Sample unit Mathematics Stage 5 - STEM Advanced Pathway

Stage 5: Mathematics STEM Advanced Pathway

Sample concentrated study unit: Precision

(sample programming proforma with some direction)

The Concentrated Studies focus on skills from within one Strand.

Duration: 5 weeks

Outcomes

A student:

 performs operations with surds and indices (MA5.3-6NA)

 generalises mathematical ideas and techniques to analyse and solve problems efficiently (MA5.3-2WM)

Content

Students:

 Define rational and irrational numbers and perform operations with surds and fractional indices (ACMNA264)

Common misconceptions

Students may:

 ‘read’ numbers in surd notation in the same way that they ‘read’ numbers in fraction form, for example 2√3 as 2 and √3 rather than 2 multiplied by √3

 not be transferring learning from previous study of factors and factorisation

 not be transferring learning from previous study of expansion of binomials

 not be transferring learning from previous study of fractions and hence, lack confidence in the equivalence of fractions that they have multiplied by a fraction of the form ^{(𝑎 + √𝑏)}

(𝑎 + √𝑏)

Link to calculus-based courses

Operating in surd form prepares students for graphing and trigonometry topics in Stage 6. The ability to express surds in index form prepares students for:

 differentiation

 integration

Teaching and Learning plan

STEM platform

Pre-test – test students’ algebraic skills:

 operating with like and unlike terms

 factorisation of mixed terms

 simplification of algebraic fractions involving indices

 expansion of binomials

Introduction Activities

Different tasks or calculations require different degrees of precision.

1. Simple class activity – cordial taste test

Some brands produce a concentrated and ‘normal’ strength cordial. Invite students to compare the importance of mixing cordials to suggested ratios for each of these by performing taste tests of accurate and inaccurate measurements.

Link to Learning

Some things are ‘higher stakes’ than others and demand greater precision. Consider medications.

A milligram can be the difference between life and death. Before commencing any calculation, students should consider the ‘stakes’. What degree of accuracy is required of their answer?

2. Construction activity – assisted bowling (Could be set as a homework task.) Show or have students search for images of bowling ramps designed to allow people with disabilities to participate in the sport.

Students construct a ramp suitable for aiming a marble at a small target. Students then test their ramps with the targets set at increasing distances.

Link to Learning

Students will observe that a small error in angle will not affect their ability to hit a target at close range, but as the distance to the target increases, so will the impact of an error in the ‘launch angle’. Consider landing a module on Mars. Not only is Mars effectively ‘marble sized’ when considered from Earth and an average of 225 million km away, but both the target and launch pad are in their own orbits and rotating on their own axes. And yet, highly precise calculations have made a Mars landing possible.

3. Thinking activity – the numbers we round all the time

Brainstorm: What kind of ‘rounding off’ do we do all the time, inside and outside the Maths

classroom? For example: pi, ⅓ as a percentage, ‘five minutes’, ‘all my friends are going’, answer to 2 decimal places, etc.

Link to Learning

Revise rounding rules and the different language used to describe rounding such as, ‘answer to the nearest km’. Describe fractions as an effective form of retaining accuracy, and the

appropriateness of answering in terms of π. Introduce surds as a method of retaining exact values.

4. Observe and wonder activity – the 1, 1, √2 triangle

Students apply Pythagoras’ theorem to calculate the hypotenuse of a right-angled isosceles triangle with equal side lengths 1. Note the calculator’s final answer.

Link to Learning

Discuss the concept of decimal places that continue infinitely without repetition. Students ponder – the hypotenuse has a very definite beginning and end, and yet we cannot describe its exact length in decimal terms.

● Recognise that a surd is an exact value that can be approximated by a rounded decimal.

● Use surds to solve problems where a decimal answer is insufficient, eg find the exact perpendicular height of an equilateral triangle. (Problem Solving)

Consolidation for Skill Development

Explicit teaching

Content Teaching and learning/ Differentiation

Perform operations with surds and fractional indices (from ACMNA264)

 define √𝑥 as the positive square root of 𝑥 for 𝑥 > 0

 demonstrate that √𝑥 is undefined for 𝑥 < 0 and that √𝑥 = 0 for 𝑥 = 0

 use the following results for 𝑥 > 0 and 𝑦 > 0:

(√𝑥)²= 𝑥 = √𝑥²

√𝑥𝑦 = √𝑥 × √𝑦

√_𝑦^𝑥=^√𝑥

√𝑦

 apply the four operations of addition, subtraction, multiplication and division to simplify expressions involving surds

 explain why a particular sentence is incorrect, eg explain why √3 + √5 ≠

√8(Communicating, Reasoning)

 expand expressions involving surds, eg expand (√3 + 5)², (2 − √5)(2 + √5)

 connect operations with surds to algebraic techniques (Communicating)

 rationalise the denominators of surds of the form ^𝑎√𝑏

𝑐√𝑑

 investigate methods of rationalising surdic expressions with binomial denominators, making appropriate connections to algebraic techniques (Problem Solving)

 establish that (√(𝑎))²= √𝑎 × √𝑎 =

√𝑎 × 𝑎 = √𝑎²= 𝑎

 apply index laws to demonstrate the appropriateness of the definition of the fractional index representing the square root,

eg (√𝑎)²= 𝑎 and (𝑎¹²)²= 𝑎

Content Teaching and learning/ Differentiation

∴ √𝑎 = 𝑎¹²

 explain why finding the square root of an expression is the same as raising the expression to the power of a half (Communicating, Reasoning)

 apply index laws to demonstrate the appropriateness of the following definitions for fractional indices:

𝑥^1𝑛= √𝑥^𝑛 , 𝑥^𝑚𝑛 = √𝑥^{𝑛 𝑚}

 translate expressions in surd form to expressions in index form and vice versa

 use the 𝑥^𝑦1 or equivalent key on a scientific calculator

 evaluate numerical expressions involving fractional indices, eg 27²³

 define real numbers: a real number is any number that can be represented by a point on the number line

 define rational and irrational numbers:

a rational number is any number that can be written as the ratio 𝑎: 𝑏 of two integers 𝑎 and 𝑏 where 𝑏 ≠ 0; an irrational number is a real number that is not rational

 recognise that all rational and irrational numbers are real (Reasoning)

 explain why all integers, terminating decimals and recurring decimals are rational numbers (Communicating, Reasoning)

 distinguish between rational and irrational numbers

 write recurring decimals in fraction form using calculator and non-

calculator methods, eg 0. 2̇, 0. 2̇3̇, 0.23̇

Resources overview

Teaching and Learning URLs of linked resources: