ComparePhysicalQuantities

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Supported Response Area Types

This evaluation function is supported by the following Response Area components:

• TEXT
• EXPRESSION
• NUMERIC_UNITS

Evaluation function which proveds some basic some dimensional analysis functionality.

• DEPRECATED Comparing physical quantities RECOMMENDED ALTERNATIVE: CompareExpressions with the physical_quantity parameter set to true
• Substitutions of symbols before comparison of expressions is done
• Checking if a comma separated list of expressions can be interpreted as a set groups that satisfies the Buckingham Pi theorem

Note: When the quantities grading parameter is set, this function cannot handle short form symbols for units. Thus when defining quantities all units must be given with full names in lower-case letters. For example Nm/s or Newton*metre/SECOND will not be handled correctly, but newton*metre/second will.

Note: Prefixes have lower precedence than exponentiation, e.g. 10*cm**2 will be interpreted as \(10 \cdot 10^{-2}~\mathrm{metre}^2\) rather than \(10 \cdot (10^{-2}~\mathrm{metre})^2\).

Note: This function allows omitting * and using ^ instead of ** if the grading parameter strict_syntax is set to false. In this case it is also recommended to list any multicharacter symbols (that are not part of the default list of SI units) expected to appear in the response as input symbols.

Note: Only the short forms listed in the tables below are accepted. Not all units that are supported have short forms (since this leads to ambiguities).

Note: When using the short forms the following convention is assumed: - Long form names takes precedence over sequences of short forms, e.g. e.g. mN will be interpreted as milli newton, Nm as newton metre, mmN as milli metre newton, mNm as milli newton metre and Nmm as newton milli metre. - Short form symbols of prefixes will take precedence over short form symbols of units from the left, e.g. mug will be interpreted as micro*gram instead metre*astronomicalunit*gram. - If there is a short form symbol for a prefix that collides with the short form for a unit (i.e. m) then it is assumed the that unit will always be placed to the right of another unit in compound units, e.g. mN will be interpreted as milli newton, Nm as newton metre, mmN as milli metre newton, mNm as milli newton metre and Nmm as newton milli metre. - Longer short form symbols take precedence over shorter short forms, e.g. sr will be interpreted as steradian instead of second radian.

Note: setting elementary_functions to true will disable using short forms symbols for units.

Inputs

All input parameters need to be supplied via the Grading parameters panel.

There are seven optional parameters that can be set: elementary_functions, substitutions, quantities, strict_syntax, rtol, atol and comparison.

custom_feedback

Custom feedback can be set on a per-task basis. Note: Custom feedback only supports fixed strings, this means that for some situations the custom feedback cannot be as detailed as the default feedback.

The parameter must be set as a dictionary with keys from the feedback tags listed below. The value for each key can be any string.

Feedback tags for all comparisons

• PARSE_ERROR_WARNING Response cannot be parsed as an expression or physical quantity.
• PER_FOR_DIVISION Warns about risk of ambiguity when using per instead / for division.
• STRICT_SYNTAX_EXPONENTIATION Warns that ^ cannot be used for exponentiation when strict_syntax is set to true.
• QUANTITIES_NOT_WRITTEN_CORRECTLY Text in error message that appears if list of quantities could not be parsed.
• SUBSTITUTIONS_NOT_WRITTEN_CORRECTLY Text in error message that appears if list of substitutions could not be parsed.

Feedback tags for buckinghamPi comparison

• VALID_CANDIDATE_SET Message that is displayed when a response is found to be a valid set of groups. Note: setting this will not affect the Correct/Incorrect message, it will only add further text.
• NOT_DIMENSIONLESS Message displayed when at least one groups is not dimensionless.
• MORE_GROUPS_THAN_REFERENCE_SET Message displayed when the response contains more groups than necessary.
• CANDIDATE_GROUPS_NOT_INDEPENDENT Message displayed when the groups in the response are not independent.
• TOO_FEW_INDEPENDENT_GROUPS Message displayed when the response contains fewer groups than necessary.
• UNKNOWN_SYMBOL Message displayed when the response contains some undefined symbol.
• SUM_WITH_INDEPENDENT_TERMS Message displayed when the response has too few groups but one (or more) of the groups is a sum with independent terms.

