Mathematics – Teacher support material
Introduction
How to use this teacher support material
This teacher support material is designed to accompany the MYP Mathematics guide
(published July 2007). It contains examples of assessed student work, and is intended
to give practical help to support teachers’ understanding and implementation of the
theory presented in the guide.
The teacher support material is divided into two sections.


Assessed student work
Appendices
The assessed student work section contains examples of assessed work for years 4
and 5. Please note that these are examples only. They have been included to
demonstrate what teacher tasks and student work may look like and do not form part of
a mandatory curriculum for schools.
Teachers may wish to use these examples as a guide to creating appropriate
mathematics tasks, or as an indication of the standard expected of students by the end
of the final year of the programme.
The examples included are authentic student work and are presented in their original
styles, which may include spelling, grammatical and any other errors. These examples
have been anonymized where necessary (names may have been changed or deleted)
and some have been retyped to make them easier to read.
The examples of student work presented are divided into the three types of assessment
tasks required for moderation.



Broad-based classroom tests/examinations
Mathematical investigations
Real-life problems
The appendices contain a series of frequently asked questions and examples of
completed moderation and monitoring of assessment forms to assist teachers in
preparing for moderation and monitoring of assessment.
Please note that the MYP Mathematics teacher support material exists in four
languages (English, French, Spanish and Chinese). If teachers are familiar with more
than one of these languages, it may be worthwhile for them to look at the other
language versions, as some examples of assessed student work are different for each
language.
Thanks are due to the schools and students who allowed the use of their work in this
document, and to the experienced MYP practitioners who worked so carefully on all of
the content.
Please note that the assessment criteria used in this material correspond to the MYP
Mathematics guide (published July 2007), and are for first use in final assessment in the
2008 academic year (southern hemisphere) and the 2008–2009 academic year
(northern hemisphere).
Assessed student work
To view the various elements of this example, please use the icons at the side of the
screen.
Overview
Number
Task type
Title
Criteria
End of year exam
A
Broad-based classroom test
A
Average limits
B, C
Example 4
Egyptian triangulation
B, C, D
Example 5
Enlarging areas and volumes
B, C, D
Should we start melting our
coins?
A, C, D
Broad-based test
A
Extended mathematics test
A
Standard mathematics
Example 1
Broad-based classroom
test
Example 2
Example 3
Example 6
Mathematical investigation
Real-life problem
Extended mathematics
Example 7
Example 8
Broad-based classroom
test
Example 9
Mathematical investigation
Example
10
Example
11
Real-life problem
Example
12
Geometric average limits
B, C
Cutting corners
B, C
Medicine and mathematics
A, C, D
London Eye
A, C, D
Standard mathematics
Example 1
Teacher task
Student work
Moderator comments
End of year exam
Broad-based classroom test
Standard mathematics
Branches of the framework: Number, algebra, geometry and trigonometry
MYP year: 4
Criterion
A
B
C
D
Level achieved
7
–
–
–
Background
This was an end-of-year examination based on all the work covered during MYP
year 4.
Students were familiar with the four operations and were able to work out
percentages, fractions, proportions, ratios and indices. Within the branch of
algebra, students were capable of solving equations algebraically and using
graphs. They were also familiar with solving linear, simultaneous and quadratic
equations. As part of geometry and trigonometry, students could find the
perimeter, area and volume of regular and irregular two-dimensional and
three-dimensional shapes. They were also capable of using the Cartesian plane
and triangle properties, including Pythagoras’ theorem, to solve problems.
The exam covered topics of three of the five branches of mathematics and as
such it was considered a broad-based exam. The exam consisted of a series of
questions of different degrees of complexity. It started with short, simple
questions and finished with longer and more challenging ones to give all
students the opportunity to reach the level appropriate to their ability. The
unfamiliar questions were clearly identified in the exam.
Students were given two weeks of class time and homework to revise all of the
topics covered during the year. The exam lasted two hours and was carried out
under strict examination conditions. Students were allowed to use scientific
calculators.
Assessment
Criterion A: Knowledge and understanding
Maximum 8
Achievement
level
7–8
Descriptor
The student consistently makes appropriate deductions when
solving challenging problems in a variety of contexts including
unfamiliar situations.
This work achieved level 7 because the student:


