Mathematics - K-10 Yr 7-10 Syllabus - Stage 5 PDF

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NESA Course Guidline which will help students plan their learning...

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CONTENTS Content ..................................................................................................................................... 4 Stage 5 ...................................................................................................................................... 4

Mathematics K–10 Syllabus

3

STAGE 5.1 NUMBER AND ALGEBRA

FINANCIAL MATHEMATICS OUTCOMES A student:

›

uses appropriate terminology, diagrams and symbols in mathematical contexts MA5.1-1WM

›

selects and uses appropriate strategies to solve problems MA5.1-2WM

›

provides reasoning to support conclusions that are appropriate to the context MA5.1-3WM

›

solves financial problems involving earning, spending and investing money MA5.1-4NA

Related Life Skills outcomes: MALS-12NA, MALS-13NA, MALS-14NA, MALS-15NA, MALS-16NA, MALS-17NA

CONTENT Students: Solve problems involving earning money •

calculate earnings from wages for various time periods, given an hourly rate of pay, including penalty rates for overtime and special rates for Sundays and public holidays !

use classifieds and online advertisements to compare pay rates and conditions for different positions (Problem Solving)

!

read and interpret examples of pay slips (Communicating)

•

calculate earnings from non-wage sources, including commission and piecework

•

calculate weekly, fortnightly, monthly and yearly earnings

•

calculate leave loading as 17.5% of normal pay for up to four weeks !

research the reasons for inclusion of leave loading provisions in many awards (Reasoning)

•

use published tables or online calculators to determine the weekly, fortnightly or monthly tax to be deducted from a worker's pay under the Australian 'pay-as-you-go' (PAYG) taxation system

•

determine annual taxable income by subtracting allowable deductions and use current tax rates to calculate the amount of tax payable for the financial year

•

!

determine a worker's tax refund or liability by comparing the tax payable for a financial year with the tax already paid under the Australian PAYG system (Problem Solving)

!

investigate how rebates and levies, including the Medicare levy and Family Tax Benefit, affect different workers' taxable incomes (Problem Solving)

calculate net earnings after deductions and taxation are taken into account

Mathematics K–10 Syllabus

4

Solve problems involving simple interest (ACMNA211) •

calculate simple interest using the formula where is the interest, is the principal, is the interest rate per time period (expressed as a fraction or decimal) and is the number of time periods

•

apply the simple interest formula to solve problems related to investing money at simple interest rates

•

!

find the total value of a simple interest investment after a given time period (Problem Solving)

!

calculate the principal or time needed to earn a particular amount of interest, given the simple interest rate (Problem Solving)

calculate the cost of buying expensive items by paying an initial deposit and making regular repayments that include simple interest !

investigate fees and charges related to 'buy today, no more to pay until …' promotions (Problem Solving)

!

compare the total cost of buying on terms to paying by cash (Problem Solving)

!

recognise that repossession does not remove financial debt (Reasoning)

Connect the compound interest formula to repeated applications of simple interest using appropriate digital technologies (ACMNA229) •

calculate compound interest for two or three years using repetition of the formula for simple interest !

connect the calculation of the total value of a compound interest investment to repeated multiplication using a calculator, eg a rate of 5% per annum leads to repeated multiplication by 1.05 (Communicating)

!

compare simple interest with compound interest in practical situations, eg to determine the most beneficial investment or loan (Communicating, Reasoning)

!

compare simple interest with compound interest on an investment over various time periods using tables, graphs or spreadsheets (Communicating, Reasoning)

Background Information Pay-as-you-go (PAYG) is the Australian taxation system for withholding taxation from employees in their regular payments from employers. The appropriate level of taxation is withheld from an employee's payment and passed on to the Australian Taxation Office. Deduction amounts will reduce the taxation debt that may be payable following submission of a tax return, or alternatively will be part of the refund given for overpayment. Simple interest is commonly used for short-term investments or loans. Calculations can involve an annual simple interest rate with a time period given in months or even days. Internet sites may be used to find commercial interest rates for home loans and to provide 'home loan calculators'.

Language Students may have difficulty interpreting the language of financial problems. For example, references to 'hourly rate', 'weekly earnings', 'monthly pay', etc need to be interpreted as the amount earned in one hour, one week, one month, etc. When solving financial problems, students should be encouraged to write a few key words on the left-hand side of the equals sign to identify what is being found in each step of their working, and to conclude with a statement in words.

Mathematics K–10 Syllabus

5

STAGE 5.1 NUMBER AND ALGEBRA

INDICES OUTCOMES A student:

›

uses appropriate terminology, diagrams and symbols in mathematical contexts MA5.1-1WM

›

provides reasoning to support conclusions that are appropriate to the context MA5.1-3WM

›

operates with algebraic expressions involving positive-integer and zero indices, and establishes the meaning of negative indices for numerical bases MA5.1-5NA

CONTENT Students:

Extend and apply the index laws to variables, using positive-integer indices and the zero index (ACMNA212)

•

use the index laws previously established for numerical bases with positive-integer indices to develop the index laws in algebraic form, eg

!

•

establish that

!

•

explain why a particular algebraic sentence is incorrect, eg explain why incorrect (Communicating, Reasoning)

explain why

using the index laws, eg

(Reasoning)

simplify expressions that involve the zero index, eg

Simplify algebraic products and quotients using index laws (ACMNA231) •

simplify expressions that involve the product and quotient of simple algebraic terms containing positive-integer indices, eg

Mathematics K–10 Syllabus

6

is

!

2

2

compare expressions such as 3a × 5a and 3a + 5a by substituting values for a (Communicating, Reasoning)

Apply index laws to numerical expressions with integer indices (ACMNA209) •

establish the meaning of negative indices for numerical bases, eg by patterns

•

evaluate numerical expressions involving a negative index by first rewriting with a positive index, eg

•

write given numbers in index form (integer indices only) and vice versa

Mathematics K–10 Syllabus

7

STAGE 5.1 NUMBER AND ALGEBRA

LINEAR RELATIONSHIPS OUTCOMES A student:

›

uses appropriate terminology, diagrams and symbols in mathematical contexts MA5.1-1WM

›

provides reasoning to support conclusions that are appropriate to the context MA5.1-3WM

›

determines the midpoint, gradient and length of an interval, and graphs linear relationships MA5.1-6NA

Related Life Skills outcomes: MALS-32MG, MALS-33MG, MALS-34MG

CONTENT Students:

Find the midpoint and gradient of a line segment (interval) on the Cartesian plane using a range of strategies, including graphing software (ACMNA294) •

determine the midpoint of an interval using a diagram

•

use the process for calculating the 'mean' to find the midpoint, M, of the interval joining two points on the Cartesian plane !

explain how the concept of mean ('average') is used to calculate the midpoint of an interval (Communicating)

•

plot and join two points to form an interval on the Cartesian plane and form a right-angled triangle by drawing a vertical side from the higher point and a horizontal side from the lower point

•

use the interval between two points on the Cartesian plane as the hypotenuse of a rightangled triangle and use the relationship

to find the gradient of the interval

joining the two points

•

!

describe the meaning of the gradient of an interval joining two points and explain how it can be found (Communicating)

!

distinguish between positive and negative gradients from a diagram (Reasoning)

use graphing software to find the midpoint and gradient of an interval

Find the distance between two points located on the Cartesian plane using a range of strategies, including graphing software (ACMNA214) •

use the interval between two points on the Cartesian plane as the hypotenuse of a rightangled triangle and apply Pythagoras' theorem to determine the length of the interval joining the two points (ie 'the distance between the two points')

Mathematics K–10 Syllabus

8

!

•

describe how the distance between (or the length of the interval joining) two points can be calculated using Pythagoras' theorem (Communicating)

use graphing software to find the distance between two points on the Cartesian plane

Sketch linear graphs using the coordinates of two points (ACMNA215) •

construct tables of values and use coordinates to graph vertical and horizontal lines, such as , , ,

•

identify the x- and y-intercepts of lines

•

identify the x-axis as the line y = 0 and the y-axis as the line x = 0 !

•

explain why the x- and y-axes have these equations (Communicating, Reasoning)

graph a variety of linear relationships on the Cartesian plane, with and without the use of digital technologies, eg , !

•

,

,

,

compare and contrast equations of lines that have a negative gradient and equations of lines that have a positive gradient (Communicating, Reasoning)

determine whether a point lies on a line by substitution

Solve problems involving parallel lines (ACMNA238) •

determine that parallel lines have equal gradients !

use digital technologies to compare the graphs of a variety of straight lines with their respective gradients and establish the condition for lines to be parallel (Communicating, Reasoning)

!

use digital technologies to graph a variety of straight lines, including parallel lines, and identify similarities and differences in their equations (Communicating, Reasoning)

Background Information The Cartesian plane is named after the French philosopher and mathematician René Descartes (1596–1650), who was one of the first mathematicians to develop analytical geometry on the number plane. He shared this honour with the French lawyer and mathematician Pierre de Fermat (1601–1665). Descartes and Fermat are recognised as the first modern mathematicians.

Mathematics K–10 Syllabus

9

STAGE 5.1 NUMBER AND ALGEBRA

NON-LINEAR RELATIONSHIPS OUTCOMES A student:

›

uses appropriate terminology, diagrams and symbols in mathematical contexts MA5.1-1WM

›

provides reasoning to support conclusions that are appropriate to the context MA5.1-3WM

›

graphs simple non-linear relationships MA5.1-7NA

CONTENT Students:

Graph simple non-linear relations, with and without the use of digital technologies (ACMNA296) •

complete tables of values to graph simple non-linear relationships and compare these with graphs drawn using digital technologies, eg , ,

Explore the connection between algebraic and graphical representations of relations such as simple quadratics, circles and exponentials using digital technologies as appropriate (ACMNA239) •

use digital technologies to graph simple quadratics, exponentials and circles, eg

!

describe and compare a variety of simple non-linear relationships (Communicating, Reasoning)

!

connect the shape of a non-linear graph with the distinguishing features of its equation (Communicating, Reasoning)

Purpose/Relevance of Substrand Non-linear relationships, like linear relationships, are very common in mathematics and science. A relationship between two quantities that is not a linear relationship (ie is not a relationship that has a graph that is a straight line) is therefore a non-linear relationship, such as where one quantity varies directly or inversely as the square or cube (or other power) of the other quantity, or where one quantity varies exponentially with the other. Examples of non-linear relationships familiar in everyday life include the motion of falling objects and projectiles, the stopping distance of a car travelling at a particular speed, compound interest, depreciation, appreciation and inflation, light intensity, and models of population growth. The graph of a non-linear relationship could be, for example, a parabola, circle, hyperbola, or cubic or exponential graph. 'Coordinate geometry' facilitates exploration and interpretation not only of linear relationships, but also of non-linear relationships.

Mathematics K–10 Syllabus

10

STAGE 5.1 MEASUREMENT AND GEOMETRY

AREA AND SURFACE AREA OUTCOMES A student:

›

uses appropriate terminology, diagrams and symbols in mathematical contexts MA5.1-1WM

›

selects and uses appropriate strategies to solve problems MA5.1-2WM

›

calculates the areas of composite shapes, and the surface areas of rectangular and triangular prisms MA5.1-8MG

Related Life Skills outcome: MALS-29MG

CONTENT Students:

Calculate the areas of composite shapes (ACMMG216) •

calculate the areas of composite figures by dissection into triangles, special quadrilaterals, quadrants, semicircles and sectors !

•

identify different possible dissections for a given composite figure and select an appropriate dissection to facilitate calculation of the area (Problem Solving)

solve a variety of practical problems involving the areas of quadrilaterals and composite shapes !

apply properties of geometrical shapes to assist in finding areas, eg symmetry (Problem Solving, Reasoning)

!

calculate the area of an annulus (Problem Solving)

Solve problems involving the surface areas of right prisms (ACMMG218) •

identify the edge lengths and the areas making up the 'surface area' of rectangular and triangular prisms

•

visualise and name a right prism, given its net !

recognise whether a diagram represents a net of a right prism (Reasoning)

•

visualise and sketch the nets of right prisms

•

find the surface areas of rectangular and triangular prisms, given their net

•

calculate the surface areas of rectangular and triangular prisms !

•

apply Pythagoras' theorem to assist with finding the surface areas of triangular prisms (Problem Solving)

solve a variety of practical problems involving the surface areas of rectangular and triangular prisms

Mathematics K–10 Syllabus

11

Background Information It is important that students can visualise rectangular and triangular prisms in different orientations before they find their surface areas. Properties of solids are treated in Stage 3. Students should be able to sketch different views of an object.

Language When calculating the surface areas of solids, many students may benefit from writing words to describe each of the faces as they record their calculations. Using words such as 'top', 'front', 'sides' and 'bottom' should also assist students in ensuring that t...