Draw a coordinate system on graph paper. Math lesson "coordinate plane"
2. GRAPHING
In a laboratory workshop and when performing computational and graphic (semester) work in physics, it often becomes necessary to build graphical dependencies. When drawing up charts, you must follow the rules listed below.
1. Graphs are built on millimeter paper with a format of at least 1416 mm(page of a standard notebook). The finished graph should be glued to the lab report.. As an exception, it is allowed to build dependencies using standard computer programs - but even in this case, the graphs must comply with all the requirements set forth here (in particular, they must have a scale-coordinate grid).
2. On the coordinate axes, the designations of the values being plotted and their units of measurement must be indicated.
3. The origin of coordinates, unless otherwise stated, may not coincide with the zero values of the quantities. It is chosen in such a way that the drawing area is used as much as possible.
4. Experimental points are depicted clearly and large: in the form of circles, crosses, etc.
5. Scale divisions on the coordinate axes should be applied evenly. The coordinates of the experimental points on the axes are not indicated, and the lines defining these coordinates are not drawn.
6. The scale is chosen so that:
a) the curve was evenly stretched along both axes (if the graph is a straight line, then the angle of its inclination to the axes should be close to 45);
b) the position of any point could be determined easily and quickly (a scale at which the reading of the graph is difficult is considered unacceptable *).
7. If there is a significant spread of experimental points, then the curve (straight line) should be drawn not by points, but between them - so that the number of points on both sides of it is the same. The curve must be smooth.
Example 7 Let it be required to plot a path dependency graph S from time t with uniform body movement. Experimental data are given in table. 4. Two variants of the dependency graph S(t) - issued with errors and correct - are shown in Fig. 4 and 5.
Table 4
|
S, m |
The main, most typical mistakes made by students when plotting graphs (Fig. 4):
incorrectly chosen directions of the coordinate axes: time t is the independent variable (argument) and should be plotted along the abscissa (horizontal) axis, and the dependent variable (function) is the path S– along the y-axis (vertical);
the ordinate value is not indicated on the y-axis (time t) and units of its measurement ( With), and on the x-axis - the units of the path S (m) - see item 2;
the area of the drawing is not fully used (since it does not follow from the condition of the example that the coordinate axes should start from zero values, then the origin of coordinates should be shifted and thereby increase the scale of the graph) - see item 3;
experimental points are not highlighted - p. 4;
scale divisions on the time axis are plotted unevenly (if there are divisions 0 and 5, then the next should be 10, etc.) - item 5;
the axis of the path is marked not with scale divisions, but with the coordinates of the experimental points; extra dotted lines are drawn - see also item 5;
the graph is compressed along the x-axis due to two reasons: an incorrectly chosen origin (item 3) and an unsuccessful (too small) scale - item 6, a;
an extremely inconvenient time scale was chosen, and therefore it is difficult to read the graph - item 6, b;
the experimental points are incorrectly connected: the dependence of the path on time with uniform motion is obviously linear, and the graph should be a straight line - item 7.
The correct chart is shown in Fig. 5.
* The scale is convenient for reading the graph if the unit of the value plotted along the axis contains one (or two, five, ten, twenty, fifty, etc.) linear unit - a millimeter or a centimeter. The inconvenient but often used scale of 15 or 30 should be avoided. mm per unit of magnitude.
MOU "Lyceum No. 7 named after Shura Kozub with. Novoivanoskoe»
Teacher: Russ Elena Nikolaevna
Subject: maths
Class: 6 - general education
Software and methodological support: planning is made based on the author's planning by N. Ya. Vilenkin according to the textbook "Mathematics - Grade 6". Textbook: Vilenkin N. Ya.
Mathematics 6th grade Proc. for general education institutions. Moscow: Mnemosyne, 2014.
Module:"Coordinate plane"
Lesson topic: "Coordinate plane"
Lesson type: generalization lesson
Methods: illustrative and explanatory, partially exploratory
Learning technology: modular.
Training
element
Teaching material indicating tasks
Management
on the assimilation of the material
UE 0
Target:
be able to build points according to given coordinates using graph paper;
be able to find the coordinates of points using graph paper;
be able to determine the location of points on the coordinate plane without constructions.
UE 1
Target: enhance students' knowledge of the topic.
The cheerful bell rang
Is everyone ready? All is ready?
We don't rest now
We start working
Guys, we have guests at the lesson today, greet them.
What's unusual in our class today?
Why is it called rectangular?
Who invented it?
Where can we use it?
How many numbers must be specified to specify the position of a point on the coordinate plane? (two)
What is the name of the rays that form the coordinate plane?
What is the name of the first number that specifies the position of a point on the coordinate plane? (abscissa)
What is the ordinate of point A (- 1; - 4)?
Answer the questions in writing in a notebook.
Mutual verification.
UE 2
Target: learn how to find the coordinates of points using graph paper
? Draw points on the coordinate plane
A (4; 6); B (1.2; - 3.4); C (- 3.25; - 4.75).
What problem are you facing? (it is inconvenient to mark fractional coordinates on a notebook sheet)
What exit can be found? (use graph paper)
What will be discussed in today's lesson?
(about the coordinate plane)
What are we going to learn in class? (mark points by given coordinates and find coordinates of points on graph paper)
Conversation
What is the unit line equal to?
Into how many parts is a single segment divided?
What is one part equal to?
Find the coordinates of the points.
A (1.3; 2); B (- 1; 2.2); C (- 1.3; 1.2); D(-1.7; 0);
E(-1.3; -2.4); F(-0.8; -1.7); M (1.5; - 1.8); K(0;-2.7)
Students complete the assignment in their notebooks.
Answer verbally.
