# 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 $7$ kinds of fruit. How many deifferent kinds of fruit salads with $4$ kinds of fruits can we make?

Rule:

We have a certain number of different elements, $n$, and the number of place, $r$, in which the spaced elements, but the order of elements isn't important. The allocation of $n$ elements on $r$ place are variation. In doing so too often considered the allocation of the same elements at different place. The arrangement $r$ elements on $r$ place is $r!$. 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.

How are combinations different from the variations?

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 $1$, $27$, $14$, $2$, $5$, $36$, $20$ represents the same sequence as $1$, $2$, $5$, $14$, $20$, $27$, $36$. In Lotko, the sequence $135788$ and $815783$ represent two different combinations.

Solution of the example:

Scheduling $7$ elements in $4$ places are variations. There are $7 \cdot 6 \cdot 5 \cdot 4 = 840$ of them.

Strech of the angular function

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

In this case, the rotating speed, which tells us the number of waves on length $2 \pi$ changes.

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

EXAMPLE:

$y = \sin 2x$ What is the period of the function $y = \sin 2x$?

How do you calculate the period with $\omega$?

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

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

Calculate it with $p = \frac{2 \pi}{\omega}$.

Conjugation

For each complex number $z=a+bi$ we can find a conjugate number $\bar{z}=a-bi$, by changing the sign in front of the imaginary part. Conjugation properties:

1. $\bar{\bar{z}}=z$,
2. $\overline{z+w}=\overline{z}+\overline{w}$,
3. $\overline{zw}=\overline{z}\cdot \overline{w}$.

Calculate the product of $a+bi$ and $a-bi$.

What you get by conjugating the number $z=a+0i$?

What you get by conjugating the number $z=0+bi$?

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 $a+bi$ and $a-bi$. Their product equals to $(a+bi)(a-bi)=a^2-(bi)^2=a^2-b^2i^2=a^2+b^2$. We get the form for a split sum of the squares of two numbers: $a^2+b^2=(a+bi)(a-bi)$.

# Conjugation numbers

If conjugation number $z=a+0i$ or $z=a$, we get the same number, $\bar{z}=a$. In this case is $\bar{z}=z$, while is $z$ real number.

# Conjugation numbers

If conjugation number $z=0+bi$ or $z=bi$ in the imaginary part changes sing and we get $\bar{z}=-bi$. In this case is $\bar{z}=-z$, while is $z$ 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? Answer: We can compose different breakfasts.

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.

Limit in one point

Does there exist a limit of a given function (see graph), when $x$ converges to a given value? $\lim_{x \to -3} f(x)$
$\lim_{x \to -2} f(x)$
$\lim_{x \to -1} f(x)$
Yes
No

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

The answer is incorrect. Limit doesn't exist at the point where the graph is broken on the $y$-axis. In a given graph this only happens when $x \to -2$. Central and circuferential angle

Under what angle do see the straight line $AB$ 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. # Wrong The answer is incorrect. You only need to calculate the size of the central angle. As the circumferential angle above $AB$ is equal to $125^{\circ}$, the central angle is equal to $2 \cdot 125^{\circ}$, which is equal to $250^{\circ}$.

Local extreme

For which $a$ has the function $f(x) = ? x^2+? x +?$ 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 $x=$ .

The value of the function at its extreme is a:

# Wrong

The answer is incorrect. Extreme is reached in the summit, with the first coordinate being $\frac{-?}{2\cdot ?} =$ (to two decimal places). When the leading coefficient is positive, extreme is a minimum. When it is negative, then we have a maximum.

Indicate which of the following is a graph of a quadratic function $f(x)=x^2-6x+6$.

Pictures and graphs are also possible for anwser options!

# Wrong

Exercises in n!

 Which statements correctly describe $4!$? $4! = 4 \cdot 3 \cdot 2 \cdot 1$ $4! = 24$ $4! = \frac{5!}{5}$ $4! = 4 \cdot 4 \cdot 4 \cdot 4$ $4! = 256$ $4! = 4 \cdot 4$ The answer is incorrect. $4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24$. We may leave $1$ out as it does not affect the result. We get $4!$ also when we divide $5!$ with $5$, since $\frac{5!}{5} = {5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{5} = 4 \cdot 3 \cdot 2 \cdot 1 = 4!$

Rules for calculating integrals

Which rule for calculating integrals is incorrect?

The answer is incorrect. The rule $\int (f(x) \cdot g(x)) dx = \int f(x) dx \cdot \int g(x) dx$ is not valid.

Example: $\int x \cdot x^2 dx$

• We know that: $\int x \cdot x^2 dx = \int x^3 dx = \frac{x^4}{4} + C$
• If the above rule is true, then this could be calculated also in a different way: $\int x \cdot x^2 dx = \int x dx \cdot \int x^2 dx = \frac{x^2}{2} \cdot \frac{x^3}{3} + C= \frac{x^5}{6} + C$

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

See the trail on the correct solve task. Depending from the answer of the student, different lerning paths can be given.

Vectors with equal length

Define $m\in\mathbb{R}$ so that the vector $\vec{a}=(m,5,-3)$ and $\vec{b}=(1,-6,-1)$ equal length.

# Correct

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

# Wrong

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

Angle in a triangle

$A(-1,-1,2)$, $B(4,-6,4)$, $C(7,2,-1)$ are vertices of the triangle. Calculate angle $ABC$.

Length of a vector

Define $m\in\mathbb{R}$ so that the length of vector $\vec{a}=(m-1,m+1,4)$ equal $6$.

# Correct

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

# Wrong

The task is solved incorrect. Solve the other, a less difficult. 0%