Math Games

7..1

Counting • Listing outcomes of experiments in a systematic way.

7..2

Concepts of probability • The probability of an event occurring: student progress from informal to formal descriptions of probability. • Predicting and determining probabilities. • Difference between experimental and theoretical probability.

7..3

Outcomes of simple random processes • Finding the probability of equally likely outcomes.

7..4

Statistical reasoning with an aim to becoming a statistically aware consumer • The use of statistics to gather information from a selection of the population with the intention of making generalisations about the whole population. They consider situations where statistics are misused and learn to evaluate the reliability and quality of data and data sources.

7..5

Finding, collecting and organising data • Formulating a statistics question based on data that vary allows for distinction between different types of data.

7..6

Representing data graphically and numerically • Methods of representing data. Students develop a sense that data can convey information and that organising data in different ways can help clarify what the data have to tell us. They see a data set as a whole and so are able to use fractions, quartiles and median to describe the data. • Mean of a grouped frequency distribution.

7..7

Analysing, interpreting and drawing conclusions from data • Drawing conclusions from data; limitations of conclusions.

7..8

Synthesis and problem-solving skills • explore patterns and formulate conjectures • explain findings • justify conclusions • communicate mathematics verbally and in written form • apply their knowledge and skills to solve problems in familiar and unfamiliar contexts • analyse information presented verbally and translate it into mathematical form • devise, select and use appropriate mathematical models, formulae or techniques to process information and to draw relevant conclusions.

7..1

AF.1 investigate patterns and relationships (linear, quadratic, doubling and tripling) in number, spatial patterns and real-world phenomena involving change so that they can: a. represent these patterns and relationships in tables and graphs b. generate a generalised expression for linear and quadratic patterns in words and algebraic expressions and fluently convert between each representation c. categorise patterns as linear, non-linear, quadratic, and exponential (doubling and tripling) using their defining characteristics as they appear in the different representations

7..2

Graphing functions • Interpreting and representing linear, quadratic and exponential functions in graphical form.

7..3

Synthesis and problem-solving skills • explore patterns and formulate conjectures • explain findings • justify conclusions • communicate mathematics verbally and in written form • apply their knowledge and skills to solve problems in familiar and unfamiliar contexts • analyse information presented verbally and translate it into mathematical form • devise, select and use appropriate mathematical models, formulae or techniques to process information and to draw relevant conclusions

7..1

Number Systems; N: the set of natural numbers, N = {1,2,3,4...}; Z: the set of integers, including 0; Q: the set of rational numbers; R: the set of real numbers; R/Q: the set of irrational numbers • The binary operations of addition, subtraction multiplication and division and the relationships between these operations, beginning with whole numbers and integers. They explore some of the laws that govern these operations and use mathematical models to reinforce the algorithms they commonly use. Later, they revisit these operations in the context of rational numbers and irrational numbers (R/Q) and refine, revise and consolidate their ideas. Students learn strategies for computation that can be applied to any numbers; implicit in such computational methods are generalisations about numerical relationships with the operations being used. • Students articulate the generalisation that underlies their strategy, firstly in the vernacular and then in symbolic language. • Problems set in context, using diagrams to solve the problems so they can appreciate how the mathematical concepts are related to real life. Algorithms used to solve problems involving fractional amounts.

7..2

Indices • Binary operations of addition, subtraction, multiplication and division in the context of numbers in index form.

7..3

Applied arithmetic • Solving problems involving, e.g., mobile phone tariffs, currency transactions, shopping, VAT and meter readings. • Making value for money calculations and judgments. • Using ratio and proportionality.

7..4

Applied measure • Measure and time. • 2D shapes and 3D solids, including nets of solids (two-dimensional representations of three-dimensional objects). • Using nets to analyse figures and to distinguish between surface area and volume. • Problems involving surface area and volume. • Modelling real-world situations and solve a variety of problems (including multi-step problems) involving surface areas, and volumes of prisms and cylinders. • Students learn about the circle and develop an understanding of the relationship between its circumference, diameter and pi.

7..5

Sets • Set language as an international symbolic mathematical tool; the concept of a set as being a well-defined collection of objects or elements. They are introduced to the concept of the universal set, null set, sub-set, cardinal number; the union, intersection, set difference operators, and Venn diagrams. They investigate the properties of arithmetic as related to sets and solve problems involving sets.

7..6

Synthesis and problem-solving skills • explore patterns and formulate conjectures • explain findings • justify conclusions • communicate mathematics verbally and in written form • apply their knowledge and skills to solve problems in familiar and unfamiliar contexts • analyse information presented verbally and translate it into mathematical form • devise, select and use appropriate mathematical models, formulae or techniques to process information and to draw relevant conclusions

7..1.1

GT.1 calculate, interpret, and apply units of measure and time

7..1.2

GT.2 investigate 2D shapes and 3D solids so that they can: a. draw and interpret scaled diagrams b. draw and interpret nets of rectangular solids, prisms (polygonal bases), cylinders c. find the perimeter and area of plane figures made from combinations of discs, triangles, and rectangles, including relevant operations involving pi d. find the volume of rectangular solids, cylinders, triangular-based prisms, spheres, and combinations of these, including relevant operations involving pi e. find the surface area and curved surface area (as appropriate) of rectangular solids, cylinders, triangular-based prisms, spheres, and combinations of these

7..1.3

GT.3 investigate the concept of proof through their engagement with geometry so that they can: a. perform constructions 1 to 15 in Geometry for Post-Primary School Mathematics (constructions 3 and 7 at HL only) b. recall and use the concepts, axioms, theorems, corollaries and converses, specified in Geometry for Post-Primary School Mathematics (section 9 for OL and section 10 for HL) I. axioms 1, 2, 3, 4 and 5 II. theorems 1, 2, 3, 4, 5, 6, 9, 10, 13, 14, 15 and 11, 12, 19, and appropriate converses, including relevant operations involving square roots III. corollaries 3, 4 and 1, 2, 5 and appropriate converses c. use and explain the terms: theorem, proof, axiom, corollary, converse, and implies d. create and evaluate proofs of geometrical propositions e. display understanding of the proofs of theorems 1, 2, 3, 4, 5, 6, 9, 10, 14, 15, and 13, 19; and of corollaries 3, 4, and 1, 2, 5 (full formal proofs are not examinable)

7..1.4

GT.4 evaluate and use trigonometric ratios (sin, cos, and tan, defined in terms of right-angled triangles) and their inverses, involving angles between 0° and 90° at integer values and in decimal form

7..1.5

GT.5 investigate properties of points, lines and line segments in the co-ordinate plane so that they can: a. find and interpret: distance, midpoint, slope, point of intersection, and slopes of parallel and perpendicular lines b. draw graphs of line segments and interpret such graphs in context, including discussing the rate of change (slope) and the y intercept c. find and interpret the equation of a line in the form y = mx + c; y – y1 = m(x – x1 ); and ax + by + c = 0 (for a, b, c, m, x1 , y1 ∈ ℚ); including finding the slope, the y intercept, and other points on the line

7..1.1

Students should be able to: N.1 investigate the representation of numbers and arithmetic operations so that they can: a. represent the operations of addition, subtraction, multiplication, and division in ℕ, ℤ, and ℚ using models including the number line, decomposition, and accumulating groups of equal size b. perform the operations of addition, subtraction, multiplication, and division and understand the relationship between these operations and the properties: commutative, associative and distributive in ℕ, ℤ, and ℚ and in ℝ\ℚ , including operating on surds c. explore numbers written as ab (in index form) so that they can: I. flexibly translate between whole numbers and index representation of numbers II. use and apply generalisations such as ap aq = ap+q ; (ap )/(aq ) = ap–q ; (ap ) q = apq ; and n1/2 = √n, for a ∈ ℤ, and p, q, p–q, √n ∈ ℕ and for a, b, √n ∈ ℝ, and p, q ∈ ℚ III. use and apply generalisations such as a0 = 1 ; ap/q = q √ap = (q √a)p ; a–r= 1/(ar ); (ab)r = ar br ; and (a/b)r = (ar )/(br ), for a, b ∈ ℝ ; p, q ∈ ℤ; and r ∈ ℚ IV. generalise numerical relationships involving operations involving numbers written in index form V. correctly use the order of arithmetic and index operations including the use of brackets d. calculate and interpret factors (including the highest common factor), multiples (including the lowest common multiple), and prime numbers e. present numerical answers to the degree of accuracy specified, for example, correct to the nearest hundred, to two decimal places, or to three significant figures f. convert the number p in decimal form to the form a × 10n, where 1 ≤ a < 10, n ∈ ℤ, p ∈ ℚ, and p ≥ 1 and 0 < p < 1

7..1.2

N.2 investigate equivalent representations of rational numbers so that they can: a. flexibly convert between fractions, decimals, and percentages b. use and understand ratio and proportion c. solve money-related problems including those involving bills, VAT, profit or loss, % profit or loss (on the cost price), cost price, selling price, compound interest for not more than 3 years, income tax (standard rate only), net pay (including other deductions of specified amounts), value for money calculations and judgements, mark up (profit as a % of cost price), margin (profit as a % of selling price), compound interest, income tax and net pay (including other deductions)

7..1.3

N.3 investigate situations involving proportionality so that they can: a. use absolute and relative comparison where appropriate b. solve problems involving proportionality including those involving currency conversion and those involving average speed, distance, and time

7..1.4

N.4 analyse numerical patterns in different ways, including making out tables and graphs, and continue such patterns

7..1.5

N.5 explore the concept of a set so that they can: a. understand the concept of a set as a well-defined collection of elements, and that set equality is a relationship where two sets have the same elements b. define sets by listing their elements, if finite (including in a 2-set or 3-set Venn diagram), or by generating rules that define them c. use and understand suitable set notation and terminology, including null set, Ø, subset, , complement, element, ∈, universal set, cardinal number, #, intersection, ∩, union, ∪, set difference, \ , ℕ, ℤ, ℚ, ℝ, and ℝ\ℚ d. perform the operations of intersection and union on 2 sets and on 3 sets, set difference, and complement, including the use of brackets to define the order of operations e. investigate whether the set operations of intersection, union, and difference are commutative and/or associative

7..6

Expressions • Using letters to represent quantities that are variable. • Arithmetic operations on expressions; applications to real life contexts. • Transformational activities: collecting like terms, simplifying expressions, substituting, expanding and factoring.

7..7

Equations and Inequalities • Using a variety of problem solving strategies to solve equations and inequalities. They identify the necessary information, represent problems mathematically, making correct use of symbols, words, diagrams, table and graphs.

7..8

Synthesis and problem-solving skills • explore patterns and formulate conjectures • explain findings • justify conclusions • communicate mathematics verbally and in written form • apply their knowledge and skills to solve problems in familiar and unfamiliar contexts • analyse information presented verbally and translate it into mathematical form • devise, select and use appropriate mathematical models, formulae or techniques to process information and to draw relevant conclusions