Formulate the topic and objectives of the lesson. Write the topic of the lesson in a notebook.
They answer questions.
Perform the task (Appendix 1).
The coordinates of points A, B, C are found by commenting, the coordinates of the remaining points are independently
One student completes the task on the back of the board.
Checking is carried out frontally.
UE 3
Target: determine the location of points on the coordinate plane without constructions.
Conversation
What are the coordinates of point A? (positive)
What quadrant is point A in? (in the first)
Mark one more point (point T) in the first coordinate quarter. What are the coordinates of this point? (positive)
What can be seen? (points lying in the first coordinate plane have positive coordinates)
Explore the points located in the II, III and IV coordinate quarters on your own.
Make a conclusion.
Conclusion:
For points located in the second quarter, the abscissa is negative, and the ordinate is positive;
The points located in the third quarter of the abscissa and the ordinate are negative;
For points located in the fourth quarter, the abscissa is positive, and the ordinate is negative.
Students answer questions.
The dependence of the location of points on the coordinate plane on the sign of the coordinates is revealed.
Make their own conclusion.
EC 4
Target: teach how to build points according to given coordinates using graph paper.
Plot Coordinates of Points (1; - 2,2); (2; 4,2); (3; - 0,6); (4; 2,3); (5; 1,1)
Mark them on the coordinate plane shown on graph paper.
Evaluation standards.
"5" - for 5 correctly marked points
"4" - for 4 correctly marked points
"3" - for 3 correctly marked points
"2" - for 2 or less marked points
Independently mark the received coordinates.
Sample self-test.
Independent troubleshooting.
The sheet of graph paper on which the task was performed is handed over to the students for verification.
Fizminutka
The game
UE 5
Starry sky video clip
I see you are ready to travel. So, imagine that you are lying under the starry sky on one of the beautiful, warm summer evenings. And before you stretched out an immense, sparkling sky.
On a cloudless clear evening, the whole sky is strewn with many stars. They appear as small sparkling dots. But in fact, these are huge hot gas balls. If certain stars are connected on the map with conditional white lines, then fabulous figures will appear before us - constellations, each of which has its own name. The entire sky is divided into 88 constellations, of which 54 can be seen on the territory of our country.
Many constellations retain their name from ancient times. They were invented in Ancient Greece. The Greeks, excellent navigators, determined the path by the celestial constellations. The names of the constellations are very beautiful: Cassiopeia, Andromeda, Perseus, Dragon and others.
Are you curious to know why they are called that?
Let's split into groups. Each group gets a task
Do you want to see the end of this legend?
Cartoon demonstration.
UE 5
Target: sum up the lesson, set marks, ask d / z.
You are just great today. Very beautiful constellations turned out, everyone actively cooperated. At the end of the lesson, I want you to say one sentence at a time, but start with the words on the board.
Grading.
D / z The name of some constellations is associated with the objects they resemble: Arrow, Triangle, Libra and others. There are constellations named after animals: Leo, Cancer, Scorpio. Draw on coordinate plane
Plotting
When performing experiments in laboratory work, it is often necessary to build graphs of functional dependencies of the form Y=f(X).
In this case, the following rules should be followed:
1. The values of the independent variable (X) are plotted along the abscissa (horizontal axis), and the values of the function (Y) are plotted along the ordinate.
2. The dimensions of the graph, the thickness of the dots and connecting lines should provide the necessary reading accuracy, as well as ease of use of the graph.
3. All points on which the graph is built must be marked on the graph. In this case, one should not specially set aside the values corresponding to the points on the axes.
4. The plotted points are connected by a smooth curved line, that is, when constructing the line, smoothing should be applied, taking into account the general nature of the resulting dependence. In this case, some points plotted on the graph may not fit into the resulting curve (due to measurement inaccuracies at these points). Due to the fact that the measurement is carried out at several points, the use of smoothing reduces the influence of these inaccuracies. Figure 1 shows examples of plotting graphs for the same points, correct (Fig. 1, a) and - incorrect (Fig. 1, b). The thickness of the points in the example is chosen large for clarity of presentation.
5. On the coordinate axes, the values \u200b\u200bof X and Y must be plotted, units of measurement in convenient values \u200b\u200bare indicated. To express a measured value with a numerical value, it is advisable to use decimal multiples and submultiples derived from the base unit and expressed as numerical values between 0.1 and 1000. This approach provides the most convenient perception of numerical data.
For example: instead of 50,000 Hz, it is more convenient to use 50 kHz, instead of 2 10 -3 A - 2 mA.
6. If two dependencies are plotted on one graph Y 1 \u003d f 1 (x) and Y2= f 2 (x) and the intervals of values in which the values of Y1 and Y2 are located differ from each other by more than 1.5 times, for each of these functions on the y-axis one should plot its own scale (otherwise the graph errors for each of the dependencies will be very different from each other ). Figure 2, a shows an example of the correct construction of a graph, in Figure 2, b - incorrect (the thickness of the points in the example is chosen large for clarity).
5. The graph must be provided with a caption, which contains information about which dependence is built and for which device.
Graph Scale Calculation
The accuracy of the reading depends on the size of the graph, however, the usability of it may suffer. Therefore, the scale of the graph is calculated based on real conditions.
When plotting instrument calibration graphs, the error introduced by the graph (δ gr) is chosen less than the error of the instrument itself (δ pr) by about 5 times. In this case, the total error δ Σ (taking into account the error introduced by the graph) will not differ significantly from the error of the device itself:
Plotting on graph paper.
In the case of plotting on graph paper, the absolute error of the graph in units of length is chosen equal to Δl=0.5 millimeters (half the division value of the millimeter grid). Then, taking into account the accepted conditions, the scale of the graph can be calculated by the formula
