middle and high school - The International School in Genoa

Transcription

2012-2016
ISG
MIDDLE AND HIGH SCHOOL
CURRICULUM | MATHEMATICS v1
This Curriculum Document was reviewed by:
Mrs. Elizabeth Rosser Boiardi
Ms. Alice Careddu
Ms. Louise Sawyer
Dr. Matteo Merlo
Mr. Samer Khoury
The International School in Genoa
Badia Benedettina della Castagna
11A, Via Romana della Castagna
16148 Genova
Italy
Phone: +39 – 010 – 386528
Fax: +39 – 010 – 398700
www.isgenoa.it
secretary@isgenoa.it
Last revision: April 10, 2013
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TABLE OF CONTENTS
ISG MISSION STATEMENT ............................................................................................................................................................... 5
MIDDLE AND HIGH SCHOOL MATHEMATICS AT ISG............................................................................................................. 7
AIMS AND OBJECTIVES .................................................................................................................................................................... 9
AIMS….. .................................................................................................................................................................................................................... 9
OBJECTIVES ........................................................................................................................................................................................................... 9
KEY KNOWLEDGE AREAS ..............................................................................................................................................................11
TEACHING METHODOLOGIES, MATERIALS AND RESOURCES .........................................................................................13
EVALUATION OF STUDENT PROGRESS ....................................................................................................................................15
ASSESSMENT POLICY ..................................................................................................................................................................................... 15
ASSESSMENT CRITERIA ................................................................................................................................................................................ 16
MATEMATICA IN ITALIANO .........................................................................................................................................................17
CURRICULUM REVISION POLICY ................................................................................................................................................17
REVISION PROCESS AND TIMETABLE ................................................................................................................................................... 17
SYLLABUS BY GRADE......................................................................................................................................................................19
GRADE 6 MATHEMATICS SYLLABUS ...................................................................................................................................................... 20
GRADE 6 PROGRAMMA DI MATEMATICA ............................................................................................................................................ 24
GRADE 10 MATHEMATICS SYLLABUS ................................................................................................................................................... 43
SYLLABUS BY KEY KNOWLEDGE AREA ....................................................................................................................................47
NUMBERS, SETS AND ALGEBRA ............................................................................................................................................................... 47
FUNCTIONS ......................................................................................................................................................................................................... 51
GEOMETRY AND TRIGONOMETRY .......................................................................................................................................................... 53
PROBABILITY AND STATISTICS ................................................................................................................................................................ 56
SOURCES .............................................................................................................................................................................................59
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ISG MISSION STATEMENT
OUR SCHOOL'S MISSION IS FOR EVERYONE TO DEMONSTRATE
THE ISG COMMUNITY THEMES OF
RESPECT,
RESPONSIBILITY AND
REACHING FOR EXCELLENCE
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MIDDLE AND HIGH SCHOOL MATHEMATICS AT ISG
MATHEMATICS KNOWS NO RACES OR GEOGRAPHIC BOUNDARIES; FOR
MATHEMATICS, THE CULTURAL WORLD IS ONE COUNTRY.
DAVID HILBERT (1862−1943)
The role played by mathematics is essential, both in school and in society: it promotes universality of language
and analytical reasoning, which in turn help develop logical and critical thinking. Correct understanding and use
of mathematics build confidence for problem-solving and decision-making in everyday life. Mathematics also
serves as a foundation for the study of sciences, engineering and technology, economics and other social sciences.
ISG Middle and High School (IMHS) Mathematics is a school-based curriculum articulated over five years; it aims
to equip all students with the knowledge, understanding and intellectual capabilities to address further courses in
mathematics at the International Baccalaureate (IB) Diploma Programme (DP) level, as well as to prepare
students to use mathematics in their workplace and life in general. It combines themes from the IB Middle Years
Programme, the requirements for the Italian national examination at the end of 8th grade, and tested practices
developed at ISG over the years by the mathematics faculty.
The four main objectives of IMHS Mathematics support the IB learner profile, promoting the development of
students who are knowledgeable, inquirers, communicators and reflective learners.
Knowledge and understanding promotes learning mathematics with understanding, allowing students to
interpret results, make conjectures and use mathematical reasoning when solving problems in school and in reallife situations.
Investigating patterns supports inquiry-based learning. Through the use of investigations, teachers challenge
students to experience mathematical discovery, recognize patterns and structures, describe these as relationships
or general rules, and explain their reasoning using mathematical justifications and proofs.
Communication in mathematics encourages students to use the language of mathematics and its different forms
of representation, to communicate their findings and reasoning effectively, both orally and in writing.
Reflection in mathematics provides an opportunity for students to reflect upon their processes and evaluate the
significance of their findings in connection to real-life contexts. Reflection allows students to become aware of
their strengths and the challenges they face as learners.
IMHS Mathematics builds on experiences in mathematics learning that students have gained in their time in the
International Primary Curriculum (IPC). At the end of the five-year course, students continuing on to the IB
Diploma Programme will have acquired concepts and developed skills which they will be able to apply and extend
in further DP mathematics courses. In particular, the IMHS Mathematics syllabus reflects the concepts and skills of
the presumed knowledge for the DP courses to allow a smooth transition to DP Mathematics.
The present document contains all the general information relevant to the teaching and learning of Middle and High
School Mathematics at the International School in Genoa. In it, ideas and concepts from the best educational
programs worldwide are adapted to the ISG context and background, and enriched to better match the needs of our
learners.
Further information related to individual mathematics courses and materials can be found in the Course Outlines
published each year and handed out to parents during Open House and to students at the beginning of September.
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AIMS AND OBJECTIVES
AIMS…..
The aims state in a general way what the teacher may expect to teach or do, and what the student may expect to
experience or learn. In addition, they suggest how the student may be changed by the learning experience.
The aims of the teaching and study of IMHS Mathematics are to encourage and enable students to:
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appreciate the usefulness, power and beauty of mathematics, and recognize its relationship with other
disciplines and with everyday life
develop a positive attitude toward the continued learning of mathematics
appreciate the international dimension in mathematics and its varied cultural and historical perspectives
gain knowledge and develop understanding of mathematical concepts
acquire the ability to communicate mathematics with appropriate symbols and language
cultivate logical, critical and creative thinking, and patience and persistence in problem solving
develop power of generalization and abstraction
promote the ability to reflect upon and evaluate the significance of their work and the work of others
develop and apply information and communication technology skills in the study of mathematics
acquire the knowledge, skills and attitudes necessary to pursue further studies in mathematics.
OBJECTIVES
The objectives state the specific targets and expected outcomes that are set for learning in the subject. They define
what the student will be able to accomplish as a result of studying the subject. These objectives relate directly to
the assessment criteria found in the EVALUATION OF STUDENT PROGRESS section.
Learning outcomes, in terms of acquired concepts and developed skills, are detailed in the SYLLABUS BY GRADE
and SYLLABUS BY KEY KNOWLEDGE AREA sections.
A Knowledge and understanding
IMHS Mathematics promotes learning mathematics with understanding, allowing students to interpret results,
make conjectures and use mathematical reasoning when solving problems in school and in real-life situations.
At the end of the course, students should be able to:
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know and demonstrate understanding of the concepts from the four KEY KNOWLEDGE AREAS of
mathematics (number, sets and algebra, functions, geometry and trigonometry, probability and statistics)
use appropriate mathematical concepts and skills to solve problems in both familiar and unfamiliar
situations, including those in real-life contexts
select and apply general rules correctly to make deductions and solve problems, including those in reallife contexts.
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B Investigating patterns
IMHS Mathematics supports inquiry-based learning. Through the use of investigations, teachers challenge
students to experience mathematical discovery, recognize patterns and structures, describe these as relationships
or general rules, and explain their reasoning using mathematical justifications and proofs.
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select and apply appropriate inquiry and mathematical problem-solving techniques
recognize patterns
describe patterns as relationships or general rules
draw conclusions consistent with findings
justify or prove mathematical relationships and general rules.
C Communication in mathematics
IMHS Mathematics encourages students to use the language of mathematics and its different forms of
representation, to communicate their findings and reasoning effectively, both orally and in writing.
At the end of the course, students should be able to communicate mathematical ideas, reasoning and findings by
being able to:
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use appropriate mathematical language in both oral and written explanations
use different forms of mathematical representation
communicate a complete and coherent mathematical line of reasoning using different forms of
representation when investigating problems.
Students are encouraged to choose and use information and communication technology (ICT) tools as appropriate
and, where available, to enhance communication of their mathematical ideas. Some of the possible ICT tools used
in mathematics include spreadsheets, graph plotter software, dynamic geometry software, computer algebra
systems, mathematics content-specific software, graphic display calculators (GDC), word processing, desktop
publishing, graphic organizers and screenshots.
D Reflection in mathematics
IMHS Mathematics provides an opportunity for students to reflect upon their processes and evaluate the
significance of their findings in connection to real-life contexts. Reflection allows students to become aware of
their strengths and the challenges they face as learners.
