# Dr. Costella’s Mathematics Notes

This page contains scans of all 823 overhead transparencies that I created when I was a high school mathematics teacher in Melbourne, Australia, from 1997 to 2006.

The pages are numbered in the order that I created them, and so represent a mixture of topics from Year 8 to Year 12 level. I actually wrote each number on the bottom corner of each transparency, which let me figure out where to find each of them (stored in a number of pocket binders with plastic sleeves). I added letters (‘a’, ‘b’, etc.) when I found that I needed to cut a transparency into parts, so that I could use just a part of it (one subtopic) in a course different from the one that it was originally created for.

Hence, the only way to find any particular topic is to search for it on this page! Sorry. (When I was teaching, I would create a page for each module of work for each subject, which linked to these notes in the correct order.)

Anything in a title between square brackets was added at the time that I compiled this collection page, and represents text not in the title of the overhead transparency itself.

001
Intercepts of graph
002a
Intercepts of a linear grap
002b
Drawing the [linear] graph from the intercept
003a
Transformations of sine graphs
003b
Transformation [of sine graphs] in the vertical direction
003c
Horizontal compression [of sine graphs]
004
Vertical expansion of sine graph
005
Probability (Chance) [introduction]
006
Parabolas: inversion [cup/hat intro]
007
Stratified samples [construction of]
008
Stratified samples [construction of], continued
009
Empirical probabilities
010a
Equally likely outcomes
010b
Combinations of equally likely outcomes
011
Tree diagrams to show possibilities
012
Sampling with replacement
013
Sampling without replacement
014a
Conditional probability
014b
Multiplication Rule [of probability]
015
Independent events
016
Stem-and-leaf plots [introduction]
017
Having separate stems for 0-4 and 5-9
018
Ordered stem-and-leaf plots
019
Back-to-back stem-and-leaf plots
020
Pie graphs [pie charts]
021
Mean [introduction]
022
Mode [introduction]
023
Median [introduction]
024
Quartiles [introduction]
025a
Range
025b
Interquartile range
026
Order of operations
027
Consecutive numbers
028
Sum, product, difference, quotient
029
Factors
030a
Prime numbers
030b
Composite numbers
031
Types of data
032
Recurring decimals
033a
Index laws summary (Years 11, 12)
033b
Index laws summary (Years 9, 10)
034a
Scientific notation [standard form]
034b
To convert a number to scientific notation
034c
To convert scientific notation to a normal number
035
Simultaneous equations [introduction]
036
Simultaneous equations: substitution method 1
037
Simultaneous equations: elimination method 1
038
Simultaneous equations: elimination method 2
039
Simultaneous literal equations
040
Partial fractions
041
042
043
Drawing a straight line from the equation
044
[Getting the] Straight-line equation from the graph
045a
Equation from one point and gradient
045b
Equation from two points
046
Solving simultaneous equations graphically
047a
Finding the angle of a slope
047b
Perpendicular lines
048a
Distance between two points
048b
Formula for distance between two points
049
Midpoint of a line
050
Angle between intersecting lines
051
Three simultaneous equations
052
Negative numbers
052
Integers
053
The number line
054
The number plane
055a
Greater than
055b
Less than
055c
Greater than or equal to
055d
Less than or equal to
056
Plotting ranges of integers
057
058
Multiplying integers
059
Leaving out the multiply sign
060a
Subtracting integers
060b
The negative of a number
061
Dividing integers
062a
Minor and major arcs
062b
Angle subtended by an arc
063
Angles standing on the same arc
064
Angles on same arc at centre and circumference
065
Angles in a semicircle
066
Opposite angles in a cyclic quadrilateral
067a
067b
Chords of equal length
068a
Tangents to a circle
068b
068c
Lengths of tangents from the same external point
069a
Area of a sector
069b
Area of a segment
070
Breaking a number into its prime factors
071a
A set
071b
Equal sets
071c
Equivalent sets
072a
Null set
072b
The universal set
072c
Subsets
072d
Disjoint sets
073a
Union of two sets
073b
Intersection of two sets
073c
Complement
073d
Finite and infinite sets
073e
Number of elements
074
Venn diagrams to illustrate sets
075a
The set of natural numbers
075b
The set of integers
076a
The set of rational numbers
076b
The set of real numbers
077a
077b
Surds of the n-th power
078a
Simplest form of a surd
078b
Rationalising the denominator
079
The discriminant
080
Solving inequations
081
082
083
Turning-point form for any quadratic relation
084
Formulas for sketching any quadratic relation
085
Parabola from intercepts and symmetry
086
087
Powers
088
Exponential graphs
089
Multiplying powers
090
Dividing powers
091a
Raising a power to a power
091b
Raising a negative number to a power
092
Powers and brackets
093
Raising to the power of zero
094
Negative powers
095
Ratios
096
Simplifying ratios
097
Equivalent ratios
098
Sharing ratios
099
Value for money
100
Scale diagrams
101
Rates
102
Units of speed
103
Average speed
104
Direct variation
105
Inverse variation
106
Fitting data
107
Joint variation
108
Part variation
109
Venn diagrams to illustrate probability
110a
Intersection and union
110b
Compound event law
111a
Mutually exclusive events
111b
Complementary events
112
Odds
113
Winning streaks
114
Polynomials
115
Rational functions
116
Finding quadratic relations from the graph 1
117
Rectangular hyperbolae
118
The truncus
119
The equation of a circle
120
Right-angled triangles
121
Pythagoras’s theorem
122
Finding a shorter side
123
Pythagorean triples
124
Converting odds to probability
124
Even money
125
Partial fractions: more complicated cases
126
improper rational functions
127
Partial fractions: more complicated examples I
128
Partial fractions: more complicated examples II
129
Equating coefficients: some examples
130
Simultaneous parabola and line
131
Simultaneous circle and line
132
Simultaneous hyperbola and line
133
Partial fractions: more complicated examples III
134
Using SIMULT on the CAS
135
Set difference
136
Intervals of real numbers
137
Special subsets of the real numbers
138a
Relations
138b
Domain and range
139a
Relations defined by a rule
139b
Implied domain
140
Graphing relations
141
Mapping diagrams
142
Types of relations
143a
Functions
143b
Vertical line test
144
Function notation
145
Using mapping notation for functions
146
The meaning of percentage
147
Converting percentages to fractions
148
Converting fractions to percentages
149
Converting percentages to decimals
150
Converting decimals to percentages
151
Percentage of
152
Making comparisons using percentages
153a
Points percentage
153b
Wins percentage
154a
Mark-up
154b
Discount
155
Simple interest
156
Percentage growth
157
Gradient-intercept form of a linear equation
158a
Acute angles
158b
Obtuse angles
158c
Reflex angles
158d
Right angles
158e
Straight angles
158f
Revolution
159
Linear programming
160a
Measuring angles with a protractor
160b
Naming angles
161a
Complementary angles
161b
Supplementary angles
162
Bisecting an angle
163
Bisecting a line
164
Vertically opposite angles
165
Parallel lines
166
Corresponding angles
167
Alternate angles
168
Co-interior angles
169
Congruent triangles
170
Sum of angles in a triangle
171
172
173a
Constant function
173b
Linear function
174
Horizontal line test
175
Restrictions of a function
176
Hybrid functions
177
Inverse functions
178
Inverse functions - examples
179
180
181
182a
Definitions of sine and cosine
182b
Symmetry properties of sine and cosine
183a
Definitions of tangent
183b
Relation of tangent to sine and cosine
184
Trigonometric ratios [SOH CAH TOA]
185
186
Symmetry properties of trig functions
187a
Signs of trig functions
187b
Negative angles
188
Exact values of trig functions
189
Sine and cosine graphs
190
Dilation of sine and cosine graphs
191
Reflection of sine and cosine graphs
192
Translations of sine and cosine graphs
193
Solving trig equations
194
Solving multiple-angle trig equations
195
General trig graph: an example
196
Symmetry properties of trig functions
197
198
Graph of the tan function
199
Frequency tables
200
Histograms
201
Frequency polygons
202
Box-and-whisker plots
203
Surveys (census)
204
Samples [data]
205
Perimeter
206
Circumference [circles]
207
Arc length
208
Perimeter of a composite figure
209
Units of area
210
211
Areas of triangles
212
Heron’s formula [triangles]
213
Areas of circles
214
Areas of sectors of circles
215
Surface area of solids
216
Volume of prisms
217
Fractional powers
218a
Multiplying surds
218b
Dividing surds
219
220
Cubic functions
221
Simple cubic functions
222
Points of inflection
223
Remainder theorem
224
Factor Theorem
225a
Difference of perfect cubes
225b
Sum of perfect cubes
226
Solving cubic equations
227
Graphing cubic functions
228
Solving cubic inequations
229
Finding a cubic equation from the graph
230
Factorising functions numerically
231
Sequences
232
Sequences generated by polynomials
233
Difference sequences
234
Finite differences for polynomials
235
Difference tables for general polynomials
236
Finding the rule from a difference table
237
Reciprocal trigonometric functions
238
Reciprocal trig function identities
239
Pythagorean trigonometric identity
240
241
Double-angle formulae
242
Examples of trigonometric proofs
243
Simplifying a sin x + b cos x
244
Lattice diagrams
245
Expected number of events
246
Sample space
247
Events
248
Complementary events
249
250
Interior and exterior angles
251
Constructing an angle of 60 degrees
252
Constructing an angle of 120 degrees
253
Constructing an angle of 90 degrees
254
Copying an angle
255
Sum of angles in polygons
256
Nets of three-dimensional shapes
257
Areas of composite shapes
258
Surface area of a sphere
259
Volumes of spheres
260
Volumes of pyramids
261
Commission
262
Finding 100%
263
Depreciation
264
Appreciation
265
Rates of change
266
Constant rates
267
Non-constant rates
268
Tangent to a curve
269
Finding the gradient at a point on a curve
270
Rates of change of simple functions
271
Rates of change using graphics calculators
272
Displacement
273
Velocity
274
Acceleration
275
Pronumerals or variables
276
Factors
277
Terms
278
Expressions
279
Equations
280
Simplifying terms
281
Labelling triangles
282
The sine rule
283
The cosine rule
284
Trig rule for area of a triangle
285
286
Chord length
287
Area of sector with radian angle
288
Area of segment with radian angle
289a
Angle of elevation
289b
Angle of depression
290
Compass bearings
291a
Planes
291b
Common line between two planes
292a
Line of greatest slope
292b
Angle between two planes
293
294
Expanding brackets
295
Expanding and simplifying
296
Expanding binomial times binomial
297
Difference of perfect squares
298
Expanding perfect squares
299
Simplifying algebraic fractions
300
Factorising
301
Substituting into rules
302
Debits, credits and balance
303
Compound interest formula
304
Two-two grouping factorisation
305
306
Exponential functions
307
Solving simple exponential equations
308
Definition of logarithm
309
Logarithm laws
310
Solving exponential equations using logarithms
311
Logarithm graph
312
Definition of a vector
313
Vector notation
314
315
Multiplication of vector by a scalar
316a
Zero vector
316b
Negative of a vector
317
Subtraction of vectors
318
Polygons of vectors
319
Parallel vectors
320
Position vectors
321
Linear combinations of non-parallel vectors
322
Components of vectors
323
Magnitude of a vector
324a
Unit vectors
324b
Unit vector in the direction of a given vector
325
Vector equations
326
Vectors in three dimensions
327
Translations of parabolas
328a
Dilations of parabolas
328b
Inversion of a parabola
329
330
331
Square roots
332
Cube roots
333
Rational and irrational numbers
334
Alternate segment theorem
335
Limit
336
Gradient at a point in the limit
337
Derivative of an expression
338a
Differentiation
338b
Differentiation by first principles
339
The derived function
340a
Derivative of a constant multiple
340b
Derivative of a sum
341
Deltas and dy/dx
342
Operator notation for derivative
343
Estimating the derivative from the graph
344
Differentiability
345
Limit of a function
346a
Limits from the left and right
346b
Limit at infinity
347a
Continuity
347b
Continuity and differentiability
348
Absolute value function
349
Formal definition of limit
350
Limit laws
350
Null factor law
352
Solving quadratic equations using the NFL
353
Solving quadratic equations in turning-point form
354
Finding r on a TI-83 calculator
355
Finding a quadratic equation from a difference table
356a
Solving an equation using inverse operations
356b
357
Examples of inverse operations
358
Transposing formulae
359
Differentiating positive powers of x
360
Differentiating negative powers of x
361
Differentiating integral powers of x
362
The chain rule
363
Examples of the chain rule
364
Function of function notation
365
Differentiating rational powers of x
366
Obtaining dx/dy from dy/dx
367
The second derivative
368a
Sequences
368b
Rule for a sequence
369
Difference equations
370
Arithmetic sequences
371
Arithmetic mean
372
Arithmetic series
373
Geometric sequence
374
Geometric mean
375
Compound interest as a geometric sequence
376
Geometric series
377
Infinite geometric series
378
Equation solving tricks 2a
379
380
Similar figures
381
Similar triangles
382
Naming sides in a right-angled triangle
383
SOH CAH TOA
384
Examples of SOH
385
Examples of CAH
386
Examples of TOA
387
Inverse trigonometric functions
388
Finding angles in right-angled triangles
388
Degrees, minutes and seconds on a CAS
390
Getting the five-figure summary from the ogive
391
Periodic graphs
392
Sine graphs
393
Distance
394
Average speed
395
Using calculus in kinematics
396
Formulae for constant acceleration
397
Examples of constant acceleration formulae
398
Using calculus for velocity-time graphs
399
Discrete random variables
400
Probability distribution function
401
The hypergeometric distribution
402
The binomial distribution
403
Normal to a curve
404
Finding the tangent using the derivative
405
Examples of rates of change from derivative
406
Stationary points
407
Sketching graphs from the stationary points
408
Types of stationary points
409
Minimax problems using derivatives
410
Sketching arbitrary non-polynomials
411
Antidifferentiation
412
Indefinite integrals
413
Antiderivative of polynomials
414a
Antidifferentiation of rational powers
414b
Using a piece of data to fix c
415
Area under a curve
416
Definite integrals
417
Definite integral laws
418
Numerical integration
419
Left endpoint integration
420
Right endpoint integration
421
Fundamental theorem of integral calculus
422
Trapezoidal integration
423
Integrals as the limits of sums
424
Sum notation
425
426
Multiplication counting rule
427
Permutations
428
Factorials
429
Permutation notation
430
Combinations
431
Combination notation
432
Pascal’s Triangle
433
Examples of permutations
434
Examples of combinations
435
Constructing graph scales
436
Contour maps
437
438
Naïve profit and loss
439
Rounding
440
Decimal places
441
Significant figures
442
Converting units
443
Types of triangle
444
Similar triangle tests
445
Two-mean regression line
446a
Density
446b
Concentration
447a
Forces
447b
Equilibrium
448
Normal force
449
Triangle of forces
450
Triangle of forces—examples I
451
Triangle of forces—examples II
452
Resolution of forces
453
Equilibrium in terms of force components
454
Resolution of forces—examples
455
Cartesian coordinates
456
Polar coordinates
457
Polar coordinate notation
458
Converting polar to Cartesian coordinates
459
Converting Cartesian to polar coordinates
460
Curve sketching in polar coordinates
461
Rooting negative numbers
462a
Complex numbers
462b
Real part of a complex number
462c
Imaginary part of a complex number
463
Equality of two complex numbers
464
Functions of complex numbers
465
466
Multiplying a complex number by a real
467
Powers of i
468
Multiplying complex numbers
469
Multiplying complex numbers—examples
470
Complex conjugate
471
Sum of perfect squares
472
473
Dividing complex numbers
474
Dividing complex numbers—examples
475
Argand diagrams
476
477
Multiplying on the Argand plane
478
Complex conjugate on the Argand plane
479a
Amplitude of a complex number
479b
Phase of a complex number
480
Example factorisation over C
481
Complex number in polar form
482
Multiplying and dividing in polar form
483
Euler’formula
484
The Binomial Theorem
485
The Binomial Theorem: Examples
486
Sign diagrams
487
The product rule
488
The quotient rule
489a
Integral of a sum of functions
489b
Integral of a constant multiple of a function
490
Integral of powers of linear functions
491a
integral of 1/x
491b
Integral of 1/(ax+b)
492
Area between two curves
493
Converting a recurring decimal to a fraction
494
Differentiation of trig functions
495
Integration of trig functions
496
Equation of an ellipse
497
Conic hyperbolae
498
The exponential function
499
500
Determining exponential rules
501
Determining logarithmic rules
502
Change-of-direction transformation
503
Change-of-direction transformation (cont.)
504
Derivative of the exponential function
505
Derivative of the logarithm function
506
Integration of exponential functions
507
Integration of 1/(ax+b)
508
Linear approximation
509
Odd and even functions
510a
Matrix notation
510b
Dimensions of a matrix
511
Equal matrices
512
Matrix elements
513a
513b
Subtracting matrices
514
Multiplying a matrix by a scalar
515
Multiplying matrices
516
Determinant of a 2 x 2 matrix
517
Inverse of a 2 x 2 matrix
518
Simultaneous equations using matrices
519
Translations using matrices
520
Dilations using matrices
521
Rotations using matrices
522
Reflections using matrices
523
Composition of mappings
524
Mean value of a distribution
525
Variance of a distribution
526
Standard deviation of a distribution
527
Mean and variance of a linear function
528
Statistics of the binomial distribution
529
Statistics of the hypergeometric distribution
530
Continuous random variables
531
The normal distribution
532
Standardised normal variable
533
Evaluating normal probabilities
534
Approximating the binomial with the normal
535
Change-of-direction transformation III
536
Cylindrical image transformation
537a
State of a system
537b
State vectors
538
Probabilistic state vectors
539
Markov processes
540
Transition matrices
541
Transition matrices (continued)
542
Transition matrices examples 1
543
Transition matrices examples 2
544
Transition matrices examples 3
545
Distribution of run lengths
546
Distribution of run lengths—Bernoulli example
547
Markov run length example
548
Outliers for univariate data
549
Modern box plots
550
Creating random data on CAS
551
Median of a continuous random variable
552
Quartiles and percentiles of a random variable
553
Mode of a random variable
554
Expected profit for a random event
555
556
Class intervals
557
Bar charts
558
Segmented bar charts
559
Shapes of histograms and stem plots
560
Symmetric and skewed distributions
561
Outliers from a histogram
562
563
Statistics symbols
564
Calculating the mean from frequencies
565
Standard deviation
566
The 68–95–99.7% rule
567
Dependent and independent variables
568
Ordered back-to-back stem plots
569
Parallel box plots
570
Two-way frequency tables
571
Percentaged two-way tables
572
Pearson’s r correlation coefficient
573
Correlation and causation
574
The coefficient of determination
575
Numerical and categorical data
576
Scatterplots
577
Positive and negative relationships
578
Strengths of relationships
579
580
Substitution tricks
581
Factorising using substitution tricks
582
Cross method: more easy examples
583
584
Introduction to parabolas
585
Equation solving tricks 1
586
Equation solving tricks 2b
587
Equation solving tricks 3a
588
Equation solving tricks 3b
589
Problem solving using linear equations
590
Sums of algebraic fractions with repeated factors
591
Simplifying products of algebraic fractions
592
Simplifying quotients of algebraic fractions
593
Finding quadratic relations from the graph 2
594
Finding quadratic relations from the graph 3
595
Finding quadratic relations from the graph 4
596
Finding quadratic relations from the graph 5
597
598
Fitting a straight line by eye: equal-points
599
Residual errors from a predicted line
600
Fitting a straight line by eye: equal-errors
601
Fitting a straight line: three-median method 1
602
Fitting a straight line: three-median method 2
603
Fitting a straight line: least-squares regression
604
Interpreting a straight-line fit
605
Interpolation and extrapolation
606
Making a table of values from a rule
607
Plotting a graph from a table of values
608
Getting the gradient from the rule
609
Calculating the y-intercept
610
Calculating the x-intercept
611
Travel graphs
612
Reading speed off a travel graph
613
Set notation for inequation solutions
614
The equation of a semicircle
615
Calculating residuals
616
Analysing residuals
617
Non-linear relationships
618
Parabolic transformation
619
Logarithmic transformation
620
Reciprocal transformation
621
Simultaneous equations: substitution method 2
622
Simultaneous equations: substitution method 3
623
Simultaneous equations: elimination method 3
