Mathematics teaching plan for the first grade

As a teacher, it is very necessary to carefully design teaching plans, which will help us to accurately grasp the key points and difficulties of teaching materials, and then choose appropriate teaching methods. How to write a good lesson plan? The following is the first-grade math teaching plan of the Western Normal University Edition that I collected and sorted out. Welcome to share.

Mathematics Teaching Plan for Senior One in West Normal University Edition 1 teaching content;

7 pages 0 understanding, example 1 example 3, 8 pages classroom activities.

Teaching purpose:

1, through practical activities, let students realize that none of them are represented by 0.

2. Through group cooperative learning and independent inquiry. 0 indicates the starting point on the ruler.

3. Cultivate students' good habits of cooperative learning and active exploration.

4. Write 0 correctly

5, can contact the practical application of life.

Teaching emphases and difficulties:

1 `, where 0 represents the starting point on the ruler.

2. Write 0 correctly

Teaching process:

First, introduce a conversation

People always fly many beautiful balloons during festivals. Have you seen them? (slide show)

Observation: What did you find?

How many were there before the reverse flight? What is the number? After the flight, the children have no balloons. What do you mean?

This will use the new number 0 (blackboard writing) that we learned today.

Second, the teaching example 1

The children don't have any balloons. What does 0 mean? None of them are represented by anything.

Show me the picture of eating fish: (1) Who said this picture? What are you in a hurry to do?

(2) How many fish are there? What about now?

(4) There are three fish. How many people are there? Not anymore. How much do you mean?

Three. Teaching example 2

1, show me the ruler map.

Observation: What's on the ruler? Where is 0? What does the 0 on the ruler mean?

Discuss at the same table and communicate with the whole class.

Summary: The beginning of this paragraph on the ruler is called the starting point, which is indicated by 0. It is also on the ruler, and 0 indicates the starting point.

Fourth, teaching examples 3

1, the teacher demonstrates writing 0.

Students have free books.

Verb (abbreviation of verb) classroom activities

1, page 8 12

2. Teachers show the enlarged grid diagram to guide students to understand the grid. What does 0 mean here?

Counting from the left 3 to the right, what do you find?

What else can I see?

The teaching goal of the second edition of mathematics teaching plan for senior one in western normal university;

1. In the case of visiting the park, the calculation methods of 8 plus a few, 7 plus a few and 6 plus a few are explored, and the calculation can be flexible.

2. Through operation, discussion and communication, cultivate the ability of independent inquiry and transfer reasoning, and optimize the algorithm.

3, stimulate interest in learning, feel that you want to learn, like to learn, you will learn.

Teaching focus:

Can correctly calculate 8 plus several, 7 plus several, 6 plus several, and master the oral calculation method.

Teaching difficulties:

Develop the ability of transfer reasoning.

Teaching preparation:

Everyone has a 10 package of injection package, stick learning package, answer sheet, exercise paper, and three turntables with 8 plus several, 7 plus several and 6 plus several.

Explain the design description:

The teaching in this section is divided into two parts. Part of it is the oral calculation of 8, 7 and 6. The key to this part of teaching is to master the oral calculation method and be flexible in oral calculation. In the design, through the deepening of teaching links, students can feel "seeking differences from many, seeking excellence from the same". For example, when playing roulette, I realized the convenience of "ten-point method", and then I felt that I chose flexible methods to optimize the oral calculation method according to different topics in "using my brain". The teaching of oral arithmetic is quite boring. In the design, we can stimulate the interest in oral calculation through vivid practice forms and be proficient in oral calculation at the same time.

The second part is "using mathematics", and the teaching design of this part strives to embody: ① making full use of situation diagram to let students learn and use mathematics; (2) Guide students to carefully observe the meaning of pictures and experience the same problems. Different observation angles will lead to different formulas; In the process of solving practical problems, I further realized the means of collecting information.

Teaching process:

First, create a problem situation.

Xiaohong gave you a test: 9+5=

This paper focuses on the process of "supplementing ten methods". Why do you divide 5 into 1 and 4?

Today's weather is really good. Xiaohong and her friends go to the children's park to play. Tell me what you see. The courseware dynamically shows the scene of buying tickets on page 103 of the textbook.

What math questions will you ask? (It is estimated that students can come up with a * * * How many people will buy tickets? This leads to the formula 8+5=? )

Second, explore new knowledge.

