Overview
The Numerical Question component is a new feature within Pivot Interactives that allows instructors to:
Give questions that require students to answer with a number.
Define an answer using an Answer Formula that evaluates the student's response and automatically grades it.
Instructors can manually input values such that all students see the same values and all have the same correct answers. For example, instructors could ask What is 3 + 2?, and all students would need to answer 5 for the question to be marked as correct.
Additionally, instructors can define variables such that each student would see a randomized, different set of values and, thus, would have different correct answers depending on the student. One student would see What is 4 + 1? with a correct answer of 5, but another student could see What is 10 + 42? with a correct answer of 52.
Insert a Numerical Question component
When editing an activity, select Numerical Question and then click Add Component.
Defining variables
Variables aren't required for the Numerical Question component, but they greatly expand the power of the tool.
To begin defining variables for your Numerical Question, first click any one of the [var] or [+var] buttons shown below. You can also click Variables at the very top of the screen.
This will bring up the Variables window. To define a new variable, select + Add New Variable, fill in the appropriate information, then click Save.
Random Variables
Name of the variable
Minimum and maximum values for the variable.
Increments between each value of the variable. For example, for a variable ranging from 100 to 300 with a step of 2, the variable can take values 100, 102, 104, 106, etc.
How many decimal places for the variable.
Check this box if you want the value to be displayed in scientific notation.
Previous Student Response
This variable type allows you to use a student's response to a previous question as a variable in another question.
Name of the Variable
Select the question that you'd like to pull a student's response from.
More on variables
Values for each question are generated for a student the first time they open the assignment. Those values cannot be changed, even if their assignment is reset, sections are locked or unlocked, etc.
Variable names carry through the entire assignment. For example, if you define a variable called $pressure and use it in question 1, then use the same variable $pressure in question 2, both variables will have the same value for an individual student. If you would like question 2 to have a different value, you will need to give it a different variable name, such as $pressure 2.
Create the question and correct answer(s)
Type in the question you'd like to ask. Instead of putting in specific numbers, input the variable names within brackets and with a $ symbol in front. For example, for the variable named distance you would type [$distance]. You can also click the [var] button, select the variable, and then click Insert.
Units that the student's answer should be in. This has no effect on the answer formula and is used only as a display to the student
This must be enabled for the question to be automatically graded.
Allows you to add conditions that give different types of feedback based on the student's response, including grading it as correct or incorrect automatically.
Default feedback given to student if none of the conditionals are met.
Access already existing variables, or define new ones.
To add a new condition, click Add Condition+, and the following appears:
Input the formula required to meet this condition. Note that the brackets aren't necessary here.
Input a +/- percentage here. For example, for a tolerance of 5%, students must input a numerical answer that is within 5% of the value calculated using the Answer Formula as defined in box 1.
Click if this condition provides the correct answer.
Feedback delivered to student upon meeting this condition.
Numbers can be mixed with variables in an Answer Formula. For example, to input the equation for kinetic energy, you could type (1/2)*$mass*$velocity^2.
When inputting a formula, the order of operations matters! For example, 6/3*2 will evaluate to 4 whereas 6/(3*2) will evaluate to 1.
What students see
Using the example above, here are examples of what three different students answering this question from the same activity would see.
136/6 = 22.67, so an tolerance of +/- 5% gives a range of correct answers between 21.53 to 23.80.
Range of correct answers is between 37.53 and 41.48
Range of correct answers is between 16.63 and 18.38.
Student answers
Students can only input numbers for an answer. If an answer is supposed to be pi, for example, they cannot enter pi as an answer. Furthermore, unless a tolerance is defined for that answer, it will be marked as incorrect if they input 3.14 since pi doesn't have an exact decimal representation. In this case, setting a tolerance of at least 1% will allow 3.14 to be accepted.
The same goes for, say, sqrt(3)/2 or sin(pi/3). The Answer Formula can be written as sqrt(3)/2 (or sin(pi/3) but students will have to enter a decimal representation. And like the example with pi above, if a tolerance of 0% is when creating the question, values even out to 5 decimal places such as 0.86602 will not be accepted as correct.
In general, using the tolerance setting can allow for decimal representations of otherwise exactly entered answers into an Answer Formula to be accepted.
More on Answer Formulas
The Answer Formula feature allows you to also use a student's response along with Boolean expressions such as greater than, less than, equal to, etc. For example, for the question:
A car travels [$distance] feet in [$time] hours. What is the car's average speed in miles per hour?
The correct Answer Formula would be ($distance/5280)/$time.
However, perhaps you know that some students will incorrectly convert feet into miles by multiplying by 5,280 instead of dividing. This means their answer will be greater than the correct one. You could define an Answer Formula to be:
$response > (($distance/5280)/$time)
...with the feedback: Did you do your unit conversions correctly? Make sure you used 5,280 ft in 1 mi and not 5,280 mi in 1 ft!
Or perhaps you know that some students will simply input the distance as their answer instead of calculating average speed. You could use an Answer Formula of
$response == $distance
...with the feedback: It looks like you input the distance the car travelled, which is [$distance]. You need to calculate and input the average speed. Check your equation sheet.
When there are multiple conditions, each with an answer formulas, the conditions are evaluated in order, and only the feedback for the first condition that evaluates as true will be presented to the student.
The table below shows the operations that will most commonly be used with Answer Formulas.
Operation | Syntax | Example |
Addition | + | $time 1 + $time 2 |
Subtraction | - | $v 0 - $v 1 |
Multiplication | * | $mass*$velocity |
Division | / | $momentum/$velocity |
Grouping | ( ) | 1/(2*pi) |
Exponentiation | ^ | $velocity^2 |
Square root | sqrt(x) | sqrt(3/2) |
Pi | pi | 2*pi |
Euler's number | e | e^3 |
Logarithm | logBase(operand) | log10(100), log2(8), log3(9) |
Natural log | log(x), ln(x) | log(e), ln(e) |
Trig function (incl. inverse) (note: trig functions use degrees) | sin(x), asin(x/y) | sin($theta), asin(2/3) |
Greater than, less than, equal to | >, <, == | $response == 2*$velocity, $response < (1/2)*mass*velocity^2 |
If statements. If $variable is equal to value, then this statement is true if the student enters correctResponse. Otherwise, continue evaluating according to the answer formulas below. | $variable==value?correctResponse:false | $trial==2?16:false |
Numerical questions use math.js to evaluate expressions. For more advanced users, see the math.js documentation for a wider range of possibilities. Note that not everything within math.js will work with Numerical Questions, such as defining objects.