correspondence between the shooting location and the photos.
2. By observing a series of continuously taken photos, one can determine the chronological order of the photo shoots.
【Key Knowledge Points】
1. "Careful Calculation" - Division with a whole number divisor
(1) Meaning of decimal division: The meaning of decimal division is the same as that of integer division, which is to find the other factor when the product of two factors and one of the factors are known.
(2) Method of dividing decimals by integers: Decimal division with a whole number divisor is similar to integer division, except that the decimal point of the quotient should align with the decimal point of the dividend.
2. "Visiting the Museum" - Division where the quotient is a decimal
Division method for integers divided by integers resulting in a decimal: Follow the rules of integer division first; if there is still a remainder at the end of the dividend, append 0s and continue dividing.
3. "Who Talks Longer on the Phone?" - Division with a decimal divisor
(1) Unchanging rule of quotients: When both the dividend and divisor are expanded or reduced by the same multiple (excluding 0), the quotient remains unchanged.
(2) Division method for decimals with a decimal divisor: Expand both the dividend and divisor by the same multiple to make the divisor a whole number, then perform the division following the rules of dividing decimals by whole numbers.
4. "Currency Exchange" - Approximate values of products and quotients
Method for finding approximate values: For products, calculate precisely first, then take the approximate value according to the problem's requirements. For quotients, directly divide one extra digit based on the requirements, then take the approximate value accordingly. Note: In some cases, rounding down below four or not rounding up above five may occur, so approximate values should be determined based on the characteristics of the problem.
5. "Who Climbs Faster?" - Recurring decimals
(1) Recurrent phenomena: Many events in life recur repeatedly. For example: sunrise and sunset, time...
(2) Recurring decimals: From a certain digit after the decimal point, one or more digits repeat indefinitely. Such decimals are called recurring decimals.
(3) Use the rounding method to find approximate values of recurring decimals, using the same method as for decimals. Retain the required number of decimal places by looking at the next digit.
6. "TV Advertisement" - Mixed operations with decimals
(1) The order of mixed operations with decimals, such as consecutive division and multiplication/division combinations, is the same as for integers.
(2) The order of mixed operations with decimals is completely the same as for integers.
Passionate Olympics
(1) Through various information provided by "Olympics," comprehensively apply the knowledge and methods learned to solve relevant problems.
(2) By solving Olympic-related problems, appreciate the connection between mathematics and sports, further understand the value of mathematics.
Understanding Equations Fair Possibilities
1. Through game activities, experience events with equal possibilities.
2. Analyze game activities to judge the fairness of game rules.
3. Can formulate fair game rules.
4. Can feel randomness in actual life through experiments.
Unequal Possibilities
1. Can experience events with unequal possibilities through game activities.
2. Can distinguish whether game possibilities are equal.
3. Can modify game rules through their own analysis and thinking to make them fair, with multiple methods.
Who Goes First? (Judging the fairness of rules, designing fair rules)
【Key Knowledge Points】
1. Experience events with equal possibilities. Understand that games are fair when possibilities are the same and unfair when possibilities differ.
2. Feel the role of rules in games, establish a sense of rules, and formulate fair game rules.
3. Further experience the characteristics of randomness in games.
"Numbers and Algebra"
I. Understanding decimals and addition/subtraction:
1. Meaning of decimals
2. Measurement activities (conversion of named numbers)
3. Comparing sizes (comparing sizes of decimals)
4. Shopping receipts (decimal addition/subtraction – no carryover addition, no borrowing subtraction)
5. Weighing (decimal addition/subtraction – carryover addition, borrowing subtraction)
6. Singing competition (mixed operations and simplified calculations for decimal addition/subtraction)
II. Decimal Multiplication:
1. Stationery store (multiplying decimals by integers)
2. Moving the decimal point (rules for changing the size of decimals due to moving the decimal point)
3. Central park (relationship between the number of decimal places in the multipliers and the product)
4. Packaging (vertical calculation for decimal multiplication)
5. Slowest mammal (vertical calculation for decimal multiplication and decimal estimation)
6. Hand-in-hand (mixed operations and simplified calculations for decimal multiplication)
III. Decimal Division:
1. Careful calculation (division with whole number divisors)
2. Visiting the museum (integer divided by integer with a decimal quotient)
3. Who talks longer on the phone? (division with decimal divisors)
4. Currency exchange (approximate values of products and quotients)
5. Who climbs faster? (repeating decimals)
6. TV advertisement (mixed operations and simplified calculations for decimal division)
IV. Recognizing equations:
1. Letters representing numbers (using letters to represent numbers, expressions, relationships, laws, and geometric formulas)
2. Equations (meaning of equations)
3. Balance game (part 1) (properties of equality 1, solving equations of the form X±a=b)
4. Balance game (part 2) (properties of equality 2, solving equations of the form X×a=b, X÷a=b)
5. Guessing game (solving equations of the form aX±b=c)
6. Number of stamps (solving application problems using equations, solving equations of the form aX±X=b)
"Space and Graphics"
I. Recognizing shapes:
1. Shape classification
2. Triangle classification
3. Sum of interior angles of triangles
4. Relationship between triangle sides
5. Quadrilateral classification
6. Pattern appreciation
II. Observing objects:
1. Festival gifts (changes in the observed object's image due to height changes from near to far)
2. Tiananmen Square (identifying and judging the correspondence between shooting locations and photos, determining the sequence of continuous photo shoots)
"Probability and Statistics"
Fair Games: (fairness of game rules, designing fair game rules)
"Comprehensive Applications"
1. Knowledge of counting figures
2. Passionate Olympics
3. Patterns in graphics
"Numbers and Algebra" Knowledge
I. Understanding decimals and addition/subtraction
【Key Knowledge Points】
Meaning of decimals
1. Meaning of decimals: Dividing the unit "1" into 10, 100, 1000... parts and taking one or several parts represents tenths, hundredths, thousandths... These numbers are called decimals.