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explain whether their results make sense in the context of the problem
explain the importance of their findings in connection to real life where appropriate
justify the degree of accuracy of their results where appropriate
suggest improvements to the method when necessary.
KEY KNOWLEDGE AREAS
NUMBER, SETS AND ALGEBRA
The ability to work with numbers is an essential skill in mathematics. Students are expected to have an
understanding of number concepts and to develop the skills of calculation and estimation. Students should
understand that the use of numbers to express patterns and to describe real-life situations goes back to
humankind’s earliest beginnings, and that mathematics has multicultural roots.
Algebra is an abstraction of the concepts first used when dealing with number and is essential for further learning
in mathematics. Algebra uses letters and symbols to represent number, quantity and operations, and employs
variables to solve mathematical problems.
FUNCTIONS
The concept of function is rightly considered as one of the most important in all of mathematics. It arose as the
necessary mathematical tool for the quantitative study of natural phenomena, and today it is an instrument for the
study of the phenomena and situations of biological sciences, human and social sciences, business,
communications, engineering, and technology.
Mathematics constitutes an essential mean of description, explanation, prediction, and control: for all these
applications, the notions of model and function are vital. Formulas from geometry, physics, and from other
sciences can be taken as examples of functions and explored from diverse viewpoints.
Several recent technological developments may have a very significant role in the study of functions. Especially
important are graphic calculators and computers with appropriate software such as spreadsheets, graph plotters.
GEOMETRY AND TRIGONOMETRY
The study of geometry and trigonometry enhances students’ spatial awareness and provides them with the tools
for analysing, measuring and transforming geometric quantities in two and three dimensions .
PROBABILITY AND STATISTICS
This branch of mathematics is concerned with the collection, analysis and interpretation of quantitative data and
uses the theory of probability to estimate parameters, discover empirical laws, test hypotheses and predict the
occurrence of events.
Through the study of statistics, students should develop skills associated with the collection, organization and
analysis of data, enabling them to present information clearly and to discover patterns. Students will also develop
critical-thinking skills, enabling them to differentiate between what happens in theory (probability) and what is
observed (statistics).
Students should understand both the power and limitations of statistics, becoming aware of their legitimate use in
supporting and questioning hypotheses. Students should use these skills in their investigations and are
encouraged to use information and communication technology (ICT) whenever appropriate.
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TEACHING METHODOLOGIES, MATERIALS AND RESOURCES
METHODOLOGIES
Teachers at ISG adopt a variety of teaching methodologies in order to cater for different learning styles. The
various approaches to learning are a means to provide students with the tools that will enable them to take
responsibility of their own learning. This involves articulating, organizing and teaching the skills, attitudes and
practices that students require to become successful learners.
Teaching methodologies
Skill area
Student learning expectations
Learning activities
Instructional practices
Long-term projects
Organization
Time management, self management
Collaboration
Group work
Group investigations
Communication
Mathematical literacy: mathematics-specific language and
forms of representation
Lectures in various forms (whiteboard,
ActiveBoard, presentations)
Communicating ideas clearly and logically
Formal demonstrations/proofs
Information literacy
Reflection
Thinking
Transfer
Individual investigations
Resourceful collection of information from a variety of
sources using a range of technologies
Individual and group investigations
Use of mathematics software
Use of mathematics reference books
Formative assessment tasks
Evaluation of results and methods
Practice exercises
Evaluation of one’s own learning
Self- and peer-assessment
Understanding and applying knowledge and concepts
Lectures
Identifying and selecting strategies to solve problems
Practice exercises
Using mathematical skills and knowledge in real-life
contexts and making connections with other subject areas
Applications across knowledge areas
Applications across disciplines
The teaching of mathematics at ISG is structured to reflect the IB learner profile in accordance with the current
educational thinking.
Increased focus on:
Decreased focus on:
connecting concepts across key knowledge areas
mathematics as a collection of uncorrelated facts
making mathematics more meaningful to students
rote practice, memorization and formal symbol manipulation
solving relevant real-life mathematics problems
word problems as problem-solving
several strategies for possible multiple solutions
classification of problems, one method per problem
student speculation, independent formation of ideas
teaching by authority
clear explanation of processes, reflection upon results
finding answers
team-work within and across disciplines
teachers working in isolation
multiple resources for learning
a textbook-driven course
investigations, questions and discussions
the use of exercise sheets
a broad range of assessment strategies
multiple-choice assessment
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RESOURCES
ISG offers several resources to facilitate student learning in mathematics. They include:
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a computer lab with 20+ laptops
free mathematics software (Gnuplot, LaTeX, Padowan Graphing Software)
licensed mathematics software (Mathematica, Geometer’s Sketchpad, LoggerPro, MS Excel, Maths300)
ActiveBoards
a library section with reference textbooks.
MATERIALS
During IMHS math classes, all students are expected to have with them the following materials:
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textbook
writing instruments
scientific calculator
notebook
ruler, compass, protractor
A Graphic display calculator (GDC) is required starting from 10th grade. The recommended model is Texas
Instruments (TI) 84 Plus.
More details on the required material can be found in each teacher’s Course Outline.
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EVALUATION OF STUDENT PROGRESS
ASSESSMENT POLICY
Assessment in IMHS Mathematics is
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designed so that students can
a.
demonstrate their learning of concepts in authentic contexts
b.
apply acquired skills to familiar and unfamiliar problems.
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structured to examine the achievement levels in each of Objectives A, B, C and D.
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meant to provide teachers with feedback that is used to adapt the teaching and learning strategies
with the aim of meeting each learner's needs.
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criterion-referenced as opposed to norm-referenced. Please see the ASSESSMENT CRITERIA section
below.
Assessment tasks for mathematics are divided into:
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Informal assessment, consisting of class worksheets, homework, projects, investigations, presentations,
class participation, etc.
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Formal assessment, consisting of tests and quizzes under examination conditions.
This reflects the IB Diploma Programme division into Internal Assessment – student investigations developed
over the two-year course – and External Assessment – a series of externally set exams taken at the end of the
second IBDP year.
Assessment is carried out formatively throughout each course: the purpose of formative assessment is to provide
students, parents and teachers with objective and timely feedback on the learner’s progress. Formative
assessment tasks, both informal and formal, are graded on a percent scale based on the assessment criteria listed
in the next section. They contribute to quarter average grades according to the following weighting matrix.
6th grade
7th grade
8th grade
9th grade
10th grade
IBDP
Informal
40%
40%
30%
30%
25%
Internal
20%
Formal
60%
60%
70%
70%
75%
External
80%
100%
100%
100%
100%
100%
100%
Quarter grades are then converted into IB grades according to the ISG Secondary School grading system below.
ISG Comment
Grade
Percent
Excellent work: the student consistently and almost faultlessly demonstrates sound understanding of
concepts and successful application of skills in a wide variety of contexts and consistently displays
independence, insight, autonomy and originality.
7
90-100
Very good work: the student consistently demonstrates sound understanding of concepts and successful
application of skills in a wide variety of contexts and generally displays independence, insight, autonomy and
originality.
6
80-89
Good work: the student consistently demonstrates sound understanding of concepts and successful
application of skills in a variety of contexts and occasionally displays independence, insight, autonomy and
originality.
5
70-79
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Satisfactory performance: the student generally demonstrates understanding of concepts and successful
application of skills in normal contexts and occasionally displays independence, insight, autonomy and
originality.
4
60-69
Mediocre work (conditional pass): the student demonstrates a limited understanding of the required
concepts and only applies skills successfully in normal situations with support. Partial achievement against
most of the objectives.
3
50-59
Poor work: the student has difficulty in understanding the required concepts and is unable to apply skills
successfully in normal situations even with support. Very limited achievement against all the objectives.
2
20-49
1
0-19
Very poor work: Minimal achievement in terms of the objectives.
High school students are also assessed summatively. Summative assessment consists of formal benchmarks at the
end of significant portions of each course – i.e. semester finals. A score out of 7 is given to all summative
assessment tasks.
Please see the document “Secondary school grading systems” for further clarification on the calculation of
semester and end-of-year averages and for GPA and letter grade conversions.
ASSESSMENT CRITERIA
The assessment criteria relate directly to the OBJECTIVES as listed in the previous sections.
The approximate weighting of the Objectives is listed below. For a coherent approach to assessment practices
over the entire programme, weights are adjusted from grade level to grade level to match the increased
expectations in terms of mathematical maturity. This means for instance that the relative importance of
reflection and communicative skills grows with respect to pure factual knowledge and recall.
Objective
Typical assessment tasks
6th gr.
7th gr.
8th gr.
9th gr.
10th
gr.
