Movement of graphics (1) (2)

Symmetry, translation, and rotation are all important Euclidean geometric transformations. The so-called geometry is actually the study of the properties of geometric figures that remain unchanged during corresponding geometric transformations. Topological geometry is the study of the properties of geometric figures that remain unchanged during topological transformations (for example: proximity, separation, closure, continuity, etc.). Projective geometry studies the properties of geometric figures that remain unchanged during projective changes, and Euclidean geometry studies the properties that remain unchanged during Euclidean geometric transformations. Symmetry, translation, and rotation are the most important Euclidean geometric transformations.

Whether it is the concept of symmetry or the concepts of translation and rotation, they are all background concepts for children at this stage. At this time, the concept of symmetry in the mind only remains at the level of purely operational experience. For translation transformation, it is just regarded as repeated drawing, which is both mechanical and boring for children. For rotation transformation, because I have rich experience in daily life, I can complete game tasks quickly. However, what he presented was only the rudimentary rotation transformation experience he had, rather than the mature concept of rotation in his mind. Of course, from another perspective, children at this stage already have rich action experience, which lays a good foundation for them to formally begin to construct the concept of generating graphic transformations.

Symmetry: exists in daily game activities, such as origami activities, folding clothes, sheets... However, the concept of symmetry in mathematics is not separated from these daily game activities.

Translation: Children walk from point A to point B, the movement of toy cars on the ground, private cars running on the road, trains running on rails... Children have rich experience in translation movements.

Rotation: turn the rattle, turn the key chain, run around in a circle... The rotation experience accumulated in children's lives is also very rich.

Children at this stage have rich graphic movement experience, but these experiences exist unconsciously in specific game situations and have not yet consciously entered the child's conscious thinking field.

? For a long time, children have not been able to use the transformation ideal of a specific Euclidean geometric figure to study its geometric properties that remain unchanged during the geometric transformation process.

The first stage: the amusement park

The first section: the overall perception of the movement of the amusement projects

? The amusement projects in this amusement park: cable cars, sightseeing elevators , big pendulum, slide, rotating plane, small train, dragonfly kite, butterfly kite, etc.

The movement characteristics of each amusement item: the cable car moves along the track (describe it in words and use your little hands to draw it) ), the small train moves forward along the straight track (all students use their bodies or small hands to indicate the movement pattern), and the sightseeing elevator also moves up and down along a straight line. The big pendulum and the rotating plane rotate rapidly (all use their bodies or small hands to demonstrate the movement of rotation). The slide also moves along a straight line, but this straight line is oblique.

The second section: Classification of amusement items, naming three types of graphic movements

? Classification according to the movement mode: big pendulum, rotating plane, clock is rotating; cable car, sightseeing elevator, The train moves in a straight line.

? Mathematicians name this movement phenomenon rotation

? They all move along straight lines, called translation

? This The wings on both sides of the two kites are exactly the same. If they are folded in half, the two parts will completely overlap. This phenomenon is called symmetry transformation.

The third section: Share three movement phenomena in life

(Sharing movement methods throughout life, other students use their bodies or small hands to draw. For example, translation phenomenon: the orthosis can move up and down , curtains moving left and right, people walking in a straight line, movement of the blackboard in the classroom, opening and closing of drawers, etc.; rotation phenomena: rotation of revolving doors in hotels, rotation of the earth around the sun, rotation of the earth, rotation of fan blades, etc.; symmetry phenomena: rectangle , square and other figures of origami, kites, etc.)

? The door that can move left and right is translation, and the skipping rope, the blackboard without words, and the square are symmetrical phenomena.

The second stage: the movement of graphics: symmetry

? The first section: defining axisymmetric graphics

? How to judge symmetry? If the two parts of the figure can completely overlap after being folded, it means that it is symmetrical. Use words to describe an axially symmetrical figure: a figure with exactly the same size and shape on both sides of the figure is an axially symmetrical figure.

? The second section: hands-on operation to explore axisymmetric graphics

? Among the common plane graphics, square, rectangle, isosceles triangle, equilateral triangle, circle, and isosceles trapezoid All are axially symmetrical figures.

Fold it in half to see if the two parts can completely overlap. If they overlap, it is an axially symmetrical figure; if they cannot completely overlap, it cannot be called an axially symmetrical figure.

Fold it by hand to verify that the rectangle is an axially symmetrical figure. (Fold in half along the length and width, not along the diagonal) Use a pencil to draw the middle crease. This crease is called the symmetry axis of the axisymmetric figure.

? A rectangle has two axes of symmetry

A square has four axes of symmetry

Because the circle can be folded in half, the two parts can completely overlap, so A circle has countless axes of symmetry.

If this crease does not pass through the center of the circle, the two parts cannot completely overlap. That is, when folding in half, it must pass through the center of the circle to ensure that the two parts completely overlap.

An equilateral triangle has three axes of symmetry

An isosceles triangle has one axis of symmetry, and any triangle is not an axially symmetrical figure.

A triangle is an axially symmetrical figure. This sentence is unreasonable. Because a triangle with three sides of the same length is an axially symmetrical figure, or a triangle with two sides of the same length is also an axially symmetrical figure, and a parallelogram is not an axially symmetrical figure.

? The third section: comprehensive application

This is half of the axially symmetrical figure. Please complete the other half to make it an axially symmetrical figure.

As long as the two parts of the figure completely overlap after being folded along a straight line, it can be verified whether the supplemented figure is an axially symmetrical figure.

It can also be supplemented as shown below:

It can be supplemented by taking any side of the triangle as the axis of symmetry.

? You can also supplement the graphics from other directions with any straight line as the axis of symmetry.

Take a point as the symmetry point, and a straight line passing through this point as the symmetry axis.

? The third stage: the movement of graphics: translation

The fourth stage: the movement of graphics: rotation

A-level goal: let students make preliminary progress through life examples Perceptual rotation is a common graphic movement in life.

Level B goal: Through students' discussion and hands-on operation, they can understand the changes and invariances during the rotation process and establish a mathematical model of "rotation".

? C-level goals: Through observation and operation activities, develop students’ spatial concepts, cultivate students’ observation and hands-on abilities, discover the beauty of graphic transformation, feel the charm of mathematics, and stimulate interest in learning mathematics.

Section 1: Perceiving changes and changes in rotation

Observe this amusement ride (dynamic picture) and draw how it moves with your hands.

(Two students hold hands to demonstrate spinning.) They serve as two rotating seats.

You can also have one child stand still in the middle (acting as a pillar in the ride) while another student (acting as a rotating seat in the ride) rotates around him.

Use body language to demonstrate how a spinning plane moves. (The index finger of the right hand acts as the plane rotating)

The plane rotates around a pillar (the index finger of the left hand acts as the pillar, and the right hand rotates around the left index finger)

? Language description: The plane rotates around a pillar A pillar is spinning.

That is, the phenomenon of rotation is that an object rotates around a fixed point. During the rotation, the shape, size, center point position of the aircraft, the direction of the aircraft's rotation, and the distance from the aircraft to the middle pillar remain unchanged. (After the airplane seat rotates once, its trajectory is a circle) and the position of the airplane changes.

Rotation in life: revolving doors in some hotels, doors in classrooms (rotating around a straight line). During the rotation of the door, the size, shape, and center of rotation of the door do not change, but the position of the door changes.

Clocks are rotating phenomena, the position of the rotation center remains unchanged, and the shape and size of the second hand remain unchanged (share various rotation phenomena, describe who rotates around whom, who changes and who does not change during the rotation. )

Part 2: Hands-on operation, understanding rotation

Challenge to draw the rotation process. Requirements: First, operate the circle, observe carefully, and then draw a scene of the circle rotating once, and draw 4 to 6 different WeChat messages.

Error: The size and shape of the circle and the length of the rope (i.e. the distance from the circle to the center of rotation) cannot be changed during the rotation process. The size of the circle in the above picture keeps changing.

Challenge the rotation process of drawing an equilateral triangle

During the rotation process, the size and shape of the equilateral triangle have not changed, and the distance from the equilateral triangle to the center point has not changed. The location has changed. But the equilateral triangle at each position should not look like this. Initially, the thin line is connected to the top vertex of the triangle. At the second position after rotation, the vertex should be connected to the thin line. The third and fourth positions should be connected to the thin line. The same is true for this location. (The teacher demonstrated simultaneously on the projector)

The circle is a particularly perfect figure with countless axes of symmetry. When we draw the rotation of a circle, we only need to note that the size and shape of the circle remain unchanged at different times, and the distance from the circle to the center of rotation remains unchanged. However, an equilateral triangle is not so perfect. We must not only pay attention to the equilateral The size and shape of the triangle remain unchanged, the distance from the equilateral triangle to the center of rotation remains unchanged, and the orientation of each corner remains the same.

? The third section: Lines move into surfaces, and surfaces move into bodies

(Animation demonstrates the rotation of a line segment around one of its endpoints)

A line segment rotates around it One endpoint of is rotating and becomes an acute angle after rotation. One side of the acute angle is the starting position of the linear rotation, and the other side is the ending position. If you continue to rotate, it will become a right angle, an obtuse angle, a semicircle, or a circle (linear motion becomes a surface)

? A rectangle becomes a cylinder after rotation. A cylinder can be obtained by rotating the rectangle around one of its sides (the upper base of the cylinder is obtained around a point)

Euclidean geometry studies the geometric properties that remain unchanged in all geometric transformations, here The symmetry, translation, and rotation learned are very important rigid Euclidean geometric transformations

? The fifth stage: the movement of graphics: synthesis

How to supplement the image of this bunny whole?

Obtained based on the axis of symmetry, (Draw the axis of symmetry of the bunny, and draw the other half. When drawing, pay attention to the size and shape of the left and right sides. The two parts can completely overlap after folding.)

Use a square piece of paper to make it according to the following steps. The basis for making it is: axial symmetry

? The first step is to fold the square in half once. Basis: axial symmetry (the crease is the axis of symmetry)

? The second step is to fold the square paper in half and then fold it in half. This is axisymmetric motion. If you cut it like this, you will get four petals of the same size.

The petals are made based on axial symmetry. In this flower, there is rotational motion, and any petal rotates around the stamen.

If you fold the shape you just folded in half again, the cut flower will have 8 petals. That is, fold it in half once to get 2 petals, fold it in half twice to get 4 petals, fold it in half 3 times to get 8 petals...

Please use a piece of paper to make this pattern of the word "中" and explain it. The basis for its production.

First find the axis of symmetry of the word "中", then fold the colored paper in half, draw half of the word "中" along the symmetrical axis of the colored paper, cut it out, and the expanded shape will be a "中" Character.

Fold the colored paper in half twice in a row, then draw a little person on the twice-folded colored paper, cut out the little person and unfold it, and the conjoined little person is completed.

If you only fold it once, you need to draw two little people on the folded colored paper.

When making the conjoined minifigures, it reflects the axial symmetrical movement and translation

? The sixth stage: thinking brain map

? The first section: Teacher-student dialogue

The motion phenomena we know: axial symmetry, rotation and translation.

? Folding in half is not a movement phenomenon, but it can be used to determine whether a figure is axially symmetrical.

? If a figure is folded in half along a straight line and the two parts can completely overlap, we call this straight line the axis of symmetry of the figure.

Characteristics of the axis of symmetry: When an axially symmetrical figure is folded in half along the axis of symmetry, the two parts can completely overlap.

Axisymmetric graphics: square, rectangle, circle, special triangle.

A square has 4 axes of symmetry, a rectangle has 2 axes of symmetry, an equilateral triangle has 3 axes of symmetry, an isosceles triangle has 1 axis of symmetry, and a circle has countless axes of symmetry (a circle is a perfect figure , when folded in half, you can get the axis of symmetry of the circle as long as it passes through the center of the circle). A parallelogram is not an axially symmetrical figure.

A figure moves along a straight line. This kind of figure movement is called translational movement. For example, walking, a small train, a car traveling in a straight line, a sightseeing car, a straight line translated to the right to form a rectangle or square, a rectangular piece of paper translated upward to obtain a cuboid.

? During the translation process, the size and shape of the graphics do not change, but the position of the graphics changes.

Common rotation phenomena in life: doors, windmills, clocks...

Rotation means that a figure rotates around a fixed point. During this process, the shape and size of the graphics, the position of the rotation center, the distance between the rotation center and the rotating object have not changed, but the WeChat of the object has changed.

? Second section: Sharing and communication

? Evaluate this work:

There are no examples for each branch. The axis of symmetry should be written on the axis of symmetry. On the branches, the picture on the left belongs to the translational phenomenon of the circle, and should be drawn next to the translational branch

What is worth learning from the following work, and what needs to be improved?

Every kind of graphic movement is given examples from life, clearly explaining the nature of each graphic movement, what has changed and what has not changed. The learning of graphic motion has not stopped and will continue in the future.

Instead of accurately learning rotation, we have an overall understanding of three types of graphic movements: classifying them according to the characteristics of the amusement projects, such as the Ferris wheel rotating, the small train translating, and the kite being axially symmetrical. Understand the three modes of movement as a whole, then learn them separately, and then make the character "中" according to the axis of symmetry, a conjoined figure

The third section: Share and display works