(a) for example, 8, 7, 6 plus a few (preliminary perception calculation method)

1, teach 8+5.

(1) Group discussion: How to know the number the fastest. Talk to each other, and then write your thoughts on the answer sheet. If you have difficulty, you can use an injection bag. Teachers participate in group discussions. )

(2) When students report, there are many kinds of oral calculation methods, with the emphasis on "adding ten methods" and naming answers.

8+5= 13 Why do you want to divide 5 into 2 and 3?

1023

(3) Summary:

Just now, the students came up with a quick and good method to calculate 8+5= 13. The teacher is really happy for you, and the students are really amazing!

2. Small contest: Turn the turntable (teaching example 2-highlighting the advantages of the ten-point method)

1, teach 7 plus a few and 6 plus a few, the first feeling of the benefits of ten methods.

(1) Look at a large green meadow. Flowers nod to us and birds smile at us. Sit down and have a rest! What math problems can you ask from this picture? How to solve it?

There are seven birds in the sky, and five others are flying. How many birds are there in the sky?

There are six flowers on one side and five on the other. How many flowers are there on the grass?

Blackboard writing formula: 7+56+5

(2) Now please use the fastest method to calculate 7+5 and 6+5.

(3) Ask students to introduce oral calculation methods.

7+5= 12 Why do you want to divide 5 into 3 and 2?

1032

6+5= 1 1 Why should 5 be divided by 4 and 1?

104 1

(4) Summary: It seems that the ten-point method can not only add a few points to 9, but also add a few points to 8, 7 and 6.

(2) teach 8, 7 and 6 to add a few.

1. Grouping carousel. (8 minutes)

(2) The number of reports made by the group that writes more and writes faster (only 8+4 and 8+8, 7+6 and 7+8, 6+6 and 6+8 are needed),

(3) Tell me how to calculate orally quickly and accurately.

3. Use your head (Teaching Example 3)

(1) can be added to ten to calculate 8+9. Can you think of a faster way? Students can answer freely.

①8+2= 10 10+7= 17

②9+ 1= 10 10+7= 17

③9+8= 178+9= 17

This paper focuses on the calculation method ③. When two addends are the same, the positions of addends can be interchanged and the sum is unchanged.

(2) Can you use the fastest method to calculate 7+9 and 6+9?

Third, consolidate the practice.

1, circle, count.

Textbook page 104, title 1.

2. Say and calculate.

Textbook page 104, question 2

Step 3 take the train

Everyone has a ticket (oral card), and three formulas on the ticket must be calculated before boarding. Then get on the train according to the number on the ticket (there are four trains, namely 15, 14, 13, 12). The teacher chooses a question from 8, 7, 6 and several points and talks about his own views.

Fourth, expand and extend.

What two kinds of carrots can a white rabbit eat? What about the gray rabbit?

Students' games include the "Star of Wisdom" and the patrol guidance of teachers.

Verb (abbreviation of verb) abstract

Today we learned "8 plus a few, 7 plus a few, 6 plus a few". What method is used in the calculation?

(Add up to ten methods; Swap the positions of addends and get the law of number invariance)

Teaching content of the third edition of mathematics teaching plan for senior one in Western Normal University;

Add up the numbers within 5.

Teaching objectives:

1. According to the actual situation, let students understand the meaning of addition, know the plus sign+,learn how to read the addition formula and calculate the addition within 5.

2. Experience the diversity of algorithms by communicating different calculation methods with peers.

3. Apply what you have learned to solve simple practical problems in specific life situations, experience mathematics around you, and initially cultivate application consciousness.

Teaching focus:

Addition calculation within 5

Teaching difficulties:

Understand the meaning of addition.

Teaching preparation:

Courseware, small disk, stick, digital card.

Teaching process:

Students, are you ready? Let's go to class

Hello, class, please sit down.

First, children, we have learned the division of numbers within 5 before. Let's clap our hands and say it together.

The students speak very well. Let's take a look at this picture together. What do you see in the picture?

This classmate saw a monkey, this classmate saw a bird and that classmate saw a peach. Son, when we look at a picture, we should observe it in a certain order, so as to find out the relevant mathematical information in the picture in an orderly way.

Who can tell the mathematical information in the picture in an orderly way?

You tell me.

Well said: 4 birds on the left, 1 bird on the right.

You tell me.

That's clear: there are four peaches on the tree.

This classmate, you tell me.

He said information about monkeys. There are two monkeys on the stone and three monkeys on the tree.

What else do you want to say?

Fortunately, he found five children on the mountain, 1 red flower and 2 yellow flowers.

Second, the students are great. They found so much mathematical information. Let's have a look. This book asks us a question: How many monkeys are there?

1. What information is needed to solve this problem?

Great, let's discuss information and problems together:

There are two monkeys on the rock and three monkeys on the tree. How many monkeys are there on the tree?

That's very complete. How to solve this problem?

Oh, you said by counting. Let's count together.

1, 2, 3, 4, 5, 5.

Is there any other way?

Let's wave it with a stick.

Use a stick to represent a monkey, two on the left to represent two little monkeys on the stone, and three on the right to represent three little monkeys on the tree. Put them together and you will know that there are five little monkeys in a * * *.

Students, do you know? The process of our pendulum just now can be expressed by the formula: 2+3=5.

Have you ever seen such a formula? Do you know what this symbol+is?

The student is great, knowing that this symbol is a plus sign.

Do you know how to pronounce this formula?

Let's read it together: 3 plus 2 equals 5.

We have solved the monkey problem. According to the above information, can you ask different math questions?

This classmate asked: A * *, how many birds are there?

That classmate suggested: How many peaches does a * * have?

How to solve these two problems?

Can you list the formulas?

Who will tell me the formula you listed?

You tell me. 4+ 1=5 2+2=4

Who can tell the meaning of 4 and 1 = 5 respectively? What does 5 mean?

You come.

That's great. Here, 4 means four birds on the left, 1 means 1 bird on the right. 5 means a * * * has five birds.

What do the numbers in 2+2=4 mean? Tell your deskmate.

Third, independent practice.

Students, look at this picture. We have solved three different problems. We know the plus sign and the addition is used for calculation. Let's take a look at autonomous exercise.

Question 1, put it on the table with your school tools. Tell your deskmate what you think.

Question 2. Please fill it out yourself. Who can tell me how you fill it out?

3+ 1=4 2+3=5

Is it different from him?

The students are great, yes.

The third question, compare, who is faster for whom.

Now let's see who can do it well.

Praise yourself for doing so well.

Fourth, summary.

Students, now let's review this lesson together. How did you find the math materials? What questions were raised? How do you operate it? Every student's performance is great. That's all for this class. Goodbye, class.

The teaching goal of the fourth grade mathematics teaching plan of West Normal University;

1. Make students go through the process of calculation method and calculate correctly.

2. Make students gradually cultivate the consciousness and habit of exploration and thinking in observation and operation. Cultivate students' innovative consciousness through algorithm diversification.

3. Enable students to use knowledge to solve practical problems in life, understand the role of mathematics, and initially cultivate the application consciousness of mathematics.

Teaching process:

First, the game guides people and stimulates interest.

Do you like playing games, children? Now let's play, shall we?

The teacher clapped his hands and said rhythmically, let me ask you, little friend, how much is 9 plus 10?

Student: Miss Shao, I tell you, 9 plus 1 equals ten.

[Comment: The relaxed and pleasant classroom atmosphere has laid a good foundation for the teaching of the new curriculum. The composition of 10 not only reviews password games, but also provides a basis for students to explore the algorithm of 8 plus 7. ]

Second, the operation of inquiry, learning new knowledge

1. Teach trumpet drawing.

(1) Question: This is a thumbnail. Who can explain the meaning of this painting?

Can you ask a question of addition calculation? How to form?

[Comment: Let students talk about ideas first and then ask questions, aiming at cultivating students' ability to collect information and ask questions. ]

(2) Question: What is 8+7? Can you see it from the picture? Talk about it in the group.

(3) Who will tell me what you think?

Students may have the following ideas when communicating:

(1) One by one.

② Eight plus two on the left is 10, 10 plus five is 15.

③ Seven plus three on the right is 10, 10 plus five is 15.

(4) There are 20 boxes * * *, and now 5 boxes are empty, which is 15.

⑤8+7=8+2+5= 15。

⑥8+7=7+3+5= 15。

Students demonstrate the process of trumpet movement through computer animation when communicating the second and third methods.

[Comment: Teachers make full use of thematic maps to let students explore 8+7 calculation strategies independently. The above different algorithms reflect students' three cognitive levels: the first algorithm shows a tendency to grasp actions, and the cognitive level needs to be improved; The second algorithm shows a tendency to grasp graphics, and such students have strong observation and imagination for graphics; The fifth algorithm shows a tendency to grasp symbols. These students have abstract thinking ability and high cognitive level. ]

2. Teaching stick figures.

(1) The children have come up with many ways to calculate 8+7= 15. Want to know what small green peppers and mushrooms think?

Put a stick on the small green pepper. Please talk in the group. What's it thinking? Say its name.

Animation demonstration, students fill in the numbers in the box.

(2) The ideas of small mushrooms and small green peppers are a little different. Please talk in the group. Communicate by name.

[Comment: Setting up the situation of helping small green peppers and mushrooms, and asking students to fill in the numbers in the box is conducive to cultivating students' virtue of helping others, and at the same time making students' cognitive level develop on the original basis. ]

(3) What's the difference between these two methods? Like what? Summary: These two methods are all ten methods.

3.( 1) Think about doing the 1 problem in teaching.

Please put it with your school tools before calculating. Students talk after they finish speaking.

(2) (The computer shows it, thinking about doing the second question) Let's play a circle ten game. Circle 10 first, then calculate.

(3) Thinking teaching. Question: Do you think so without looking at the picture or putting a stick? Please fill in this book.

Question: What other related formulas can you think of when calculating 8+9?

Somebody say something. Students may think:

① Because 9+8= 17, 8+9= 17.

② Because 9+9= 18, 8+9= 17.

③ Because 8+ 10= 18, 8+9= 17.

④ Because17-9 = 8,8+9 =17.

[Comment: Let different students show different thinking processes, let them have a positive learning experience, feel the happiness of success, and further develop their creative thinking. ]

(4) Summary: When we calculate 8+9, we can think of the formula we learned before. This method is really good. (The computer shows it, thinking about doing the fourth question) Can you calculate the number of these questions quickly?

Students answer orally.

[Comment: Through the comparison of problem groups, students realize that if a small number is added to a large number, the number can be directly calculated by the formula they have learned, and at the same time, they realize that the two numbers are added, the positions are exchanged and the sum is unchanged. ]

Third, look for laws and consolidate new knowledge.

1。 The computer shows the question of 8 plus a few, and the students answer it orally, leading them to find that if you divide the added number by 2 plus a few, you will know that the number is greater than ten. Summary: If this rule is discovered, it will be correct and soon.

[Comment: Providing students with rich learning materials, let them observe and compare, so as to find the law of 8 plus several, which can not only improve students' oral calculation speed, but also cultivate students' habit of inquiry and thinking. ]

2。 The computer shows more than 7 questions. Question: So, is there such a rule that seven plus several? Who can quickly calculate the number of these questions?

3。 Organize a verbal contest. Each boy and girl will send a representative, and the rest will make gestures.

Fourth, contact life and solve problems.

Question: It's not enough to know what it is. We should learn to use our brains and use what we have learned to solve problems in life. You see, there are three bags of bread in the bakery. The first bag has nine bags, the second bag has eight bags and the third bag has six bags. Aunt Wang in kindergarten is going to prepare snacks for the children in the class 15. Which two boxes do you think are more suitable? Organize students to communicate on the basis of independent thinking.

Conclusion: Applying mathematical knowledge can solve problems in life. Moreover, as long as you are willing to use your head, there are often more than one way to solve the problem.

[Comment: The teacher raised a challenging question from real life, which requires students to make analysis, estimation and judgment in specific situations. The process of solving problems makes students get the joy of success, at the same time, it also enhances their confidence in learning mathematics, develops their thinking of seeking differences, and cultivates their attitude of seeking truth from facts and innovative spirit. ]

General comment: There is no rigorous explanation of calculation methods and repeated standardized arithmetic language training in this course. Teachers allow students to think in a form suitable for their own thinking characteristics, explore calculation methods, and form general strategies to solve problems. Students have acquired basic mathematics knowledge and skills, and at the same time fully developed their emotions and attitudes. Students' learning activities are a lively and personalized process.

