Observing Objects
1. The range of observing objects varies at different positions.
2. The shape of objects observed from different positions is different.
[Key Knowledge Points]
Festival Gifts (the range of observing objects differs at different positions)
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 location.
Tiananmen Square (the shape of objects observed from different positions is different)
1. By observing and comparing some photos, identify and determine the correspondence between shooting locations and photos.
2. By observing a series of continuously shot photos, determine the order in which the photos were taken.
[Key Knowledge Points]
1. "Careful Calculation" —— Division with decimal dividends and integer divisors
(1) The meaning of decimal division: The meaning of decimal division is the same as that of integer division. It involves finding another factor when the product of two factors and one of the factors are known.
(2) Method for dividing decimals by integers: Decimal division with integer divisors is similar to integer division. Just align the decimal point of the quotient with the dividend’s decimal point.
2. "Visiting the Museum" —— Division where both dividend and divisor are integers but the quotient is a decimal
Method for division where the dividend and divisor are integers but the quotient is a decimal: Proceed according to the rules of integer division. If there is still a remainder after reaching the last digit of the dividend, append 0s and continue the division.
3. "Who Talks Longer?" —— Division with decimal divisors
(1) Rule of constant quotients: When the dividend and divisor are simultaneously expanded or reduced by the same multiple (excluding 0), the quotient remains unchanged.
(2) Method for dividing decimals by decimals: Expand both the dividend and divisor by the same multiple to convert the divisor into an integer, then proceed with the division following the rules for dividing decimals by integers.
4. "Currency Exchange" —— Approximate values of products and quotients
Method for finding approximate values: For products, calculate precisely first and then take the approximate value based on the question's requirements. For quotients, divide one extra place based on the requirement and then take the approximate value as per the question's requirements. Note: Sometimes, rounding down numbers less than four or not rounding up numbers equal to five may occur, depending on the characteristics of the problem.
5. "Who Climbs Faster?" —— Repeating Decimals
(1) Repetitive Phenomena: In life, many things repeat cyclically, such as sunrise and sunset, time, etc.
(2) Repeating Decimals: From a certain digit in the decimal part onwards, one digit or several digits repeat indefinitely. Such numbers are called repeating decimals.
(3) Using the rounding method to find approximate values of repeating decimals. The method is the same as for regular decimals. To retain a certain number of decimal places, look at the next digit of the decimal.
6. "TV Advertising" —— Mixed operations of the four basic arithmetic operations with decimals
(1) The order of mixed operations involving consecutive divisions and multiplication-division combinations is the same as for integers.
(2) The order of mixed operations involving decimals and integers is completely identical.
Passion Olympics
(1) Through various information provided by the "Olympics," comprehensively apply the knowledge and methods learned to solve related problems.
(2) By solving problems related to the Olympic arena, appreciate the connection between mathematics and sports, further understanding the value of mathematics.
Understanding Equations and Equal Probabilities
1. Through game activities, experience events with equal probabilities.
2. Analyze game activities to determine the fairness of game rules.
3. Be able to formulate fair game rules.
4. Experience randomness in actual life through experiments.
Unequal Probabilities
1. Be able to experience events with unequal probabilities through game activities.
2. Be able to distinguish whether game probabilities are equal.
3. Be able to modify game rules through analysis and thinking to make them fair, using multiple methods.
Who Goes First? (Determine the fairness of rules, design fair rules)
[Key Knowledge Points]
1. Appreciate the equal probability of events occurring. Understand that games are fair if the probabilities are the same and unfair if the probabilities differ.
2. Feel the role of rules in games, establish rule awareness, and be able to formulate fair game rules.
3. Further experience the random nature present in games.
"Numbers and Algebra"
I. Understanding Decimals and Addition/Subtraction:
1. Meaning of decimals
2. Measurement activities (conversion of named quantities)
3. Comparing sizes (comparing the size of decimals)
4. Shopping receipts (decimal addition/subtraction - no carryover addition, no borrowing subtraction)
5. Weighing body weight (decimal addition/subtraction - carryover addition, borrowing subtraction)
6. Singing competition (mixed operations and simplifications of decimal addition/subtraction)
II. Decimal Multiplication:
1. Stationery store (multiplying decimals by integers)
2. Decimal point relocation (rules for changing the size of decimals due to the movement of the decimal point)
3. Central park (relationship between the number of decimal places in two multipliers and their product)
4. Packaging (vertical calculation of decimal multiplication)
5. Slowest-moving mammal (vertical calculation of decimal multiplication and decimal estimation)
6. Hand-in-hand (mixed operations and simplifications of decimal multiplication)
III. Decimal Division:
1. Careful calculation (decimal division with integer divisors)
2. Visiting museums (integer divided by integer, quotient is a decimal)
3. Who talks longer? (decimal division with decimal divisors)
4. Currency exchange (approximate values of products and quotients)
5. Who climbs faster? (repeating decimals)
6. TV advertising (mixed operations and simplifications of 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. Scale game (one) (properties of equations, solving X±a=b type equations)
4. Scale game (two) (properties of equations, solving X×a=b, X÷a=b type equations)
5. Guessing game (solving aX±b=c type equations)
6. Number of stamps (solving word problems with equations, solving aX±X=b type equations)
"Space and Geometry"
I. Recognizing Figures:
1. Classification of figures
2. Classification of triangles
3. Sum of triangle angles
4. Relationship between sides of triangles
5. Classification of quadrilaterals
6. Pattern appreciation
II. Observing Objects:
1. Festival gifts (feeling the changes in the observed object's image due to changes in height and distance, experiencing the changes in observation range when observing objects from far to near)
2. Tiananmen Square (identifying and determining the correspondence between shooting locations and photos, judging the sequence of a series of continuously shot photos)
"Probability and Statistics"
Fair Games: (fairness of game rules, designing fair game rules)
"Comprehensive Application"
1. Learning about shapes in numbers
2. Passion Olympics
3. Patterns in shapes
"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 more parts represents tenths, hundredths, thousandths... These are called decimals.
