Multi-part arithmetic

Multi-part questions

Multi-part questions require more than one mathematical task to solve. Most story problems are multi-part questions. Such questions require strategic thinking to decompose the problem into subproblems and combine the partial results.

In this section, we will focus on multi-part arithmetic problems. Arithmetic problems do not require formulas or variables. We will extend to algebra and calculus in the following sections.

Example: An arithmetic multi-part question

Consider the story problem below. We will learn how to create this question using the Multistep question type.

Solution models

The table below shows a solution model for our example problem.

If you are a teacher or have designed assessment questions before, you probably are familiar with such solution models.

Solution models have the following properties:

• Every part represents something, such as "The total costs for option A." Algebrakit uses this to generate helpful hints and error feedback, as we will see later.
• Some parts depend on other parts. For example, to calculate $C_B$, you must first calculate $N_{months}$. Algebrakit will use these dependencies to determine in what order to solve the parts.
• You can assign marks to each part to allow for partial scoring.
• You do not have to specify how each part is solved or what its result is. Algebrakit handles this automatically.

Adding the solution model to Multistep

The image below shows how you define this solution model in Multistep: Figure: The overview of the solution model in Multistep. Click on the Edit button of the live example below to see this screen in your browser.

As shown in the figure, the Multistep question allows you to add multiple parts. Each part has a name, a description, and a task.

There are three types of parts:

• Given information. These are numbers or expressions given in the problem statement, and are not part of a worked solution.
• Intermediate parts. These represent subproblems relevant to solving a problem.
• The solution part. There is always exactly one solution part. It is the only required part for authoring a multistep exercise.

You can add new intermediate parts and given information by clicking the + button on the right. Parts in the overview have four pieces of information:

• The part name. Part names allow you to refer to other parts. They are never visible to the student.
• The variable. Variables are visible to students. Variables are covered further in sections Multi-part algebra and Multi-part calculus.
• The description. Students can see the description. Algebrakit uses the description in automatically generated hints, feedback and the worked solution.
• The task. The task defines what students should calculate, rewrite or simplify. Most parts have one task, but parts can have multiple tasks or no tasks at all, as we will see later.

You can build many exercises entirely using the overview interface, without needing the detail interface. Clicking a part opens its description, task and criteria editors. In those editors, you can do more advanced, detailed editing. The detail editor for complex multistep exercises is very similar to the editor for simple exercises with only one part described in the section Multistep (core). The criteria editor is covered here.

Descriptions

Descriptions consist of two parts: What is being derived and How this should be done.

• The field labelled What is being derived contains a description of what a result represents.
• The field labelled How this should be done contains a description of what students should do.

By default, Algebrakit automatically generates How this should be done by adding the word "Calculate" to What is being derived. This automatically generated sentence is often appropriate, like in our example. However, this construction is not appropriate for all situations and all supported languages, so authors can set How this should be done manually.

Algebrakit sometimes uses What is being derived and sometimes How this should be done, in places like hints and feedback.

Live example

The live example below shows the result after we created the Multistep question. Go ahead and solve the problem step by step. Or, click the Edit button to see how we made it.

Live example: The arithmetic story problem we are creating.

Multiple strategies

What if there are multiple valid strategies to solve a problem? For example, you can find the answer to our example problem as follows:

You can see that this second strategy introduces new parts. The $solution$ part exists in both approaches, though the definitions differ.

You can add this second strategy to your Multistep question by adding the new parts.

Figure: The overview of the solution model after adding a second strategy to the solution model in Multistep.

If you want a student to be able to use this strategy to arrive at the solution, you need to add a second task to the solution part. To do this:

• Click on the solution part in the overview to open up its single part interface.
• In the single-part interface, click on the + button in the top-right corner of the task definition. Doing so will open up a new task labelled "Click to define...".
• Define the solution task for the second strategy.

Figure: The single-part interface of the solution part after adding a second task.

Partial marking

The strategy tables above have a column for the marks you award for each part. Enable the configuration of the marks from the Multistep dropdown. Figure: Enabling the marking options.

The mark configuration shows all the strategies you defined. You can edit the number of marks for each part in each strategy. The total number of marks must be the same for each strategy. Figure: Configuring the marks for each part.

How Algebrakit uses the solution model

Algebrakit uses the solution model to generate worked solutions, provide personalised support, and support partial marking.

Generate structured solutions

Algebrakit uses the solution model to generate a worked solution with a clear structure. Algebrakit also adds structure to the input area when students press the hint button. It decomposes the input into sections and adds the part descriptions. This helps students connect their mathematical work to the concepts they are calculating. Figure: Algebrakit adds structure to students' input and worked solutions.

Personalised support following the student's strategy

If a solution model has multiple solution strategies, as in our example problem, Algebrakit will track the student's strategy. In the example below, the hint depends on what parts the student was working on.

Figure: Algebrakit tracks the student's strategy.

Skipped parts

What if a student finds the correct answer to the problem without calculating every part of the solution model? For instance, in our example problem, a student might calculate $D_{monthly}$ directly without first calculating $d_{monthly}$. Algebrakit will detect that the student skipped the step and award the marks. In an extreme case, a student could enter just the final answer and receive full marks.

You can specify that an intermediate part is required. Requiring an intermediate part means that a student must enter that intermediate part to receive the marks. You can read more about required parts here.

What if the student uses an unknown strategy?

Most math problems have just one or a few strategies to find the solution. What happens if a student applies a creative approach you didn't anticipate? Algebrakit will not be able to award marks for unknown steps. However, if the student's approach is correct, they will calculate the result of a known part sooner or later and receive the marks.