SHS1 Additional Mathematics · Semester 1, Week 10
Applications of Algebra
Lesson notes
Learning Objectives
Indicator: 1.1.2.LI.5 - Establish, describe and determine bijective functions and composite functions.
By the end of the lesson, learners can:
- Define bijective, injective (one-to-one), and surjective (onto) functions in their own words and identify each type from arrow diagrams and algebraic rules.
- Determine algebraically whether a given function is injective, surjective, or bijective, using the formal definitions with domain and co-domain.
- Explain why a function that is not bijective cannot have an inverse, and justify this with examples.
- Establish composite functions f ∘ g for given functions f and g, evaluate them at specific inputs, and express the result as a single algebraic rule.
- Determine whether a composite function is bijective by examining the injectivity and surjectivity of the original functions.
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Sign in with phone numberCurriculum details
- Strand
- Modelling with Algebra (Strand 1)
- Sub-strand
- Applications of Algebra (1.2)
- Content standard
- 1.1.2.CS.1 - Demonstrate knowledge and understanding of applying algebraic processes and reasoning involving sequence, functions, and linear programming. 1.1.2.LO.1 Examine, analyse, determine and predict other terms in a pattern/sequence. 1.1.2.LO.2 Distinguish among various types of relations, find the domain and range of, and evaluate functions. 1.1.2.LO.3 Show that a function is injective (into) and/or surjective (onto). Find the inverse, describe the relationship between two variables and establish composite functions. 1.1.2.LO.4 Graph linear and quadratic functions and determine the intercepts. 1.1.2.LO.5 Find graphical and algebraic solutions to a system of three linear equations in three variables and apply them to solve real life problems. 1.1.2.LO.6 Perform algebraic manipulations on polynomial functions and graph polynomial functions. 1.1.2.LO.7 Find the domain, range, and zero of a rational function and state when it is undefined. 1.1.2.LO.8 Identify and describe the order of a matrix, the identity matrix and the zero matrix; find the determinant and perform basic arithmetic operations on 2 by 2 matrices (addition and subtraction).
- Indicator
- 1.1.2.LI.5 - Establish, describe and determine bijective functions and composite functions.
- Suggested placement
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Semester 1, Week 10
(Week 10 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
- Source document note
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The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:
- exemplars - p.101: adjacent duplicated CambriaMath characters were collapsed, but this PDF also maps some equation glyphs to the wrong letter or operator; verify every mathematical expression in this row against the rendered page
- Curriculum reference
- NaCCA curriculum document, p. 101
Exemplars (from the NaCCA curriculum)
Collaborative Learning: Learners will be working in convenient groups (ability, mixed-ability, mixed gender etc.) to solve problems and present answers related to functions. Talk for Learning Approaches: Learners will brainstorm through think-pair-square and debate and discuss the bijective functions. Experiential Learning: Learners will collaboratively engage in hands-on activity (learning by doing) to create bijective functions. Problem-based Learning: Learners will collaboratively investigate one-to-one, onto and invertible functions inquiry functions. Activity 1: Injective, surjective and bijective functions: Collaborative Learning, Talk for Learning, Experiential Learning and Problem-Based Learning: Learners work in convenient groups (mixed gender, ability mixed-ability or pairs) and investigate the fact that, given two sets A and B the following definitions hold. - Definition 1: A function 𝑓: 𝐴 → 𝐵 is called surjective (or is said to map A onto B) if B = range f. - Definition 2: A function 𝑓: 𝐴 → 𝐵 is called injective (or one-to-one) if, for all a and a' in A, 𝑓(𝑎) = 𝑓(𝑎 - ) implies that 𝑎 = 𝑎'. - Definition 3: A function 𝑓: 𝐴 → 𝐵 is called bijective if it is both surjective and injective.
Collaborative learning, Experiential Learning and Problem-Based Learning: Learners show whether or not a given function is one-to-one or onto or both using mappings, diagrams and algebraic proofs. Note: If a function is both surjective and injective, then it is said to be particularly "well behaved". Activity 2: Investigate increasing and decreasing functions. Collaborative Learning, Talk for Learning, Experiential Learning and Problem-Based Learning: Learners work in groups/pairs to investigate increasing and decreasing functions by verifying graphically and algebraically: - A function 𝑓 is strictly increasing if 𝑓(𝑥) > 𝑓(𝑦) when 𝑥 > 𝑦. - A function 𝑓 is strictly decreasing if 𝑓(𝑥) < 𝑓(𝑦) when 𝑥 < 𝑦. - A function f is increasing if f(x) ≥ f(y) when x>y. - A function f is decreasing if f(x) ≤ f(y) when x<y. Graphically,
Some conclusions: - A function is one-to-one if it is either strictly increasing or strictly decreasing. - A one-to-one function never assigns the same value to two different domain elements. - For onto functions, range and co-domain are equal. If a function 𝑓 is not bijective, the inverse function of 𝑓 cannot be. Teaching and Learning Resources: - Cut out shapes of different colours and orientation - Maths Technology learning apps, tools and devices - Chalkboard illustrations - Worksheets - Cut-out geometrical shapes of different colours and orientation Assessment (1.1.2.AS.5). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 3 Strategic reasoning; Level 4 Extended critical thinking and reasoning.