elementary_functions

When using implicit multiplication function names with mulitple characters are sometimes split and not interpreted properly. Setting elementary_functions to true will reserve the function names listed below and prevent them from being split. If a name is said to have one or more alternatives this means that it will accept the alternative names but the reserved name is what will be shown in the preview.

sin, sinc, csc (alternative cosec), cos, sec, tan, cot (alternative cotan), asin (alternative arcsin), acsc (alternatives arccsc, arccosec), acos (alternative arccos), asec (alternative arcsec), atan (alternative arctan), acot (alternatives arccot, arccotan), atan2 (alternative arctan2), sinh, cosh, tanh, csch (alternative cosech), sech, asinh (alternative arcsinh), acosh (alternative arccosh), atanh (alternative arctanh), acsch (alternatives arccsch, arcosech), asech (alternative arcsech), exp (alternative Exp), E (equivalent to exp(1), alternative e), log, sqrt, sign, Abs (alternative abs), Max (alternative max), Min (alternative min), arg, ceiling (alternative ceil), floor

Note: setting elementary_functions to true will disable using short forms symbols for units.

substitutions

String that lists all substitutions that should be done to the answer and response inputs before processing.

Each substitution should be written in the form ('original string','substitution string') and all pairs concatenated into a single string. Substitutions can be grouped by adding | between two substitutions. Then all substitutions before | will be performed before the substitutions after |.

The input can contain an arbitrary number of substitutions and | symbols.

Note that using substitutions will replace all default definitions of quantities and dimensions.

quantities

String that lists all quantities that can be used in the answer and response.

Each quantity should be written in the form ('quantity name','(units)') and all pairs concatenated into a single string. See tables below for available default units.

Whenever units are used they must be written exactly as in the left columns of the tables given below (no short forms or single-character symbols) and units must be multiplied (or divided) by each other, as well as any accompanying quantities.

NOTE: Using units and predefined quantities at the same time in an answer or response can cause problems (especially if quantities are denoted using single characters). Ideally it should be clear that either predefined quantities, or units should only be used from the question.

If the comparison parameter is set to dimensions, it is not necessary to give exact units for each quantity, but the dimensions must be given instead. See tables below for available default dimensions.

If the comparison parameter is set to buckinghamPi, then quantities should be set in a different way. See the detailed description of buckinghamPi further down.

Table: Base SI units

SI base units taken from Table 1 of https://physics.nist.gov/cuu/Units/units.html

Note that gram is used as a base unit instead of kilogram.

SI base unit	Symbol	Dimension name
metre	m	length
gram	g	mass
second	s	time
ampere	A	electriccurrent
kelvin	k	temperature
mole	mol	amountofsubstance
candela	cd	luminousintensity

Table: SI prefixes

SI base units taken from Table 5 of https://physics.nist.gov/cuu/Units/prefixes.html

SI Prefix	Symbol	Factor		SI Prefix	Symbol	Factor
yotta	Y	\(10^{24}\)		deci	d	\(10^{-1}\)
zetta	Z	\(10^{21}\)		centi	c	\(10^{-2}\)
exa'	E	\(10^{18}\)		milli	m	\(10^{-3}\)
peta	P	\(10^{15}\)		micro	mu	\(10^{-6}\)
tera	T	\(10^{12}\)		nano	n	\(10^{-9}\)
giga	G	\(10^{9}\)		pico	p	\(10^{-12}\)
mega	M	\(10^{6}\)		femto	f	\(10^{-15}\)
kilo	k	\(10^{3}\)		atto	a	\(10^{-18}\)
hecto	h	\(10^{2}\)		zepto	z	\(10^{-21}\)
deka	da	\(10^{1}\)		yocto	y	\(10^{-24}\)

Table: Derived SI units

Derived SI units taken from Table 3 of https://physics.nist.gov/cuu/Units/units.html

Note that degrees Celsius is omitted.