The design idea of the fifth grade mathematics teaching plan of the Western Normal University Edition;

Using the calculation experience of continuous addition and subtraction, this paper directly tries to calculate the mixed problems of addition and subtraction. Through the transfer of learning experience, let students try to calculate boldly. Retire from the role of teacher, encourage students to try and discuss on their own, and give full play to students' subjectivity. Pay attention to discussion and exchange activities and fully develop students' different resources. Give play to the role of eugenics, help each other and learn from each other, so that every student can participate in it and get different development in mathematics learning.

Interpretation of curriculum standards;

The purpose of learning mathematics is to let students ask questions, solve problems and cultivate students' ability to solve problems with mathematical thoughts. Through mathematics learning, students' interest and motivation, self-confidence and will, attitude and habits are cultivated. Therefore, the textbook combines the teaching of calculation, arranges applied mathematics to solve problems, initially forms the consciousness of applied mathematics, and experiences the diversity of problem-solving strategies.

Content analysis:

The addition and subtraction are mixed up on page 75 of the textbook. Arrange after addition and subtraction, using the same arrangement method. The purpose is to make students learn the mixed calculation of addition and subtraction by connecting addition and subtraction. The vivid illustrations arranged in the textbook help students to gradually transition from watching the second step of the number calculated in the first step to memorizing the number calculated in the first step in their minds to complete the second step, thus gradually improving their computing ability.

Analysis of learning situation:

Last class, the students have learned the meaning and calculation method of addition and subtraction formula. For the formula that needs two steps to calculate the result, they have basically mastered the calculation of the first step, and then use the first step to add and subtract, and then proceed to the second step. The addition, subtraction and mixing of this lesson also need to be calculated in two steps. With the foundation of the last lesson, students can learn the knowledge of this lesson better. However, we can't ignore another difficulty in the mixed calculation of addition and subtraction-because the calculation methods used in the previous two examples are inconsistent, the calculation process can't be carried out smoothly. Therefore, it is necessary to create specific situations in teaching to help students understand the significance of the first step and the second step.

Teaching objectives:

1. Knowledge and skills: Through observation, operation and discussion, we can initially understand and master the significance and calculation method of addition and subtraction mixed calculation; Mastering the calculation order of addition and subtraction mixing can correctly calculate the addition and subtraction mixing within 10.

2. Mathematical thinking: cultivate students' ability of observation, comparison and abstract generalization, and experience the close relationship between addition and subtraction mixed calculation and life.

3. Problem solving: Being able to think independently, cooperate and communicate, practice and consolidate and other learning activities, initially learning to ask and understand problems from the perspective of mathematics, and cultivating the ability to cooperate with others, clearly stating their own views and applying what they have learned to solve practical problems.

4. Emotion and attitude: in learning activities, stimulate students' interest in learning, make students realize that there is mathematics everywhere in life, and establish confidence and determination to learn mathematics well.

Teaching focus:

Master the calculation method of adding and subtracting mixed questions, and be skilled in calculating numbers.

Teaching difficulties:

Remember the calculation in the first step. The key to teaching is to understand the meaning of formulas, master the operation order and cultivate memory ability.

Teaching strategies:

1, using the calculation experience of continuous addition and subtraction, directly try to calculate the mixed problem of addition and subtraction. Through the transfer of learning experience, let students try to calculate boldly.

2. Retire from the role of teacher, encourage students to try and discuss on their own, and give full play to students' subjectivity.

3. Pay attention to discussion and exchange activities, and fully develop students' different resources. Give play to the role of eugenics, carry out inter-lake mutual learning, let every student participate in it, and let every student get different development in mathematics learning.

Preparation before class:

1, students learn to prepare: recall the calculation process of addition and subtraction.

2. Teachers' teaching preparation: courseware preparation.

3. Design and layout of the teaching environment: draw the graphics of the game on the playground.

4. Design and preparation of teaching tools: courseware preparation.

Teaching process:

First, create a situation

1, review for the exam

Look at the formula and make up a story, then talk about the order of calculation.

3 + 4 + 2 = 7 - 3 - 2 =

Teacher: Look at these two formulas. Can you make up a short story? We can talk about things around us. Let the students speak freely. For example, three deer were looking for food in the forest, and four came soon, and then two came. How many deer are there now? ) Let the students say what counts first, then what counts, and then let the students say which two numbers are subtracted from the calculation of 7-3-2 =.

