Design of blackboard writing in junior high school mathematics classroom teaching
Properties of Angular bisector (2)
Teaching objectives
1, the properties of angular bisector
2. The properties of the angular bisector will be described. The points with equal distance to both sides of the angle are on the bisector of the angle? .
3. These two properties can be used to solve some simple practical problems.
Teaching focus
Properties and application of angular bisector.
Teaching difficulties
Flexible application of two properties to solve problems.
teaching process
Ⅰ. Creating situations and introducing new lessons
Take out the origami and scissors prepared before class, cut a corner, fold the cut corner in half so that the two sides of the corner overlap, and then unfold the paper. What do you see? Fold the folded paper again at will, and then unfold it. What do you see?
Analysis: the crease after the first folding is the bisector of this angle; Fold it again, and there will be two creases with the same length. This method can be done countless times, so this crease with the same length can be folded into countless pairs.
Ii. introducing new courses
What conclusions can be drawn from the properties of the angular bisector, that is, the known angular bisector?
Fold the creases PD and PE as shown.
Draw a picture:
Draw three creases of a corner in the order of origami, and measure whether the drawn PD and PE are equal in length.
Project the following two pictures for students to comment and make the concept clear.
Conclusion: The drawing method of classmate B is correct. Classmate A draws a point on the bisector to draw the vertical line of the bisector, instead of a point on the bisector as the vertical line on both sides, so his drawing method does not meet the requirements.
Question 1: How to describe the nature of drawings in written language?
The distance between points on the bisector of an angle is equal to both sides of the angle.
Question 2: Can it be translated in symbolic language? Is the distance from the point on the bisector of the angle equal to both sides of the angle? This sentence. Please fill in the following form:
Known: OC split equally? AOB，PD？ OA，PE？ OB, d and e are vertical feet.
A substance derived from a known substance: PD=PE.
So we get the properties of the angular bisector:
A point on the bisector of an angle is equal to the distance on both sides of the angle.
[T] So are all the points with equal distance to both sides of the corner on the bisector of the corner? (display projection)
Question 3: According to the numbers and known substances in the table below, guess what can be inferred from the known substances and fill in the table with symbolic language:
[Discussion] It is known that the congruence condition of right triangle is met, so RT △ peo △ PDO (HL) can be obtained? PDE=？ Pods.
Known: Where is point P? On the bisector of AOB.
From this, we can get another property: the points with equal distance to both sides of the angle are on the bisector of the angle. Is there any connection between these two attributes?
Analysis: The known conditions and derived conclusions of these two properties are interchangeable.
Thinking:
As shown in the figure, build a market in area S, so that the distance between it and roads and railways is equal, and it is 500m away from the intersection of roads and railways. Where should this market be built (mark the location on the map with the scale of 1: 20000)?
1. Where is the market built, and does it have anything to do with the nature of the bisector of this class? Which property can solve this problem?
2. What does the scale of1:20000 mean?
Conclusion:
1. The second attribute should be used. This market should be built on the bisector of the corner formed by the highway and the railway, and it is required to be 500 meters away from the apex of the corner.
2. When drawing on paper, we often take centimeters as the unit, while the distance in the topic is meters as the unit, which involves a unit conversion problem. 1m= 100cm, so the scale is 1: 20000, which actually refers to the actual distance of 200m in the drawing. The drawings are as follows:
Step 1: How to draw a ruler and a gauge? The bisector OP of AOB.
Step 2: Intercept OC=2.5cm from ray OP, and determine point C, which is where Bazaar is built.
Conclusion: By applying the properties of bisector, the steps of proving triangle congruence can be omitted and the problem can be simplified. Therefore, if you encounter the problem of bisector, prove that the line segments are equal, and you can directly use the properties to solve the problem.
Examples and exercises
For example, as shown in the figure, the angular bisectors BM and CN of △ABC intersect at point P. 。
It is proved that the distances from point P to three sides AB, BC and CA are equal.
Analysis: The lengths of vertical line segments PD, PE and PF from point P to AB, BC and CA are the distances from point P to three sides, which means that it is necessary to prove that PD=PE=PF, while BM and CN are respectively? B, is it? According to the properties of the angular bisector and the transitivity of the equation, the bisector of C can solve this problem.
Proof: a small P as PD? AB，PE？ BC，PF？ AC, the vertical feet are d, e and F.
Because BM is the bisector of △ABC, point P is on BM.
So PD=PE.
Similarly, PE=PF.
So PD=PE=PF.
That is, the distance from point P to trilateral AB, BC and CA is equal.
Exercise:
1. Textbook P 107 Exercise.
2. Textbook P 108 Exercise 13.3─2.
It is emphasized that the properties of angular bisector should be used directly under sufficient conditions, and it is not necessary to prove triangle congruence.
Four. Course summary
Today, we learned two properties of the bisector: ① the distance between a point on the bisector and both sides of the angle is equal; (2) The points with equal distance to both sides of the angle are on the bisector of the angle. They are mutual, and with the deepening of learning, it is easier and easier to solve problems. For example, we can directly use the properties of angular bisector to prove the equality of straight lines without having to prove the congruence of triangles.
ⅴ. Homework after class
1, textbook exercise13.3 ─ 3,4,5.
2. Classroom perception and exploration
Design of blackboard writing in junior high school mathematics classroom teaching II
Axisymmetric (1)
Teaching objectives
1. Understanding axisymmetric graphics in real life.
2. Analyze the axisymmetric figure and understand the concept of axisymmetric.
Teaching focus
The concept of axisymmetric figure.
Teaching difficulties
Can identify the axisymmetric figure and find its axis of symmetry.
teaching process
Ⅰ. Creating situations and introducing new lessons
We live in a symmetrical world. Many buildings are designed to be symmetrical. The creation of artistic works is often considered from the perspective of symmetry. Many animals and plants in nature also grow symmetrically. Some of China's characters are also symmetrical. What a wonderful feeling symmetry brings us! A preliminary grasp of symmetric seconds can not only help us find some graphic features, but also make us feel the beauty and harmony of nature.
Axisymmetry is an important symmetry. Starting from this lesson, let's learn Chapter 14: Axisymmetry. Today, let's learn the first section to understand what is an axisymmetric figure and what is an axis of symmetry.
Ii. introducing new courses
Show pictures of textbooks and observe what they have in common.
These figures are symmetrical. After these figures are separated from the middle, the left and right parts can completely overlap.
Summary: Symmetry is everywhere. People can find examples of symmetry from natural landscape to molecular structure, from architecture to artworks, and even daily necessities. Now students will look for some symmetrical examples from the things around us.
Our blackboard, desks, chairs, etc.
Our bodies, as well as airplanes, cars and maple leaves, are all symmetrical.
For example, as shown in the textbook figure 14. 1.2, fold a piece of paper in half, cut out a pattern (don't cut off the crease completely), then open the folded paper and cut out beautiful window grilles. What can you find about the window grilles and the characters in the picture 14. 1
Window grilles can be folded in half along the crease, so that the parts on both sides of the crease completely overlap. Window grilles can not only be folded in half along a straight line, so that both sides of the straight line overlap, but also the figures in the above figure 14. 1. 1 can be folded in half along a straight line, so that the parts on both sides of the straight line overlap.
Conclusion: If a figure is folded along a straight line, the parts on both sides of the straight line can overlap each other. This figure is called an axisymmetric figure, and this straight line is its axis of symmetry. At this time, we also say that this figure is symmetrical about this straight line (axis).
After understanding the concept of axisymmetric figure and its symmetry axis, let's do it.
Take a piece of hard paper, fold the paper in half, carve a pattern at random in the center of the paper with a knife, open the paper and smooth it. Do you have two symmetrical patterns? Communicate with your peers.
Conclusion: The patterns on both sides of the crease are symmetrical and they can overlap each other.
From this, we can get the characteristics of axisymmetric graphics: after a graphic is folded along a straight line, the graphics on both sides of the crease are completely coincident.
Next, we discuss a question about the axis of symmetry. Some axisymmetric figures have only one axis of symmetry, but some have multiple axes of symmetry, and some even have countless axes of symmetry.
Can you find the symmetry axis of the following picture?
