10 Maths Optional Content
The following is provided by ACARA to direct students interested in pursuing Methods in senior school for current year 10 students. This content is to be covered in addition to the year 10 curriculum.
You can Solve reference to this lesson material at 10 Maths Lesson Material near the bottom of the page.
Strand: Number
Suggestion 1
Operations on numbers involving fractional exponents and surds.
- Explaining that
for , generalizing to √(n&a)=a^(1/n), and evaluating corresponding expressions; for example, √10= 10^0.5≈3.162 , 2^5=32 so 32^(1/5)= √(5&32)=2 - Explaining that a^(m/n)= (a^(1/n) )^m= (√(n&a))^m= √(n&a^m )=(a^m )^(1/n) and evaluating corresponding expressions; for example, 8^(2/3)=(∛8)^2= 2^2=4 and 8^(2/3)= ∛(8^2 )=∛64=4
- Use the y^x and √(x&y)=y^(1/x) functions of technology to Evaluate expressions involving decimal exponents and surds approximately, for example, 10^3.47≈2951.21, ∛78 ≈4.273, 200^0.7≈40.8057 showing that √(a+b)≠√a+√b and √(a- b) ≠√a- √b for a,b>0, for example, √(16+9)=√25=5 but √16+√9=4+3=7 and √(16-9)=√7 ≈2.646,but √16-√9=4-3=1
- calculating with products and quotients of square roots, including simplifying square roots of natural numbers that have perfect square factors, for example,
, and rationalizing expressions involving square roots, for example, - performing the four arithmetic operations with surds of the form
where is a natural number, including rationalising the denominator of a quotient, for example,
Strand: Algebra
Suggestion 2
Simplification of combinations of linear expressions with rational coefficients and the solution of related equations
- Simplifying sums and differences of linear expressions of the form
where and are integers, and is a non-zero integer, for example, . - Solving equations involving sums and differences of linear expressions with rational coefficients, for example
, and verifying the solution
Suggestion 3
Algebraic representations of quadratic functions of the form
- connecting the expanded and transformed representations of a quadratic function by completing the square, for example,
and
Suggestion 4
where
- Deriving the quadratic formula and applying it to Solve quadratic equations, using the discriminant to Identify the number and nature of the roots of a quadratic equation, and verifying solutions
- Identifying what can be known about the Draw of a quadratic function by considering its coefficients, the discriminant and symmetry to assist sketching by hand
- Recognising conjugate pairs of irrational roots of a quadratic equation and their location with respect to the axis of symmetry of the graph of the corresponding function
Suggestion 5
The graphs of
- exploring the Use of the unit circle and animations to show the periodic, symmetric, and complementary nature of the sine and cosine functions
- graphing the sine and cosine functions over different domains of a real variable, including negative values
- establishing relationships between Pythagoras’ theorem, the unit circle, trigonometric ratios, and angles in half-square triangles and equilateral triangles
- approximating values of the sine and cosine functions from a suitably scaled diagram of the unit circle, and solving equations of the form
and over a specified interval graphically
Suggestion 6
The inverse relationship between exponential functions and logarithmic functions and the solution of related equations
- Using the definition of a logarithm and the exponent laws to establish the logarithm laws evaluating
for decimal values of and relating this to a logarithm base scale; solving exponential equations algebraically using base 10 logarithms; for example,
and connecting to the graph of the corresponding function
Strand: Measurement
Suggestion 7
The effect of increasingly small changes in the value of variables on the average rate of change and in relation to limiting values
- Using the gradient of the line segment between two distinct points as a measure of rate of change to obtain numerical approximations to instantaneous speed and interpreting ‘tell me a story’ piecewise linear position-time graphs
Space
Suggestion 8
Relationships between angles and various lines associated with circles (radii, diameters, chords, tangents)
- Identifying relationships, angles between tangents and chords, angles subtended by a chord with respect to the centre of a circle, and with respect to a point on the circumference of a circle, including using dynamic geometric software
- Exploring how deductive reasoning and diagrams are used in proving geometric theorems related to circles
Statistics
Suggestion 9
Measures of spread, their interpretation and usefulness with respect to different data distributions
- Comparing the use of quantiles, percentiles, and cumulative frequency to analyse the distribution of data
- Comparing measures of spread for different data distributions, such as mean or median absolute deviations with standard deviations, and exploring the effect of outliers
Probability
Suggestion 10
Counting principles, and factorial notation as a representation that provides efficient counting in multiplicative contexts, including calculations of probabilities
-
applying the multiplication principle to problems involving combinations including probabilities related to sampling with and without replacement, and representing these using tree diagrams
-
Understanding that a set with
elements has different subsets formed by considering each element for inclusion or not in combination, and that these can be systematically listed using a tree diagram or a table, for example, the set has subsets which are . -
Using the definition of
to represent and Calculate in contexts that involve choices from a set for example, how many different combinations of playing cards from a pack? How many if the suits are ignored? How many with and without replacement? -
Performing calculations on numbers expressed in factorial form, such as
to evaluate the number of possible arrangements of objects in a row, of which are identical, for example objects, of which are identical, can be arranged in a row in
different ways