> For the complete documentation index, see [llms.txt](https://docs.lamsfoundation.org/lams/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.lamsfoundation.org/lams/question-bank/numerical-questions.md).

# Numerical Questions

## Create a Numerical Question 

Use a numerical question when you **want students to calculate, estimate or identify a number**. LAMS lets you define one or more accepted numerical answers and, when appropriate, accepted units with conversion multipliers.

Numerical questions work well for outcomes that require students to:

* apply a formula or quantitative method;
* interpret data and calculate a result;
* convert between units;
* estimate an order of magnitude;
* derive a parameter from a scenario; or
* report a quantitative conclusion.

Do not use a numerical question when you need to assess the method, assumptions or reasoning independently of the final value. A correct number can result from incorrect reasoning, and a small transcription error can conceal otherwise sound work. Pair the numerical response with a worked solution, justification or other constructed response when the learning outcome requires evidence of the process.

### Before you create the question

Start with the learning outcome, solve the problem independently and define the acceptable result before you configure LAMS.

Decide:

* which value or values are correct;
* which units students may use;
* the required precision or number of significant figures;
* how intermediate rounding affects the final result;
* whether scientific notation is acceptable;
* whether an estimate or exact value is required; and
* which assumptions students should make.

The editor shown does not provide a visible tolerance field. Do not assume that LAMS will accept values merely because they are close to an authored answer. If several rounded values should receive credit, enter and test them explicitly, or redesign the question so that one unambiguous value is expected.

### Create the question

1. Enter a short, distinctive **Question's title**.

   Use the title to identify and manage the item in the Question Bank. Include information that distinguishes it from similar items, such as the topic and intended method. Do not put information here that students need in order to solve the problem.
2. Enter the problem in **Question's description**.

   Include all necessary data, assumptions and instructions. State the required units and precision. Use mathematical notation consistently and make clear whether students should enter the number only or select an accompanying unit.
3. Enter the accepted value in **Answer 1**.

   The initial answer fields display `0.0`. Replace the placeholder with the intended numerical answer.
4. Enter other accepted values when required.

   LAMS provides three answer fields initially. Select **Add another answer** to include another accepted numerical result.
5. Add general feedback if it will help students understand the calculation.
6. Set the default grade and any penalty, tag the relevant learning outcome or outcomes, and configure accepted units.
7. Select **Save**, then preview and test the question in the LAMS activity where students will answer it.

### Write an effective numerical problem

* Assess a meaningful quantitative decision or method rather than arithmetic for its own sake, unless arithmetic fluency is the intended outcome.
* Supply all required data and identify any constants students should use.
* Avoid irrelevant information unless selecting relevant data is part of the learning outcome.
* Specify assumptions that could otherwise produce different defensible answers.
* Use realistic values, units and contexts without adding unnecessary reading demand.
* State how students should round the final answer. For example: **Give your answer to two decimal places** or **Give your answer to three significant figures**.
* State whether students should retain unrounded intermediate values.
* Distinguish percentage from percentage-point change when relevant.
* Identify whether angles use degrees or radians and whether logarithms use a particular base.
* Check the sign convention, coordinate system, reference date, exchange rate or other frame of reference when these affect the result.
* Do not make students infer the expected unit or notation from an input box.

### Configure accepted answers

Each **Answer** field represents an accepted numerical result. Use additional answers only when more than one result is genuinely valid under the stated task.

Appropriate reasons for adding another answer include:

* more than one rounding outcome is acceptable;
* the problem deliberately permits alternative stated assumptions;
* an exact decimal and an accepted rounded decimal should both receive credit; or
* a known representation must be entered as a different numerical value because the delivery activity does not normalise it automatically.

Do not use additional answers to conceal ambiguity. If two results arise because the question omits an assumption, revise the question.

#### Precision and rounding

The accepted numerical answer and the precision requested from students must agree. For example, if the computed value is `2.71828` and you ask for two decimal places, enter and test `2.72` rather than assuming the longer value will match.

Consider how students may round intermediate results. When a defensible calculation can produce several nearby final values, either:

* instruct students to keep full precision until the final step;
* supply an agreed intermediate value;
* add every defensible final value and test it; or
* use an assessment method that supports a documented tolerance.

Avoid demanding more precision than the source data justify. Excess decimal places can reward calculator output rather than quantitative judgement.

#### Input formats

Before deployment, test the formats students are likely to enter:

* integers and decimal values;
* negative values;
* a leading zero, such as `0.5`;
* trailing zeros, such as `2.50`;
* scientific notation, such as `1.2e3`, if permitted;
* the decimal marker used by your students' locale; and
* values copied from calculators or spreadsheets.

Tell students which format LAMS accepts. A mathematically equivalent representation should not be rejected because of an undocumented interface convention.

### Add feedback for students

Enter **General feedback** that helps students understand the solution rather than merely displaying the correct number. You can include:

* the formula or quantitative model;
* a worked solution with units at every step;
* an explanation of why a particular method applies;
* a warning about a common sign, conversion or rounding error;
* a reasonableness or order-of-magnitude check; or
* a follow-up question that asks students to interpret the result.

The numerical editor provides one general feedback field rather than response-specific feedback. Avoid assuming that every incorrect result arose from the same error.

Feedback is most useful when students receive it in time to act on it. Check the settings of the activity that uses the banked question, because that activity determines when students can see feedback.

### Configure the advanced settings

#### Default question grade

Enter the overall mark or weight for the question. The interface accepts an integer, for example `1`, `5` or `10`.

Align the weighting with the learning outcome and the work required. If the final answer receives all available marks, the question does not distinguish a minor arithmetic error from a fundamental conceptual error. Use another question or an additional constructed-response step when method marks are important.

Preview the question and confirm the score produced by every accepted value and unit combination.

#### Penalty factor

Enter the penalty that should apply in a context where students can make another attempt. The field accepts a floating-point number.

Use a penalty only when it serves a clear assessment purpose. Its effect can depend on how the receiving LAMS activity handles attempts, so test the question in that activity before relying on the calculation. Explain any penalty to students in advance.

Be cautious when a formatting, rounding or unit-entry issue could cause an otherwise valid answer to be rejected. A penalty should reflect the intended assessment rule, not an undisclosed input convention.

#### Learning outcomes

Search by outcome name or code, then select every learning outcome for which the question provides meaningful evidence.

Tag the outcome at the level actually assessed. Entering a recalled constant does not demonstrate the same capability as selecting a model, completing a multi-step calculation or interpreting a quantitative result. A scenario-based numerical question may assess application, but only if the scenario requires students to decide how to use the data.

Avoid tagging every broadly related outcome. Across an assessment, review the distribution of questions by outcome, cognitive demand, quantitative method and weighting so Question Bank metadata supports meaningful blueprinting.

### Configure units and multipliers

Use the **Units** section when students may answer using specified units. Each row contains:

* **Unit:** the unit label that LAMS should recognise; and
* **Multiplier:** the factor used to convert a value in that unit to the reference value used by the accepted answer.

The first unit row defaults to a multiplier of `1.0`. Treat this as the reference unit. Select **Add another unit** to accept an equivalent unit, and use the delete button to remove an unnecessary row.

#### Multiplier example

Suppose the accepted answer is `2` metres:

| Unit | Multiplier | Equivalent response |
| ---- | ---------: | ------------------: |
| `m`  |      `1.0` |               `2 m` |
| `cm` |     `0.01` |            `200 cm` |
| `mm` |    `0.001` |           `2000 mm` |

The intended normalisation is:

`student value × unit multiplier = value in the reference unit`

Therefore, `200 × 0.01 = 2`. Because unit handling affects marks directly, verify this direction and every conversion in the actual LAMS student view before consequential use.

#### Write units carefully

* Use standard, unambiguous unit symbols.
* Observe case: `m` and `M`, or `W` and `w`, may not mean the same thing.
* Use the correct multiplier for prefixes such as kilo-, centi-, milli- and micro-.
* Add separate recognised unit labels when LAMS requires both a symbol and a written name.
* Do not use a multiplier of `0.0` as a completed conversion rule; replace the placeholder with the required factor.
* Avoid accepting units that imply a different physical quantity, even if a numerical conversion appears possible.
* Test spaces, plural forms, compound units, superscripts and Unicode symbols such as `µ`, `°` or `²` where relevant.

A quantity consists of both a number and a unit. Standard conventions place a space between the number and most unit symbols and preserve case-sensitive SI symbols ([NIST guidance](https://www.nist.gov/pml/owm/writing-si-metric-system-units)). Apply the conventions of your discipline and tell students what the LAMS input accepts.

#### Temperature and non-linear conversions

A multiplier alone supports proportional conversions. It cannot, by itself, represent a conversion that also requires an offset or another transformation.

For example, Celsius-to-Fahrenheit conversion is not a simple multiplication. Do not configure such units as though a single multiplier were sufficient. Ask for one specified unit, author separate accepted values only when the delivery behavior is clear, or redesign the question.

### Use numerical questions within a programme of assessment

Treat each numerical response as one data point. In programmatic assessment, individual assessment events should maximise learning and feedback, while higher-stakes decisions draw on multiple data points collected over time ([van der Vleuten et al., 2012](https://pubmed.ncbi.nlm.nih.gov/22364452/)).

Numerical questions efficiently assess quantitative results, but they may not reveal how students selected a method, handled uncertainty or interpreted the answer. Combine them with worked calculations, oral explanation, practical performance, data analysis or other evidence when the learning outcome includes reasoning and judgement.

Use low-stakes numerical questions to practise retrieval, calculation and estimation. Include feedback that develops quantitative sense—for example, checking dimensions, signs and orders of magnitude—rather than encouraging students to search repeatedly for an accepted number.

### Review the question after use

Review the item and response patterns before you reuse it.

* Identify unanticipated but mathematically defensible answers.
* Check whether alternative rounding paths should receive credit.
* Look for recurring unit, sign, decimal-place and order-of-magnitude errors.
* Determine whether an incorrect answer points to a misconception or merely a transcription error.
* Check whether students omitted units because the instructions or interface were unclear.
* Verify that every multiplier normalises to the intended reference value.
* Review whether the accepted precision is consistent with the data supplied.
* Correct marks through the appropriate assessment process when a valid response was rejected.
* Record substantive revisions as a new version so results from materially different rules are not treated as directly comparable.

Statistics can flag a numerical question for review, but they cannot tell you whether the accepted value, rounding rule or unit conversion is conceptually defensible. Review the mathematics and the response data together.

### Final check

Before you save or deploy the question, confirm that:

* the question measures a tagged learning outcome at the intended cognitive level;
* all required data and assumptions are present;
* the accepted answer has been independently verified;
* the requested precision and authored values agree;
* every defensible rounding result is handled deliberately;
* the required input and decimal format are clear;
* units are stated and configured consistently;
* every unit multiplier has been independently checked;
* no conversion requires an offset that a multiplier cannot represent;
* the grade and any penalty behave as intended;
* feedback explains the method and supports a reasonableness check;
* the question is accessible and mathematical notation is readable with assistive technology; and
* you have tested representative correct and incorrect responses in the receiving LAMS activity.


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