makes correct deductions when solving challenging problems in familiar
contexts
answers correctly questions 15 and 17(c), which were unfamiliar
questions.
The student would have achieved a higher level if he had consistently made
appropriate deductions when solving the variety of problems presented in the
exam.
Example 2
Teacher task
Student work
Moderator comments
Broad-based classroom test
Broad-based classroom test
Standard mathematics
Branches of the framework: Number, algebra, geometry and trigonometry,
statistics and probability
MYP year: 4
Criterion
A
B
C
D
Level achieved
6
–
–
–
Background
This test was based on topics that students had covered in the second semester.
The test was a broad-based test because it covered topics of four of the branches
of the MYP framework for mathematics.
Students were familiar with using the four number operations with integers,
decimals and simple fractions. They could also work out prime numbers and
factors including greatest common divisor and least common multiple. Students
were also able to solve equations including linear, simultaneous and quadratic
equations. As part of geometry and trigonometry, students were able to use the
Cartesian plane and triangle properties, including Pythagoras’ theorem, to solve
problems. Students could calculate the probability of simple, mutually exclusive
and combined events, and they could use tree diagrams to determine probability.
The test consisted of a series of questions of different complexities including
simple, complex and a few unfamiliar situations. Questions 18(b) and 20 were
considered unfamiliar as the context of the question was modified by the teacher
to present students with a greater challenge.
The test lasted 90 minutes and was carried out under strict exam conditions.
Students were allowed to use scientific calculators and graphic display
calculators.
Assessment
Criterion A: Knowledge and understanding
Maximum 8
Achievement
level
5–6
Descriptor
The student generally makes appropriate deductions when solving
challenging problems in a variety of familiar contexts.
This work achieved level 6 because the student:

makes appropriate deductions when solving familiar and some complex
questions.
The student would have achieved a higher level if he had answered questions 18
and 20 correctly as these represented unfamiliar situations.
Example 3
Teacher task
Student work
Moderator comments
Average limits
Mathematical investigation
Standard mathematics
Branches of the framework: Number, algebra, statistics and probability
MYP year: 5
Criterion
A
B
C
D
Level achieved
–
8
5
–
Background
Prior to this activity the students had done some work on sequences and
averages and were able to discover formulae.
For this activity students were asked to investigate what would happen to a
sequence that started with any two numbers, where the following term in the
sequence was always the average of the previous two. This task allowed students
to make basic calculations in a sequence and recognize the limit of a sequence.
Students had to tabulate results, find rules and patterns and come up with
justifications or proofs.
The time allocated for the activity was about one week of class time and
homework.
Students were expected to use a variety of mathematical language and forms of
representation such as symbols, diagrams and tables.
Assessment
Criterion B: Investigating patterns
Maximum 8
Achievement
level
7–8
Descriptor
The student selects and applies mathematical problem-solving
techniques to recognize patterns, describes them as relationships or
general rules, draws conclusions consistent with findings, and
provides justifications or proofs.
This work achieved level 8 because the student:


generates sequences, discovers patterns and finds general rules
comes up with algebraic proofs for the discoveries made.
Criterion C: Communication in mathematics
Maximum 6
Achievement
level
5–6
Descriptor
The student shows good use of mathematical language and forms
of mathematical representation. The lines of reasoning are concise,
logical and complete.
The student moves effectively between different forms of
representation.
This work achieved level 5 because the student:



uses a variety of mathematical language, symbols and diagrams when
communicating his ideas and findings
uses clear lines of reasoning for his explanations
moves between different forms of representation such as tables and
algebraic forms.
This work did not achieve level 6 because, although the lines of reasoning are
clear, they are not complete, as the student does not show how he got to the
formula
.
Example 4
Teacher task
Student work
Moderator comments
Egyptian triangulation
Mathematical investigation
Standard mathematics
Branches of the framework: Geometry and trigonometry
MYP year: 5
Criterion
A
B
C
D
Level achieved
–
7
2
2
Background
Prior to this investigation, students had covered concepts of geometry including
Pythagoras’ theorem and trigonometric ratios. Students also knew how to find
the area of a triangle; however, they were not familiar with the formula
.
This task required students to investigate the area of different triangles and to
develop the formula
for the area of a triangle. Students were also
expected to reflect upon the limitations of their formula and to evaluate the
accuracy of their findings.
Students had a double lesson of 90 minutes for the investigation and another
double lesson for completing the part on reflection. The investigation was carried
out under test conditions; students could consult resources but could not
discuss their results with other students. Students were allowed to use
calculators for this task.
This task was appropriate for year 5 students and, since it was carried out under
test conditions, it was challenging enough. This investigation was open-ended
because even though the end result of the task was a set formula, this formula
could be given in any format depending on the process followed by the students.
The last instruction asked students to model how Egyptians measured their land,
applying these concepts to a real-life context. This method, called triangulation,
is still in use today.
Assessment
Criterion B: Investigating patterns
Maximum 8
Achievement
level
7–8
Descriptor
The student selects and applies mathematical problem-solving
techniques to recognize patterns, describes them as relationships or
general rules, draws conclusions consistent with findings, and
provides justifications or proofs.
This work achieved level 7 because the student:


selects a method and consistently applies it to arrive at the formula
justifies the formula found by comparing it with the formula for the area
of the triangle
.
The student would have achieved a higher level if she had justified why all three
formulae work to find the area of a triangle.
Criterion C: Communication in mathematics
Maximum 6
Achievement
level
1–2
Descriptor
The student shows basic use of mathematical language and/or
forms of mathematical representation. The lines of reasoning are
difficult to follow.
This work achieved level 2 because the student:



shows basic use of mathematical language and forms of mathematical
representation
uses poor lines of reasoning that are sometimes difficult to follow
makes mistakes with the use of symbols, omits labelling of triangles and
fails to include any diagrams in F.
The student would have achieved a higher level if she had:



used clear lines of reasoning including words and better labelled
diagrams to explain the height in the obtuse angled triangle
explained how the formula changed when using another pair of sides
used a different method to obtain the area of the triangle in G, for
example, by using accurate forms of representation such as drawing and
counting squares or using geometry software.
Criterion D: Reflection in mathematics
Maximum 6
Achievement
level
1–2
Descriptor
The student attempts to explain whether his or her results make
sense in the context of the problem. The student attempts to
describe the importance of his or her findings in connection to real
life.
This work achieved level 2 because the student:



checks if the results make sense by using different methods
explains the limitations of the method if used in a real-life context
attempts to justify the degree of accuracy.
The student would have achieved a higher level if she had:


correctly justified the degree of accuracy used
explained in detail, using different diagrams, why the method of
triangulation works but has limitations.
Example 5
Teacher task
Student work
Moderator comments
Enlarging areas and volumes
Mathematical investigation
Standard mathematics
Branches of the framework: Number, algebra, geometry and trigonometry
MYP year: 5
Criterion
A
B
C
D
Level achieved
–
8
6
4
Background
The students were able to find the perimeter, area and volume of basic shapes
including cuboids and prisms. Students were also familiar with the concept of
enlargement prior to this activity. This task could be used as an introduction to
the idea of similar figures and Thales’ theorem.
The students were asked to find out what happens to the area of a shape if its
dimensions are doubled. Students worked on the five shapes given and had to
find out how many times bigger in area each shape was. They were asked to
scale up the areas by three, four and five times. Students were expected to find a
rule connecting the linear scale factor and the area scale factor. Students were
encouraged to check their results using shapes of their choice.
Students then had to repeat the process for volumes and try to discover a rule
connecting the scale factor and the volume factor. Finally, students had to apply
their discoveries to answer a set of questions they were given at the end of the
project (see teacher task).
This activity allows students to identify patterns, describe them as rules and
justify these mathematically. Students are encouraged to use a variety of
mathematical language and different forms of representation. Also, this task
provides students with the opportunity to reflect upon their findings and explain
the importance of their discoveries to a real-life context.
Students were given one week of class time (four hours) and homework time to
complete this activity. Students were allowed to discuss their ideas and findings
with the teacher.
Assessment
Criterion B: Investigating patterns
Maximum 8
Achievement
level
7–8
Descriptor
The student selects and applies mathematical problem-solving
techniques to recognize patterns, describes them as relationships or
general rules, draws conclusions consistent with findings, and
provides justifications or proofs.
This work achieved level 8 because the student:




recognizes patterns in the tables of results, for example, that the area
factor is the scale factor squared and that the volume factor is the scale
factor cubed
discovers the rules for enlarging areas and volumes
draws conclusions supported by her findings
uses a variety of different shapes in both two dimensions and three
dimensions to check out her results and gives a generalized justification of
why the rules work.
Criterion C: Communication in mathematics
Maximum 6
Achievement
level
5–6
Descriptor
The student shows good use of mathematical language and forms
of mathematical representation. The lines of reasoning are concise,
logical and complete.
The student moves effectively between different forms of
representation.
This work achieved level 6 because the student:




uses a wide variety of mathematical language and her lines of reasoning
are clear and complete
uses a variety of different forms of representation (for example,
diagrams, tables, formulae)
combines different forms of representation to reach a conclusion
clearly identifies that the shapes are increased by a scale factor in two
directions with plane shapes and in three directions with solid shapes.
Criterion D: Reflection in mathematics
Maximum 6
Achievement
level
3–4
Descriptor
The student correctly but briefly explains whether his or her results
make sense in the context of the problem and describes the
importance of his or her findings in connection to real life.
The student attempts to justify the degree of accuracy of his or her
results where appropriate.
This work achieved level 4 because the student:


briefly describes why the results make sense in the context of the
problem
attempts to explain the importance of her findings to a real-life context.
The work would have achieved a higher level if the student had given a more
detailed and thorough explanation of the importance of the findings to a real-life
context.
Example 6
Teacher task
Student work
Moderator comments
Should we start melting our coins?
Real-life problem
Standard mathematics
Branches of the framework: Number
MYP year: 5
Criterion
A
B
C
D
Level achieved
6
–
3
2
Background
For this real-life problem students needed to have covered number concepts
such as approximations, scientific notation, ratios, proportion and the decimal
system. They also needed to have a general awareness of the major imperial
units.
Students were given an instruction sheet and an article from a financial
newspaper to carry out this real-life problem. Students had to read and analyse
the article, which dealt with the rising price of metals and how, in a near future,
the metal value of the US dollar one cent coin would overtake its face value.
Students were expected to use a variety of number concepts to find out if the
claim of the article was also true for any Euro coins. As part of the investigation,
students also had to make future predictions. Finally, students had to
communicate their findings in the form of a newspaper article. The best articles of
the class were to be published in the school newspaper.
Students had one double lesson of 90 minutes to work on the task and one week
of homework. Students had access to the Internet and other resources such as
newspapers, magazines and books. Students had to find information on the metal
composition of the Euro coins, the price of the metals, the Euro–US dollar
exchange rate and the conversion rates between the imperial and decimal
systems.
This investigation contributed to raising students’ awareness of the possible
application of mathematics to real-life situations. It also gave students an idea of
the fluctuations of the stock market, and how and why the price of the metals and
exchange rates change as part of the economy and financial markets.
This investigation was open-ended as different students found different data, so
each project was quite original. It was an appropriate task for year 5 students
because even though the concepts were not too difficult, the task required
students to understand the stock market and to find, select and analyse
information from a range of sources, make reasoned deductions and critically
make a prediction for the future.
Assessment
Criterion A: Knowledge and understanding
Maximum 8
Achievement
level
5–6
Descriptor
The student generally makes appropriate deductions when solving
challenging problems in a variety of familiar contexts.
This work achieved level 6 because the student:


makes appropriate deductions when solving problems even when the
situation was unfamiliar
shows a good understanding of number in its different representations,
selecting and using appropriate concepts to solve problems.
The student would have achieved a higher level if he had:


tested for other coins, such as coins with a lower face value instead of
choosing the Euro 50 cent coin and making unsupported assumptions
about the Euro 1 coin
represented the metal value as a percentage of the face value.
Criterion C: Communication in mathematics
Maximum 6
Achievement
level
3–4
Descriptor
The student shows sufficient use of mathematical language and
forms of mathematical representation. The lines of reasoning are
clear though not always logical or complete.
The student moves between different forms of representation with
some success.
This work achieved level 3 because the student:


shows good use of mathematical language and units throughout the
problem
communicates his findings using clear and logical lines of reasoning.
The student would have achieved a higher level if he had also used other forms of
mathematical representation such as a table or graph to communicate his
findings.
Criterion D: Reflection in mathematics
Maximum 6
Achievement
level
1–2
Descriptor
The student attempts to explain whether his or her results make
sense in the context of the problem. The student attempts to
describe the importance of his or her findings in connection to real
life.
This work achieved level 2 because the student:


makes an attempt to justify the degree of accuracy chosen for the
answers but it is not supported by the results
briefly explains the fact that it is not yet worth melting the coins.
The student would have achieved a higher level if he had:


calculated the ratio face value/metal value for the coins tested and
commented on the difference
justified the degree of accuracy used correctly.
Extended mathematics
Example 7
Teacher task
Student work
Moderator comments
Broad-based test
Broad-based classroom test
Extended mathematics
Branches of the framework: Geometry and trigonometry, algebra, number
MYP year: 5
Criterion
A
B
C
D
Level achieved
7
–
–
–
Background
This test was considered a broad-based test as it included topics from at least
three of the branches of the framework of mathematics. The test was appropriate
for extended mathematics because it included concepts and skills from the
extended mathematics framework such as logarithms, exponential functions,
arithmetic and geometric series.
Prior to the test, students had covered concepts of Euclidean geometry of circles.
Students were also familiar with the use of surds, prime numbers, indices and
logarithms. Students were able to solve problems that included arithmetic and
geometric series and quadratic functions. Some basic calculus had also been
covered but this was not tested in this test.
The task allowed students to apply their knowledge and understanding of a range
of mathematical topics and to use deductions when solving problems. The test
consisted of a range of questions of different complexities and included some
questions that were unfamiliar to the student such as questions 2(b) and 2(c),
question 3 and question 7(b).
Students had two hours to complete the test. The test was carried out under strict
exam conditions. Students were allowed to use graphic display calculators
during the test but their working had to be shown on the test paper answer sheet.
Assessment
Criterion A: Knowledge and understanding
Maximum 8
Achievement
level
7–8
Descriptor
The student consistently makes appropriate deductions when
solving challenging problems in a variety of contexts including
unfamiliar situations.
This work achieved level 7 because the student:



