Unit 1 multiple and factor

The world of numbers

Knowledge points:

Know natural numbers and integers, and know multiples and factors related to multiplication.

Numbers like 0, 1, 2, 3, 4, 5, 6, … are natural numbers.

Numbers like -3, -2,-1, 0, 1, 2, 3, ... are all integers.

We only study multiples and factors within the range of natural numbers (except zero).

Multiplication and factor are interdependent, so it is necessary to make clear who is whose multiple and who is whose factor.

Supplementary knowledge points:

The multiple of a number is infinite.

Multiplicity characteristics of exploration activity (1) 2,5

Knowledge points:

Characteristics of multiples of 2.

Numbers in units of 0, 2, 4, 6 and 8 are multiples of 2.

Characteristics of multiples of 5.

A number with 0 or 5 is a multiple of 5.

Definition of even and odd numbers.

Numbers that are multiples of 2 are called even numbers, and numbers that are not multiples of 2 are called odd numbers.

You can tell whether a number is a multiple of 2 or a multiple of 5. It can be judged whether a non-zero natural number is odd or even.

Supplementary knowledge points:

It is a multiple of 2 and a multiple of 5. A number with a unit of 0 is a multiple of both 2 and 5.

Multiplicity characteristics of exploration activity (2)3

Knowledge points:

Characteristics of multiples of 3.

The sum of the digits of a number is a multiple of 3, and this number is a multiple of 3.

It can be judged whether a number is a multiple of 3.

Supplementary knowledge points:

Characterized by being a multiple of 2 and 3 at the same time.

The number of each digit is 0, 2, 4, 6, 8, and the sum of each digit is a multiple of 3, which is both a multiple of 2 and a multiple of 3.

Characterized by being a multiple of 3 and 5 at the same time.

The number on each digit is 0 or 5, and the sum of numbers on each digit is a multiple of 3, which is both a multiple of 3 and a multiple of 5.

Characteristics that are multiples of 2, 3 and 5 at the same time.

The number of one digit is 0, and the sum of the numbers of each digit is a multiple of 3, which is not only a multiple of 2 and 5, but also a multiple of 3.

Find a factor

Knowledge points:

Find all the factors of a natural number from 1 to 100. Methods: Use the multiplication formula to think about which two numbers are multiplied to equal this natural number.

Supplementary knowledge points:

The number of factors of a number is limited. The smallest factor is 1, and the biggest factor is itself.

Looking for prime numbers

Knowledge points:

Understand the meaning of prime numbers and composite numbers.

A number has only two factors, 1 and itself. This number is called prime number.

A number has other factors besides 1 and itself. This number is called a composite number.

1 is neither prime nor composite.

The method of judging whether a number is a prime number or a composite number;

Generally speaking, first of all, we can judge whether this number has factors 2, 5 and 3 by "the characteristics of multiples of 2, 5 and 3"; If you can't judge yet, you can try to divide by smaller prime numbers such as 7, 1 1 to see if there is a factor of 7, 1 1. As long as we find a factor other than 1 and itself, we can determine that this number is a composite number. If no other factor can be found except 1 and itself, this number is a prime number.

Equal number of people

Knowledge points:

Use the methods of "list" and "drawing schematic diagram" to discover the law;

The ship was originally on the south bank, sailing from the south bank to the north bank, and then sailing back from the north bank to the south bank, constantly going back and forth. Through the methods of "list" and "sketch", we will find the law of "odd times on the north bank and even times on the south bank"

You can use the parity of the numbers found above to solve some simple problems in life.

Through calculation, the law of parity change is found.

Even+even = even odd+odd = even.

Even+odd = odd

Area of the second unit graph (1)

Compare the areas of graphs.

Knowledge points:

With the help of grid paper, the size of graphic area can be directly judged.

There are many ways to compare the area of a plane figure:

According to the size of the graphic area, it can be directly compared; Can be compared with reference objects; Overlapping method can be used for comparison; With the help of squares, compare by several squares; Calculate the area directly and compare it.

The figures have the same area, but their shapes can be different.

Supplementary knowledge points:

Determining the size of a graphic area depends not only on the shape of the graphic, but also on the number of squares occupied by the graphic.

Graphic area on carpet

Knowledge points:

According to the given pattern on carpet, the calculation method of irregular pattern area is explored.

The area of the answer is directly obtained by square.

According to the characteristics of the pattern, the whole pattern is divided into several small patterns with the same area, and the area of the whole pattern is obtained by finding the area of the small patterns.

The method of "reducing the area in a large area" is adopted, that is, the required area is obtained by calculating the area of related graphics.

Supplementary knowledge points:

There are various strategies and methods to solve the problem.

Do this.

Knowledge points:

Know the base and height of parallelogram, triangle and trapezoid.

Draw a vertical line segment from one point on one side of the parallelogram to the other. This vertical line segment is the height of the parallelogram, and this opposite side is the base of the parallelogram.

The vertical line segment from a vertex of a triangle to the opposite side is the height of the triangle, and this opposite side is the bottom of the triangle.

Draw a vertical line segment from one point of two parallel lines of a trapezoid to the opposite side. This vertical line segment is the height of the trapezoid, and this opposite side is the bottom of the trapezoid.

The relationship between height and bottom is corresponding.

The method of drawing the height of parallelogram with triangle.

1) Overlap one right-angled side of the triangle with one side of the parallelogram, and 2) Let the other right-angled side of the triangle pass through a point on the opposite side.

3) Draw a vertical line from this point along the other right-angled side of the triangle to its opposite side. 4) This vertical line (from point to vertical foot) is the height of one side of the parallelogram. Note: the height can be drawn from any point on one side to its opposite side, 5) it can also be drawn from any point on the other side to its opposite side, 6) but primary schools do not require the height to be drawn on the extension line of the bottom edge.

The method of drawing the height of a triangle with a triangle board.

8) Align one right-angled side of the triangle with one vertex of the triangle, and 9) Align the other right-angled side with the opposite side of the vertex.

10) Draw a vertical line from this vertex to its opposite side along the other right-angled side of the triangle. 1 1) This vertical line (from the vertex to the vertical foot) is the height of one side of the triangle.

The method of drawing trapezoid height with triangle.

Similarly, draw a vertical line segment between two parallel lines of the trapezoid, which is the height of the trapezoid.

The area of exploration activity (1) parallelogram

Knowledge points:

Parallelogram area = rectangular area.

The length of a rectangle is the base of a parallelogram; The width of a rectangle is the height of a parallelogram.

Therefore: parallelogram area = base × height

If S is used to represent the area of the parallelogram, and A and H are used to represent the base and height of the parallelogram respectively, then the area formula of the parallelogram can be written as follows:

S = ah

Using the area calculation formula of parallelogram to calculate the area of related graphics, some practical problems are solved.

Supplementary knowledge points:

When the base of a parallelogram is at the same position, its area is the same.

Exploration activity (2) Area of triangle

Knowledge points:

Area of triangle = area of parallelogram composed of two identical triangles ÷2

The base and height of a triangle are those of a parallelogram.

Therefore: area of triangle = area of parallelogram ÷2= base × height ÷2.

If S is used to represent the area of the triangle, and A and H are used to represent the bottom and height of the triangle respectively, then the area formula of the triangle can be written as follows:

S=ah÷2 or s = ah.

The area formula of triangle is used to calculate the area of related graphics and solve practical problems.

Supplementary knowledge points:

The factor that determines the area of a triangle is not the shape of the figure, but the length and height of the base of the triangle. As long as the base and height are the same, the areas of triangles with different shapes are the same.

Exploration activities (3) Trapezoidal area

Knowledge points:

Trapezoidal area = the area of a parallelogram composed of two identical trapezoids ÷2.

The sum of the upper bottom and the lower bottom of the trapezoid is the bottom of the parallelogram, and the height of the trapezoid is the height of the parallelogram.

Therefore: trapezoidal area = parallelogram area ÷2= bottom × height ÷2= (upper bottom+lower bottom) × height ÷2.

If S is used to represent the area of the trapezoid, A and B are used to represent the upper and lower bottoms of the trapezoid, and H is used to represent the height of the trapezoid, then the area formula of the trapezoid can be written as follows:

S= (a+b)h

Using the formula of trapezoidal area to solve the corresponding practical problems.

Supplementary knowledge points:

The factor that determines the size of the trapezoid area is not the shape of the figure, but the sum of the heights of the upper and lower bottoms and the trapezoid. As long as the sum and height of the upper and lower bottoms are the same, the trapezoidal areas of different shapes are also the same.

Unit 3 "Fractions"

Re-understanding of fractions

Knowledge points:

For details, learn more about the score. The "whole" corresponding to the score is different, and the size or specific number of parts represented by the score is also different, that is, the score is relative.

Divide the cake (true and false)

Knowledge points:

Understand the meaning of true score, false score and score.

Scores like,,,, … are called true scores.

Features: numerator is smaller than denominator.

Scores like,,,, … are called false scores.

Features: numerator is greater than denominator, or numerator is equal to denominator.

Scores like 2 1 are called band scores.

Features: It consists of two parts: integer and true fraction.

The true score is less than 1 and the false score is greater than or equal to 1.

Pronunciation with scores: 2 Pronunciation: 2 1/4.

Supplementary knowledge points:

A false fraction whose numerator is a multiple of the denominator can be turned into an integer.

A false fraction whose numerator is not a multiple of the denominator can be changed into a fraction.

Fraction and division

Knowledge points:

Understand the relationship between fraction and division: dividend ÷ divisor = (divisor is not 0).

The denominator of a fraction cannot be 0. Because in division, 0 can't be a divisor, so according to the relationship between fraction and division, the denominator in the fraction is equivalent to the divisor in division, so the denominator can't be 0.

Using the relationship between fraction and division to solve practical problems. Commercial fraction representation of dividing two numbers.

According to the relationship between fraction and division, the false fraction is turned into a method with fraction.

Divide the numerator by the denominator, write the quotient in the integer position with the fraction, write the remainder on the numerator of the fraction, and still use the original denominator as the denominator.

The way to turn a score into a false score. (two kinds)

1) Divide the banded fraction by the sum of the integer and the true fraction, 2) Convert the integer into a false fraction with the true fraction as the denominator, 3) Add the original true fractions, and 4) Convert the banded fraction into a false fraction.

5) Multiply the numerator by the product of integer and denominator, and 6) The denominator remains unchanged.

Basic properties of fractions

Knowledge points:

Understand the basic properties of fractions.

The numerator and denominator of a fraction are multiplied or divided by the same number (except 0), and the size of the fraction remains the same.

Contact the relationship between fraction and division and the law of "quotient invariance" to understand the basic properties of fraction.

The numerator is equivalent to the dividend, the denominator is equivalent to the divisor, and the dividend and divisor are multiplied or divided by the same number at the same time (except 0), and the quotient remains unchanged. Therefore, both the numerator and denominator of a fraction are multiplied or divided by the same number (except 0), and the size of the fraction remains unchanged.

By using the basic properties of fractions, fractions are converted into fractions with a specified denominator (or numerator) and the same size.

Find the greatest common divisor

Knowledge points:

Understand the meaning of common factor and greatest common factor.

The common factor of two numbers is their common factor, and the largest is their greatest common factor.

The method of finding the common factor and the greatest common factor of two numbers.

Using the method of finding factors, we first find the respective factors of two numbers, and then find the same factor among the factors of two numbers, which is the common factor of two numbers; Let's look at the greatest common denominator. This number is the greatest common factor of two numbers.

Will find the greatest common factor of numerator and denominator.

Supplementary knowledge points:

Other ways to find the greatest common divisor.

To find the common factor and the greatest common factor of two numbers, we can first find the factor of the smaller number of two numbers, and then see which of these factors are also the factors of the larger number, so these numbers are the common factors of two numbers. The greatest is the greatest common factor of these two numbers.

For example, find the common factor and maximum common factor of 15 and 50:

You can first find the factors of 15: 1, 3,5, 15. Then, judge which of the four numbers is also a factor of 50. Only 1 and 5, 1 and 5 are common factors of 15 and 50. 5 is their greatest common divisor.

If two numbers are different prime numbers, then the common factor of these two numbers is only 1.

If two numbers are continuous natural numbers, then the common factor of these two numbers is only 1.

If two numbers have a multiple relationship, then the smaller number is the greatest common factor of these two numbers.

It is also appropriate to introduce the short division of common factors to students. It depends on the actual situation of the students. )

The greatest common factor of 4 and all odd numbers is1; The greatest common factor of 4 and multiples of 4 is 4.

Reduce (part of)

Knowledge points:

Understand the meaning of reduction.

The numerator and denominator of a fraction are divided by the common factor at the same time, and the value of the fraction remains unchanged. This process is called reduction.

Understand the meaning of the simplest score.

In this way, the common factor of numerator and denominator is only 1, which cannot be reduced any more. This score is the simplest score.

Master the method of reduction.

Generally, there are two simplification methods, one is to divide by the common factor of two numbers, and the other is to divide by the greatest common factor of two numbers directly.

Supplementary knowledge points:

When the comparison score is large, those with the same denominator and the same numerator can be compared directly, and sometimes the numerator and denominator are different, so the method of descending first and then comparing can be adopted.

For example: ○

Find the least common multiple

Knowledge points:

Understand the meaning of common multiple and minimum common multiple.

The common multiple of two numbers is called the common multiple of these two numbers, and the smallest is called the least common multiple.

Find the common multiple and the least common multiple of two numbers.

First find the multiple of two numbers (within a certain range), then find the common multiple, the most common multiple of two numbers, and see what is the smallest of these common multiples, which is the least common multiple of two numbers.

The number of common multiples of two numbers is infinite, so only the smallest common multiple has no largest common multiple.

Supplementary knowledge points:

Other methods for finding common multiple and minimum common multiple.

To find the common multiple and the minimum common multiple of two numbers, you can first find the multiple of the larger number of two numbers (within a certain range), and then see which of these multiples are also multiples of the smaller number, then these numbers are the common multiples of two numbers. The smallest is the least common multiple of these two numbers.

For example, find the common multiple and the minimum common multiple of 6 and 9. (Within 50) We can first find the multiples of 9 (within 50): 9, 18, 27, 36, 45, and then find the multiples of 6 from these numbers: 18, 36, 18 and 36 are common multiples of 6 and 9,/kloc-0.

If two numbers are different prime numbers, then the least common multiple of these two numbers is the product of these two numbers.

If two numbers are continuous natural numbers, then the least common multiple of these two numbers is the product of these two numbers.

If two numbers have a multiple relationship, then the larger number is the least common multiple of these two numbers.

The method of finding the least common multiple by short division can also be introduced to students appropriately. It depends on the actual situation of the students. )

The size of the score

Knowledge points:

Understand the meaning of general points.

The process of changing a fraction with different denominators into a fraction with the same denominator as the original fraction is called total score.

Two main points of general division:

Equal to the original score.

Numbers with the same denominator.

Score comparison.

Compared with the denominator score, the larger the numerator, the greater the score.

Compared with the molecular fraction, the smaller the denominator, the greater the score.

A method of comparing fractions with different numerators and denominators.

By general division, the fractions with different denominators are converted into fractions with the same denominator as the original fractions, and then the sizes are compared.

Is to change two fractions into fractions with the same molecule, and then compare the sizes.

Supplementary knowledge points:

Generally, the denominator is the least common multiple.

Mathematics and transportation

meet with

Knowledge points:

1. Analyze the quantitative relationship in simple practical problems.

Distance = speed × time

2. Solve simple practical problems with equations.

Emphasize the steps of solving application problems with column equations;

(1) Find the equivalence relation in the problem.

(2) Set the required quantity as x.

(3) According to the equivalence relation, the corresponding equations are listed.

(4) Solve the equation, and note that the result has no unit name.

(5) Test and answer.

Supplementary knowledge points:

Speed = distance/time/time = distance/speed

travelling expenses

Knowledge points:

1, will use the existing knowledge, 2, according to the actual situation to give a more economical plan.

3. Master the list method to solve problems.

Look at the picture and find the relationship.

Knowledge points:

Can read some charts used to express quantitative relations, can obtain relevant information from charts and experience the intuition of charts.

