Chapter 15 Products and Factors

1

WORK PROGRAM

Chapter 15 Products and factors

Strands: Patterns and algebra, Number

Substrands and outcomes:

Number patterns PAS4.2 Creates, records, analyses and generalises number patterns using words and algebraic symbols in a variety of ways

Algebraic techniques PAS4.3 Uses the algebraic symbol system to simplify, expand and factorise simple algebraic expressions

Algebraic techniques PAS5.2.1 Simplifies, expands and factorises algebraic expressions involving fractions and negative and fractional indices

Algebraic techniques PAS5.2.2 Solves linear and simple quadratic equations, solves linear inequalities and solves simultaneous equations using graphical and analytical methods

Algebraic techniques PAS5.3.1 Uses algebraic techniques to simplify expressions, expand binomial products and factorise quadratic expressions

Multiplication and division NS2.3 (Unit 2) Uses mental and informal written strategies for multiplication and division

Operations with whole numbers NS4.1 Recognises the properties of special groups of whole numbers and applies a range of strategies to aid computation

Fractions, decimals and percentages NS4.3 Operates with fractions, decimals, percentages, ratios and rates

Section / GC tips, Investigations,
History of mathematics, Maths Quest challenge, 10 Quick Questions,
Code puzzles,
Career profiles / SkillSHEETs, WorkSHEETs, Interactive games, Testyourself, Topic tests
(CD-ROM) / Technology applications
(CD-ROM) / Learning outcomes
Are you ready? (page 560) / SkillSHEETs (page 560)
15.1: Expanding to remove grouping symbols III
15.2: Expanding and simplifying expressions
15.5: Finding the highest common factor
15.6: Factorising by finding the highest common factor
15.7: Finding a factor pair that adds to a given number
15.8: Simplifying algebraic fractions
15.9: Multiplying and dividing algebraic fractions
15.10: Adding and subtracting algebraic fractions / PAS4.3
·  recognising like terms and adding and subtracting like terms to simplify algebraic expressions
·  simplifying expressions that involve simple algebraic fractions
·  expanding algebraic expressions by removing grouping symbols (the distributive property)
·  factorising algebraic expressions by finding a common factor
PAS5.2.1
·  simplifying algebraic expressions involving fractions
NS2.3 (Unit 2)
·  determining factors for a given number
NS4.3
·  finding highest common factors
Binomial products (page561)
WE 1, 2a–d, 3
Ex 15A Binomial products (page 563) / SkillSHEET 15.1: Expanding to remove grouping symbols III (page 563)
SkillSHEET 15.2: Expanding and simplifying expressions (page 563)
SkillSHEET 15.3: Expanding to remove a double set of grouping symbols (page 563)
Game time 001 (page564) / Mathcad: Binomial products (page 564)
GC program — Casio: Expanding (page 564)
GC program — TI: Expanding (page 564) / PAS5.3.1
·  simplifying algebraic expressions
·  expanding binomial products by finding the area of rectangles
·  using algebraic methods to expand a variety of binomial products
Special products (page564)
WE 4a–d, 5a–d
Ex 15B Special products (page 567) / Investigation: Using expanding formulas to square large numbers (page 568) / Excel: Expanding (page567)
Mathcad: Expansion patterns (page 567)
Excel: Expanding (page 567) / PAS5.3.1
·  recognising and applying the special products
(a + b)(a - b) = a2 — b2
(a ± b)2 = a ± 2ab + b2
·  describing relationships between the algebraic symbol system and number properties (Reflecting, Communicating)
·  developing facility with the algebraic symbol system in order to apply algebraic techniques to other strands and substrands (Applying strategies, Communicating)
More complicated expansions (page 568)
WE 6a–c
Ex 15C More complicated expansions (page 569) / Investigation: Higher order expansions and Pascal’s triangle (page 570) / SkillSHEET 15.4: Recognising expansion patterns (page 569) / Mathcad: More complicated expansions (page 569)
GC program — Casio: Expanding (page 569)
GC program — TI: Expanding (page 569)
Excel: Expanding (page 570) / PAS5.2.1
·  expanding, by removing grouping symbols, and collecting like terms where possible, algebraic expressions
PAS5.3.1
·  using algebraic methods to expand a variety of binomial products
·  recognising and applying the special products
(a + b)(a - b) = a2 — b2
(a ± b)2 = a ± 2ab + b2
NS4.1
·  identifying special groups of numbers including numbers in Pascal’s triangle
PAS4.2
·  asking questions about how number patterns have been created and how they can be continued (Questioning)
Applications (page 571)
WE 7a–d, 8a–d
Ex 15D Applications (page 573) / WorkSHEET 15.1 (page575) / PAS5.2.1
·  generating a variety of equivalent expressions that represent a particular situation or problem (Applying strategies)
PAS5.2.2
·  using a number of strategies to solve unfamiliar problems, including: drawing a diagram, looking for patterns, working backwards, simplifying the problem and trial and error (Applying strategies, Communicating)
·  solving non-routine problems using algebraic methods (Communicating, Applying strategies)
PAS5.3.1
·  simplifying algebraic expressions, including those involving fractions
·  using algebraic methods to expand a variety of binomial products
The highest common factor (page 575)
WE 9a–b, 10a–b, 11a–d
Ex 15E The highest common factor (page578) / Maths Quest challenge: Q1–2 (page 579) / SkillSHEET 15.5: Finding the highest common factor (page 578)
SkillSHEET 15.6: Factorising by finding the highest common factor (page 578) / Mathcad: HCF (page 578)
Excel: HCF (page 578)
GC program — Casio: HCF (page 578)
GC program — TI: HCF (page 578)
Mathcad: Factorising using the HCF (page578)
Excel: Factorising (page 579) / NS4.3
·  finding highest common factors
PAS4.3
·  factorising algebraic expressions by finding a common factor
PAS5.2.1
·  factorising, by determining common factors, algebraic expressions
PAS5.3.1
·  factorising expressions: common factors
More factorising using the highest common factor (page 580)
WE 12a–b, 13a–c
Ex 15F More factorising using the highest common factor (page582) / Maths Quest Challenge: Q1–2 (page 582)
10 Quick Questions 1 (page 582) / Mathcad: Factorising using grouping (page582) / PAS5.3.1
·  factorising expressions: common factors, grouping in pairs for four-term expressions
Factorising using the difference of two squares rule (page 583)
WE 14a–e
Ex 15G Factorising using the difference of two squares rule (page 584) / Maths Quest challenge: Q1–3 (page 585)
Investigation: What has area got to do with factorising? (page 586) / Excel: Difference of two squares rule (page 584)
Mathcad: Difference of two squares rule (page584) / PAS4.3
·  simplifying algebraic expressions that involve multiplication and division
PAS5.3.1
·  factorising expressions: common factors, difference of two squares
·  generating a variety of equivalent expressions that represent a particular situation or problem (Applying strategies)
Quadratic trinomials (page587)
WE 15a–d
Ex 15H Quadratic trinomials (page 589) / Investigation: Mouse pad dimensions (page 590)
10 Quick Questions 2 (page 590) / SkillSHEET 15.7: Finding a factor pair that adds to a given number (page589) / Excel: Factorising (page 589)
Mathcad: Factorising (page 589) / PAS5.3.1
·  factorising expressions: common factors, trinomials
·  generating a variety of equivalent expressions that represent a particular situation or problem (Applying strategies)
More quadratic trinomials (page 591)
WE 16a–b, 17a–c
Ex 15I More quadratic trinomials (page 593) / Maths Quest challenge: Q1–2 (page 593)
Code puzzle (page 594) / WorkSHEET 15.2 (page593) / Excel: Factorising (page 593)
Mathcad: Factorising (page 593) / PAS5.3.1
·  factorising expressions: common factors, trinomials
·  using a variety of methods, including combinations of the above, to factorise expressions
Mixed factorising practice (page 595)
Ex 15J Mixed factorising practice (page 595) / Game time 002 (page 595) / Mathcad: Mixed factorising (page 595) / PAS5.3.1
·  factorising expressions: common factors, difference of two squares, perfect squares, trinomials, grouping in pairs for four-term expressions
·  using a variety of methods, including combinations of the above, to factorise expressions
Simplifying algebraic fractions — multiplication and division (page 596)
WE 18a–b, 19a–b, 20a–b
Ex 15K Simplifying algebraic fractions — multiplication and division (page599) / Investigation: Equal or not equal? (page 600)
Investigation: What’s the problem? (page 600) / SkillSHEET 15.8: Simplifying algebraic fractions (page 599)
SkillSHEET 15.9: Multiplying and dividing algebraic fractions (page599) / Mathcad: Simplifying algebraic fractions (page599) /

PAS5.2.1

·  explaining why an algebraic expansion or factorisation is incorrect (Reasoning, Communicating)
PAS5.3.1
·  factorising expressions: common factors, difference of two squares, trinomials, grouping in pairs for four–term expressions
·  factorising and simplifying a variety of more complex algebraic expressions
·  check expansions and factorisations by performing the reverse process (Reasoning)
Simplifying algebraic fractions — addition and subtraction (page 601)
WE 21, 22a–b
Ex 15L Simplifying algebraic fractions — addition and subtraction (page 602) / SkillSHEET 15.10: Adding and subtracting algebraic fractions (page 602)
WorkSHEET 15.3 (page603) / Mathcad: Adding and subtracting algebraic fractions (page 603) / PAS5.3.1
·  factorising expressions: common factors, difference of two squares, trinomials, grouping in pairs for four-term expressions
·  factorising and simplifying a variety of more complex algebraic expressions
Summary (page 604)
Chapter review (page 605) / ‘Test yourself’ multiple choice questions (page606)
Topic tests (2)