2. Fractions with denominators of 10, 100, 1000... can be expressed as decimals. Decimals representing tenths have one decimal place, those representing hundredths have two decimal places, and those representing thousandths have three decimal places...
3. Composition of decimals: Decimals consist of an integer part and a fractional part separated by a decimal point.
4. Decimal places, units of calculation, and ratios:
① Units of decimal calculation are tenths, hundredths, thousandths... written as 0.1, 0.01, 0.001... Like integers, each adjacent decimal unit has a ratio of 10.
② The largest unit of decimal calculation is tenths, and there is no smallest unit in the fractional part.
③ The number of decimal places is infinite.
④ In a decimal, the number of decimal places after the decimal point determines how many decimal places it has. The zeros at the end of the fractional part also count.
5. Reading and writing decimals: Read decimals from left to right, reading the integer part as an integer (reading "zero" for zero), read the decimal point as "point", and sequentially read each digit in the fractional part, even if they are consecutive zeros. Writing decimals is also from left to right, writing the integer part as an integer (writing "0" for zero), placing the decimal point in the lower right corner of the ones place, and sequentially writing each digit in the fractional part.
6. Understanding the differences and connections between 0.1 and 0.10: Differences: 0.1 represents one 0.1, 0.10 represents ten 0.01, different meanings. Connection: 0.1 = 0.10, these two numbers are equal in size. Using the basic properties of decimals, you can rewrite or simplify decimals without changing their size.
7. Pure decimals have an integer part of 0; decimals with a non-zero integer part are called mixed decimals.
Measurement Activities (Conversion of Named Numbers)
1. 1 decimeter = 0.1 meter, 1 centimeter = 0.01 meter, 1 gram = 0.001 kilogram... Learn to convert between lower and higher units (length, area, weight...). Convert lower unit single names to higher units by rewriting the lower unit number as fractions with denominators of 10, 100, 1000..., then write these fractions as decimals and add the name of the higher unit being converted to.
2. Converting compound names to single names: Copy the same, change the different (copy the same units in the integer part, and convert the different units as described above in the fractional part).
3. Other conversion methods: Single-name conversions ① Lower unit name ÷ rate = higher unit name. ② Higher unit name × rate = lower unit name. Conversion between compound names and single names: Copy the same, change the different (same as single-name conversions). Example: 3 meters 2 centimeters = ( ) meters. Copy the same unit meters in the integer part, making it 3; convert differently: 2 centimeters ÷ 100 = 0.02 meters (the rate between centimeters and meters is 100).
Comparing Sizes (Comparing Decimal Sizes)
1. Method for comparing two decimal sizes: First look at the integer part; the decimal with the larger integer part is greater; if the integer parts are the same, then compare the tenths place in the fractional part; the decimal with the larger digit in the tenths place is greater...
2. Arranging several decimals in order: Compare their sizes first. Then arrange them according to the problem's requirements. When comparing quantities with different units, unify the units of these quantities first, then compare the decimal sizes using the method described, and finally answer according to the original numbers given in the problem.
Decimal Addition/Subtraction
1. Meaning of decimal addition/subtraction: The meaning of decimal addition/subtraction is the same as that of integer addition/subtraction. ① Meaning of decimal addition: Combining two numbers into one. ② Meaning of decimal subtraction: Knowing the sum of two addends and one of the addends, finding the other addend.
2. Basic property of decimals: Adding or removing "0" at the end of a decimal does not change its size.
3. Decimal addition/subtraction calculation rules: Align the decimal points; follow the rules of integer addition/subtraction. Start from the last digit; if the sum of any digit reaches ten, carry over to the previous digit. If there aren't enough digits in the minuend’s fractional part, add "0" and continue subtracting; if there aren't enough digits in a particular place to subtract, borrow from the previous place, adding ten to the current place before subtracting; the decimal point in the result should align with the decimal point in the horizontal line.
4. The order of mixed operations with decimals is the same as for integers. Same-level operations proceed from left to right; for operations involving parentheses, compute inside first, then outside.
5. The operational laws of integer addition/subtraction also apply to decimal addition/subtraction.
II. Decimal Multiplication
【Knowledge Framework】
1. Stationery Store (Multiplying decimals by integers)
2. Moving the Decimal Point (Rules for changing the size of decimals due to moving the decimal point)
3. Central Park (Relationship between the number of decimal places in the multipliers and the product)
4. Packaging (Vertical calculation for decimal multiplication)
5. Slowest Mammal (Vertical calculation for decimal multiplication and decimal estimation)
6. Hand-in-Hand (Mixed operations and simplified calculations for decimal multiplication)
【Key Knowledge Points】
Meaning of Decimal Multiplication
1. The meaning of multiplying decimals by integers is the same as that of multiplying integers. It can be seen as finding the sum of several identical addends or finding how much the integer multiple of this decimal is. For example, 2.3 × 5 indicates finding the sum of five 2.3s or how much five times 2.3 is.
2. Changes in multiplication: ① In multiplication, if one factor expands to m times its original value (m ≠ 0) and another factor expands to n times its original value (n ≠ 0), the product expands to m × n times its original value. ② In multiplication, if one factor shrinks to (m ≠ 0) times its original value and another factor shrinks to (n ≠ 0) times its original value, the product shrinks to times its original value. ③ In multiplication, if one factor expands to n times its original value (or shrinks to ) (n ≠ 0), and another factor shrinks to (or expands to n times), the product remains unchanged.
3. If one factor is less than "1", the product is less than the other factor. If one factor is greater than "1", the product is greater than the other factor. If one factor equals "1", the product equals the other factor.
Rules for Changing the Size of Decimals Due to Moving the Decimal Point
1. Rules for changing the size of decimals due to moving the decimal point: Moving the decimal point one, two, three... places to the left makes the number shrink to , , ... of its original value. Moving the decimal point one, two, three... places to the right makes the number expand to 10 times, 100 times, 1000 times... its original value.
2. If there aren’t enough digits when moving the decimal point to the right, add "0"s to fill in. After moving the decimal point, remove leading "0"s before the highest digit of the integer part; if there aren’t enough digits when moving the decimal point to the left, use "0"s to fill in, place the decimal point, use "0" to indicate if there is no integer part, and remove trailing "0"s in the fractional part based on the nature of decimals.
3. Relationship between the number of decimal places in the product and the number of decimal places in the multipliers: In decimal multiplication, the total number of decimal places in the two multipliers determines the number of decimal places in the product.
Rules for Decimal Multiplication
1. To calculate decimal multiplication, first calculate the product as if they were integers, then count the total number of decimal places in the factors, and starting from the rightmost digit of the product, move the decimal point leftward by that many places. Simplify the result if possible.
2. Estimating decimal multiplication: First round both factors to the nearest integer, then multiply them.
3. The order of mixed operations with decimals is the same as for integers: For same-level operations, proceed from left to right; for two-level operations, do multiplication/division first, then addition/subtraction; for operations involving parentheses, compute inside first, then outside.
4. The operational laws of integers also apply to decimal operations, such as the associative law, commutative law, and distributive law of multiplication.
III. Decimal Division
【Key Knowledge Points】
Decimal Division and Calculation Rules
1. Meaning of decimal division: The meaning of decimal division is the same as that of integer division, which is to find the other factor when the product of two factors and one of the factors are known.
2. Rule for decimal division with a whole number divisor: Perform the division as if they were integers, but align the decimal point of the quotient with the decimal point of the dividend; if there is still a remainder after dividing the entire dividend, append "0"s and continue dividing. If the integer part of the dividend is smaller than the divisor, use "0" to occupy the integer part of the quotient. If a certain place cannot be divided, place "0" in the quotient at that place.
3. Unchanging rule of quotients: When both the dividend and divisor are multiplied or divided by the same number (excluding 0), the quotient remains unchanged.
4. Rule for decimal division with a decimal divisor: According to the unchanging rule of quotients, convert decimal division to integer division. First, move the decimal point of the divisor to make it an integer; move the decimal point of the dividend the same number of places to the right (add "0"s if necessary), then perform the division as if they were integers.
5. Method for comparing the size of the quotient and the dividend: The key to comparing the size of the quotient and the dividend in a division expression lies in the divisor. If the divisor is greater than 1, the quotient is smaller than the dividend; if the divisor (not 0) is smaller than 1, the quotient is greater than the dividend; if the divisor equals 1, the quotient equals the dividend.
6. The order of mixed operations with decimals is the same as for integers. The order of calculating mixed operations with decimals is completely the same as for integers.
Currency Exchange
1. Method for exchanging currency between RMB and foreign currencies: RMB ÷ exchange rate = foreign currency; foreign currency × exchange rate = RMB.
2. In currency exchange, since the smallest unit of currency is "fen", when using "yuan" as the unit, the first decimal place represents "jiao" and the second decimal place represents "fen", while the third decimal place has no meaning. Therefore, in problems involving RMB, even without special requirements, generally use the "rounding" method to retain two decimal places, obtaining approximate values for the product and quotient.
3. Method for finding approximate values of products: Generally, calculate the exact product first, then use the "rounding" method to obtain the approximate value based on the problem's requirements or common practices, i.e., look at the next digit after the desired retention place to decide whether to round up or down.
4. Method for finding approximate values of quotients: First determine the retention place, during calculation, just divide one extra place based on the desired retention place, then round to obtain the approximate value.
5. Other methods for finding approximate values: ① Truncation method. ② Ceiling method. ③ Remainder of decimal division: The decimal point of the remainder in decimal division should align with the decimal point of the dividend.
Repeating Decimals
1. Repeating decimal: A repeating decimal is a decimal where one digit or a group of digits repeats indefinitely starting from a certain position after the decimal point.
2. Related concepts of repeating decimals: ① A decimal with a finite number of decimal places is called a terminating decimal; a decimal with an infinite number of decimal places is called a non-terminating decimal. Repeating decimals are non-terminating decimals. ② The group of digits that repeats indefinitely in the fractional part of a repeating decimal is called the repeating cycle of that repeating decimal. ③ A repeating decimal whose repeating cycle starts from the first digit after the decimal point is called a pure repeating decimal; a repeating decimal whose repeating cycle does not start from the first digit after the decimal point is called a mixed repeating decimal.
3. Use the rounding method to find approximate values of repeating decimals. The method is the same as for decimals, retaining the specified number of decimal places by looking at the next digit.
IV. Recognizing Equations
【Key Knowledge Points】
Using Letters to Represent Numbers
1. Letters or expressions containing letters can represent quantities and quantity relationships.
2. Using letters to represent relevant geometric calculation formulas:
① Perimeter formula for rectangles: C=2(a+b).
② Area formula for rectangles: S=ab.
③ Perimeter formula for squares: C=4a.
④ Area formula for squares: S=a².
3. Using letters to represent operation laws: If a, b, c represent three numbers, then
① Commutative law of addition: a+b=b+a
② Associative law of addition: (a+b)+c=a+(b+c)
③ Commutative law of multiplication: a×b=b×a
④ Associative law of multiplication: (a×b)×c=a×(b×c)
⑤ Distributive law of multiplication: (a±b)×c=a×c±b×c
⑥ Subtraction operation property: a-b-c=a-(b+c)
⑦ Division operation property: a÷b÷c=a÷(b×c)
4. In expressions containing letters, the multiplication sign between letters and letters or between letters and numbers can be represented by "•" or omitted. Numbers are usually written before letters. When 1 is multiplied by a letter, 1 is omitted, and letters are arranged in order. For example, a×b=ab, 5×a=5a, 1×a=a, a×a=a².
5. Distinguish between a² and 2a.
Meaning and Properties of Equations
1. Meaning of equation: An equation is an equality containing unknowns.
2. Relationship and difference between equations and equalities: An equation is an equality, but not all equalities are equations.
3. Property of equality 1: Adding (or subtracting) the same number to both sides of an equality keeps the equality valid.
4. Property of equality 2: Multiplying both sides of an equality by a number (or dividing by a nonzero number) keeps the equality valid.
5. Writing format for solving equations: Before solving the equation, write "Solution:" first; perform one step per line, aligning each equal sign vertically; place the letter representing the unknown generally on the left side of the equal sign.
6. The value of the unknown that makes both sides of an equation equal is called the solution of the equation. The process of finding the solution of an equation is called solving the equation.
7. Use the relationships between the components of subtraction and division to find equations where the unknown is the subtrahend or the divisor.
8. The key to listing equations based on diagrams is understanding the diagram, identifying the equal relationships, and then listing the equation based on the equal relationships. Place the unknown generally on the left side of the equal sign when listing the equation.
9.