Combined with practical problems, the relationship between quantity and quantity is analyzed.

Determine or describe the change of behavior and events according to the change of the diagram.

Unit 4 Fractional addition and subtraction

Origami (fractional addition and subtraction 1)

Knowledge points:

1, addition and subtraction of fractions with different denominators.

To add and subtract fractions with different denominators, first divide them into the same denominator, and then add and subtract.

2. The calculation result can be simplified to the simplest score.

Sunday Arrangement (Fraction Addition and Subtraction II)

Knowledge points:

Understand that fractions are added and subtracted in the same order as integers and decimals.

When calculating the mixed operation of addition and subtraction, the method should be flexible. You can divide all the points first and then calculate. You can also calculate two of the three numbers and then divide them; Others are divided into parts first, and the results of some parts are calculated, and then divided for the second time. Note: through the analysis of specific problems, the calculation process is simpler.

Supplementary knowledge points:

Integer additive commutative law and associative law also apply to fractional addition.

Time spent reading extracurricular books (fractions and decimals)

Knowledge points:

1. There are two ways to divide the fraction by the decimal: one is to use the relationship between the fraction and division; The other is numerator divided by denominator; One is to change the fraction into decimals first and then divide it into decimals.

Note: The first method is a general method, which is applicable to all fractions being decimals, while the latter method is a special method, which needs to be determined according to the denominator value.

4. Method of converting a finite decimal into a fraction: decimal fraction, 5. There are several decimals in the original text, 6. Write a few zeros after 1 as the denominator, 7. Remove the decimal point from the original decimal point as a molecule; After the number of components is 8, the number of quotation points can be reduced.

Area of the fifth unit graph (2)

Combined graphic area

Knowledge points:

Understand composite graphics: There are several simple graphics that are spelled out, which we call composite graphics.

There are many ways to calculate the area of combined graphics. Common methods are "division" and "addition".

Segmentation is to divide this figure into several basic figures. The simpler the sub-graph, the simpler the problem-solving method will be, and the relationship between the sub-graph and the given conditions should be considered.

Supplementary method, that is, by adding a simple graph, makes the whole graph become a big regular graph.

Apply what you have learned to solve the practical problems of combined graphics in life.

Exploration: the Footprint of Growth

Knowledge points:

The size of irregular graphic area can be estimated correctly.

The area of irregular graphics can be calculated by calculating the grid.

The content of irregular figure area calculation is mainly based on the grid diagram, such as Beijing, so the grid diagram can help to establish the calculation method of irregular figure area.

Try to guess.

The chicken and the rabbit are in the same cage.

Knowledge points:

With the help of the carrier of "chicken and rabbit in the same cage", let students go through the process of listing, trying and constantly adjusting, and realize the general strategy of solving problems-listing.

Laws in lattice

Knowledge points:

In the observation activities, we can find the hidden rules in the grid and realize the connection between figures and numbers.

In the activity of "rules in the grid", we can infer the number of points in the subsequent graphics by observing the changing rules of the points before and after.

Unit 6 The size of the possibility

Touch the ball game (using scores to indicate the size of the possibility)

Knowledge points:

Use scores to indicate the size of the possibility.

In objective events, the "impossible" phenomenon is represented by data as "possibility is 0", while in objective events, the "certain" phenomenon is represented by data as "possibility is 1", and when the possibilities are equal, it is represented by data as "".

Gradually realize the simplicity and objectivity of data representation.

Design activity plan

Knowledge points:

Use scores to express possibilities, and you can design some activities independently.

Be able to use the knowledge of possibility to reasonably explain the events and phenomena in real life.

Mathematics and life

Welcome the new year.

Knowledge points:

Review the understanding of fractions and the knowledge of addition and subtraction through activities.

Through activities, we can deepen our understanding of the possibility, express the possibility with scores, and design the scheme according to the specified possibility.

Can synthesize the knowledge learned and solve some simple practical problems.

firebrick

Knowledge points:

Learn to comprehensively use the knowledge of graphic area, multiplication and division, equations and so on to solve simple practical problems.

Knowledge network diagram: