Mathematics Curriculum for the Adolescent Aged Twelve to Fifteen Years
Jump to section:
Arithmetic
4.MA.010 Whole numbers
4.MA.020 Rational & irrational numbers
4.MA.030 Fractions
4.MA.040 Powers of numbers - Indices
4.MA.050 Percentages
4.MA.060 Ratio and proportion
4.MA.070 Number patterns
4.MA.080 Squaring and cubing
Algebra
4.MA.090 Algebraic rules
4.MA.100 Linear Equations
4.MA.110 Structure
4.MA.120 Probability and Statistics
4.MA.130 Cartesian graphs
4.MA.140 Functions
Geometry
4.MA.150 Geometry I
4.MA.160 Geometry II
4.MA.170 Patterns and design
4.MA.180 Integrating geometry and algebra to solve problems
Measurement
4.MA.190 Units of Measurement
4.MA.200 Time
4.MA.210 Length and Perimeter
4.MA.220 Study of Area
4.MA.230 Study of Volume
4.MA.240 Angles
Mathematics Across The Curriculum
Introduction
In the Montessori adolescent curriculum the discipline of mathematics, incorporating arithmetic, algebra, geometry and measurement, combines both theoretical study, in other words, exploring relations between abstract concepts, and applying knowledge to solve problems in the context of the occupations of the adolescent community. Whether theoretical or applied, the study of mathematics involves working with patterns and relationships. For this reason, theoretical and applied studies in mathematics complement each other. For example, mathematics can be applied to the investigation of scientific problems, while the symbol systems of theoretical mathematics help scientists organise their data. The study of mathematics also provides the structure that underpins the laws and formulae of science. In addition, it can be used to improve understanding of technology, while technology provides new opportunities for mathematical exploration.
Twenty-first century society and culture is more dependent than ever on mathematics, but the teaching of mathematics in schools does not always appeal to adolescents enough for them to engage with this study at the level demanded by contemporary society. The role of mathematics in contemporary society and the task of the school in this context are described by Montessori (1976 [1948] 116) in the following way:
Mathematics are necessary because intelligence today is no longer natural but mathematical, and without development and education in mathematics it is impossible to understand or take any part in the special forms of progress characteristic of our times. A person without mathematical training today is like an illiterate in the times when everything depended on literary culture. But even in the natural state the human mind has a mathematical bent, tending to be exact, to take measurements and make comparisons, and to use its limited powers to discover the nature of the various ‘effects’ that nature presents to man while she conceals from him the world of causes. Because of this vital importance of mathematics the school must make use of special methods for teaching it and make clear and comprehensible its elements with the help of plenty of apparatus that demonstrates the ‘materialized abstractions’ of mathematics.
The Montessori mathematics curriculum is designed to renew the appeal of mathematics, and to make it accessible to students in a way that promotes both understanding and engagement. For this reason, the Montessori mathematics methodology is activity-, inquiry- and project-based, linked to the history of ideas and real life problems, and is based on the use of manipulative materials (materialised abstractions).
In the Montessori environment the study of mathematics is not organised according to year groups; instead students work through the curriculum at their own pace and their own level over a period the three to four years. When needed, assistance and/or extension activities are provided within individual or small group tutorial programs.
Students apply their mathematical knowledge to the occupations, which include the micro-enterprises the students are responsible for within the adolescent and wider community. For example, while living and working in a Montessori rural community, students might use their knowledge of measurement to analyse the rate animals consume feed and the relation of livestock to the area of pasture, or create a variety of spreadsheets and graphs to plot and record productivity/profit and loss. For enterprises such as the guesthouse and market stall, students might apply their knowledge of mathematics to business and budget planning, inventory, bookkeeping and calculating interest, commission or discounts. Students in an urban adolescent program might apply their mathematical knowledge, skills and understanding to planning, purchasing, preparing, serving school lunches, as well as cleaning up after lunch. If the school has space for a garden on the grounds or rooftop, or in a greenhouse, students can also plan food cultivation, purchase and plant seeds, and maintain and harvest foods for school lunches.
Through these applied activities students both develop mathematical knowledge, understanding and skill, and learn to appreciate the value of this knowledge. Most importantly, they are not rushed through the topics of the curriculum, but have the opportunity to explore real life application in some depth.
The Aims of the Montessori Mathematics Curriculum for Adolescents from Twelve to Eighteen Years
The aims of the Montessori mathematics curriculum for adolescents aged from twelve to eighteen years include the following:
to develop both theoretical and applied mathematical knowledge and skills
to build connections between mathematical knowledge and scientific knowledge, and between mathematical knowledge and real life problem solving
to apply mathematical knowledge, reasoning and skill to the occupations of the adolescent community
to model phenomena in the real world using a variety of mathematical functions
A framework for the Montessori Mathematics Curriculum has been developed that reveals to students how mathematical knowledge in the three components of the discipline, Arithmetic, Algebra and Geometry, has developed over time through the work of mathematicians from many times and places.
Whole numbers 4.MA.010
Knowledge, skills and understanding
Typically, individuals will:
01. Review memorisation, whole number operations, and the order of operations
02. Use a range of strategies to perform mental arithmetic
03. Demonstrate skills in estimating and rounding in all operations
04. Work with a range of number base systems (in addition to the decimal system), including number base systems used in other times and cultures
05. Solve word problems
06. Apply the associative, commutative, and distributive laws to aid mental computation
07. Compare, order, add and subtract integers
08. Investigate and calculate ‘best buys’ with and without digital technologies
＋ Materials and Activity
Activities include:
- Flash cards
- Review exercises
- Mental math games
- Quizzes
Resources include:
- word problem cards
- bead frames for a range of bases, with charts
- charts, card material and artefacts
- research and reference materials (paper-based, digital, web-based, multimedia)
- variety of relevant commercial resources
Rational and irrational numbers 4.MA.020
Knowledge, skills and understanding
Typically, individuals will:
01. Work with rational and irrational numbers
02. Carry out operations with rational numbers, using mental and written strategies
03. Investigate recurring and terminating decimals, including pi
＋ Materials and Activity
Activities include:
- Find payable tax for income
- How to share a pizza
- Interest rates on savings, loans & mortgages
- Occupations projects, e.g. Finance
Resources include:
- Finance occupation
- Bank websites
- Fair Work Australia website
＋ Links to Australian Curriculum (ACv9)
Fractions 4.MA.030
Knowledge, skills and understanding
Typically, individuals will:
01. Review and extend knowledge of fractions:
equivalent fractions
comparison of fractions
simplifying fractions
adding and subtracting fractions and mixed numbers
multiplying and dividing fractions
02. Review and extend knowledge about decimals
converting fractions to decimals
comparison of decimals and rounding
multiplying and dividing decimals by multiples of 10
03. Apply knowledge about numbers between zero and one (fractions, decimals) and ratios to occupations
＋ Materials and Activity
Activities include:
- Practise operations using fractions.
- Use fractions in everyday life such as cooking and building.
- Practise decimals related to money in occupations.
- Currency conversions
Resources include:
- metal fraction insets (to tenths)
- box of manipulable cut-out fractions, and words and symbols on labels
- insets of equivalence
- divided skittles for division of fractions
- materials for graphing
- yellow board of decimal hierarchies, colour-coded beads, charts and colour-coded cards
- decimal checkerboard
- constructive triangles
- variety of relevant commercial resources
＋ Links to Australian Curriculum (ACv9)
Powers of numbers - Indices 4.MA.040
Knowledge, skills and understanding
Typically, individuals will:
01. Review and extend knowledge of powers of numbers
powers 1 to 10
investigate index notation
index laws with positive indices & zero index
apply index laws to integers with indices
operations with numbers expressed as powers
decomposition of numbers using exponential notation
using exponential notation in science
Percentages 4.MA.050
Knowledge, skills and understanding
Typically, individuals will:
01. Review and extend knowledge of percentages:
estimating percentages
converting percentages to fractions and decimals
converting fractions and decimals to percentages
expressing an amount as a percentage of another amount
complementary percentages
percentage increase and decrease
calculating a whole number or fractional percentage of an amount
using decimals to find a percentage of an amount
02. Use a calculator to find percentages
03. Apply knowledge of percentages to occupations e.g. banking and simple interest, use of credit cards
＋ Materials and Activity
Activities include:
- Find discount or sale price for products
- Find best buys
- Occupations:
- The shop
- Financial management
- Land
- Dining
Resources include:
- golden bead material
- centesimal circle inset
- interest, time and principal command cards
- calculators
- variety of relevant commercial resources
＋ Links to Australian Curriculum (ACv9)
Ratio and proportion 4.MA.060
Knowledge, skills and understanding
Typically, individuals will:
01. Review and extend knowledge of ratio and proportion:
recognise and solve simple ratios
similarity and proportion
direct proportion & rate problems
profit and loss
direct variation
gradient
Number patterns 4.MA.070
Knowledge, skills and understanding
Typically, individuals will:
01. Review and extend knowledge of prime numbers
02. Work with puzzles and patterns related to prime numbers e.g. Fibonacci numbers, Goldbach’s conjecture
03. Introduce, consolidate and/or review knowledge of prime factors and factor trees
04. Introduce, consolidate and/or review knowledge of square numbers and square roots
05. Introduce and/or review knowledge of signed numbers and compare them
06. Work with number sequences, polynomials, substitution/evaluations
Squaring and cubing 4.MA.080
Knowledge, skills and understanding
Typically, individuals will:
01. Review and extend knowledge of squaring and cubing
02. Review and extend knowledge of Pythagorean Theorem and its Euclidean demonstration
03. Review and extend knowledge of square root and cube root
＋ Materials and Activity
Activities include:
- Apply Pythagorean theorem to word problems and real-life scenarios
Resources include:
- material for working with squares of numbers
- insets of equivalence
- wooden cubing material
- command cards
- calculators
- variety of relevant commercial resources
＋ Links to Australian Curriculum (ACv9)
Algebraic rules 4.MA.090
Knowledge, skills and understanding
Typically, individuals will:
01. Review and extend algebraic rules and their application:
changing rules to formulae
using pronumerals (adding, subtracting, multiplying, dividing)
substituting values into formulae
using number patterns to solve problems
learning the language of algebra
using the distributive law to expand brackets, including binomials and collecting like terms
using algebra and algebraic formulae in word problems
factorise algebraic expression using factors
simplify algebraic expressions involving four operations
Linear Equations 4.MA.100
Knowledge, skills and understanding
Typically, individuals will:
01. Work with number sentences and equations:
using pronumerals in equations
using a flowchart to build expressions
using backtracking to solve equations
using substitution to solve equations
balancing equations
identifying equivalent equations
performing the same operation on both sides to solve equations
solving equations involving two operations
deriving an equation from a word problem to solve the problem
＋ Materials and Activity
Resources include:
- charts
- word problem command cards
- bead material
＋ Links to Australian Curriculum (ACv9)
Structure 4.MA.110
Knowledge, skills and understanding
Typically, individuals will:
01. Work with different types of algebraic structure:
operations and their order
geometric and graphic models of operations
laws of exponents
rules for radicals
distributive law: minus and parentheses
identities
multiplying, dividing, factoring
inverses: opposites and reciprocals, inverse operations, inverse functions
algebraic functions: equivalent fractions, lowest terms
abstract algebra
＋ Materials and Activity
Resources include:
- graphing calculators
- relevant computer software
Probability and Statistics 4.MA.120
Knowledge, skills and understanding
Typically, individuals will:
01. Study and apply knowledge of probability and statistics:
surveys; collecting statistics
construct sample spaces for single-step experiments with equally likely outcomes
frequency tables and averages
bar, pie and line graphs
histograms, stem-and-leaf plots and dot plots
experimental and theoretical probability
use sum of probabilities to solve problems
union and intersection
02. Work with data: (calculate, interpret & describe)
graphs and tables
averages, centre of spread
effect of outliers on mean and median
two-way tables and Venn diagrams
investigate techniques to collect data including census, sampling and observations
chance experiments using tree diagrams
relative frequencies
＋ Materials and Activity
Activities include:
- collect data through surveys and observations
- create tables and graphs on excel spreadsheets
- Occupations:
o The shop
o Financial management
o Land
o Dining
Resources include
- a variety of commercial resources
- graphing software
＋ Links to Australian Curriculum (ACv9)
AC9M7ST02
AC9M8ST01
AC9M8ST02
AC9M7ST03
AC9M7P01
AC9M8P01
AC9M9ST01
AC9M9ST02
AC9M9P01
AC9M9P02
AC9M9P03
AC9M8P02
AC9M7P02
AC9M8P03
AC9M7ST01
AC9M8ST03
AC9M8ST04
AC9M9ST03
AC9M9ST04
AC9M9ST05
Cartesian graphs 4.MA.130
Knowledge, skills and understanding
Typically, individuals will:
01. Study and apply knowledge of Cartesian graphs:
Cartesian plane: first quadrant
interpreting points on a graph (scatter and line graphs)
using tables to plot relationships
conversion graphs
find distance between two points located on the Cartesian plane
find mid-point and gradient
finding a rule for relationships
sketch linear graphs & graph non-linear relations
＋ Materials and Activity
Activities include:
- describe the position of any object on a map
- map projections
- latitude and longitude
- Occupations: Land
Resources include:
- interactive whiteboard
- string lines, tape measures
- graphing resources
＋ Links to Australian Curriculum (ACv9)
AC9M7SP03
AC9M8A02
AC9M7A04
AC9M7A05
AC9M7A06
AC9M9A03
Functions 4.MA.140
Knowledge, skills and understanding
Typically, individuals will:
01. Study and apply knowledge of functions:
input-output tables
cartesian graphs: intercepts, effects of parameters, rate of change, ordered pairs, equations
linear functions: gradient-intercept form, standard form, constant sum, constant difference
quadratic functions: intercept form, standard form, vertex form
other functions: exponential functions, constant products, rational functions, step functions, absolute value functions, iterating linear functions
definitions of function, domain, range
＋ Materials and Activity
Activities include:
- calculate the distance between landmarks
- design the roof of a house
- design and create a bridge
- Occupations: o Land o Community service
Resources include:
- a variety of commercial resources
＋ Links to Australian Curriculum (ACv9)
AC9M8A03
AC9M8A04
AC9M9A04
AC9M9A05
AC9M9A06
Geometry I 4.MA.150
Knowledge, skills and understanding
Typically, individuals will:
01. Review and extend knowledge of parts of triangles, quadrilaterals and polygons (height, base, surface, perimeter, vertices, diagonals, apothem, centre)
02. Explore diagonals and angles to find formulae
03. Explore rotation, reflection and enlargement
04. Identify corresponding, alternate, and co-interior angles when two straight lines are crossed by a transversal
05. Investigate conditions for two lines to be parallel and solve simple numerical problems using reasoning
＋ Materials and Activity
Activities include:
- Occupations: o Land o Community service
Resources include:
- geometry cabinet, constructive triangles, metal plates for squares and triangles
- Montessori material for measuring height of figures
＋ Links to Australian Curriculum (ACv9)
AC9M9SP03
Geometry II 4.MA.160
Knowledge, skills and understanding
Typically, individuals will:
01. Review and extend knowledge of angles:
describing angles
measuring angles
relations between angles
02. Investigate angles in triangles, quadrilaterals and polygons (size and sum of angles)
03. Congruency
define congruency of plane shapes using transformations
develop conditions for congruency of triangles
establish properties of quadrilaterals using congruent triangles and angle properties
＋ Materials and Activity
Activities included designing and building wooden structures using angles. such as portable enclosure for chicks, benches and planter boxes.
Resources include:
- protractors 180° and 360°, set squares, compasses, rulers
- graphing resources
- nets of solids
- a variety of commercial resources.
＋ Links to Australian Curriculum (ACv9)
AC9M7SP01
AC9M7SP02
AC9M8SP01
AC9M8SP02
AC9M8SP04
Patterns and design 4.MA.170
Knowledge, skills and understanding
Typically, individuals will:
01. Investigate line designs in angles
02. Investigate patterns and designs involving plane figures and geometric solids
03. Visualise and draw 3D shapes
04. Investigate and apply Trigonometry
05. Investigate nets of solids and Euler’s rule for polyhedra
＋ Materials and Activity
Activities include:
- Occupations: o Land o Woodworking o Community service
Resources include materials for model making and technical drawing
＋ Links to Australian Curriculum (ACv9)
AC9M7SP04
AC9M9M03
AC9M9SP01
AC9M9SP02
Integrating geometry and algebra to solve problems 4.MA.180
Knowledge, skills and understanding
Typically, individuals will:
01. Select appropriate operations to solve a variety of application problems using real numbers
02. Identify coordinates of a point in a plane or in space
03. Find the length and the midpoint of a segment in two or three dimensions to solve problems
04. Use definitions, properties and theorems of lines, angles and polygons to solve problems:
05. Recognise, identify and model regular and non-regular polyhedra and use coordinate geometry to confirm properties
06. Use formulae to solve problems related to
the perimeter of a geometric figure and circumference of a circle
the area of a triangle, parallelogram, rhombus, trapezoid, square, rectangle, regular polygons, and circles
arc lengths and the area of sectors of a circle
the ratio of the perimeters, areas, and volumes of similar geometric figures, scale factor
the lateral area, surface area, and volume of a right prism, pyramid, right circular cylinder, cone, and sphere
.07 Use gradients to determine if two lines are parallel or perpendicular
.08 Write equation of a line parallel or perpendicular to a given line through a given point
.09 Transform (translate, reflect, rotate, dilate) polygons in the coordinate plane, describing the transformation in simple algebraic terms
＋ Materials and Activity
Activities include:
- Occupations: o Land o Workshop o Textiles
- Find latitude and longitude of places on a map
- Calculate how steep a road or hill is
- Locate the position of an aircraft in space
Materials include a variety of commercial resources, paper-based and digital
＋ Links to Australian Curriculum (ACv9)
AC9M7M06
AC9M8M07
AC9M8SP03
Units of Measurement 4.MA.190
Knowledge, skills and understanding
Typically, individuals will:
.01 Review prefixes and suffixes used to name units of the metric system
.02 Review units of measurement
.03 Choose appropriate units of measurement and convert
＋ Materials and Activity
Activities include:
- convert from Metric to Imperial system
- investigate Roman, Egyptian and Babylonian systems
- Humanities projects -ancient civilisations
Resources include teacher made metric system charts, strip and card materials
＋ Links to Australian Curriculum (ACv9)
AC9M8M05
AC9M9M04
Time 4.MA.200
Knowledge, skills and understanding
Typically, individuals will:
01. Review the base systems of time e.g. 24 hours/day, 60 sec/minute, 60 min/hr
02. Apply knowledge of time to reading timetables and working with time zones
03. Investigate very small and very large time scales and intervals
＋ Materials and Activity
Activities include:
- Calculate time difference in different time zones.
- Create an itinerary to travel around the world and calculate travel times.
- Land occupation
- Dining occupation
Resources include:
- variety of clocks: analogue and digital
- variety of timetable formats
- time zone globe
＋ Links to Australian Curriculum (ACv9)
AC9M8M04
AC9M9M02
Length and Perimeter 4.MA.210
Knowledge, skills and understanding
Typically, individuals will:
01. Review and extend techniques, rules, formulae, operations and word problems related to the measurement of length and calculation of perimeter
02. Perimeter of all quadrilaterals
03. Investigate the relationship between features of a circle such as circumference, radius and diameter
＋ Materials and Activity
Activities include:
- Land occupations:
- calculate perimeter of the chicken coup or bee enclosure
- design a round pizza oven and calculate its perimeter and cost
- Textiles occupation
- Community service occupation
Resources include chalk, string, tape measures, rulers and graphing resources
＋ Links to Australian Curriculum (ACv9)
AC9M7M03
AC9M8M03
Study of Area 4.MA.220
Knowledge, skills and understanding
Typically, individuals will:
01. Review and extend the study of area to abstract calculation of area
02. Investigate the calculation of area of a range of composite geometric shapes, including different types of triangles and quadrilaterals, and circles
03. Apply knowledge of area to tessellation
04. Build theorems for the calculation of the area of a range of shapes
05. Surface area of cylinders, prisms
06. Investigate and compare total and lateral surface area
＋ Materials and Activity
Activities include:
- Land occupations:
- calculate the surface area of a wooden fence to paint
- calculate the surface area of a water tank
Resources include:
- area of a circle material
- area of parts of circle material
＋ Links to Australian Curriculum (ACv9)
AC9M7M01
AC9M8M01
AC9M9M01
Study of Volume 4.MA.230
Knowledge, skills and understanding
Typically, individuals will:
01. Review and extend the study of volume:
solid geometry review
from the solid to its volume
02. Investigate the volume of a range of solid shapes, including different types of prisms, pyramids and cylinders
＋ Materials and Activity
Activities include:
- Land occupations:
- calculate the volume of water in the water tank
- calculate the volume of concrete required to fix a pole into the ground
- Workshop occupation
- Textiles occupation
Resources include:
- hollow volume material
- volume of a sphere material
＋ Links to Australian Curriculum (ACv9)
AC9M7M02
AC9M8M02
Angles 4.MA.240
Knowledge, skills and understanding
Typically, individuals will:
01. Review and extend study of the measurement of angles
02. Apply knowledge about measuring angles to occupations
＋ Materials and Activity
Activities include:
- Occupations:
- Land
- Textiles
- Community service
- Workshop
Resources include Montessori concrete materials, protractors and equipment for measuring angles in the field.