Design intention: through reading the questions and telling stories by yourself, students' understanding of the meaning of the formula is further strengthened, and abstraction and concreteness are combined to facilitate students' understanding and combination with reality. At the same time, it is helpful for students to tell the steps of calculation.

2. Questioning passionately.

The computer shows forests and lakes, and swans are playing happily.

Look! What's here?

According to the students' descriptions of the pictures, educate students to love nature and protect the environment.

Second, explore new knowledge.

1, introducing new knowledge.

Teacher: In order to welcome us, the cygnets danced a swan dance.

(1) Teacher's computer demonstration: 4 swans are on the lake and 3 swans are coming. Can you ask a math question when you see this scene?

Health: There are four swans in the lake, and three swans fly in. How many swans are there in the lake? problem

Teacher: How are the swans arranged in the lake? The teacher wrote on the blackboard according to the answers: 4+3.

(2) The teacher operates the screen, and there is a scene where seven swans fly away from two in the lake.

Teacher: Two swans in the lake took a bath and flapped their wings and flew away. How many swans are there now? What should we do next? According to the students' answers, the teacher writes -2 after 4+3, completes the formula and reads the formula.

(3) Observation: What is the difference between this formula and the addition and subtraction we just learned? Can you give it a name? (Students speak freely. )

(4) Guiding topic: There are addition and subtraction operations like this, which we call addition and subtraction mixing. (blackboard writing: addition and subtraction. Students read the questions together.

2. Analogy query algorithm.

(1) Who can read this question again and tell the meaning of 4+3-2 by comparing the pictures? (Name)

(2) So how to calculate 4+3-2?

① Students discuss.

② Student report. According to the students' narrative process, mark the calculation order in Formula 4+3-2 on the screen.

③ Teacher's summary. Write five on the blackboard. )

Design intention: Through discussion, let students find the breakthrough point of new knowledge according to existing knowledge and effectively break through the key points of new knowledge.

3. Research Example 2.

Teacher: On the beautiful small lake, the swan has undergone new changes. (Computer demonstration modification 2: There are five swans in the lake. Two of them flew away first, and then three came. Can you tell us the meaning of this painting?

(1) Students say pictures and mean things.

(2) What questions can I ask?

(3) Who will make the presentation? The student answers, and the teacher writes it on the blackboard. )

(4) Try to calculate 5-2+3.

(5) Calculation process of exchange report.

Health: Count 5-2 first. According to the students, the teacher flashed on the screen.

Teacher: Why count 5-2 first?

Health: Because in the picture, five swans fly away first and two swans fly away. Only by subtraction can we figure out how many swans are left in the lake after three swans fly away?

Teacher: What are the two numbers to add in the second step?

Health: It's 5 minus 2, and 3 plus 3. The teacher writes 2 in the box in front of the formula with the students' answers and flashes it on the screen.

4. Teachers and students * * * the same summary: add and subtract mixed calculation order. (from left to right)

Design intention: the combination of support and release, observation and generalization, addition and subtraction in teaching can cultivate the ability to observe, summarize and solve practical problems.

Third, feedback exercises.

1, by car (computer demonstration: there are 6 passengers in the car. When they arrived at the station, they got off the No.2 bus, but got on the No.3 bus again. )。

(1) means meaning.

(2) Put forward mathematical problems.

(3) Formula calculation.

2. Guide students to place sticks. Put seven on the table first, then take three, and then add four. Ask the students to complete Formula 7-()+()= () according to the process of swinging. Let the students talk about the meaning and calculation order of formula 7-3+4=8 in combination with the process of swinging.

3. Outdoor games.

(1) The teacher drew a picture on the playground in advance.

(2) Lead the students to the playground, and select three students to divide into two groups for activities to see who can calculate correctly and quickly.

5+3 -2 +4 -5 +3=

8-6 +4 +2 -3 +5=

7+2 -3 -4 +5 +3=

1+4 -5 +7 +2 -5=

8- 1 -3 +6 -7 +3=

2+6 -6 +5 +3 -8=

This will help students feel the process of addition, subtraction, multiplication and division and improve their computing ability. ]

4. Go back to the classroom and continue to practice.

(1) Display: Do it.

(2) student column calculation.

Fourth, class summary.

Guide students to summarize the operation order of addition and subtraction and addition and subtraction.