Using the capabilities of the Computer Algebra System (CAS) behind Grasple it is possible to create exercises that allow your students to practice with mixed numbers. However, mixed numbers are not interpreted as such in the CAS by default, so they need special attention. Below we explain all ins-and-outs of the support of mixed numbers within the platform.
Reading expressions with mixed numbers
A mixed number is defined as a number consisting of an integer and a fraction. So the mixed number
2 1/2 represents two and a half, or
3/2. However, in the default mode, the CAS system behind Grasple will interpret the expression
2 1/2 as
2 x 1/2, so two times a half, which is equivalent to
In order to make sure the CAS interprets
2 1/2 as a mixed number (and thus equal to
5/2) you can use the
Read Expression With Mixed Numbers parameter type. This parameter type takes an expression and assumes that constructs such as
2 1/2 represent mixed numbers, instead of a multiplication.
For more information about parameters, check What are the possibilities of parameters in an exercise and How do I create a parameterised exercise.
Important: in the default mode the CAS does not interpret expressions as mixed numbers. If you want to interpret an answer from a student as a mixed number you need to explicitly read it as such. See the Using mixed fractions in exercises below for how this can be done.
Writing expressions with mixed numbers
By default the CAS will always visualise expressions with improper fractions instead of mixed numbers. If you want to display an expression with mixed numbers you can use the
Write Expression With Mixed Numbers parameter type. This parameter takes an expression and will visualise this expression with mixed numbers instead of improper fractions.
Using mixed fractions in exercises
So how can these parameter types be used to generate an exercise which allows students to rewrite improper fractions to mixed numbers? Let's see!
Defining the exercise parameters
As a start, we define some parameters to generate an improper fraction. For this we use two range-parameters which will give us some random integers. We also define the improper fraction we want to rewrite:
After this, we can also calculate both the whole number and the numerator we expect to see in our simplified mixed number:
Note that we have added a condition for the
num parameter to be unequal to zero, as we want to make sure we actually have a fraction in our mixed number.
As a last step we define the exact mixed number we expect using the
Read Expression With Mixed Numbers parameter type:
Defining the question
With these parameters defined we can define our actual question:
Defining the answer model
All that is left to do is the definition of the answer model. We want to define the following rules:
if the given student answer is exactly equal to the mixed number, it is correct
if the given student answer is algebraically equal to our mixed number we see it as incorrect, but explain that it can be simplified more.
otherwise the answer is incorrect
As a first step in our definition, we need to make that the answer of the student is read with mixed numbers. This is important, as by default the CAS will not recognise mixed numbers.
To do this, we define an answer parameter of the
Read Expression with Mixed Numbers type. As the expression we simply give the expression which is entered by the student, which is in the form of
student.answerX, where X is the number of question in the exercise.
In our case we only have a single question, so the expression is
student.answer1. We can defined this parameter by clicking the
edit answers and specific feedback button in the
Math/Equation tab of the exercise:
We can define the answer parameter as follows:
Based on this parameter we can express our answer rules relatively straight-forward:
It is important to note here that the left-hand-side of the answer rules have our previously defined answer parameter selected. This ensures that the expression given by the student is parsed and evaluated as a mixed number instead of a multiplication.
Defining the detailed solution
As a last step we need to make sure that we provide students with valuable feedback in the scenario in which they only gave incorrect answers. A regular question will show the correct answer to the student in this scenario, but because of the way in which the answer rules are defined this does not happen in this case. Because of this it is important to also provide feedback in the
Detailed solution field:
And we are done!
Important caveats to take into account
In the creation of the exercises with mixed numbers there are two important things to keep in mind:
student answers are normally not interpreted with mixed numbers, they need to be explicitly processed as such.
the default feedback of the correct answer is not being shown, this type of feedback needs to be explicitly provided in the
Questions, comments, remarks?
Hopefully this support allows you to create meaningful exercises for your students. As always, if you have any questions please reach out to us via the chat icon in the bottom right.