A
Knowledge and
understanding
classroom tests, examinations, real-life problems
and investigations that may have a variety of
solutions
75%
75%
60%
60%
50%
B
Investigating
patterns
mathematical investigations of some complexity
which should allow students to choose their own
mathematical techniques to investigate problems
10%
10%
15%
15%
20%
C
Communication
in mathematics
real-life problems, tests, examinations and
investigations designed to allow students to show
complete lines of reasoning using mathematical
language
10%
10%
15%
15%
15%
D
Reflection in
mathematics
mathematical investigations or real-life problems
designed to provide students with opportunities to
use mathematical concepts and skills to solve
problems in real-life contexts
5%
5%
10%
10%
15%
100%
100%
100%
100%
100%
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MATEMATICA IN ITALIANO
Il programma di matematica in italiano nasce dall’esigenza di preparare gli studenti secondo le metodologie e i
programmi di studio delle scuole italiane, in vista degli esami di idoneità alla terza media (8th grade).
I nostri programmi sono ministeriali, perciò validi in qualunque scuola italiana, e sono approvati da anni dalle
varie scuole statali italiane.
Il corso prevede l’integrazione del programma internazionale, insistendo sulle differenze di metodo e di curricula
italiani. Gli studenti dovranno impadronirsi dei concetti fondamentali, dovranno saper risolvere problemi e
svolgere esercizi su tutti gli argomenti (talvolta esercizi di ragionamento, talvolta più meccanici), e dovranno
acquisire una solida base per le conoscenze future.
Saranno valutati i compiti a casa, la costanza e l’impegno nel loro svolgimento, i test in classe (sempre con
preavviso), l’attenzione durante le lezioni, l’ordine e la completezza del quaderno, l’impegno in generale.
CURRICULUM REVISION POLICY
A curriculum revision process is established at ISG to ensure that the mathematics syllabus is
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adequate to current students’ needs
in line with current educational thinking
pursuant to the current IBDP Mathematics Curriculum and to the Italian State Examinations.
To this effect, the results of student assessment – both internal (e.g. ISG tests) and external (e.g. ISA testing, IBDP
scores) – will be carefully evaluated to identify areas of weakness and strength in the delivery of the curriculum.
A four-year revision cycle is established for each curricular area on a rotation basis, with two curricula revised
each academic year.
REVISION PROCESS AND TIMETABLE
The present document will become effective at the beginning of the academic year 2012-2013. During its first year
of validity, it will be completed and updated in all its parts as a work-in-progress process. It will then be in place in
its definite form for the academic years 2013-2014 and 2014-2015. The next year will be a curriculum review
year, with the new document entering into effect by September 2016.
academic year
curriculum in place
action
2012-2013
Mathematics 2012-2016 v1
(present document)
creation of
curriculum
update and
completion
2013-2014
none
2014-2015
none
2015-2016
curriculum review
2016-2017
update and
completion
2011-2012
next
cycle
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SYLLABUS BY GRADE
The following section contains the details of the syllabus. Each grade level syllabus is split in the four key
knowledge areas and the corresponding sub-topics; concepts and skills are indicated, and possible
amplifications/extensions are highlighted in red.
Example:
sub-topic
Representations
Function notation
Functions
Use of a variety of function
notations (for example,
mapping, f(x)=,y=, etc.)
Rational/reciprocal/other
Reciprocal function: domain
and range, equations of
asymptotes
Graphing different types of
functions and understanding
their characteristics
extension
Quadratic
Domain and range
Quadratic functions in the form y=a(x-h)2 +k and in
intercept form
Quadratic inequalities
Quadratics in disguise
Solution of quadratic equations by factoring and the
quadratic formula (use of the GDC is also encouraged)
Sketching and interpreting graphs of quadratic
functions
Transformations
Effects of parameters a,h,k on the graph of y=a(x-h)2 +k
Transformations of quadratic functions
concepts
key knowledge area
skills
More information on the syllabus, including the sequence of topics, can be found in each teacher’s Course Outline.
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GRADE 6 MATHEMATICS SYLLABUS
Number
Integers
Using integers
Rule of order
Adding and subtracting integers
Multiplying and dividing integers
Number theory
Odd and even, prime and composite
Divisibility
Factors
Multiples
Prime factorisation
Triangular numbers
Square numbers
Square roots
HCF by listing factors
LCM by listing multiples
Fractions
Fractions of shapes
One number as a fraction of another
Equivalent fractions
Simplifying fractions
Mixed numbers and improper
fractions
Fractions and decimals
Decimals
Adding and subtracting
Multiplying decimals by whole
numbers
Multiplying decimals by decimals
Dividing decimals by whole
numbers
Comparing fractions
Adding and subtracting fractions
“Fractions of” whole numbers
Multiplying and dividing integers by
fractions
Word problems
Number, sets and algebra
Number
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Percentages, ratios and interest
Estimation and error
Out of 100
Percentages to fractions and decimals
Fractions and decimals to percentages
Percentages of – mentally
Percentages of – using calculator
Whole number place value
Decimal place value
Putting numbers in order
Placing numbers on the number line
Reading whole numbers and decimals
Rounding
Rounding to the nearest 10,100,1000, to the
first and second decimal place, to the nearest
whole number
Idea of direct method
Writing ratios
Equivalent ratios
Ratio and proportion
Dividing in a given ratio
Standard
form
scientific
notation
Multiplication and division by
10,100,1000
Units of measurement
Units for length, mass, time,
capacity
Metric and imperial
equivalents
Conversions
Reading scales
red=amplifications/extensions
Functions
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Algebra
Sequences and series
Recognizing and describing number patterns
Finding the next term in a sequence
Expressions, exponents and logarithms
Notation
Equations
Writing equations from words
Solving equations by inspection
Writing sequences from rules
Writing sequences using the nth term rule
Making sequences using matches and finding the
rule
Writing the rule using nth term notation
Writing expressions in best algebraic notation
Collecting like terms
Substituting into expressions and formulae
Solving equations using inverse operations on flow diagrams
Functions and their representations
Finding the output number
Finding the rule given the input and output
Graphs
Point interpretation
Graphing real life data to form lines
Using the inverse function to find the input number
Reading and interpreting real life graphs
Geometry
Geometry and trigonometry
Quadrilaterals
Names of all types
Lines of symmetry of each
Tangrams
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Polygons
Names of each
Definition of “regular”
Tessellations
3D shapes
Nets of prisms and pyramids
Vertices, faces and edges of prisms and pyramids
Euler’s Rule
Isometric drawings
Geometry
Trigonometry
Coordinate geometry
Coordinates in 4 quadrants
Reflection and symmetry
Rotation and symmetry – include order
Translation
Include above on the coordinate plane
Lines and angles
Measure lines
Estimate, measure, draw and name angles,
include reflex
Classify angles – acute, right, obtuse, straight,
reflex, revolution
Calculating angles – on a straight line, at appoint,
vertically opposite
Define parallel and perpendicular lines
Recognise and name the types of angles on
parallel lines
Perimeter, area, volume
Triangle, rectangle, parallelogram, trapezium
Perimeter and area of shapes made from
rectangles
Surface area from nets
Volume by counting cubes
Volume of a rectangular prism
Triangles
Naming triangles
Classifying triangles:
sides- scalene, isosceles, equilateral
angles- acute, right, obtuse
Angle sum of a triangle
Constructing triangles, including use of compass
Probability and statistics
Probability
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Probability of events
The language of probability – likely, unlikely, certain, impossible, even chance, equally likely
Probability scale
Listing outcomes
Calculating probability
Probability from experiments
Statistics
Data collection
Survey
Tally chart
Grouped data
Statistical representations
Draw and interpret the following:
Bar charts include grouped (discrete)
Line graphs
Frequency diagrams
Pie charts
Statistical measures
Mean, mode, median, range – not grouped
GRADE 6 PROGRAMMA DI MATEMATICA
Aritmetica
Aritmetica, insiemi e algebra
Numeri e rappresentazione sulla retta
Concetto di numero
Numeri cardinali e numeri ordinali
La numerazione decimale: migliaia, centinaia,
decine, unità, decimi, centesimi, millesimi
Valore posizionale delle cifre
Valore assoluto e valore relativo dei numeri
Notazione polinomiale dei numeri
Insieme dei numeri naturali,
rappresentazione sulla retta
Numeri decimali
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Ordinare su una retta numeri naturali e
numeri decimali
Scrivere un numero dato in forma
polinomiale, come somma di prodotti con
potenze di 10
Scomporre un numero dato in migliaia,
centinaia, decine, ecc
Stabilire se un numero dato appartenga
all’insieme dei numeri naturali
Le quattro operazioni
Concetto di operazione binaria Proprietà
delle operazioni di “essere interne” ad un
insieme
Concetto di elemento neutro
Addizione, sottrazione, moltiplicazione e
divisione: definizione, terminologia,
proprietà e eventuale elemento neutro
Moltiplicazione e divisione per 10, 100, 1000
Operazioni inverse e loro significato:
addizione/sottrazione e
moltiplicazione/divisione
Eseguire le quattro operazioni in riga e in
colonna con la “prova” dell’operazione
inversa
Saper operare con i numeri decimali
Utilizzare le proprietà delle operazioni per
semplificare i calcoli
Risolvere semplici espressioni aritmetiche
contenenti le quattro operazioni
Frazioni
Concetto di unità frazionaria e di
frazione come operatore
Terminologia e caratteristiche dei
vari tipi di frazione (propria,
impropria, apparente)
Concetto di frazione
complementare e di frazioni
equivalenti
I numeri razionali
Concetto di numero razionale,
rappresentazione di un
numero razionale
Operazioni (addizione,
sottrazione, moltiplicazione,
divisione ed elevamento a
potenza) con numeri razionali
Concetto di frazioni inverse
Individuare unità frazionarie e
frazioni, riconoscere e scrivere i
vari tipi di frazione
Scrivere frazioni equivalenti
Applicare il concetto di frazioni
equivalenti per ridurre ai minimi
termini o al minimo comune
denominatore
Confrontare due o più frazioni,
saperle scrivere in ordine crescente
e decrescente, confrontarle con
numeri e saper scrivere la frazione
complementare di una frazione
data
Scrivere e rappresentare un
numero razionale
Eseguire le operazioni con i
numeri razionali
Espressioni con numeri
razionali Risoluzione di
problemi del tipo: “Calcola il
numero sapendo che i suoi
2/3 valgono 44”,
“Calcola i numeri e
sapendo che
e
che è la metà di ”
Aritmetica, insiemi e algebra
Aritmetica
25
Divisori, multipli, MCD e mcm
Concetto di divisore e multiplo di un
numero
Definizione di numero primo
Scomposizione in fattori primi Criteri di
divisibilità
Concetto e definizione di MCD e di mcm,
concetto di numeri primi tra loro, calcolo
di MCD e mcm, problemi con essi
Potenze
Terminologia delle potenze, concetto di
elevamento a potenza
Proprietà delle potenze
Potenze di 1 e di 0
Espressioni aritmetiche con le potenze
Notazione esponenziale
Dato un numero trovare i suoi divisori
Stabilire se un numero è divisore o
multiplo di un altro, o se è divisibile per
un altro
Riconoscere i numeri primi, scomporre i
numeri in fattori primi
Calcolare MCD e mcm tra due o più
numeri, anche nei casi particolari (ad
esempio se uno è multiplo dell’altro),
applicarli per risolvere problemi del tipo:
“Quante parti si ottengono tagliando
quattro corde lunghe rispettivamente
144cm, 180cm, 126cm e 108cm in parti
uguali e della massima lunghezza
possibile?”