makes appropriate deductions when solving problems in a variety of
contexts
answers most questions correctly and shows working
attempts and answers correctly most of the unfamiliar questions.
The student would have achieved a higher level if he had:


not made a mistake in question 4 regarding the sum of roots of a
quadratic
used the correct formula for question 7(b), which was an unfamiliar
situation.
Example 8
Teacher task
Student work
Moderator comments
Extended mathematics test
Broad-based classroom test
Extended mathematics
Branches of the framework: Number, algebra, geometry and trigonometry
MYP year: 5
Criterion
A
B
C
D
Level achieved
8
–
–
–
Background
This test was considered a broad-based test because it covered at least three
branches of the MYP framework for mathematics. The test is appropriate for an
extended mathematics course as it covers a variety of topics from the extended
mathematics framework.
The students were familiar with solving arithmetic and geometric series. Within
the branch of algebra, students were capable of solving exponential equations
numerically and graphically. Students had also seen logarithmic functions in
class and were able to solve logarithmic equations. Students were familiar with
the laws of sines and cosines and could solve trigonometric equations
algebraically and graphically.
The test included a series of questions ranging from simple and familiar to more
complex ones. The teacher included a few unfamiliar questions (questions 1, 4, 5
and 8) where the context of the problem had been modified to make it unfamiliar
to the students.
The exam lasted two hours and was carried out under strict examination
conditions. Students were allowed to use scientific calculators.
Assessment
Criterion A: Knowledge and understanding
Maximum 8
Achievement
level
7–8
Descriptor
The student consistently makes appropriate deductions when
solving challenging problems in a variety of contexts including
unfamiliar situations.
This work achieved level 8 because the student:


consistently makes appropriate deductions when solving challenging
problems
solves problems posed in an unfamiliar context such as those in
questions 1, 4, 5, and 8.
Example 9
Teacher task
Student work
Moderator comments
Geometric average limits
Mathematical investigation
Extended mathematics
Branches of the framework: Number, algebra
MYP year: 5
Criterion
A
B
C
D
Level achieved
–
7
6
–
Background
For this investigation, students required prior knowledge on rules of indices,
fractional indices, number patterns and formulae. Students also needed to be
competent in the use of spreadsheets.
This investigation follows on from example 3 (Average limits) but it can also be
done as a separate activity. The task was appropriate for extended mathematics
as the topics covered were part of the extended mathematics framework. The
investigation allowed students to identify patterns, look for general rules, come
up with general formulae and justify their findings.
Students were given a worksheet and were asked to complete the first sequence
and show that it has a limit of 4. Students then generated the sequences and had
to find a rule connecting the first two values and the limit of each sequence.
Students were encouraged to use algebraic approaches to solve the problem but
could use other methods to generate the sequences such as using programmable
calculators or Microsoft Excel®.
Students had one week to complete the investigation including class and
homework time. Students were allowed to discuss their findings with other
students and the teacher. However, this investigation could also be adapted to be
done under test conditions.
Assessment
Criterion B: Investigating patterns
Maximum 8
Achievement
level
7–8
Descriptor
The student selects and applies mathematical problem-solving
techniques to recognize patterns, describes them as relationships or
general rules, draws conclusions consistent with findings, and
provides justifications or proofs.
This work achieved level 7 because the student:




correctly generates the sequences using a spreadsheet software and
finds the limit of each sequence
analyses the table of results and finds a general rule that fits the
generated data
tests the general formula and shows that it fits the given data
justifies/proves the same result using an exhaustive algebraic approach.
The work would have achieved a higher level if the student had checked that the
formula worked for other values not given in the list.
Criterion C: Communication in mathematics
Maximum 6
Achievement
level
Descriptor
5–6
The student shows good use of mathematical language and forms
of mathematical representation. The lines of reasoning are concise,
logical and complete.
The student moves effectively between different forms of
representation.
This work achieved level 6 because the student:



uses a variety of terminology, notation and symbols
uses different forms of representations such as tables, formulae and
equations
moves between the different forms of representation to solve the
problem.
Example 10
Teacher task
Student work
Moderator comments
Cutting corners
Mathematical investigation
Extended mathematics
Branches of the framework: Number, algebra
MYP year: 5
Criterion
A
B
C
D
Level achieved
–
8
6
–
Background
This investigation required students to be familiar with arithmetic and geometric
progressions. Students were also expected to be able to work with square
numbers and powers of 2.
This investigation allows students to generate data and identify patterns, and to
describe these as general rules. This task also gives students the opportunity to
use a range of mathematical language including notation and symbols, and
different forms of representation such as tables and algebraic forms.
Students were given a worksheet and had to follow the instructions given to fold
a piece of paper and cut off its corners; this procedure allowed them to generate
a number pattern. Students then had to analyse the number patterns generated
and suggest rules or a general formula to explain them. Finally, students were
expected to provide a justification or proof of why their rule worked for all cases.
The students had three classes of 80 minutes each to complete the investigation;
one class to generate the data and two to find the patterns. The investigation was
done individually in class and the students were allowed to consult the teacher.
The task could be done under test conditions so long as the correct equipment is
provided, for example, paper and scissors.
Assessment
Criterion B: Investigating patterns
Maximum 8
Achievement
level
7–8
Descriptor
The student selects and applies mathematical problem-solving
techniques to recognize patterns, describes them as relationships or
general rules, draws conclusions consistent with findings, and
provides justifications or proofs.
This work achieved level 8 because the student:




generates a complex sequence that may require the use of geometric,
square and powers sequences from the extended mathematics framework
finds formulae that relate to the sequence identified for even and odd
numbers of folds
justifies her results and explains the sequence’s dependence on
doubling as being related to the number of folds
explains that there are different formulae for odd and even numbers of
folds and that the formula is only valid up to
because she can only fold
paper seven times.
Criterion C: Communication in mathematics
Maximum 6
Achievement
level
5–6
Descriptor
The student shows good use of mathematical language and forms
of mathematical representation. The lines of reasoning are concise,
logical and complete.
The student moves effectively between different forms of
representation.
This work achieved level 6 because the student:



uses a wide variety of mathematical language and symbols
communicates her findings using clear and logical lines of reasoning
moves effectively between different forms of representation.
Example 11
Teacher task
Student work
Moderator comments
Medicine and mathematics
Real-life problem
Extended mathematics
Branches of the framework: Algebra
MYP year: 5
Criterion
A
B
C
D
Level achieved
8
–
6
5
Background
Prior to this task, students had been working with exponential functions and
logarithms. They were able to move between these two forms and were aware
that one was the inverse function of the other.
For this task, students had to investigate the absorption of drugs into the
bloodstream. They were guided into the investigation by first working with a
given set of data about insulin in the bloodstream. Then, they had to create a
model for penicillin. Using this model, they were asked to investigate what
happens when the dosage is administered at six-hour intervals.
During an introductory lesson, the task was read in class and expectations were
clarified. This included the use of technology to produce graphs. Students were
advised to do research on the Internet and also to use technology to produce
graphs. All students have graphic display calculators and graphing software in
the school computer lab was available for this investigation. Students had one
lesson of 50 minutes to discuss the investigation in class and then the task was
set for homework. The students had one week to work on this investigation,
including a weekend.
This assignment allowed students to understand some of the possible
applications of mathematics to a real-life problem.
This task was appropriate for year 5 students as it was adequately challenging in
the application of exponential functions, which are covered in the extended
mathematics framework.
Assessment
Criterion A: Knowledge and understanding
Maximum 8
Achievement
level
7–8
Descriptor
The student consistently makes appropriate deductions when
solving challenging problems in a variety of contexts including
unfamiliar situations.
This work achieved level 8 because the student:



selects a method and consistently makes appropriate deductions
is able to move between different forms to solve a challenging problem
is able to apply the rules of logarithms to unfamiliar situations

makes appropriate deductions when using piecewise functions to obtain
the final model.
Criterion C: Communication in mathematics
Maximum 6
Achievement
level
5–6
Descriptor
The student shows good use of mathematical language and forms
of mathematical representation. The lines of reasoning are concise,
logical and complete.
The student moves effectively between different forms of
representation.
This work achieved level 6 because the student:



uses appropriate mathematical language including verbal explanations
and uses correct mathematical notation
uses appropriate forms of mathematical representation and moves easily
from one form of representation to another
uses correctly labelled graphs to support his reasoning.
Criterion D: Reflection in mathematics
Maximum 6
Achievement
level
5–6
Descriptor
The student critically explains whether his or her results make
sense in the context of the problem and provides a detailed
explanation of the importance of his or her findings in connection to
real life.
The student justifies the degree of accuracy of his or her results
where appropriate.
The student suggests improvements to the method when
necessary.
This work achieved level 5 because the student:



verifies if the results make sense by using different methods
provides an explanation of his findings in a real-life context by explaining
what would happen if the dosage was administered at shorter or longer
intervals
attempts to justify the degree of accuracy used.
The student would have achieved a higher level if he had:


correctly justified the degree of accuracy used
explained the importance of his findings to drug administration in real
life by including a spreadsheet and piecewise function for the
administration of the drug at different intervals, for example, at four-hour
intervals.
Example 12
Teacher task
Student work
Moderator comments
London eye
Real-life problem
Extended mathematics
Branches of the framework: Geometry and trigonometry
MYP year: 5
Criterion
A
B
C
D
Level achieved
7
–
6
6
Background
Students had learned trigonometric functions in MYP year 4. The unit on function
transformations was covered in class and students were familiar with the model
and the significance of the constants in this function.
For this task, students had to use a set of data to model the motion of a Ferris
wheel, the London eye. Students were expected to use trigonometric functions
and function transformations, and to look into piecewise functions to solve this
task.
Students had two weeks to complete the task but they were allowed to work in
class during the first week. Students were allowed to use computer software with
graphing applications and graphic display calculators for this task.
This is an appropriate task for year 5 extended mathematics students because it
involves finding an appropriate mathematical model. The task starts off with a
simple trigonometric model but it encourages students to think about the real
motion of the Ferris wheel with stops at specific intervals.
This task reinforces knowledge about functions and transformations and it also
extends the model to piecewise functions. The application to real life makes the
study of trigonometric functions more tangible to students.
Assessment
Criterion A: Knowledge and understanding
Maximum 8
Achievement
level
7–8
Descriptor
The student consistently makes appropriate deductions when
solving challenging problems in a variety of contexts including
unfamiliar situations.
This work achieved level 7 because the student:


consistently makes appropriate deductions to adjust the model in a
variety of contexts
identifies the limitations of the model when applied to an unfamiliar
situation, and improves it accordingly.
The student would have achieved a higher level if he had:


obtained better values for the constants before using regression
written the functions used in the graph for the last questions.
Criterion C: Communication in mathematics
Maximum 6
Achievement
level
5–6
Descriptor
The student shows good use of mathematical language and forms
of mathematical representation. The lines of reasoning are concise,
logical and complete.
The student moves effectively between different forms of
representation.
This work achieved level 6 because the student:




uses mathematical language and forms of representation appropriately
explains work at every stage and his lines of reasoning are clear, logical
and complete
moves confidently between different forms of representation
makes use of appropriate information and communication technology
tools.
Criterion D: Reflection in mathematics
Maximum 6
Achievement
level
5–6
Descriptor
The student critically explains whether his or her results make
sense in the context of the problem and provides a detailed
explanation of the importance of his or her findings in connection to
real life.
The student justifies the degree of accuracy of his or her results
where appropriate.
The student suggests improvements to the method when
necessary.
This work achieved level 6 because the student:

critically explains the choice of parameters used to obtain results




provides detailed explanations of the importance of his findings in
connection to real life
mentions other situations in real life where this mathematical model can
be used
justifies the new model by using data obtained from reliable sources
justifies the degree of accuracy chosen.
Appendices
Frequently asked questions
Which types of tasks are best suited to assess students against criterion A and
which for criterion B?
Although criteria A and B complement each other, it can sometimes become a
challenge for teachers to design a single task to assess students against both
criteria.
Criterion A requires students to apply their knowledge and demonstrate their
understanding to make deductions when solving challenging problems, including
problems in unfamiliar situations. Criterion B, on the other hand, requires
students to discover patterns and thus create new knowledge.
Therefore, suitable tasks to be assessed against criterion A are those that require
students to use deductive reasoning, that is, reasoning from general mathematic
rules to their applications in particular situations. On the other hand, suitable
tasks to be assessed against criterion B are those that stimulate inductive
reasoning, that is, reasoning from specific examples to a general rule.
Successful criterion B tasks provide students with a specific situation or facts,
and expect students to identify a pattern and describe it as a general rule.
Experience has shown that mathematical investigations are generally the most
appropriate tasks for assessing students against criterion B since students will
need a variety of problem-solving techniques to analyse the data and discover
new knowledge.
Should workings and calculations be included in student work responses?
It is good teaching and learning practice to encourage students to write out all
the workings and calculations of their responses. In this way, teachers can fully
understand the path a student takes to reach the final result. Complete working
out and explicit calculations enable teachers to spot trivial errors that might lead
to an erroneous final result.
What does a “real-life problem” mean in MYP mathematics?
A “real-life problem” in MYP mathematics refers to a task set in a real-life context,
that is, a situation that actually exists outside of the classroom. The purpose of
using real-life situations is to make the learning of mathematics relevant and
meaningful to all students so that they can understand the importance of the
relationship between mathematics and other areas of knowledge, as well as the
applications of mathematics to real life.
Examples of contexts can be obtained from the identification of mathematical
patterns in the real world (for example, Fibonacci numbers, geometrical figures in
architecture), but also in the numerous applications of mathematics to other
areas of knowledge such as the sciences, business, engineering, data
processing, statistics, and so on. It is important that teachers include in tasks
problems and resources that relate to real life in order to engage students’
curiosity and motivate their desire to find a solution.
Should tasks be designed to require the use of calculators?
Not necessarily. Many problems may not need a calculator in order to be solved.
Students should be able to recognize when the use of a calculator is appropriate
and when it is not, so it may be useful to set some work that does not require the
use of a calculator.
Could I send more than three tasks in the moderation sample?
The prescribed minimum requirement for moderation is three tasks including one
investigation, one real-life problem and one broad-based exam. However, it may
be difficult to provide two judgments per criterion using only three tasks.
Therefore, four or a recommended maximum of five tasks may be sent in the
sample in order to meet the moderation requirements.
What does the phrase “when necessary” mean in relation to suggested
improvements to the method in level 5–6 of criterion D?
At level 5–6 for criterion D, students are expected to reflect upon the method
chosen. They should suggest improvements to the method, based on their
findings, and assess whether these improvements would have significantly
altered the reliability of the results or their degree of accuracy.
If the student has chosen the most efficient method, or if the task can only be
solved using one method, then this part of the criterion can be ignored.
How many topics of the extended section of the framework have to be included in
any task for it to be considered extended?
It is not a question of the number of topics. For a task to be considered extended
it must have been designed to address the expectations of extended mathematics
and should contain evidence of at least one of the topics (concepts and skills)
specified as extended in the framework for mathematics.
This should not be confused with the concept of a broad-based test. For a test to
be considered broad-based, at least three branches of the framework of
mathematics need to be addressed with a variety of questions.
What is the difference between a “variety of contexts” and an “unfamiliar
situation” in MYP broad-based tests?
An “unfamiliar situation” refers to a new situation, one that students have not
faced before. It requires students to apply their knowledge and/or skills in a way
that they have not experienced previously in class or in earlier work. Unfamiliar
situations include challenging questions set in a way that makes the problem
appear unfamiliar to the students. On the other hand, a “variety of contexts”
refers to questions worded in different ways, or using new parameters. These will
not appear unfamiliar to students but will allow them to move from one context to
another using their powers of deduction.
How many unfamiliar situations are expected to be included in a test for it to be
appropriate for moderation?
There is no definite number. However, there should be at least one clearly
identified unfamiliar situation present in the test to allow students to reach the
higher achievement levels of criterion A. It is important that teachers ensure that
tests include problems in a variety of contexts including unfamiliar situations.
What is the difference between an MYP test and any other traditional mathematics
test that assesses students’ knowledge and understanding?
There are no major differences. However, a traditional non-MYP maths test may
be based only on exercises that require direct application of knowledge without
any reference to applications to real-life situations. An MYP maths test should
include a range of questions that allow students to make deductions, but also
problems set in real-life contexts and unfamiliar situations that require students
to understand the connections between mathematics and real life. (Please refer to
the preamble of criterion A in the section “Mathematics assessment criteria” in
the MYP Mathematics guide for more information on this issue).
In criterion C, what does “the lines of reasoning are clear though not always
logical or complete” mean in the context of communications in mathematics?
This is evident in students’ work when mathematical procedures or calculations
lack verbal explanations or when steps are left out, making it difficult for the
reader to follow the line of mathematical reasoning.
Why does one of the investigations sent for moderation have to be completed
under test conditions?
The reason for this requirement is twofold. Firstly, to ensure authentic and
individual student work and, secondly, to encourage the use of concise
mathematical investigations that can be completed during class time. The
intention behind this decision was to encourage teachers to design shorter and
time-limited mathematical investigations.
How can I keep up-to-date with current issues in the assessment of MYP
mathematics?
You can read the latest MYP General Report for mathematics and extended
mathematics, which can be accessed from the OCC. Go to the MYP mathematics
page and click on Assessment and moderation. The report contains analyses of
the most common problems currently seen in schools submitting assessed work
for moderation, as well as useful advice on assessment and how to use the MYP
mathematics assessment criteria.