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4. GRADE GEOMETRY TESTS
PROPERTIES OF THE TESTS *
We try to prepare three groups of tests.
The first group is directed to teach. We expect the children to learn the subject as they solve the problems. These contain tips and helps (with figures). Their number changes according to the question. There is no time limit. The pupil or student may browse the web and gather the lacking information meanwhile.
For the 4th Grade we have (almost) completed this group. Some questions are beyond level (directed to the gifted) and we will make arrangements on the way. They may be skipped by the average learners for the time being.
The second group of our tests are time limited with 2 minutes per question. They are a step towards the realtime tests which normally permit a minute per question. They are for the level and may contain tips and help depending on the situation.
The third group consists of real time tests as they normally are offered.
* In the followings you find our evaluations of our tests. We know we are here to be evaluated and not to evaluate. We look forward for your feedback and contributions. As they arrive, via our
forum, email etc.
we will publish them and this page will then change.
TOPICS and FIRST GROUP of TESTS
BASICS OF GEOMETRY
T1.1: Basics of Geometry (Tests 4th Grade 1.1)
Basic geometrical concepts like point, line, plane, angle and their measures (if measurable) are subjects of the first 6 questions of the tests. Question 12 is about congruence and similarity. Question 20 is something seen everyday though it is mostly not correctly answered. The questions (14 , 15 and 22) related with the angles between intersecting lines within the region between two parallel lines deserve attention.
T1.2: Basics of Geometry (Tests 4th Grade 1.2)
Questions 1 and 2, both give really an equation with an unknown in a very simple way. Angles including vertical angles are mostly the subject of questions, most of the questions. Question 7 creates an illusion and happens to be hard to see. Question 22 is an important question that should be learned and desreves special attention .
T1.3: Basics of Geometry (Tests 4th Grade 1.3)
Question 1 and 2 are asked to derive the necessary equations among 1 or two unknowns and to solve them. They may at the first look be evaluated as hard but they are not. Most of the questions around 9 are related with the fact that there are a redundant set of basic facts of geometry out of which some may be chosen as axioms and some other proven based on these axioms. Alternate interiors and exteriors are defined meanwhile. Question 18 and 22 deserve spacial attention in addition.
PLANE GEOMETRY (BASICS)
T1.1: Basics of Plane Geometry (Tests 4th Grade 2.1)
The first 12 questions are about triangles because they are the simplest polygons that are contained in all of them. Question 22 can be solved with the help of the triangles within the polygon. Triangle inequality (question 6) must be learned. Then square and regular polygons take their places.
T1.2: Basics of Plane Geometry (Tests 4th Grade 2.2)
Questions 1 and 2 are worth attention. With the help of regular polygons having a large number of sides we will try to attain the number π in the future. Question 11 is interesting; it should be solved if possible. Question 22 deserves special attention. Anyone who solves it can solve most of the geometry problems that are to be encountered during the next classes.
T1.3: Basics of Plane Geometry (Tests 4th Grade 2.3)
The first 2 problems are interesting. Question 7 is worth special attention. Be careful with 8. Question 10 shold be understood; so comes the Pythagorian Thorem, one of the milestones in geometry. Questions with figures out of matches are entertaining and brain training. Question 22 is once more a simple two equationstwo unkowns problem .
SOLID GEOMETRY (BASICS)
T1.1: Basics of Solid Geometry (Tests 4th Grade 3.1)
There are two basic shapes that are candidates for being the basic shape in solid geometry: The cube and the sphere. We adopt the cube because we need shelters. So we begin with the cube. Each of the first 12 questions is somewhat related with it. Nothing interesting.
Then there comes one of the best known puzzles : The Rubik's Cube. Qustions 1318 are about it. Then we introduce the rectangular prism. The only interesting quetion for practical minds is question 22. The ones who can solve it have learned more than enough for the grade.
T1.2: Basics of Solid Geometry (Tests 4th Grade 3.2)
Euclidean Geometry is so intuitive that the only mystery of plane geometry is the number π. The mysteries of solid geometry correspondingly are the volumes of the shapes like the pyramid, the cone and the sphere. Thes mysteries can be properly handled after having learned calculus. They can also contribute to the understanding of the calculus.
To help learning of the volume of a pyramid we ask two questions and give helps. The first pyramid we get by dividing a brick into two halves and getting two triangular prisms. We than chip one of these and get a pyramid whose approximate volume we calculate via simple averaging. That is question 7.
In questions 12 and 13 we calculate the volume with the help of slices having shape of rectagular prisms. We compare our results via calculation of accuracies.
When we have enough silces and plot the volume vs. the number of slices taken into account we will see that ... . We will see this in a couple of years. It is called the Fundemental Theorem of the Calculus foundations of which we owe to Newton and Leipnitz.
T1.3: Basics of Solid Geometry (Tests 4th Grade 3.3)
Here we repeat the same excercise with a cone. With a sphere we can not yet, we do not know trig. Our new exercise is our question 2.
Question 12 of this test is also interesting. We see how to use different methods to obtain the same result.
Then comes the sphere. After a couple of exercises with sphere and sphere in a cylinder we ask question 22. If you can solve it and understand the solution you are a really gifted student beyond Grade4.
SYMMETRY
T1.1: Symmetry (Tests 4th Grade 4.1)
Symmetry may be naive but it has a formal structure also. On top of this formal base come the advanced topics used in particle and solid state physics. We ask relatively simple questions in this test.
In the last part we make an introduction to the coordinate geometry via symmetry.
T1.2: Symmetry (Tests 4th Grade 4.2)
The test is simple. Questions 8 and 9 are a 4th graders introduction to cryptography. Some basic shapes of Solid State Physics are introduced. Some concepts like group and group generators are implicitly implied. Question 22 is a naive step towards Fourier Analysis.
T1.3: Symmetry (Tests 4th Grade 4.3)
Rotational symmetry is the simplest. Test is almost trivial. Spherical symmetry and distribution of charges on a spherical conductor is just mentioned. This is question 13. Just before it question 12 mentions the 1/r
^{2}
law of radiation. Pentaminos and hexiamaonds are introduced.
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