Calcolare le potenze, riconoscere i vari
termini (base, esponente, potenza)
Saper applicare le proprietà delle potenze
in modo opportuno
Espressioni con le potenze
Insiemi
Algebra
Insiemistica
Definizione di insiemi e di insiemi
matematici
Cardinalità di un insieme
Rappresentazione di un insieme
per caratteristica, per elencazione e
mediante diagrammi di EuleroVenn
Insieme finito, infinito e vuoto
Concetto di sottoinsieme e sua
rappresentazione grafica
Le operazioni con gli insiemi:
unione, intersezione e differenza
Determinare se un elemento
appartiene o no ad un insieme dato
Determinare se un insieme è finito
o infinito e, se finito, determinarne
la cardinalità
Rappresentare un insieme dato
(per caratteristica, per elencazione,
con diagramma di Eulero-Venn)
Stabilire se un insieme è un
sottoinsieme di un insieme dato
Dati due insiemi determinarne (e
rappresentare con i diagrammi di
Eulero-Venn) unione, intersezione
e differenza
Soluzione di problemi
Comprensione, analisi e soluzione
di un problema
Fasi della risoluzione: lettura e
comprensione, traduzione in
linguaggio matematico, analisi,
ipotesi, verifica dell’ipotesi,
soluzione
Risolvere problemi del tipo: “I
risparmi di Mr X ammontano a
28000 euro, quelli di Mr Y sono la
metà Se Mr Y compra
un’automobile da 4500 euro, quanti
soldi gli rimangono?”
Individuare l’algoritmo migliore per
risolvere ogni problema
Usare le espressioni se necessario
Geometria e trigonometria
Geometria
Enti geometrici
Significato di geometria,
introduzione degli enti
fondamentali (punto, retta,
piano)
Concetto di semiretta e di
segmento
Assiomi euclidei
Rette parallele e perpendicolari
26
Individuare e rappresentare gli
enti fondamentali della
geometria
Riconoscere e disegnare punti,
rette (eventualmente parallele o
perpendicolari), semirette,
segmenti e spezzate
Riconoscere e disegnare
segmenti consecutivi, adiacenti,
incidenti e coincidenti
Confrontare i segmenti e
operare su essi
Poligoni
Concetto di poligono, vari
tipi di poligono
(convesso, concavo,
equilatero, equiangolo,
regolare, irregolare)
Proprietà generali dei
poligoni
Concetto di congruenza e
isoperimetria tra
poligoni
Quadrilateri
Definizione e vari tipi di
quadrilatero: scaleno,
trapezio, parallelogramma
(rettangolo, rombo,
quadrato), deltoide Alcune
proprietà di essi
Perimetro dei quadrilateri
Riconoscere e disegnare
un poligono, un poligono
convesso, un poligono
concavo
Riconoscere poligoni
equilateri, equiangoli e
regolari
Individuare le proprietà
generali di un poligono,
riconoscere poligoni
congruenti e
isoperimetrici
Riconoscere i vari tipi di
quadrilateri e individuarne
le proprietà
Riconoscere i vari tipi di
parallelogramma
Individuare le proprietà di
quadrati, rettangoli, rombi
Risolvere problemi sul
perimetro dei quadrilateri
Trigonometria
Gli angoli
Concetto di angolo, definizione
Introduzione ai vari tipi di
angoli (retto, piatto, giro, acuto,
ottuso)
Angoli tra loro consecutivi,
adiacenti, complementari,
supplementari, esplementari
Concetto di bisettrice di un
angolo
Confronto, somma, differenza e
multipli di angoli
Riconoscere un angolo e
individuarne i vari tipi
Disegnare la bisettrice di un
angolo
Confrontare angoli
Riconoscere angoli
complementari, supplementari
ed esplementari e saperli
disegnare
Riconoscere angoli consecutivi
e adiacenti
Triangoli
Concetto di triangolo e sue proprietà
Elementi di un triangolo
Concetto di altezza, bisettrice,
mediana e asse di un triangolo e
proprietà
Punti notevoli di un triangolo
(baricentro, incentro, ortocentro,
circocentro)
Triangoli particolari e loro proprietà
(isoscele, equilatero, rettangolo)
Perimetro dei triangoli
Riconoscere e disegnare i vari tipi di
triangolo e individuarne le proprietà
Disegnare altezze, bisettrici, mediane
e assi e individuare i punti notevoli
Risolvere problemi sul perimetro dei
triangoli
Number
27
Integers
Rule of order
Number theory
Divisibility
Factors
Multiples
Prime factorisation
Squares and square roots
Cubes and cube roots
Fractions
Fractions of shapes
Ordering fractions
HCF by prime factorisation
LCM by prime factorisation
Multiplying and dividing fractions
Word problems
Using the calculator
Decimals
Multiplying by decimals
Dividing by decimals
Converting fractions to decimals
using division (including recurring
decimals)
Number
Percentage and increase or decrease
Unitary method for direct proportion
Solving ratio and proportion problems
Putting numbers in order
Placing numbers on the number line
Rounding
Estimation and accuracy
Standard form - scientific notation
Using indices to write powers of 10
Names of very large numbers
Multiplying and dividing by powers
of 10
Units for length, mass, time,
capacity
Metric and imperial equivalents
Rounding to powers of 10 (order of
magnitude)
Rounding to decimal places
Multiplying and dividing by
multiples of the powers of 10
Conversions including area and
volume
Functions
Algebra
28
Continue number sequences
Notation
Index laws
Equations
Making sequences using matches and finding the rule
Collecting like terms – integer answers
Multiplying and dividing simple terms that
involve indices
Using distributive property to remove brackets
Substituting into expressions that involve
indices
Solving linear equations using inverse operations on flow
diagrams
Solving linear equations using algebra – variable on one side
only
Using x and y to describe functions
Graphs
Linear
Graphing lines of the form y=mx+c
Informal discussion of gradient of lines
Intercepts of lines
Lines parallel to the axes
Geometry
Quadrilaterals
Names of all types
Definitions using sides and
angles
Properties of each including
diagonal properties
Constructing quadrilaterals
using compass and
protractor
29
Polygons
Angle sum of polygons
Angle size of regular
polygons
3D shapes
Euler’s Rule for prisms and pyramids
Platonic solids – identify these
Existence of semi-regular polyhedra
Isometric drawings
Plans and elevations
Cross sections
Geometry
Trigonometry
Coordinate geometry
Transformations on the coordinate plane
Symmetry
Congruence and transformations
Enlargement
Scale drawings
Reading scale drawings
Lines and angles
Estimate, measure, draw and name angles,
include reflex.
