Showcase - Mathematics

Showcase - Mathematics

Author: EduPlexor

Combinations

When the order of the elements chosen from the set is not impotrant, we call this a combination. Putting it differently, the combinations are subsets of a given set.

Example:

We have kinds of fruit. How many deifferent kinds of fruit salads with kinds of fruits can we make?

Rule:

We have a certain number of different elements, , and the number of place, , in which the spaced elements, but the order of elements isn't important. The allocation of elements on place are variation. In doing so too often considered the allocation of the same elements at different place. The arrangement elements on place is . With this number divided number of variations to obtain the number of combinations. The form is:

This type of selection is also known as combinations without repetition.

Solution example

THINK ABOUT IT:

How are combinations different from the variations?

Answer

 
Theory can be made interesting and interactive using picture gallery!

In variations the order is important, in the combinations it isn't!

EXAMPLE:

Compare games LOTO and LOTKO. In Loto 7 balls with numbers are selected. The order in which the numbers were selected is not important and the grand prize winner is who has all numbers on his ticket. In the game Lotko 6 numbers are selected and the grand prize winner is the one, who has numbers on his ticket in exactly the same order as they were selected.

In Loto game the sequence , , , , , , represents the same sequence as , , , , , , . In Lotko, the sequence and represent two different combinations.


Solution of the example:

Scheduling elements in places are variations. There are of them.

Strech of the angular function

By stretching the function in the direction of -axis, zeros, minimums and maximums all change.

In this case, the rotating speed, which tells us the number of waves on length changes.

EXAMPLE:
Stretch the sinus function in the direction of the -axis:



EXAMPLE:


(/files/101/raztegx_sin.png)

THINK ABOUT IT:

What is the period of the function ?

Answer

How do you calculate the period with ?

Answer



 
Further questions can be made in a form of a pop-up window!

Answer

Period of a function is . In the picture we can see in what interval the wave repeats itself!

Answer

Calculate it with .

Conjugation

For each complex number we can find a conjugate number , by changing the sign in front of the imaginary part.

(/files/101/konjugiranjea.png)
When conjugating a number, we flip it over the real axis. The imaginary part of the conjugae is exactly the opposite to the imaginary part of the original complex number.

Conjugation properties:

  1. ,
  2. ,
  3. .

THINK ABOUT IT:

Calculate the product of and .

Answer

What you get by conjugating the number ?

Answer

What you get by conjugating the number ?

Answer



 
We can include graphs in our materials to further explain the topic!

The product of a complex number with a conjugate number

We have numbers and . Their product equals to . We get the form for a split sum of the squares of two numbers: .

Conjugation numbers

If conjugation number or , we get the same number, . In this case is , while is real number.

Conjugation numbers

If conjugation number or in the imaginary part changes sing and we get . In this case is , while is imaginary number.

Product rule

Solve the task with the help of the fundamental theorem of combinatorics.
For breakfast we have bread with spreading and some drinks. You can choose between 2 types of drinks, 3 types of bread and 5 types of spreadings. How many different breakfasts can we compose?

(/files/101/kruh in sok.png)

Answer: We can compose different breakfasts.

Check

The answer is correct!

Naprej

The answer is incorrect. Because the choice of the spreding is independant from the choice of both, bread and drinks, and the choice of dring is as well independent from the choice of other ingredients, we can multiply all the numbers to get the result.

Next

Limit in one point

Does there exist a limit of a given function (see graph), when converges to a given value?

(/files/101/lim2.png)



Yes
No

Check

 
We can use questions with anwsers in the form of matching. We can have multiple repetitions of an anwser.

The answer is correct.

Next

The answer is incorrect. Limit doesn't exist at the point where the graph is broken on the -axis. In a given graph this only happens when .

(/files/101/lim2.png)

Next

Central and circuferential angle

Under what angle do see the straight line from the center of the circle?

 
A very convinient form for question is the one where there is a single correct option for the answer.
(/files/101/vg_vaja18.png)


Correct

The answer is correct.

Next

Wrong

(/files/101/vg_vaja18a.png)

The answer is incorrect. You only need to calculate the size of the central angle. As the circumferential angle above is equal to , the central angle is equal to , which is equal to .

Next

Local extreme

For which has the function a local extreme? Is it a maximum or a minimum?

 
We can generate questions, with the same text and different imput of the values. In this way we can create different material for every student!

Extreme is at .

The value of the function at its extreme is a:

minimum
maximum

Correct

The answer is correct.

Naprej

Wrong

The answer is incorrect. Extreme is reached in the summit, with the first coordinate being (to two decimal places). When the leading coefficient is positive, extreme is a minimum. When it is negative, then we have a maximum.

Next

Graph of a quadratic function

Indicate which of the following is a graph of a quadratic function .

 
Pictures and graphs are also possible for anwser options!
(/files/101/graf1_1.png)
(/files/101/graf1_2.png)
(/files/101/graf1_3.png)
(/files/101/graf1_4.png)

Correct

The answer is correct.

Next

Wrong

The answer is incorrect.

Next

Exercises in n!

Which statements correctly describe ?

(/files/101/student.png)

Check

The answer is correct!

Naprej

The answer is incorrect. . We may leave out as it does not affect the result. We get also when we divide with , since

Next

Rules for calculating integrals

Which rule for calculating integrals is incorrect?

The answer is correct!

Naprej

The answer is incorrect. The rule is not valid.

Example:

  • We know that:
  • If the above rule is true, then this could be calculated also in a different way:

We got a different result and thus proved that the product rule does not apply here.

Next

Structured exercises about vectors

See the trail on the correct solve task.

(/files/101/koraki.png)

Start



 
Depending from the answer of the student, different lerning paths can be given.

Vectors with equal length

Define so that the vector and equal length.

Check

Correct

The task is solved correct. Solve the other, a little more difficult.

Next

Wrong

The task is solved incorrect. Solve the other, a less difficult.

Next

Angle in a triangle

, , are vertices of the triangle. Calculate angle .

Check

Length of a vector

Define so that the length of vector equal .

Check

Correct

The task is solved correct. Solve the other, a little more difficult.

Next

Wrong

The task is solved incorrect. Solve the other, a less difficult.

Next

0%
0%