Results: The graph (1) has four symmetry axes. Figure (2) has four symmetry axes; Figure (3) has numerous symmetry axes; Figure (4) has two symmetry axes; Figure (5) has seven axes of symmetry.
( 1) (2) (3) (4) (5)
Show the wall chart. Think about it. What did you find?
In this way, a graph is folded along a straight line. If it can overlap with another graph, it is said that the two graphs are symmetrical about this straight line, which is called the symmetry axis, and the overlapping point after folding is the corresponding point, which is called the symmetry point.
ⅲ. Classroom exercises
(a) textbook P 1 17 exercises (B) P 1 18 exercises.
Ⅳ. Class summary
This lesson mainly knows the axisymmetric figure, understands the axisymmetric figure and related concepts, further discusses the characteristics of axisymmetric, and distinguishes the axisymmetric figure from the axisymmetric formed by two figures.
Verb (short for verb) homework
(1) textbook exercise 14. 1─ 1, 2, 6, 7, 8.
Homework after class:>
ⅵ. Activities and surveys
Reflections on the textbook P 1 18.
Are two symmetrical figures congruent? If an axisymmetric figure is divided into two figures along the axis of symmetry, are the two figures the same? Are these two figures symmetrical?
Process: Draw two symmetrical figures on the cardboard, and then cut them out with scissors to see if they overlap. Draw another symmetrical figure on the cardboard, then cut it out and cut it along the symmetrical axis to see if the two parts can completely overlap.
Conclusion: Two axisymmetric figures are congruent. If an axisymmetric figure is divided into two figures along the axis of symmetry, the two figures are congruent and axisymmetric.
Axisymmetric refers to the positional relationship between two graphs, while axisymmetric graphs refer to graphs with special shapes.
Two axisymmetric graphics and axisymmetric graphics should be folded along a straight line and then overlapped; If an axisymmetric figure is divided into two parts along the axis of symmetry, then these two figures are axisymmetric about this straight line; On the contrary, if two axisymmetric figures are regarded as a whole, they are axisymmetric figures.
blackboard-writing design
? 14. 1. 1 axis symmetry (1)
1. Axisymmetric: If a graph is folded along a straight line, the parts on both sides of the straight line can completely overlap. This graph is called an axisymmetric graph, and this straight line is called an axis of symmetry.
2. Two graphs are symmetrical: one graph is folded along a straight line, and if it can overlap with another graph, then the two graphs are said to be symmetrical about this straight line.
Design of blackboard writing in junior high school mathematics classroom teaching III
Axisymmetric (2)
Teaching objectives
1. Understand the essence of axial symmetry formed by two figures, and understand the essence of axisymmetric figures.
2. Explore the nature of the vertical line.
3. Go through the process of exploring the properties of axisymmetric graphics, further experience the characteristics of axisymmetric, and develop space observation.
Teaching focus
The properties of 1. axial symmetry.
2. The nature of perpendicular bisector.
Teaching difficulties
Experience the characteristics of axial symmetry.
teaching process
Ⅰ. Creating situations and introducing new lessons
Last class, we discussed axisymmetric graphics, knowing that the world has become very beautiful because of axisymmetric graphics in real life. Well, let's think about it. What is an axisymmetric figure?
Today we will continue to study the properties of axial symmetry.
Ii. introducing new courses
Watch the projection and think.
As shown in the figure, △ABC and △A? b？ c？ On the symmetry of line MN and point A? 、B？ 、C？ Are the symmetrical points of points A, B and C, and the line segment AA? 、BB？ 、CC？ What does it have to do with linear MN?
A, a in the picture? It's a symmetry point, AA? Vertical to MN, BB? And CC? Also perpendicular to MN.
AA？ 、BB？ And CC? Apart from vertical, is it related to MN?
△ABC and△ a? b？ c？ On the symmetry of line MN and point A? 、B？ 、C？ Is the symmetry point of point A, point B and point C, let AA be? The symmetry axis MN is at point P, while △ABC and △A? b？ c？ After being folded in half along MN, it is divided into a and a? Coincidence, so there is AP=A? p，？ MPA=？ MPA？ =90? So AA? 、BB？ And CC? In addition to being perpendicular to MN, MN also crosses the line segment AA? 、BB？ And CC? The midpoint of.