2. Fractions with denominators of 10, 100, 1000... can be represented as decimals, where tenths are one-digit decimals, hundredths are two-digit decimals, thousandths are three-digit decimals...
3. Composition of decimals: Decimals consist of an integer part and a decimal part, separated by a decimal point.
4. Decimal places, units, and progression:
① The counting units of decimals are tenths, hundredths, thousandths..., written as 0.1, 0.01, 0.001... Like integers, the progression between adjacent decimal counting units is 10.
② The largest counting unit in the decimal part is tenths; there is no smallest counting unit in the decimal 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. Trailing zeros in the decimal part are also counted.
5. Reading and writing decimals: Read decimals from left to right. The integer part is read as an integer (read "zero" if it is 0), the decimal point is read as "point", and each digit in the decimal part is read sequentially, even if there are consecutive zeros. When writing decimals, write from left to right. The integer part is written as an integer (write "0" if it is zero), the decimal point is placed below the ones place, and each digit in the decimal part is written sequentially.
6. Understanding the difference and connection between 0.1 and 0.10: Difference: 0.1 represents one 0.1, 0.10 represents ten 0.01, different meanings. Connection: 0.1 = 0.10, these two numbers have the same size. Use the fundamental property of decimals to rewrite or simplify decimals without changing their size.
7. A decimal with an integer part of 0 is called a pure decimal; a decimal with a non-zero integer part is called a mixed decimal.
Measurement Activities (Conversion of Named Quantities)
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 units, area units, weight units...). Convert lower unit single names to higher units by rewriting the lower unit number as a fraction with a denominator of 10, 100, 1000..., then write the fraction as a decimal form and add the name of the higher unit being converted.
2. Converting compound names to single names: Copy the same, change the different. (Copy the same unit to the integer part, and rewrite the different unit to the decimal part according to the above conversion method.)
3. Other conversion methods: Single name mutual conversion ① Lower unit quantity ÷ progression = higher unit quantity. ② Higher unit quantity × progression = lower unit quantity. Mutual conversion between compound names and single names: Copy the same, change the different (same as single name mutual conversion method). For example, 3 meters 2 centimeters = ( ) meters. Copy the same unit meters to the integer part, which is 3; Rewrite differently: 2 centimeters ÷ 100 = 0.02 meters (progression between centimeters and meters is 100).
Comparing Sizes (Comparing the Size of Decimals)
1. Method for comparing two decimal sizes: Look at the integer part first; the larger integer part makes the decimal larger; if the integer parts are the same, then compare the tenth place in the decimal part, the larger digit in the tenth place makes the decimal larger...
2. Arranging several decimals in order: First compare their sizes. Then arrange them in order according to the requirements of the question. When comparing the sizes of several quantities with different units, unify the units of these quantities first, then compare them according to the method for comparing decimal sizes, and finally answer according to the original numbers given in the question.
Decimal Addition and Subtraction
1. Meaning of decimal addition and subtraction: The meaning of decimal addition and subtraction is the same as that of integer addition and subtraction. ① The meaning of decimal addition: Combining two numbers into one number. ② The meaning of decimal subtraction: Given the sum of two addends and one of the addends, find the other addend.
2. Basic property of decimals: Adding or removing "0" at the end of a decimal does not change its size.
3. Rules for decimal addition and subtraction: Align the decimal points; perform calculations according to the rules of integer addition and subtraction. Start from the last digit; if the sum of the digits in any place exceeds ten, carry over to the previous place. If the number of decimal places in the minuend is insufficient, append "0" before subtracting, borrow from the previous place if there are insufficient digits in a place, add ten to the current place, and then subtract; the decimal point of the result should align with the decimal point on the line.
4. The order of mixed operations for decimal addition and subtraction is the same as for integer addition and subtraction. Perform same-level operations from left to right; perform two-level operations first high then low; for operations with parentheses, do inside first then outside.
5. The operational laws of integer addition and subtraction also apply to decimal addition and subtraction.
II. Decimal Multiplication
[Knowledge Framework]
1. Stationery Store (Multiplying Decimals by Integers)
2. Decimal Point Relocation (Rules for Changing Decimal Size Due to Decimal Point Movement)
3. Central Park (Relationship Between the Number of Decimal Places in Two Multipliers and Their Product)
4. Packaging (Vertical Calculation of Decimal Multiplication)
5. Slowest-Moving Mammal (Vertical Calculation of Decimal Multiplication and Decimal Estimation)
6. Hand-in-Hand (Mixed Operations and Simplifications of 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 considered 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 the 5 times of 2.3 is.
2. Changes in multiplication: ① In multiplication, if one factor expands to m times (m ≠ 0) and the other factor expands to n times (n ≠ 0), the product expands to m × n times the original product. ② In multiplication, if one factor shrinks to (m ≠ 0) times and the other factor shrinks to (n ≠ 0) times, the product shrinks to times the original product. ③ In multiplication, if one factor expands to n times (or shrinks to ) (n ≠ 0) and the other factor shrinks to (n ≠ 0) (or expands to n times), the product remains unchanged.
3. When one factor is less than "1", the product is less than the other factor. When one factor is greater than "1", the product is greater than the other factor. When one factor equals "1", the product equals the other factor.
Rules for the Change of Decimal Place Value Due to Decimal Point Movement
1. Rules for the change of decimal place value due to decimal point movement: Moving the decimal point one, two, three... places to the left reduces the number to , , ... times the original number. Moving the decimal point one, two, three... places to the right increases the number to 10, 100, 1000... times the original number.
2. When moving the decimal point to the right, if there are not enough digits, append "0" to fill in. After moving the decimal point, remove the leading "0" before the highest digit of the integer part; when moving the decimal point to the left, if there are not enough digits, use "0" to fill in, place the decimal point, if there is no digit in the integer part, use "0" to indicate, if there are trailing "0"s in the decimal part, remove them according to the nature of decimals.
3. The 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. Calculate decimal multiplication, first follow the rules of integer multiplication to find the product, then see how many decimal places are in the factors, count that many places from the end of the product, and place the decimal point. Simplify the result if possible.
2. Decimal multiplication estimation: Round the two factors to integers using the "round half up" method, then multiply.
3. The order of operations for decimal four-operation mixed operations is the same as for integer four-operation mixed operations: Same-level operations go from left to right; two-level operations go first high then low; operations with parentheses go inside first then outside.
The operational laws of integers also apply to decimal operations, such as associative law, commutative law, distributive law, etc.
III. Decimal Division
[Key Knowledge Points]
Decimal Division and Calculation Rules
1. The meaning of decimal division: The meaning of decimal division is the same as that of integer division, both involve knowing the product of two factors and one of the factors, seeking the other factor.
2. Rules for decimal division with integer divisors: Calculate decimal division with integer divisors according to the rules of integer division, align the decimal point of the quotient with the decimal point of the dividend; if there is still a remainder after dividing to the end of the dividend, append "0" to continue dividing. If the integer part of the dividend is smaller than the divisor, use "0" to occupy the place in the integer part of the quotient. If there is not enough to divide in a place, place "0" in the quotient for that place.
3. Constant Quotient Rule: When the dividend and divisor are multiplied or divided by the same number (except 0), the quotient remains unchanged.
4. Rules for decimal division with decimal divisors: According to the constant quotient property, convert decimal division to integer division for calculation. 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 (if there are not enough digits, append "0" to the end of the dividend), then calculate according to the rules of decimal division with integer divisors.
5. Method for comparing the size of the quotient and the dividend: To compare the size of the quotient and the dividend in a division expression, focus on 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 larger than the dividend; if the divisor equals 1, the quotient equals the dividend.
6. The order of operations for continuous decimal division and mixed multiplication/division operations is the same as for integers. The order of operations for decimal four-operation mixed operations is completely the same as for integer four-operation mixed operations.
Currency Exchange
1. Method for exchanging currency between RMB and foreign currencies: RMB ÷ exchange rate = foreign currency; foreign currency × exchange rate = RMB.
2. When exchanging currency, 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 questions about RMB, even without special requirements, generally use the "round half up" method to retain two decimal places and find the approximate product or quotient.
3. Method for finding the approximate value of a product: Generally, calculate the correct product first, then round to the required number of decimal places according to the question requirements or daily habits, i.e., look at the next digit of the retained number of decimal places for rounding.
4. Method for finding the approximate value of a quotient: First determine the number of decimal places to retain, during calculation, based on the number of decimal places to retain, just divide one more place, then round to find the approximate value.
5. Other methods for finding approximate values: ① Truncate 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 decimal where, starting from a certain digit in the decimal part, one digit or several digits repeat endlessly is called a repeating decimal.
2. Related Concepts of Repeating Decimals: ① A decimal whose decimal part has a finite number of digits is called a finite decimal; a decimal whose decimal part has an infinite number of digits