Classify angles – acute, right, obtuse, straight,
reflex, revolution
Calculating angles – on a straight line, at appoint,
vertically opposite
Recognise and name the types of angles on
parallel lines
Calculate angles on parallel lines including any of
the above angle
Review triangle, rectangle, parallelogram,
trapezium
Circumference and area of a circle
Surface area and volume of prisms
Triangles
Naming triangles
sides- scalene, isosceles, equilateral
angles- acute, right, obtuse
Constructing triangles, including use of compass
Probability
The language of probability, include mutually exclusive events
Estimating probability from relative frequency
Comparing experimental and theoretical probabilities
Tree diagrams
Calculating probability by listing outcomes
Statistics
Data collection
Surveys, questionnaires
Two way tables
30
Draw and interpret bar charts (include
grouped continuous), line graphs, frequency
diagrams, pie charts
Stem and leaf diagrams
Misleading graphs
Mean, mode, median, range – include grouped
ARITMETICA
Numeri
31
Frazioni e numeri decimali
Concetto di numero decimale
Frazioni generatrici di numeri decimali
I numeri decimali illimitati periodici
Dal numero periodico alla frazione
Confronto di frazioni
La radice quadrata
Significato di estrazione di radice
quadrata, proprietà della radice
(radice del prodotto e radice del
quoziente)
Algoritmo per l’estrazione
Radice quadrata esatta e
approssimata, radice di un numero
decimale
Uso delle tavole numeriche,
definizione di numero irrazionale ed
esempi
Rapporti e proporzioni
Concetto di rapporto numerico,
concetto di proporzione e
terminologia (medi, estremi)
Proprietà delle proporzioni
(comporre, scomporre, invertire,
permutare, proprietà fondamentale),
catena di rapporti
Concetto di percentuale
La proporzionalità
Significato di grandezze
direttamente e inversamente
proporzionali
Concetto di funzione di
proporzionalità diretta e inversa
Riconoscere un numero decimale limitato e
illimitato
Riconoscere un numero periodico semplice e
periodico misto
Saper trasformare un numero decimale in
frazione e viceversa
Operazioni ed espressioni con i numeri decimali
Calcolare la radice quadrata di un
numero
Calcolare radici quadrate esatte e
approssimate di un numero naturale
e razionale
Applicare le proprietà delle radici
quadrate, usare le tavole numeriche
per il calcolo
Espressioni con radici quadrate
Scrivere il rapporto tra due numeri,
individuare e scrivere proporzioni
Applicare le proprietà ad una
proporzione
Risolvere una proporzione e
risolvere catene di rapporti
Problemi con la percentuale, ad
esempio: “Calcola lo sconto applicato
ad un articolo da 15 euro che a
prezzo pieno costava 25 euro”
Applicazione delle proporzioni alle
carte geografiche in scala
Riconoscere grandezze
direttamente ed inversamente
proporzionali
Scrivere e rappresentare una
funzione di proporzionalità
diretta e inversa
Risolvere problemi del tre
semplice, ad esempio: “Per
comprare 12 kg di frutta si
spendono 15.48 euro Quanto si
spenderebbe per comprare 18 kg
dello stesso tipo di frutta?”
Geometria
Area di poligoni
Figure piane, concetto di equivalenza
Area di: triangolo, quadrato, rettangolo, rombo,
parallelogramma, trapezio
Area di un poligono qualsiasi
Formula di Erone (area di un triangolo
qualsiasi conoscendo solo la misura dei lati)
GEOMETRIA
Individuare e disegnare figure equivalenti,
applicare il principio di equiscomponibilità per
riconoscere figure equivalenti
Calcolare l’area di triangoli e quadrilateri
Problemi con perimetro e area
32
Circonferenza e cerchio
Significato di circonferenza e cerchio
Parti di circonferenza e cerchio (raggio, diametro, arco,
corona circolare, segmento circolare, settore circolare)
Posizioni reciproche tra retta e circonferenza e tra due
circonferenze (secanti, tangenti, esterne)
Angoli al centro e angoli alla circonferenza e loro
proprietà
Poligoni inscritti e circoscritti ad una circonferenza
Area dei poligoni circoscritti, area dei poligoni regolari
Individuare, riconoscere e disegnare circonferenze e
cerchi, riconoscerne caratteristiche, proprietà e parti
Individuare e applicare proprietà di circonferenze in
particolari posizioni con una retta o con un’altra
circonferenza
Disegnare angoli al centro e alla circonferenza e
utilizzare le loro proprietà
Disegnare poligoni inscritti e circoscritti ad una
circonferenza, calcolare l’area di poligoni circoscritti
Calcolare l’area di poligoni regolari utilizzando il
numero fisso
Teorema di Pitagora
Enunciato e significato del
Teorema di Pitagora
Significato di terna
pitagorica ed esempi
Applicazioni del teorema
Similitudine
Concetto di similitudine
Criteri di similitudine dei
triangoli
Teoremi di Euclide
Riconoscere e scrivere una
terna pitagorica
Applicare il Teorema di
Pitagora per calcolare i lati
di un triangolo rettangolo
Applicare il Teorema di
Pitagora alle figure piane
studiate
Problemi risolvibili
mediante l’uso del Teorema
di Pitagora
Costruire figure simili
secondo un rapporto di
similitudine assegnato
Risolvere problemi sulla
similitudine
Applicare i teoremi di
Euclide
Number
Rates as a comparison of unlike quantities
Increasing and decreasing by a ratio and
percentage
Proportional change
Functions
Algebra
Use of metric units for measuring
Use of metric units and conversion
when solving problems (including
area and volume)
Sets
Simplifying algebraic expressions by adding,
subtracting, multiplying and dividing
Factorising a common factor
Review of index notation
Simplify expressions using the index laws
Define the zero index and negative indices
Simplify expressions with negative indices
33
Rounding answers to a specified number
of significant figures
Equations
Solving equations using inverse operations
Solving equations with the variable on both
sides
Solving and graphing one-variable
inequalities
Graphs
Binomial theorem
Squaring a binomial
Set theory
Basic ideas
Venn diagrams
Linear
Graphing linear functions
Intersections of lines through graphical means
Gradient of a line
Gradient – intercept form of a line
Horizontal and vertical lines as special cases
Distance time graphs
Geometry
Quadrilaterals
Properties of quadrilateral –
include the diagonal
properties
Solving angle sum of a
quadrilateral problems
using algebra
34
Polygons
Develop and use the angle
sum of interior and exterior
angles of polygons
3D shapes
Cones, prisms, spheres
Review perimeter and area of all shapes and extend to composite figures
Review circumference and area of a circle and extend to calculation of
arc length and area of a sector
Surface area of prisms and composite shapes
Surface area of a cone
Volume of prisms and composite shapes
Volume and pyramids and cones
Surface area and volume of spheres
Geometry
Trigonometry
Coordinate geometry
Symmetry
Enlargement
Scale drawings
Lines and angles
Solving parallel lines angle problems using
algebra
Constructing angle bisectors, perpendicular
bisector of a line segment, 300, 600, 900, 1200,
450 (and others), parallel and perpendicular lines
Draw the locus of points
Solving problems using locus
Triangles
Develop and use the exterior angle of a triangle
theorem
Develop and use the Pythagorean theorem to
solve for the unknown side in right triangles
Probability
Defining and calculating probability for complementary events
Tree diagrams
Statistics
Data collection
35
Draw and interpret frequency diagrams – the polygon and histogram and
cumulative frequency graphs
Calculate the median from the c f graph
Mean, mode, median, range – include frequency tables
Mean (by approximating the interval by the mid-point)
modal class, median class, for grouped data
ALGEBRA
Algebra
36
Numeri reali
Monomi e polinomi
Identità ed equazioni
Funzioni
Formalizzazione degli
insiemi numerici: da N a
Z, da Z a Q, da Q a R
Procedimenti di calcolo
tra numeri razionali,
notazione esponenziale e
scientifica
Significato di espressione letterale
Principali nozioni sul calcolo letterale
Significato di monomio, terminologia
(coefficiente e parte letterale) e
caratteristiche (monomi simili, opposti,
uguali)
Grado di un monomio
Definizione di polinomio, procedimenti di
calcolo
Prodotti notevoli (quadrato di un binomio
e differenza di quadrati)
Concetto di identità e di equazione
Concetto di equazioni equivalenti
Principi di equivalenza
Risoluzione delle equazioni di
primo grado ad una incognita con
verifica
Equazioni determinate,
indeterminate e impossibili
Concetto di funzione, differenza tra funzione empirica e
funzione matematica
Esempi di funzioni e non-funzioni
Introduzione al piano cartesiano: punti, coordinate,
punto medio e distanza tra due punti (caso semplice
con valore assoluto e caso generale con il teorema di
Pitagora)
Rappresentazione delle funzioni mediante una tabella
di coordinate
Funzioni di proporzionalità diretta, inversa e
quadratica e loro grafici
Cenni sulla risoluzione grafica delle equazioni
Distinguere i numeri reali
e rappresentarli su una
retta
Eseguire operazioni tra
numeri razionali,
calcolarne potenze e
radici quadrate