The line where the axis of symmetry lies passes through the midpoint of the line segment connected by the symmetrical points and is perpendicular to the line segment. We call the straight line passing through the midpoint of the line segment and perpendicular to the line segment as the middle perpendicular of the line segment.
Draw an axisymmetric figure by yourself, find out two symmetrical points, and see the relationship between the symmetrical axis and the connecting line of the two symmetrical points.
We can see that an axisymmetric figure is just like two figures are symmetrical about a straight line, and the straight line where the axis of symmetry lies passes through the midpoint of the line segment connected by symmetrical points and is perpendicular to this line segment.
Summarize the properties of symmetry of graphs;
If two graphs are symmetrical about a straight line, then the symmetry axis is the middle perpendicular of the line segment connected by any pair of symmetrical points. Similarly, the axis of symmetry of an axisymmetric figure is the perpendicular bisector of line segments connected by arbitrary symmetrical points.
Let's discuss the nature of the median vertical line.
[Explore 1]
As shown below, the battens L and AB are nailed together, and L vertically bisects AB, P 1, P2, P3,? Is the point on L, measuring point P 1, P2, P3,? What do you find is the distance from A to B?
1. Transform the above problems with the plan. First, make a line segment AB, and the midpoint passing through AB is the median line L of AB, and take P 1, P2, P3? , links AP 1, AP2, BP 1, BP2, CP 1, CP2?
2. After drawing, measure AP 1, AP2, BP 1, BP2, CP 1, CP2? Discuss what kind of rules are found.
Query results:
The distance between the point on the vertical line of a line segment and the two endpoints of the line segment is equal, that is, AP 1=BP 1, AP2=BP2,?
certificate
Proof 1: used to judge the congruence of two triangles.
As shown below, in △APC and △BPC,
△APC?△BPC PA = PB。
Proof 2: Using the axisymmetric property.
Since point C is the midpoint of line AB, if line AB is folded in half along line L, then line PA and line PB coincide, so they are equal.
With the conclusion of exploring 1, let's look at the following questions.
[Survey 2]
As shown on the right, make a simple one with a wooden stick and an elastic rubber band. Bow? ,? Arrow? How can I keep the arrow perpendicular to the stick when shooting through the hole in the center of the stick? Why?
Activities:
1. Transform the above problems with plane graphics. As the line segment AB, take the midpoint P, the crossing point P is L, and take the points P 1 and P2 on L to connect AP 1, AP2, BP 1 and BP2. There are two possibilities.
2. Discussion: What conditions should be met to make L perpendicular to AB, AP 1, AP2, BP 1 and BP2?
Query process:
1. As shown in Figure A above, if AP 1? BP 1, then after the graph is folded along l, A and B cannot overlap, that is? APP 1BPP 1, that is, l is not perpendicular to AB.
2. As shown in Figure B above, if AP 1=BP 1, then after the graph is folded along L, A and B just coincide, and? APP 1=？ BPP 1, that is, l coincides with AB. The same is true when AP2=BP2.
Query conclusion:
The point with equal distance from the two endpoints of a line segment is on the middle vertical line of this line segment. That is to say, in Figure [Question 2], as long as the distance between the two ends of the arrow and the two ends of the bow is equal, the direction of the arrow can be kept perpendicular to the wooden stick.
[T] The results of the above two questions give the nature of the midline of the line segment, that is, the distance between the point on the midline of the line segment and the two endpoints of the line segment is equal; On the other hand, the points with equal distance from the two endpoints of this line segment are all on the vertical line, so the vertical line of this line segment can be regarded as the set of all points with equal distance from the two endpoints of this line segment.
ⅲ. Classroom exercises
Textbook P 12 1 exercise 1 2.
Ⅳ. Class summary
By exploring the symmetry process of axisymmetric figures, this lesson understands the properties of the vertical line in the line segment, and students should use these properties flexibly to solve problems.
ⅴ. Homework after class
(1) Textbook Exercise14.1-3,4,9.
Homework after class:>
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