＋ Links to Australian Curriculum (ACv9)
AC9M7M04
AC9M7M05
Mathematics Across the Curriculum
As with the study of English language in the Montessori adolescent curriculum, the study of mathematics is embedded in all areas of the curriculum, wherever mathematics skills are needed to engage productively with curriculum content. How mathematics is embedded in specific curriculum areas is summarised under the curriculum areas below. Click each heading to learn more.
The Domain of Development
+ Moral Education (4MA270)
Apply knowledge of mathematics to contribute to community projects, including fund-raising, surveys, managing food distribution with and beyond the adolescent community
+ Language: English and LOTE (4MA280)
Use knowledge of mathematics to assist with layout and presentation of texts in digital environments
The Domain of Creative Expression
+ Mathematics for creative expression, music and visual arts (4MA290)
Apply knowledge of mathematics to:
- preparation for theatrical performance e.g. production budget, working with sound and light, ticketing, calculating margins, fund-raising, costume, set and prop design, stage directions
- the study and making of music e.g. scale patterns, rhythm, tempo
- the study and making of visual art and craft e.g. geometry, measurement, perspective, budgets for materials and equipment
- the study of poetry e.g. rhythm and metrics
The Domain of Preparation for Adult Life and Contemporary Culture
+ Science (4MA300)
Apply knowledge of mathematics to:
- collecting, collating, storing, comparing and interpreting data; tracking variability
- identifying patterns and cycles
- problem solving
- selecting and using mechanical and electronic equipment for measuring
- the study of physics e.g. exploring the laws of motion, inertia, mass and weight and calculating velocity and acceleration
- the study of energy e.g. to gauge efficiency of alternative energy sources such as solar or wind
- the study of astronomy, the solar system and the Earth e.g. measuring in light years, using the powers of numbers to measure large distances and sizes, measuring time over a day, seasons, time zones etc
- the study of geology and geography e.g. map-making, measuring comparative density, reading and making drawings to scale
- the study of water e.g. flow rate, use, conservation, measuring pollutants
- the study of health and physical education e.g. the relation between energy use and calorie expenditure, speed, distance and velocity, heart rate
+ History/Humanities (4MA310)
Apply knowledge of mathematics to:
- designing timelines
- reading and making maps
- interpreting geographical data - apply geographical knowledge and concepts to geographic data analysis represent geographical information in a variety of forms such as tables, graphs
- studying the currencies, exchange rates, economies and time zones of different countries
- reading, interpretation and analysis of economic and business data presented in a variety of formats
- exploring number base systems used by ancient civilisations
- exploring the Australian ballot system and calculation of votes relevant to the political system
- biographies of great mathematicians
- planning field trips e.g. preparing itineraries, reading public transport timetables, calculating costs, distances and times
+ Occupations (4MA320)
Apply knowledge of mathematics to:
- cooking projects e.g. shopping budgets, measuring ingredients, doubling and tripling proportions, exploring the ratio of protein, fat and carbohydrate
- construction projects e.g. estimating costs and calculating cost/unit measurement; calculating area or volume, drawing to scale, calculating angles of roof pitch
- projects involving animals e.g. budgets for housing, fencing, feed etc, administering supplements and medications, calculating productivity, monitoring prices, geometry of bee hives
- micro-enterprises e.g. preparing budgets, using money, marketing, bookkeeping, reporting