Note that the function treats radians and steradians as dimensionless values.

Unit name	Symbol	Expressed in base SI units
radian	r	1
steradian	sr	1
hertz	Hz	\(\mathrm{second}^{-1}\)
newton	N	\(\mathrm{metre}~\mathrm{kilogram}~\mathrm{second}^{-2}\)
pascal	Pa	\(\mathrm{metre}^{-1}~\mathrm{kilogram}~\mathrm{second}^{-2}\)
joule	J	\(\mathrm{metre}^2~\mathrm{kilogram~second}^{-2}\)
watt	W	\(\mathrm{metre}^2~\mathrm{kilogram~second}^{-3}\)
coulomb	C	\(\mathrm{second~ampere}\)
volt	V	\(\mathrm{metre}^2~\mathrm{kilogram second}^{-3}~\mathrm{ampere}^{-1}\)
farad	F	\(\mathrm{metre}^{-2}~\mathrm{kilogram}^{-1}~\mathrm{second}^4~\mathrm{ampere}^2\)
ohm	O	\(\mathrm{metre}^2~\mathrm{kilogram second}^{-3}~\mathrm{ampere}^{-2}\)
siemens	S	\(\mathrm{metre}^{-2}~\mathrm{kilogram}^{-1}~\mathrm{second}^3~\mathrm{ampere}^2\)
weber	Wb	\(\mathrm{metre}^2~\mathrm{kilogram~second}^{-2}~\mathrm{ampere}^{-1}\)
tesla	T	\(\mathrm{kilogram~second}^{-2} \mathrm{ampere}^{-1}\)
henry	H	\(\mathrm{metre}^2~\mathrm{kilogram~second}^{-2}~\mathrm{ampere}^{-2}\)
lumen	lm	\(\mathrm{candela}\)
lux	lx	\(\mathrm{metre}^{-2}~\mathrm{candela}\)
becquerel	Bq	\(\mathrm{second}^{-1}\)
gray	Gy	\(\mathrm{metre}^2~\mathrm{second}^{-2}\)
sievert	Sv	\(\mathrm{metre}^2~\mathrm{second}^{-2}\)
katal	kat	\(\mathrm{mole~second}^{-1}\)

Table: Common non-SI units

Commonly used non-SI units taken from Table 6 and 7 of https://physics.nist.gov/cuu/Units/outside.html

Note that the function treats angles, neper and bel as dimensionless values.

Note that only the first table in this section has short form symbols defined, the second table does not.

Unit name	Symbol	Expressed in SI units
minute	min	\(60~\mathrm{second}\)
hour	h	\(3600~\mathrm{second}\)
degree	deg	\(\frac{\pi}{180}\)
liter	l	\(10^{-3}~\mathrm{metre}^3\)
metric_ton	t	\(10^3~\mathrm{kilogram}\)
neper	Np	\(1\)
bel	B	\(\frac{1}{2}~\ln(10)\)
electronvolt	eV	\(1.60218 \cdot 10^{-19}~\mathrm{joule}\)
atomic_mass_unit	u	\(1.66054 \cdot 10^{-27}~\mathrm{kilogram}\)
angstrom	å	\(10^{-10}~\mathrm{metre}\)

Unit name	Expressed in SI units
day	\(86400~\mathrm{second}\)
angleminute	\(\frac{\pi}{10800}\)
anglesecond	\(\frac{\pi}{648000}\)
astronomicalunit	\(149597870700~\mathrm{metre}\)
nauticalmile	\(1852~\mathrm{metre}\)
knot	\(\frac{1852}{3600}~\mathrm{metre~second}^{-1}\)
are	\(10^2~\mathrm{metre}^2\)
hectare	\(10^4~\mathrm{metre}^2\)
bar	\(10^5~\mathrm{pascal}\)
barn	\(10^{-28}~\mathrm{metre}\)
curie	$3.7 \cdot 10^{10}~\mathrm{becquerel}
roentgen	\(2.58 \cdot 10^{-4}~\mathrm{kelvin~(kilogram)}^{-1}\)
rad	\(10^{-2}~\mathrm{gray}\)
rem	\(10^{-2}~\mathrm{sievert}\)

