5.7 Mathematical Formula Questions
Mathematical Formula questions allow instructors to compare a student response (generally a mathematical expression) to a specified answer. Mathematical Formula is the general term, encompassing nine different subcategories of formula type questions. Table 5.2 discusses these ten subtypes in detail and provides an example response of each.
Mathematical Formula Question Types Comparison Table
Table 5.2: Mathematical Formula Subtypes Comparison Table
Question Type | Description | Example Responses |
| The Formula subtype accepts numbers and formulas. It does not accept equations, that is, expressions consisting of two components separated by an "=" sign. | e^{x} sin(x^{2}) |
- Formula without Logs and Trig
| Formula without Logs and Trig accepts numbers and formulas that do not include trigonometric functions or logarithms. | 1/sqrt(2) |
- Formula with Physical Units
| The Formula with Physical Units subtype accepts a combination of numeric quantities, algebraic formulas, unit dimensions, and physical units. It also recognizes equivalent physical units. For example, if 500g is specified as the correct answer, 0.5kg is also graded correctly. | sin(x) 6 m 6m cos(x) m/s |
- Formula that Matches Responses to within +C
| The Formula that Matches Responses to within +C subtype accepts numbers and formulas. It also accepts two answers as equivalent if they differ by a constant number. | x ln(x) x ln(x)+ 1 |
- Formula without Simplification
| Formula without Simplification questions leave the expression in its unsimplified form. The system grades the unsimplified form as correct. | 4x + x |
| The Equation subtype specifically allows for the use of an "=" sign in the equation. It grades equivalent equations as correct. Note: The expression given as the correct answer must consist of a single variable on the left-hand side of the equation and a formula that does not involve that variable on the right-hand side. (That is, you cannot specify an implicit formula.) This restriction does not apply to students entering their answers. | y-2 = 4(x+1) |
- Unordered List of Formulas
| The Unordered List of Formulas subtype accepts an unordered list of numbers or formulas separated by semicolons. A question that has a single formula answer is graded correctly if this type is used. In other words, the answer must match the terms of the list in any order. It is recommended that you use the Unordered List of Formulas subtype for questions that have a single correct answer, but, in general, have multiple correct answers because the Formula subtype indicates to students that there is only one solution. | 4; 5x; 3 |
| The Ordered List of Formulas subtype accepts an ordered list of numbers or formulas. Parentheses accepted but not recommended. The grading is same for an Ordered List of Formulas as it is for Vector of Formulas. Explanatory text tells students to enter an ordered list of formulas or numbers separated by commas. | 2, 4, 8, 16 |
| Vector of Formulas should be used for questions about Cartesian coordinates or vectors. This question subtype accepts an ordered list of number or formulas separated by commas (for example, a Cartesian point or a vector). Parentheses are recommended but not required. Explanatory text tells students to enter a list of formulas separated by commas enclosed in parentheses. | (2, 4t, 7t+1) |
| The Chemical Equation subtype accepts the following characters in student answers: superscripts (^), subscripts (_), arrows (->), dot operator (*), + sign, ion changes (-), and physical states. | NaOH + HCl -> NaCl + H_2O 2H_2+O_2 ->2H_2O |
For a detailed example of each subtype of Mathematical Formula, see:
See Also:
Question Designer
Adaptive Questions in Question Designer
Student responses must contain a mathematical expression including numbers or mathematical formulas. The system grades a student response by comparing it with the correct answer. If the student response and the correct answer are algebraically or numerically equivalent, the response is graded correctly. To require that the response have a specified form, use the form subtype.
To create a Formula question:
- From the Class Homepage, click the Content Repository.
- From the Create New drop-down menu, select Question/Text.
- On the Question Designer screen:
- Enter a title for the question under the Question Name panel.
- Enter the Question Text: input the question statement. To include complicated mathematical expressions, click the Equation Editor icon () in the toolbar. For more details, see Editing with the Equation Editor.
- Click Response Area ().
- Under Question Type, select Mathematical Formula.
- Weighting: specify the weight of this response area in the overall question. (By default, the Weighting is set to 1).
- From the Sub-type drop-down list, select the formula math grader you would like to use.
- Enter the Answer: input the correct answer in symbolic math. Your answer must be in the correct format. For example, for a restricted formula question, you cannot use sin.
- On the Question Summary pane,
- Click Save or Preview to see your question.
See Also:
Authoring Mathematical Questions
Mathematical Formula Question Types Comparison Table
Understanding the Math Capabilities
Example 2: Formula without Logs and Trig Subtype
When using the formula without logs and trig subtype, a response is graded correctly if it is algebraically equivalent to the correct answer. The correct answer is a formula that uses only basic operations (for example, arithmetic operations and sqrt). If you want the answer to contain more advanced operations (for example, trig, log, etc.) you have to use the formula subtype.
To create a Formula without Logs and Trig subtype question:
- From the Class Homepage, click the Content Repository.
- From the Create New drop-down menu, select Question/Text.
- The Question Designer screen appears:
- Enter the following for Question Name: Evaluate trig expression.
- Enter the following for Question Text: Evaluate
- Weighting: specify the weight of this response area in the overall question. (By default, the Weighting is set to 1).
- From the Sub-type drop-down menu, select Formula without logs and trig.
- Enter the following for Answer: 1/3*3^(1/2)
- Click Save to save the question, then click Preview to view it. See Figure 5.11.
Example 3: Formula with Physical Units Subtype
When using the formula with physical units subtype, correct answers can include algebraic formulas, unit dimensions, numeric responses, or any combination of these. Use this question type for any formula where you require the student to specify the physical units for the expression (for example, , where the formula is not an equation. When you give the units with the answer, you must use the units that are recognized by the system. The system can convert between units of the same type. For example, if you type the student can give the answer as: hour. In each case, the system grades the response correct.
To create a Formula with Physical Units subtype question:
- From the Class Homepage, click Content Repository in the top menu.
- From the Create New drop-down menu, select Question/Text.
- The Question Designer screen appears:
- Enter the following for Question Name: Particle's velocity.
- Enter the following for Question Text: A particle moves along the x-axis such that its position after seconds ( is meters ( Give the expression for the particle's velocity at time
- Click Response Area ().
- Under Choose Question Type, select Mathematical Formula.
- Enter the following for Answer: 2t m/s
- From the Sub-type list, select Formula with physical units then click OK.
- Click Save to save the question or Preview to view it. For more details, see Figure 5.12.
Example 4: Formula that Matches Responses to within +C Subtype
The formula that matches responses to within +C subtype should be used when you want the question to have an answer with an additive, numeric constant. This enables the question to have many valid answers that differ by the addition of a constant quantity. This subtype accepts all such answers as equivalent.
To create a Formula that Matches Responses to within +C subtype question:
- From the Class Homepage, click Content Repository in the top menu.
- From the Create New drop-down menu, select Question/Text.
- The Question Designer screen appears:
- Enter the following for Question Name: Integral of polynomial.
- Enter the following for Question Text: Evaluate
- At the bottom of the page, click the add icon () and enter the following Algorithm:
$r=rint(2, 10);
$s=rint(2, 10);
$t=rint(2, 10);
condition: gt($r,$s) gt($s,$t);
$rplus=int($r+1);
$splus=int($s+1);
$tplus=int($t+1);
$a=rint(2,20);
$b=rint(2,20);
$c=rint(2,20);
- Click Refresh algorithm preview to preview the variables you defined, as shown in Figure 5.13.
- Click Response Area ().
- Under Choose Question Type, select Mathematical Formula.
- Weighting: specify the weight of this response area in the overall question. (By default, the Weighting is set to 1).
- From the Sub-type drop-down menu, select Formula that Matches Responses to within +C.
- Enter the following for Answer: ($a/$rplus)x^$rplus + ($b/$splus)x^$splus + ($c/$tplus)x^$tplus
- Click Save to save or Preview to view the question. See Figure 5.14.
Example 5: Formula without Simplification
Formula without Simplification should be used when the answer must be left in its unsimplified form to be graded correctly. In other words, if the instructor states that they would like the answer to be left in unsimplified form, the student would leave their answer in the form , rather than simplify to
To create a Formula without Simplification subtype question:
- From the Class Homepage, click Content Repository in the top menu.
- From the Create New drop-down menu, select Question/Text.
- This brings you to the Question Designer screen:
- Enter a title for the question under the Question Name panel.
- Enter a question in the Question Text. Optionally, enter the question using symbolic math by clicking the Equation Editor ()icon. The Equation Editor opens. See Formatting Tips.
- Click Response Area ().
- In the Edit Response Area dialog:
a. Weighting: specify the weight of this response area in the overall question. (The default Weighting is set to 1).
b. Under Sub-type, select Formula without simplification.
c. Answer: enter the correct answer. (Remember: Do not simplify). For more details, see Figure 5.15 below.
d. Click OK.
- Click Save to save the question, then click Preview to view it.
See Also:
Mathematical Formula Question Types Comparison Table
Example 6: Equation Subtype
The equation subtype question requires responses in mathematical equation form. An equation question is different from a formula question because it contains an "=" sign in the response. Any equation that is algebraically equivalent to the correct answer is graded correctly.
Authoring Note: For a formula-response question with the equation subtype, for the correct answer, one side of the equation must be in the form of a single-variable. The student response does not need to be in this form. Any equivalent equation is graded as correct.
To create an Equation subtype question:
- From the Class Homepage, click Content Repository in the top menu.
- From the Create New drop-down menu, select Question/Text.
- The Question Designer screen appears:
- Enter the following for Question Name: Equation of a straight line.
- Enter the following for Question Text: What is the equation of the straight line passing through the point with a slope ?
- Click the add icon () and enter the following Algorithm:
$xone = rint(-10,10);
$yone = decimal(1, rand(-10,10));
$m = rint(2, 10);
- Click Refresh algorithm preview to preview the variables you defined, as shown in Figure 5.16.
- Click Response Area ().
- Under Choose Question Type, select Mathematical Formula.
- Weighting: specify the weight of this response area in the overall question. (By default, the Weighting is set to 1).
- From the Sub-type drop-down menu, select Equation.
- Enter the following for Answer: y = $m*(x-$xone) + $yone
- Click Save to save or Preview to view the question. See Figure 5.17.
Example 7: Unordered List of Formulas Subtype
The unordered list of formulas subtype accepts a list of numbers or formulas, separated by semicolons. The response is graded correctly if the list of formulas matches the list in the correct answer when ignoring the ordering. If the correct answer is 1;2;3, then any of the 6 permutations of the formulas, for example, 2;1;3, 3;2;1, and 1;2;3, is graded correctly. To accept an ordered list of formulas, use the ordered lists of formulas subtype. For more details, see Ordered List of Formulas Subtype.
To create an Unordered Lists of Formulas subtype:
- From the Class Homepage, click Content Repository in the top menu.
- From the Create New drop-down menu, select Question/Text.
- The Question Designer screen appears:
- Enter the following for Question Name: Find the roots.
- Enter the following for Question Text: What are the roots of the quadratic equation ? Enter only the expressions for the roots, omitting "".
- Click the add icon () and enter the following Algorithm:
$a = rint(2, 5);
$b = rint(10, 15);
$c = rint(2, 5);
- Click Refresh algorithm preview to preview the variables you defined, as shown in Figure 5.18.
- Click Response Area ().
- Under Choose Question Type, select Mathematical Formula.
- Weighting: specify the weight of this response area in the overall question. (By default, the Weighting is set to 1).
- From the Sub-type drop-down menu, select Unordered list of formulas.
- Enter the following for Answer: (-$b + sqrt($b^2 - 4*$a*$c))/(2*$a); (-$b - sqrt($b^2 - 4*$a*$c))/(2*$a)
- Click Save to save the question, then click Preview to view it. See Figure 5.19.
Example 8: Ordered List of Formulas Subtype
The ordered list of formulas subtype accepts a list of numbers or formulas, separated by commas. The response is graded correctly if the list of formulas matches the exact order of the list in the correct answer. If the correct answer is 1,2,3, then only 1,2,3 is accepted. To accept an unordered list of formulas, use the unordered list of formulas subtype. For more details, see Unordered List of Formulas Subtype.
To create an Ordered Lists of Formulas question:
- From the Class Homepage, click Content Repository in the top menu.
- From the Create New drop-down menu, select Question/Text.
- The Question Designer screen appears:
- Enter the following for Question Name: Compute exact values.
- Enter the following for Question Text: Compute the exact values of
- Click Response Area ().
- Under Choose Question Type, select Mathematical Formula.
- Weighting: specify the weight of this response area in the overall question. (By default, the Weighting is set to 1).
- From the Sub-type drop-down menu, select Ordered lists of formulas.
- Enter the following for Correct answer: sqrt(3)/2, 1/2, sqrt(3)
- Click Save to save the question, then click Preview to view it. See Figure 5.20.
Example 9: Vectors of Formulas Subtype
The vectors of formulas subtype requires a vector of formulas (a sequenced list) in correct responses. Use this for questions with vectors of formulas or numbers. You can use the full range of functions in the formulas, that is, all trigonometric functions, log, ln, abs, and sqrt.
To create a Vectors of Formulas question:
- From the Class Homepage, click Content Repository in the top menu.
- From the Create New drop-down menu, select Question/Text.
- The Question Designer screen appears:
- Enter the following for Question Name: Find the minimum point.
Enter the following for Question Text:
Find the minimum point of the given function below.
$plot
- Click the add icon () and enter the following Algorithm:
$a = rint(1,6);
$b = rint(2,10);
$bottomrange = $a-2;
$toprange = $a+2;
$plot = plotmaple("plot((x-$a)^2 + $b, x=$bottomrange..$toprange), plotoptions='width=250, height=250'");
- Click Refresh algorithm preview to preview the variables you defined, as shown in Figure 5.21.
- Click Response Area ().
- Under Choose Question Type, select Mathematical Formula.
- By default, the Weighting is set to 1.
- From the Sub-type drop-down menu, select Vectors of formulas.
- Enter the following for the Answer: ($a,$b)
- Click Save to save or Preview to view the question. See Figure 5.22
Example 10: Chemical Equation Subtype
In the chemical equation subtype, instructors can use the following characters in their answers: superscripts (^), subscripts (_), arrows (->), dot operator (*), + sign, ion changes (-), and physical states. Table 5.3 describes the keys in more detail. In the chemical equation question type, student responses require a formula that matches the correct answer.
Table 5.3: Rules and Keys for Entering Chemistry Expressions
Expression | Keys |
---|
Superscripts and Subscripts | Enter superscripts using the caret ^ character, and subscripts using the underscore _. |
Arrows in Equations | Use the text ->, <-, <=> for arrows. |
Other Operators | Use * for the center dot operator. |
Use the + sign in equations. |
Use the - or + signs to indicate ion charges. |
No other operations are allowed in equations. |
Physical States,
Ion charges, and Parentheses | Be sure to include physical states (in parentheses) if your equation requires them. Use the + and - characters for polarity and ion charges Use parentheses to clarify interpretation of groups of characters |
To create a Chemical Equation question:
- From the Class Homepage, click Content Repository in the top menu.
- From the Create New drop-down menu, select Question/Text.
- The Question Designer screen appears:
- Enter the following for Question Name: Reaction.
- Enter the following for Question Text: Enter the reaction of sodium hydroxide (NaOH) with hydrochloric acid (HCl).
- Click Response Area ().
- Under Choose Question Type, select Mathematical Formula.
- Weighting: specify the weight of this response area in the overall question. (By default, the Weighting is set to 1).
- From the Sub-type drop-down menu, select Chemical equation.
- Enter the following for the Answer: NaOH + HCl -> NaCl + H_2O
- Click Save to save the question, then click Preview to view it. See Figure 5.23.
See Also:
Entering Chemistry Expressions
See Also:
Mathematical Formula Question Types Comparison Table