# Tutorial

A principle of the Algebra interaction is that the author does not specify the answer but the problem. This allows AlgebraKiT to evaluate intermediate steps, support the student with automatic hints, and generate worked solutions.

The problem is specified through one or more tasks. Each task represents a mathematical procedure.

Tasks fall under one of the following categories. Click on a category for more details.

Simplify

The default task of the algebra interaction type is Simplify. With this task, the goal is to simplify the expression as much as possible, using the mathematical rules that are available for the student.

Examples
##### Example

With a task $4+5$, AlgebraKiT will evaluate the sum, because $9$ is a simpler expression than $4+5$.

Try it yourself:

• $2x+4x+4$
AlgebraKiT will combine the like terms.
• $x(x+1)$
AlgebraKiT will do nothing. Expanding the brackets is not necessarily ‘simpler’.
• $x(x+1)-x^2$
Now AlgebraKiT will expand the brackets, because it can then cancel terms and reach a simple result.
• $\frac{2x+4}{x^2+2x}$
Using the mathematical knowledge that AlgebraKiT has, it decides to factor both numerator and denominator and cancel factors to reach a simpler result.

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Let’s create an exercise with the following algebra interaction:

The interactive result is shown below:

Interactive student exercise:

Worked solution:

AlgebraKiT uses the audience to know what mathematical rules can be used.

Note: To make automatic simplification possible, AlgebraKiT must assess if one expression is more ‘simple’ than another. See the examples above for more details.

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Numerical

Combination

The goal of is this task is to calculate the numeric combination $\binom{n}{k}$, also known as n choose k. In the total field you fill in the total size n of the set of items, in the select field you fill in the number k items to select. So to calculate $\binom{4}{2}$, you fill in 4 in the total field and 2 in the select field.

Factorial

The goal of this task is to calculate the factorial $n!$ of a natural number n. You can filled in the natural number in the number field.

Greatest Common Divisor

The goal of this task is to calculate the greatest common divisor of two or more numbers. By default, the number 1 and number 2 fields are present. To add more number fields, you click on the + sign at the left of a number field. If you have more than two number fields and you want to remove a specific number field, you can click on the corresponding remove button on the left.

Least Common Multiplier

Within the Numerical task, you can find the Least Common Multiplier task. The goal of this task is to calculate the least common multiplier of two or more numbers. By default, the number 1 and number 2 fields are present. To add more number fields, you click on the + sign at the left of a number field. If you have more than two number fields and you want to remove a specific number field, you can click on the corresponding remove button on the left.

Round

Round Up

The goal of this task is to round a specific number up. You can fill in this number at the expression field. This task supports step-by-step guidance for students to hint that the number should be rounded up.

Round Down

The goal of this task is to round a specific number down. You can fill in this number at the expression field. This task supports step-by-step guidance for students to hint that the number should be rounded down.

Write decimal number as a fraction

The goal of this task is to write a specific decimal number as a fraction. You can fill in this decimal number at the number field. Both mixed fractions and basic fractions will be correct, unless specific otherwise using a mode. The correct answers to the below example with decimal number $1.2$ are $1 \frac{1}{5}$​ and $\frac{6}{5}$.

Write in scientific notation

The goal of this task is to write a specific number in scientific notation. You can fill in this number at the number field.

Convert units

This task is deprecated. Refer to the section on unit settings on how to convert a unit.

The goal of this task is to convert a number with a unit to another unit. You can fill in the number with its unit in the quantity field. You can fill in the final unit in the target unit field.

The goal of this task is to add a specific percentage of a certain number to that number. You can fill in the percentage at the percentage field and the number at the value field.

Percentage of

The goal of this task is to calculate the value of a certain percentage of a given number. You can fill in the requested percentage at the percentage field and the given number at the value field.

Refer to the following interactive exercise to test and inspect the behavior of the task(s).

Interactive student version:

Worked solution:

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Rewriting expressions
Expand brackets

The goal of this task is to expand the brackets of a specific expression. You can fill in this expression at the expression field.

Factor

The goal of this task is to factorise a specific expression. You can fill in this expression at the expression field.

Combine fractions

The goal of this task is to write a given expression as a single fraction. You can fill in this expression at the expression field.

Complete the square

The goal of this task is to write a specific expression in the completed square form relative to a specific variable. You can fill in this expression at the expression field and the variable at the variable field.

Write into linear form

The goal of this task is to write a specific expression into the linear form relative to a specific variable. You can fill in this expression at the expression field and the variable at the variable field.

Write as a power function

The goal of this task is to write a specific expression into a power function relative to a specific base value. You can fill in the expression in the expression field and the base value in the base field.

Write into standard exponential form

The goal of this task is to write a specific expression into standard exponential form. You can fill in this expression at the expression field.

Write cartesian vector into polar coordinates

The goal of this task is to write a specific 2D cartesian vector into 2D polar coordinates. You can fill in the x-coordinate of the cartesian vector at the x field and the y-coordinate of the cartesian vector at the y field.

Write polar vector into cartesian coordinates

The goal of this task is to write a specific polar vector into cartesian coordinates. You can fill in the radius of the specific polar vector at the r field and the angle of the specific polar vector at the angle field. If the angle is in degrees, you have to add the degrees symbol at the angle field.

Refer to the following interactive exercise to test and inspect the behavior of the task(s).

Interactive student version:

Worked solution:

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Solving equations
Algebraical, one variable, exact

The goal of this task is to solve a specific equation for one specific variable. By default, AlgebraKiT will require an exact answer. You can fill in the equation at the relation field, the variable at the variable field. If you want to add a specific domain for the variable, you can tick the domain box and fill in the domain at the appearing field.

Algebraical, one variable, rounded

The goal of this task is to solve a specific equation for one specific variable, and the answer must be rounded to a certain amount of decimal points.

Algebraical, one variable, with calculator

This task is deprecated. Refer to the item on accuracy settings for more information on rounding. Use the mode allow_calculator to enable the calculator hint functionality.

The goal of this task is to solve a specific equation for one specific variable, the student may use their calculator and the answer must be rounded to a certain amount of decimal points.

Algebraical, system

The goal of this task is to solve a system of two equations. You can fill the two equations in the equations fields. You can fill in the two variables for which the system of equations must be solved in the variables fields. If you want to add a specific domain for the variable, you can tick the domain box and fill in the domain at the appearing field.

If you want to change the preferred method of solving the system of equations, you can tick the method box and select the preferred method.

Make one side equal to zero

The goal of this task is to rewrite the equation such that one side is equal to $0$. You can fill in the equation at the equation field and the variable at the variable field.

Refer to the following interactive exercise to test and inspect the behavior of the task(s).

Interactive student version:

Worked solution:

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Calculus
Differentiate

The goal of this task is to differentiate a specific expression according to a specific variable. You can fill in this expression at the expression field and the variable at the variable field. If you want the student to find the $n$-th derivative, you can tick the order box and fill in the number at the appearing field. The default value of this number is $1$.

Integrate, indefinite

The goal of this task is to calculate a specific indefinite integral with respect to a specific variable. You can fill in this integral (without integral sign) in the expression field and the variable in the variable field.

Integrate, definite

The goal of this task is to calculate a specific definite integral with respect to a specific variable. You can fill in this integral (without integral sign and without boundaries) in the expression field, the lower bound at the lower bound field, the upper bound at the upper bound field and the specific variable at the variable field.

Refer to the following interactive exercise to test and inspect the behavior of the task(s).

Interactive student version:

Worked solution:

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Other

Currently, only one task is available in the Other group:

Evaluate function

The goal of this task is to evaluate a given expression for a specific value.

The expression you want the student to evaluate should be entered in the “expression” field. The variable which should be replaced by the specific value should be entered in the “variable” field respectively in the “value” field.

Refer to the following interactive exercise to test and inspect the behavior of the task(s).

Interactive student version:

Worked solution:

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AlgebraKiT

With this task selected, you are able to provide a task definition using AlgebraKiT Syntax. This task exists due to historical reasons.

Currently, the AlgebraKiT task is deprecated. That means we will not provide support for new questions that use this task selection. Questions created before the other types of tasks were available (early 2020) are still supported.