Table: Imperial units

Commonly imperial units taken from https://en.wikipedia.org/wiki/Imperial_units

Unit name	Symbol	Expressed in SI units
inch	in	\(0.0254~\mathrm{metre}\)
foot	ft	\(0.3048~\mathrm{metre}\)
yard	yd	\(0.9144~\mathrm{metre}\)
mile	mi	\(1609.344~\mathrm{metre}\)
fluid ounce	fl oz	\(28.4130625~\mathrm{millilitre}\)
gill	gi	\(142.0653125~\mathrm{millilitre}\)
pint	pt	\(568.26125~\mathrm{millilitre}\)
quart	qt	\(1.1365225~\mathrm{litre}\)
gallon	gal	\(4546.09~\mathrm{litre}\)
ounce	oz	\(28.349523125~\mathrm{gram}\)
pound	lb	\(0.45359237~\mathrm{kilogram}\)
stone	st	\(6.35029318~\mathrm{kilogram}\)

strict_syntax

If strict_syntax is set to true then the answer and response must have * or / between each part of the expressions and exponentiation must be done using **, e.g. 10*kilo*metre/second**2 is accepted but 10 kilometre/second^2 is not.

If strict_syntax is set to false, then * can be omitted and ^ used instead of **. In this case it is also recommended to list any multicharacter symbols (that are not part of the default list of SI units) expected to appear in the response as input symbols.

By default strict_syntax is set to true.

rtol

Maximum relative error allowed when comparing expressions.

atol

Maximum absolute error allowed when comparing expressions.

comparison

Parameter that determines what kind of comparison is done. There are four possible options:

• expression Converts the expression to base SI units and checks that the units are the same and that the value of the answer and response is sufficienty close (as specified by the atol and rtol parameters).
• expressionExact Converts the expression to base SI units and checks that the units are the same and that the value of the answer and response is identical to as high precision as possible.
• dimensions Checks that the answer and response have the same dimensions, does not compare the values of the physical quantities.
• buckinghamPi Checks that the set of quantities in the response matches the set of quantities in the sense given by the Buckingham Pi theorem.

For more details on each options see the description below and the corresponding examples.

If comparison is not specified it defaults to expression.

expression

Converts the expression to base SI units and checks that the units are the same and that the value of the answer and response is sufficienty close.

How big the difference is between the value of the answer and the value of the response is decided by the rtol and atol parameters. If neither atol nor rtol is specified the function will allow a relative error of \(10^{-12}\). If atol is specified its value will be interpreted as the maximum allowed absolute error. If rtol is specified its value will be interpreted as the maximum allowed relative error. If both atol and rtol the function will check both the absolute and relative error.

expressionExact

Converts the expression to base SI units and checks that the answer and response are identical to the highest precision possible (note that some unit conversions are not exact and that using decimal numbers in the answer or response limits this to floating point precision).

dimensions

Checks that the answer and response have the same dimensions, but does not compare the values of the physical quantities.

With this option the quantities (specified by the quantities parameter) can be given either dimension only, or units.

buckinghamPi

Checks that the set of quantities in the response matches the set of quantities in the sense given by the Buckingham Pi theorem.

There are three different ways of supplying this function with the necessary information. - In the answer, provide an example set of groups as a comma seprated list. When used this way the function assumes that the given list is correct and contains at least the minimum number of groups. - In the quantities parameter, supply a list of what the dimensions for each quantity is and set answer to -. The function will then compute a list of sufficiently many independen dimensionless quantities and compare to the response. - In the quantities parameter, supply a list of what the dimensions for each quantity is and in the answer, supply a list of groups as in the first option. The function will then check that the supplied answer is dimensionless and has a sufficient number of independent groups before comparing it to the response.

Note that in lists of groups the items should ideally be written on the form \(q_1^{c_1} \cdot q_2^{c_2} \cdots q_n^{c_n}\) where \(q_1, q_2 \ldots q_n\) are quantities and \(c_1, c_2 \ldots c_n\) are integers, but the function can also handle item that are sums with terms written on the form \(a \cdot q_1^{c_1} \cdot q_2^{c_2} \cdots q_n^{c_n}\) where \(q_1, q_2 \ldots q_n\) are quantities, \(c_1, c_2 \ldots c_n\) rational numbers and \(a\) a constant. If the total number of groups is less than required the set of groups is considered invalid, even if there is a sufficient number of terms with independent power products in the response.

Examples

Implemented versions of these examples can be found in the module 'Examples: Evaluation Functions'.

1 Checking the dimensions of an expression or physical quantity

DEPRECATED

RECOMMENDED ALTERNATIVE: CompareExpressions with the physical_quantity parameter set to true

This example will check if the response has dimensions \(\frac{\mathrm{length}^2}{\mathrm{time}^2}\).

a)

To check an expression there needs to be some predefined quantities that can be used in the expression. Since only dimensions will be checked units are not necessary (but could be used as well).

Here a response area with input type TEXT and two grading parameters, quantities and comparison, will be used.

quantities is defined as follows:

('d','(length)') ('t','(time)') ('v','(length/time)')

comparison is set to dimensions.

The answer is set two some expression with the right dimensions, e.g. v**2.

With default settings it is required to put * (or /) between each part of the response and answer. To remove this requirement the grading parameter strict_syntax is set to false.

In the example given in the example problem set, the following responses are tested and evaluated as correct:

Strict syntax	Relaxed syntax
v**2	v^2
5*v**2	5v^2
(d/t)**2+v**2	(d/t)^2+v^2
d**2/t**2	d^2/t^2
d**2*t**(-2)	d^2 t^(-2)
d/t*v	vd/t

b)

Checking the dimensions of a quantity directly, i.e. the dimensions of an expression of the form number*units, no predefined quantities are necessary.

Here a response area with input type TEXT and one grading parameter,comparison, will be used.

comparison is set to dimensions.

The answer is set two some expression with the right dimensions, e.g. length**2/time**2.

With default settings it is required to put * (or /) between each part of the response and answer. To remove this requirement the grading parameter strict_syntax is set to false. Since only default SI units are expected in the answer we do not need to set any input symbols.

In the example given in the example problem set, the following responses are tested and evaluated as correct:

Strict syntax	Relaxed syntax	Using symbols
metre**2/second**2	metre^2/second^2	m^2/s^2
(centi*metre)**2/hour**2	(centimetre)^2/h^2	(cm)^2/h^2
246*ohm/(kilo*gram)*coulomb**2/second	246 ohm/(kilogram) coulomb^2/second	246 O/(kg) c^2/s

2 Checking the value of an expression or a physical quantity

DEPRECATED

RECOMMENDED ALTERNATIVE: CompareExpressions with the physical_quantity parameter set to true

This examples checks if your expression is equal to \(2~\frac{\mathrm{kilometre}}{\mathrm{hour}}\).

a)

Here an expression with predefined quantities is checked as exactly as possible. This is done with a TEXT response area with the following parameters: quantities is set to:

('d','(length)') ('t','(time)') ('v','(length/time)')

Note that short form symbols cannot be used when defining quantities.

comparison is set to expressionExact.

The response area answer is set to 2*v but there are many other expressions that would work just as well. Note that we cannot write 2*kilo*metre/second as response or answer since the predefined quantity t will substitute the t in metre which results in unparseable input.

With default settings it is required to put * (or /) between each part of the response and answer. To remove this requirement the grading parameter strict_syntax is set to false. Since only default SI units and single character symbols are expected in the answer we will not set the grading parameter symbols.

In the example given in the example problem set, the following responses are tested and evaluated as correct:

Strict syntax	Relaxed syntax
2*v	2v
2000/3600*d/t	2000/3600 d/t
1/1.8*d/t	d/(1.8t)
v+1/3.6*d/t	v+d/(3.6t)

b)

Checking if a quantity is equal to \(2~\frac{kilometre}{hour}\) with a fixed absolute tolerance of \(0.05 \frac{metre}{second}\) can be done with a TEXT response area with atol set to 0.05 and the answer set to 2*kilo*metre/hour.

Note: atol is always assumed to be given in the base SI units version of the expression. This is likely to change in future versions of the function.

The comparison parameter could also be set to expression but since this is the default it is not necessary.

With default settings it is required to put * (or /) between each part of the response and answer. To remove this requirement the grading parameter strict_syntax is set to false. Since only default SI units are expected in the answer no input symbols are necessary.

In the example given in the example problem set, the following responses are tested and evaluated as correct:

Strict syntax	Relaxed syntax	Using symbols
0.556*metre/second	0.556 metre/second	0.556 m/s
0.560*metre/second	0.560 metre/second	0.560 m/s
0.6*metre/second	0.6 metre/second	0.6 m/s
2*kilo*metre/hour	2 kilometre/hour	2 km/h
1.9*kilo*metre/hour	1.9 kilometre/hour	1.9 km/h
2.1*kilo*metre/hour	2.1 kilometre/hour	2.1 km/h

In the example given in the example problem set, the following responses are tested and evaluated as incorrect:

Strict syntax	Relaxed syntax	Using symbols
0.61*metre/second	0.61 metre/second	0.61 m/s
2.2*kilo*metre/hour	2.2 kilometre/hour	2.2 km/h

c)

Checking if a quantity is equal to \(2~\frac{kilometre}{hour}\) with a fixed relative tolerance of \(0.05\) can be done with a TEXT response area with rtol set to 0.05 and the answer set to 2*kilo*metre/hour.

The comparison parameter could also be set to expression but since this is the default it is not necessary.

In the example given in the example problem set, the following responses are tested and evaluated as correct:

Strict syntax	Relaxed syntax	Using symbols
0.533*metre/second	0.533 metre/second	0.533 m/s
2.08*kilo*metre/hour	2.08 kilometre/hour	2.08 km/h

With default settings it is required to put * (or /) between each part of the response and answer. To remove this requirement the grading parameter strict_syntax is set to false. Since only default SI units are expected it is not necessary to set any input symbols.

In the example given in the example problem set, the following responses are tested and evaluated as incorrect:

Strict syntax	Relaxed syntax	Using symbols
0.522*metre/second	0.522 metre/second	0.522 m/s
2.11*kilo*metre/hour	2.11 kilometre/hour	2.11 km/h

3 Checking if a set of quantities match the Buckingham pi theorem

a)

In this example the task is: Given \(U\), \(L\) and \(\nu\), suggest a dimensionless group.

For this problem we do not need to predefine any quantities and give exact dimensions. The algorithm assumes that all symbols in the answer (that are not numbers or predefined constants such as \(\pi\)) are quantities and that there are no other quantities that should appear in the answer.

Note: This means that the algorithm does not in any way check that the stated answer is dimensionless, ensuring that that is left to the problem author.

For this example a TEXT response area is used with comparison set to buckinghamPi and answer set to ['U*L/nu']. It is not necessary to use this specific answer, any example of a correct dimensionless group should work.

With default settings it is required to put * (or /) between each part of the response and answer. To remove this requirement the grading parameter strict_syntax is set to false. Since nu is a multicharacter symbol it needs to be added as an input symbol.

b)

In this example the task is: Given \(U\), \(L\), \(\nu\) and \(f\), determine the necessary number of dimensionless groups and give one example of possible expressions for them.

This task is similar to example a) with two significant differences. First, adding \(f\) means that there are now two groups required, and second the problem will constructed by defining the quantities and let the function compute the rest on its own instead of supplying a reference example.

For this example a TEXT response area is used with comparison set to buckinghamPi, quantities set to ('U','(length/time)') ('L','(length)') ('nu','(length**2/time)') ('f','(1/time)') and answer set to -.

With default settings it is required to put * (or /) between each part of the response and answer. To remove this requirement the grading parameter strict_syntax is set to false. Since nu is a multicharacter symbol it needs to be added as an input symbol.

c)

In this example the task is: Suppose we are studying water waves that move under the influence of gravity. We suppose that the variables of interest are the acceleration in free fall \(g\), the velocity of the wave \(v\), the height of the wave \(h\) and the wave length \(\ell\). We also suppose that they are related by a dimensionally consistent equation \(f(g,v,h,l) = 0\). Determine the minimum number of dimensionless \(\pi\)-variables needed to describe this problem according to the Buckingham pi-theorem and give one example of possible expressions for the dimensionless quantities.

For this problem two dimensionless groups are needed, see the worked solution for a terse solution that gives the general form of the dimensionless quantities.

For this example a TEXT response area is used with comparison set to buckinghamPi and then give a list of correct group expressions formatted as the code for a python list. For this example the answer ['g**(-2)*v**4*h*l**3', 'g**(-2)*v**4*h**2*l**4'] was used (this corresponds to \(p_1 = 1\), \(p_2 = 2\), \(q_1 = 3\), \(q_2 = 4\) in the worked solution). The feedback was costumized by setting the custom_feedback parameter too: "custom_feedback": { "VALID_CANDIDATE_SET": "Your list of power products satisfies the Buckingham Pi theorem.", "NOT_DIMENSIONLESS": "At least one power product is not dimensionless.", "MORE_GROUPS_THAN_REFERENCE_SET": "Response has more power products than necessary.", "CANDIDATE_GROUPS_NOT_INDEPENDENT": "Power products in response are not independent.", "TOO_FEW_INDEPENDENT_GROUPS": "Candidate set contains too few independent groups.", "UNKNOWN_SYMBOL": "One of the prower products contains an unkown symbol.", "SUM_WITH_INDEPENDENT_TERMS": "The candidate set contains an expression which contains more independent terms that there are groups in total. The candidate set should ideally only contain expressions written as power products." }

With default settings it is required to put * (or /) between each part of the response and answer. To remove this requirement the grading parameter strict_syntax is set to false. Since nu is a multicharacter symbol it needs to be added as an input symbol.

4 Defining costum sets of units

In this problem it is demonstrated how to use substitutions to define costum units.

a)

In this problem currencies will be us as units, and thus the quantities will no longer be physical.

Here the substitutions parameter will be set so that the evaluation function can be used to compare. Note that using substitutions this way means that the default SI units can no longer be used.

The following exchange rates (from Bank of England 1 August 2022) will be used:

Currency	Exchange rate
\(1\) EUR	\(1.1957\) GBP
\(1\) USD	\(1.2283\) GBP
\(1\) CNY	\(8.3104\) GBP
\(1\) INR	\(96.943\) GBP

To compare prices written in different currencies a reference currency needs to be chosen. In this case GBP will be used. To substitute other currencies for their corresponding value in GBP the following grading parameter can be used:

"substitutions":"('EUR','(1/1.1957)*GBP') ('USD','(1/1.2283)*GBP') ('CNY','(1/8.3104)*GBP') ('INR','(1/96.9430)*GBP')"

Since these conversion are not exact and for practical purposes prices are often not gives with more than two decimals of precision we also want to set the absolute tolerance, atol, to \(0.05\).

With default settings it is required to put * (or /) between each part of the response and answer. By setting the grading parameter strict_syntax to false the * can be omitted and ^ can be used instead of **. To ensure that this works correctly it is necessary to list the multicharacter symbols that are expected to appear in the answer and response as input symbols. For this example this means setting EUR, USD, CNY and INR as codes for inut symbols.

In the example given in the example problem set, the answer set to 10*GBP and the following responses are tested and evaluated as correct:

Strict syntax	Relaxed syntax
11.96*EUR	11.96 EUR
12.28*USD	12.28 USD
83.10*CNY	83.10 CNY
969.43*INR	969.43 INR