Risolvere espressioni in
Q
Riconoscere un’espressione letterale e
calcolarne il valore, riconoscere i monomi,
le loro parti, e individuarne le
caratteristiche
Eseguire operazioni con i monomi
Riconoscere i polinomi e individuarne le
caratteristiche
Eseguire le operazioni tra poolinomi e tra
monomi e polinomi
Espressioni con polinomi e prodotti
notevoli
Riconoscere identità ed equazioni
Scrivere un’equazione equivalente
ad una data
Risolvere un’equazione di primo
grado ad una incognita, riconoscere
le equazioni determinate,
indeterminate e impossibili
Riconoscere una funzione, distinguere funzioni
empiriche da funzioni matematiche
Operare nel piano cartesiano: rappresentare punti,
trovare punto medio e lunghezza dei segmenti
Rappresentare funzioni di cui si ha la tabella dei valori,
costruire la tabella dei valori data l’espressione
algebrica di una funzione, rappresentare funzioni di
proporzionalità diretta, inversa e quadratica
Geometria
GEOMETRIA
Lunghezza della circonferenza
e area del cerchio
Significato e calcolo di lunghezza di una
circonferenza e di un arco di circonferenza
Storia di π
Calcolo dell’area di un cerchio, di una corona
circolare, di un settore circolare e di un
segmento circolare
37
Calcolo della lunghezza di una circonferenza e di
un suo arco, dell’area di un cerchio e delle sue
parti
Problemi con circonferenze e cerchi
I poliedri
I solidi di rotazione
Volume e peso specifico
Concetto di geometria nello spazio
Classificazione dei solidi: solidi a superficie
curva e poliedri
Terminologia (vertici, spigoli, facce)
Poliedri regolari (i 5 esempi), poliedri non
regolari (prismi, piramidi)
Prismi retti e parallelepipedi
Sviluppo dei solidi, solidi equivalenti
Concetto di volume di un solido e di
superficie laterale e totale
Formule di superficie e volume di prismi e
piramidi e loro significato
Classificare un solido dato, riconoscere
solidi equivalenti, disegnare lo sviluppo di
un solido
Calcolare la superficie laterale e totale e il
volume dei prismi e delle piramidi, e di
solidi composti
Concetto di solido di rotazione;
definizione e caratteristiche di cono,
cilindro, sfera
Procedimento di calcolo della
superficie laterale e totale e del
volume dei solidi di rotazione
Solidi di rotazione, ottenuti dalla
rotazione di altri poligoni (triangoli
non rettangoli, trapezi, )
Concetto di peso e di peso
specifico di un corpo
Relazione tra volume (V),
peso (P) e peso specifico (ps):
ps=P/V
Calcolare superficie laterale, totale e
volume di sfera, cilindro e cono e di
altri solidi di rotazione, anche
composti da due o più solidi
Risolvere problemi a riguardo
Applicare la relazione tra V, P
e ps per risolvere problemi
inerenti il calcolo del volume
dei solidi studiati
PROBABILITÀ E STATISTICA
Probabilità e Statistica
Probabilità
Concetto di eventi dipendenti ed indipendenti e di evento composto
Calcolo della probabilità composta
Definizione e significato di probabilità classica, frequentista e soggettiva
Statistica
Significato di dati discreti e continui
Concetto di problema del campionamento
Elaborazione di dati continui: raggruppamento in classi, frequenza assoluta,
relativa e percentuale, classe modale, mediana, media aritmetica e deviazione
standard; rappresentazione grafica dei dati
Frequenza cumulata
Grafici: istogrammi, ideogrammi, aerogrammi, diagrammi cartesiani
Distinguere eventi semplici da eventi composti e saper individuare gli eventi
semplici che costituiscono un evento composto Riconoscere eventi dipendenti ed
indipendenti
Calcolare la probabilità di un evento composto
Riconoscere le differenze tra probabilità classica, frequentista e soggettiva
Applicazioni in esercizi di genetica
Elaborare i dati di un’indagine statistica
Calcolare le frequenze, la moda, la media e la mediana
Rappresentare i risultati in un grafico e saperli interpretare dal punto di vista
statistico
FISICA
Fisica
38
Moto rettilineo uniforme
Introduzione al moto rettilineo uniforme, confronto con altri moti non uniformi
Formula che lega velocità, spazio percorso e tempo impiegato: s=vt
Le leve
Concetto di leva e suoi elementi costitutivi, utilizzo delle leve da parte dell’uomo
Tipi di leve (primo, secondo e terzo tipo) e concetto di leva vantaggiosa e
svantaggiosa
Risolvere semplici problemi sul moto rettilineo uniforme, come applicazione di
una legge di proporzionalità diretta (tra s e t)
Riconoscere il tipo di leva e se si tratta di una leva vantaggiosa o no
Risolvere semplici problemi sulle leve
Le forze e i vettori
Definizione di forza come grandezza vettoriale, differenza
tra grandezze vettoriali e scalari
Somma, differenza e multipli di vettori
Regola del parallelogramma
Cenni sul secondo principio della dinamica: F=ma
Dati due vettori trovare il vettore somma, il vettore
differenza e alcuni multipli
Stabilire direzione e verso di un vettore dato
Risolvere semplici problemi sulle forze
Prima legge di Ohm
Introduzione alla legge di Ohm e alle sue
grandezze Esempi tratti dalla vita reale per
comprendere il significato di tale legge
Risolvere semplici problemi mediante
l’applicazione della legge di Ohm
Number
Number systems
Classification of solutions to polynomial
equations as natural, integer, rational, irrational
and/or real
Reasonableness of results,
estimation
Use of various currencies in
appropriate problems and gain a
realistic idea of their value
Use of standard form when solving
problems and writing answers to
problems
Use of metric units as a scale for
axes when graphing
Use of metric units and
conversion when solving
problems (including area and
volume)
Use of currency conversions as an
application of linear functions
39
Number
Algebra
Simple interest
Connection of linear functions
with arithmetic sequences
such as simple interest
Laws of exponents (including fractional
exponents)
Binomial theorem
Solution of problems (for example, linear
equations) involving simple interest
Algebra
Formal proof
Deductive reasoning
Use of deductive reasoning for
formal proof
e g exterior angle theorem
Sets
Matrices
Systems of equations
Set theory
Venn diagrams
Solution of systems of linear
equations by substitution,
elimination, and comparison
Representations
Functions
Circular
40
Linear
Gradient
Parallel, perpendicular
Linear equations
Graphs
Applications
Solution of systems of linear equations by
substitution, elimination, and comparison
Exponential and logarithmic
Quadratic
Factoring into linear expressions
Polynomial
Absolute Value
Composite
Inverse
Transformations
Rigid
transformations
(horizontal shift)
PreCalculus
Properties of
perpendicularity
Finding the
equation of a line
Geometry
Coordinate geometry
Coordinate plane
Parallel and perpendicular lines
Shapes
Continual reference throughout the course
Vectors
Parallel, perpendicular and skew lines
Description of possible intersections of lines and
planes
Solving triangles
Right-triangles, formulae for the area of a triangle
SOHCAHTOA
Pythagorean theorem
Identities
Distance formula
Midpoint formula
41
Trigonometry
Arcs and angles
Bearings (with right angles only)
Equations
Solutions of equations for
angles in the first
quadrant
Probability
Probability diagrams
Statistics
Terminology
42
Statistical
representations
Statistical distributions
Correlations
Chi-squared test for
independence
Use and understanding of
terminology such as
sample space, outcome,
event, etc
Number
43
Number systems
Solutions and graphs of polynomial equations
over restricted domains (ie
etc)
Reasonableness of results, estimation
Use of various currencies in
appropriate problems and gain a
realistic idea of their value
eg
for
,
quadratic (kinematics) word problems
disregarding negative solution
(Calculations of absolute and
percentage error)
Number
Algebra
Compound and reducible interest
formula
Arithmetic and geometric sequences and series
Sum of finite series, apply to value of investments
Links to linear and exponential functions such as
simple and compound interest, currency
conversion
Continual reference throughout the
course where appropriate
Use of metric units as a scale for
axes when graphing
Exponential growth and decay
Applications
Solution of exponential equations using
logarithms
Laws of logarithms
Binomial theorem
Expansion of polynomials of
degree 2 and 3
Introduction to Pascal’s
triangle
Use of spreadsheets
Algebra
Sets
Formal proof
Matrices
Set theory
Continual reference to definitions and theorems
studied previously
Matrix operations
Inverses
Determinants
Solutions of systems of linear equations graphically,
algebraically and using matrices
Venn diagrams
Proof of circle geometry theorems
Representations
Function notation
Use of a variety of
function notations (for
example, mapping,
f(x)=,y=, etc )
Linear
Domain and range
Various forms of linear
equations
Coordinate plane
Solutions of systems of
linear equations,
algebraically and using the
GDC (finding the
intersection of two lines)
Quadratic
Domain and range
Solution of quadratic equations by factoring and
the quadratic formula (use of the GDC is also
encouraged)
Quadratic functions in the form y=a(x-h)2 +k
and in intercept form
Graphs of quadratic functions Quadratic
inequalities
Polynomial
Division of
polynomials using
long
and synthetic
division
Factor/remainder
theorem
Absolute Value
Solution of absolute value
equations, both graphically
and algebraically
Solution of linear and
quadratic inequalities
Exponential and
logarithmic
Domain and range
Graphs of exponential
functions
Equations of asymptotes
Circular
Rational/reciprocal/ot
her
Reciprocal function:
domain and range,
equations of asymptotes
Rational functions:
domain and range,
Transformations
Composite
Inverse
PreCalculus
Transformations of
quadratic functions
Effects of parameters
a,h,k on the graph of
y=a(x-h)2 +k
Find composite
functions in the form
f(g(x)) and (f°g)(x)
Find and graph
the inverse of
quadratic,
linear and
other types of
functions
Related
rates
Functions
Unit circle
Domain and range
Graphs of circular
functions
44
Geometry
Coordinate geometry
Shapes
Circle geometry
Vectors
Addition, subtraction and scalar multiplication of
vectors
Arcs and angles
Solving triangles
Identities
3D bearings with any angle
Solving problems using the sine and cosine rules
The ambiguous case of the sine rule
Trigonometric identities
Tan identity
Unit circle and unit circle identities
Justifying or proving simple trigonometric identities
Trigonometry
45
Probability
Definition of probability
Laws of probability (conjunction, disjunction, complement)
Conditional probability
Tree diagrams and charts
Venn diagrams
Statistics
Frequency tables and graphs, stem-and-leaf diagrams and
box-and-whisker plots
46
Range, quartiles and
percentiles
Standard deviation
Introduction of the
normal distribution curve
and
Correlations
Scatter diagrams
Line of best fit by eye
area values within one,
two and three standard
deviations from the mean
SYLLABUS BY KEY KNOWLEDGE AREA
NUMBERS, SETS AND ALGEBRA
Number
6th
Integers
Using integers
Rule of order
7th
Rule of order
Divisibility
Factors
Multiples
HCF by prime factorisation
Prime factorisation
Triangular numbers
Square numbers
Square roots
Prime factorisation
Squares and square roots
Cubes and cube roots
LCM by prime factorisation
10th
9th
8th
Number theory
Odd and even, prime and
composite
Divisibility
Factors and multiples
HCF by listing factors
LCM by listing multiples
Fractions
Fractions of shapes
Equivalent fractions
Simplifying fractions
Mixed numbers and improper fractions
Comparing fractions
Multiplying and dividing integers by fractions
Word problems
Fractions of shapes
Ordering fractions
Multiplying and dividing fractions
Word problems
Using the calculator
Decimals
Multiplying decimals by
whole numbers
Multiplying by decimals
Dividing by decimals
Multiplying decimals by
decimals
Dividing decimals by
whole numbers
Converting fractions to
decimals using division
(including recurring
decimals)
10th
9th
8th
7th
6th
Number
47
Out of 100
Idea of direct method
Writing ratios
Equivalent ratios
Percentage and increase or decrease
Unitary method for direct proportion
Solving ratio and proportion problems
Rates as a comparison of unlike quantities
Increasing and decreasing by a ratio and
percentage
Proportional change
Whole number place
value
Decimal place value
Putting numbers in
order
Rounding to the nearest
10,100,1000, to the
first and second
decimal place
Simple interest
Use of various currencies in appropriate problems
and gain a realistic idea of their value
Use of currency conversions as an application of
linear functions
Use of various currencies in appropriate problems
and gain a realistic idea of their value
(Calculations of absolute and percentage error)
9th
8th
7th
6th
Number
10th
Solution of problems (for example, linear
equations) involving simple interest
Compound and reducible interest formula
Use of spreadsheets
Placing numbers on the
number line
Reading whole numbers
and decimals
Rounding
Rounding to the nearest
whole number
Putting numbers in
order
Placing numbers on the
number line
Rounding
Estimation and accuracy
Rounding to powers of
10 (order of
magnitude)
Rounding to decimal
places
Rounding answers to a specified number of
significant figures
Number
Using indices to write powers of 10
Names of very large numbers
Multiplying and dividing by powers of 10
Multiplying and dividing by multiples of the
powers of 10
Use of standard form (scientific notation)
10th
9th
8th
7th
6th
Multiplication and division by 10,100,1000
48
Units for length, mass,
time, capacity
Metric and imperial
equivalents
Conversions
Units for length, mass,
time, capacity
Reading scales
Metric and imperial
equivalents
Conversions including
area and volume
Use of metric units for measuring
Use of metric units and conversion when solving
problems (including area and volume)
Use of standard form when solving problems
and writing answers to problems
Use of metric units as a scale for axes when graphing
Use of metric units and conversion when solving
problems (including area and volume)
Use of metric units as a scale for axes when graphing
Recognizing and describing number patterns
Finding the next term in a sequence
Writing sequences using the nth term rule
rule
Writing the rule using nth term notation
Continue number sequences
Notation
rule
Collecting like terms –
integer answers
Multiplying and
dividing simple terms
that involve indices
Connection of linear functions with arithmetic
sequences such as simple interest
Laws of exponents (including fractional exponents)
Arithmetic and geometric sequences and series
Sum of finite series, apply to value of investments
Links to linear and exponential functions such as
simple and compound interest, currency
conversion
Exponential growth and decay
Applications
Solution of exponential equations using logarithms
Laws of logarithms
10th
9th
8th
7th
6th
Algebra
Writing expressions in
best algebraic notation
Collecting like terms
Substituting into
expressions and
formulae
Notation
Index laws
Using distributive
property to remove
brackets
Substituting into
expressions that involve
indices
Simplifying algebraic expressions by adding,
subtracting, multiplying and dividing
Factorising a common factor
Review of index notation
Simplify expressions using the index laws
Define the zero index and negative indices
Simplify expressions with negative indices
49
Matrices
9th
Equations
on flow diagrams
Solving linear equations using inverse
operations on flow diagrams
Solving linear equations using algebra –
variable on one side only
Solving equations with the variable on both
sides
Solving and graphing one-variable
inequalities
Solution of systems of linear
equations by substitution,
elimination, and comparison
10th
8th
7th
6th
Algebra
Solutions of systems of linear
equations graphically,
algebraically and using matrices
Matrix operations
Inverses
Determinants
Algebra
Sets
Formal proof
Set theory
7th
6th
Binomial theorem
Basic ideas
Venn diagrams
10th
9th
8th
Squaring a binomial
50
Expansion of polynomials of degree 2 and 3
Introduction to Pascal’s triangle
Deductive reasoning
Use of deductive reasoning for formal
proof
e g exterior angle theorem
Continual reference to definitions and
theorems studied previously
Proof of circle geometry theorems
Venn diagrams
Venn diagrams
Using x and y to describe functions
Graphs
10th
9th
8th
7th
6th
FUNCTIONS
Function notation
Use of a variety of function notations (for example,
mapping, f(x)=,y=, etc )
Quadratic
Polynomial
Graphing lines of the form y=mx+c
Informal discussion of gradient of lines
Intercepts of lines
Lines parallel to the axes
Graphing linear functions
Intersections of lines through graphical
means
Gradient of a line
Gradient – intercept form of a line
Horizontal and vertical lines as special
cases
Distance time graphs
Gradient
Parallel, perpendicular
Linear equations
Graphs
Applications
Solution of systems of linear equations by
substitution, elimination, and comparison
Domain and range
Various forms of linear equations
Coordinate plane
10th
9th
8th
7th
6th
Linear
Factoring into linear
expressions
Domain and range
Solution of quadratic equations
by factoring and the quadratic
formula (with use of GDC)
The form y=a(x-h)2 +k
and the intercept form
Graphs of quadratic functions
Quadratic inequalities
Division of polynomials
using long
and synthetic division
Factor/remainder
theorem
Solutions of systems of linear equations,
algebraically and using the GDC (finding
the intersection of two lines)
51
Absolute Value
Exponential and logarithmic
Circular
Solution of absolute
value equations, both
graphically and
algebraically
Solution of linear and
quadratic inequalities
Domain and range
Graphs of exponential
functions
Equations of asymptotes
Unit circle
Domain and range
Graphs of circular
functions
10th
9th
8th
7th
6th
Transformations
Rigid transformations
(horizontal shift)
Transformations of
quadratic functions
Effects of parameters
a,h,k on the graph of
y=a(x-h)2 +k
Composite/inverse
PreCalculus
10th
9th
8th
7th
6th
52
Reciprocal function: domain and range,
Rational functions: domain and range,
Find composite functions in
the form f(g(x)) and (f°g)(x)
Find and graph the inverse of
quadratic, linear and other
types of functions
Properties of
perpendicularity
Finding the equation of
normal lines
Related rates
GEOMETRY AND TRIGONOMETRY
Polygons
Names of each
Definition of “regular”
Tessellations
3D Shapes
Vertices, faces and edges of prisms
and pyramids
Euler’s Rule
Isometric drawings
Names of all types
Definitions using sides and
angles
Properties of each including
diagonal properties
Constructing quadrilaterals
using compass and protractor
Angle sum of polygons
Angle size of regular polygons
Euler’s Rule for prisms and pyramids
Platonic solids – identify these
Existence of semi-regular polyhedra
Isometric drawings
Plans and elevations
Cross sections
Properties of quadrilateral –
include the diagonal
properties
Solving angle sum of a
quadrilateral problems using
algebra
Develop and use the angle sum
of interior and exterior angles
of polygons
Cones, prisms, spheres
9th
Continual reference
throughout the course
Continual reference throughout
the course
course
Continual reference
Continual reference throughout
the course
course
8th
7th
6th
Quadrilaterals
Names of all types
Lines of symmetry of each
Tangrams
10th
Geometry
Triangle, rectangle, parallelogram, trapezium
Perimeter and area of shapes made from
rectangles
Surface area from nets
Volume by counting cubes
Volume of a rectangular prism
Review triangle, rectangle, parallelogram, trapezium
Circumference and area of a circle
Surface area and volume of prisms
Review perimeter and area of all shapes and extend to composite figures
Review circumference and area of a circle and extend to calculation of arc length and area of a sector
Surface area of prisms and composite shapes
Surface area of a cone
Volume of prisms and composite shapes
Volume and pyramids and cones
Surface area and volume of spheres
10th
9th
8th
7th
6th
Geometry
53
8th
7th
6th
Geometry
Coordinate geometry
Coordinates in 4 quadrants
Reflection and symmetry
Rotation and symmetry – include order
Translation
Include above on the coordinate plane
Symmetry
Enlargement
Scale drawings
Symmetry
Enlargement
Scale drawings
10th
9th
Coordinate plane
Parallel and perpendicular lines
54
Vectors
Distance formula
Midpoint formula
Parallel, perpendicular and skew lines
Description of possible intersections of lines and
planes
Addition, subtraction and scalar multiplication of
vectors
Lines, arcs and angles
Measure lines
Estimate, measure, draw and name
angles, include reflex
Classify angles – acute, right, obtuse,
straight, reflex, revolution
Calculating angles – on a straight line, at
appoint, vertically opposite
Recognise and name the types of angles
on parallel lines
Triangles
Naming triangles
sides- scalene, isosceles,
equilateral
Angles- acute, right, obtuse
Constructing triangles,
including use of compass
Estimate, measure, draw and name
angles, include reflex
Classify angles – acute, right, obtuse,
straight, reflex, revolution
Calculating angles – on a straight line, at a
point, vertically opposite
Recognise and name the types of angles
on parallel lines
Calculate angles on parallel lines including
any of the above angle
Naming triangles
sides- scalene, isosceles,
equilateral
Angles- acute, right, obtuse
Constructing triangles,
including use of compass
Solving parallel lines angle problems
using algebra
Constructing angle bisectors,
perpendicular bisector of a line segment,
300, 600, 900, 1200, 450 (and others),
parallel and perpendicular lines
Draw the locus of points
Solving problems using locus
Develop and use the exterior
angle of a triangle theorem
Bearings (with right angles only)
Right-triangles, formulae for
the area of a triangle
SOHCAHTOA
Pythagorean theorem
9th
8th
7th
6th
Trigonometry
10th
3D bearings with any angle
Identities and equations
Develop and use the
Pythagorean theorem to
solve for the unknown side
in right triangles
Continual reference
Solving problems using the
sine and cosine rules
The ambiguous case of the
sine rule
Solutions of equations for
angles in the first quadrant
Trigonometric identities
Tan identity
Unit circle and unit circle
identities
Justifying or proving simple
trigonometric identities
55
PROBABILITY AND STATISTICS
10th
9th
8th
7th
6th
Probability
The language of probability – likely, unlikely,
certain, impossible, even chance, equally likely
Probability scale
Listing outcomes
Calculating probability
Probability from experiments
The language of probability, include mutually
exclusive events
Comparing experimental and theoretical
probabilities
Defining and calculating probability for
complementary events
Use and understanding of terminology such as
sample space, outcome, event, etc
Definition of probability
Laws of probability (conjunction, disjunction,
complement)
Conditional probability
Tree diagrams and charts
Venn diagrams
6th
Statistics
Survey
Tally chart
Grouped data
Draw and interpret bar charts (include grouped
continuous), line graphs, frequency diagrams, pie
charts
Stem and leaf diagrams
Misleading graphs
Draw and interpret frequency diagrams – the
polygon and histogram and cumulative frequency
graphs
Calculate the median from the c f graph
10th
9th
8th
7th
Surveys, questionnaires
Two way tables
Draw and interpret the following:
Bar charts include grouped (discrete)
Line graphs
Frequency diagrams
Pie charts
56
Frequency tables and graphs, stem-and-leaf
diagrams and box-and-whisker plots
Mean, mode, median, range – not grouped
Correlations
Introduction of the normal
distribution curve
Area values within one, two
and three standard
deviations from the mean
Scatter diagrams
Line of best fit by eye
Mean, mode, median, range – include
grouped
Mean, mode, median, range – include
frequency tables
Mean (by approximating the interval by
the mid-point) modal class, median class,
for grouped data
10th
9th
8th
7th
6th
Statistics
Range, quartiles and percentiles
Standard deviation
57
58
SOURCES
The present series of IMHS Curriculum Documents draws on a number of existing documents that we
acknowledge in the following list. All rights belong to the respective owners.
Documents published by the International Baccalaureate Organization are used under the following
conditions (Rules and policy for use of IB intellectual property, Copyright materials, IB World
Schools, Guidelines for permitted acts):
“b) IB teachers with authorized access to the online curriculum centre (OCC) may download to a
computer and save any IB files that are published there as programme documentation. They, or a
designated department of the school on their behalf, may then print a copy (or copies) in part or
whole. They may also extract sections from that file, for using independently or inserting into another
work for information or teaching purposes within the school community.”
Documents published by the NGA Center for Best Practices and the Council of Chief State Officers are
used under the following conditions (Public License, License grant):
“The NGA Center for Best Practices (NGA Center) and the Council of Chief State School Officers
(CCSSO) hereby grant a limited, non-exclusive, royalty-free license to copy, publish, distribute, and
display the Common Core State Standards for purposes that support the Common Core State
Standards Initiative. These uses may involve the Common Core State Standards as a whole or selected
excerpts or portions.”
Documents published by the Council of Europe are used under the following conditions (Copyright
Information):
“The Common European Framework of Reference for Languages is protected by copyright. Extracts
may be reproduced for non-commercial purposes provided that the source is fully acknowledged.”
LANGUAGE A:




MYP GUIDE Published January 2009
DP GUIDE Published February 2011
Common core standards “© Copyright 2010. National Governors Association Center for Best
Practices and Council of Chief State School Officers. All rights reserved.”
MIUR, Ministero dell’Istruzione, dell’Università e della Ricerca, “Piani Specifici di
Apprendimento – Scuola Secondaria di I grado”
LANGUAGE B :



MYP GUIDE Published March 2012
DP GUIDE Published March 2011
CEFR various documents © Council of Europe 2011
59
SOCIAL STUDIES :


MYP GUIDE Published August 2009, Published February 2012
DP GUIDE – HISTORY Published March 2008
SCIENCE:


MYP GUIDE Published February 2010
DP GUIDE – BIOLOGY, CHEMISTRY, PHYSICS Published March 2007
MATHEMATICS :


MYP GUIDE Published January 2011
DP GUIDE Published September 2006
ARTS
 MYP GUIDE Published August 2008
PE and IT
60


MYP GUIDE PE Published August 2009
MYP GUIDE TECHNOLOGY Published August 2008



https://sites.google.com/a/westlakeacademy.org/teachers/Home/MYPtechnologycourseinfo
http://www.wuxitaihuinternationalschool.org/technology.html#4
http://www.isparis.edu/page.cfm?p=406)
61
END OF DOCUMENT
Last revision: April 10, 2013

middle and high school - The International School in Genoa

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