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 July2007). 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 years4 and5. 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.
· Broadbased classroom tests/examinations
· Mathematical investigations
· Reallife 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 July2007), 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 / CriteriaStandard mathematics
Example 1 / Broad-based classroom test / End of year exam / A
Example 2 / Broad-based classroom test / A
Example 3 / Mathematical investigation / Average limits / B, C
Example 4 / Egyptian triangulation / B, C, D
Example 5 / Enlarging areas and volumes / B, C, D
Example 6 / Real-life problem / Should we start melting our coins? / A, C, D
Extended mathematics
Example 7 / Broad-based classroom test / Broad-based test / A
Example 8 / Extended mathematics test / A
Example 9 / Mathematical investigation / Geometric average limits / B, C
Example 10 / Cutting corners / B, C
Example 11 / Real-life problem / Medicine and mathematics / A, C, D
Example 12 / 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 / DLevel 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 twodimensional and threedimensional 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 broadbased 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 / Descriptor7–8 / 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 / DLevel achieved / 6 / – / – / –
Background
This test was based on topics that students had covered in the second semester. The test was a broadbased 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 / Descriptor5–6 / 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 / DLevel 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 / Descriptor7–8 / 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 / Descriptor5–6 / 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 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 / DLevel 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 openended 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 reallife context. This method, called triangulation, is still in use today.
Assessment
Criterion B: Investigating patterns
Maximum 8
Achievement level / Descriptor7–8 / 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 / Descriptor1–2 / 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 / Descriptor1–2 / 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 reallife 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 / DLevel 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.