624
Time series
625
Secular trends
626
Seasonal trends
627
Cyclic trends
628
Random trends
629
The two-mean trendline
630
Moving averages
631
3-point moving averages
632
5-point moving averages
633
Smoothing seasonal or cyclic data
634
Even-point moving averages
635
2-point moving averages
636
4-point moving averages
637
Centred even-point moving averages
638
Centred 2-point moving averages
639
Centred 4-point moving averages
640
Median smoothing
641
3-point median smoothing
642
5-point median smoothing
643
644
De-seasonalising time series 1
645
De-seasonalising time series 2
646
De-seasonalising time series 3
647
Graphing cubic functions with a calculator
648
Sums of polynomials and proper rational functions
649
Converting to an improper rational function
650
Converting to a proper rational function 1
651
Converting to a proper rational function 2
652
Converting to a proper rational function 3
653
Converting to a proper rational function 4
654
Converting to a proper rational function 5
655
Converting to a proper rational function 6
656
Converting to a proper rational function 7
657
Relative frequency
658
Number of outcomes
659
Introduction to tree diagrams
660
Compound events as intersections
661
Ratios as a percentage of the total
662
Ratios in direct proportion
663
Strength of a solution
664
Diluting a solution
665
Drug dosages depending on body mass
666
Ratio of photographic reproduction
667
Gear ratios
668
669
Arithmetic sequences
670
Checking if a sequence is arithmetic
671
Finding d for an arithmetic sequence
672
Finding the first term of an arithmetic sequence
673
Finding the n-th term of an arithmetic sequence
674
Finding the number of an arithmetic term
675
Arithmetic series
676
Arithmetic series from a, L and n
677
Arithmetic series from a, L and d
678
Arithmetic series from a, n and d
679
Finite and infinite sequences
680
Geometric sequences
681
Checking if a sequence is geometric
682
Finding r for a geometric sequence
683
Finding the first term of a geometric sequence
684
Finding the n-th term of a geometric sequence
685
Finding the number of a geometric term
686
Geometric series
687
Geometric series from a, r and n
688
Number of terms needed for a geometric series
689
Growth as a geometric sequence
690
Compound interest as a geometric sequence
691
Infinite geometric series
692
Recurring decimals as infinite geometric series
693
Graphs of arithmetic sequences
694
Graphs of geometric sequences
695
Number of payment periods of compound interest
696
Difference equations
697
First-order difference equations
698
Arithmetic sequence from a difference equation
699
Geometric sequence from a difference equation
700
Combination of arithmetic and geometric sequences
701
Explicit solutions to difference equations
702
Explicit solutions to arithmetic difference equations
703
Explicit solutions to geometric difference equations
704
Explicit solutions to combined difference equations
705
Graphing difference equations
706
Graphing difference equations on a calculator
707
Interpreting difference equation graphs
708
Finding d from Sn, n and a
709
Finding L from Sn, n and a
710
711
Dividing a line segment in a given ratio
712
Parallel lines
713
Finding the first term of an infinite geometric series
714
Finding r for an infinite geometric series
715
Polygons and regular polygons
716
Interior and exterior angles
717
Angle formulas for regular polygons
718
Perpendicular lines
719a
Line segments
719b
Rays
719c
Lines
720
Perpendicularly bisecting a line segment
721
Triangle inside a semicircle
722
Curved surface area of a cylinder
723
Curved surface area of a cone
724
Units of volume
725
Scale factors for lengths, areas and volumes
726
Pythagoras in three dimensions
727
Magic formulas for the sine rule
728
Ambiguous case of the sine rule
729
Magic formulas for the cosine rule
730
Equilateral triangles
731
Right-angled isosceles triangles
732
Degrees, minutes and seconds
733
Working with angles on a TI-83
734
Angles making up a right angle
735
Angles making up a straight angle
736
Standard compass bearings
737
Line segment graphs
738
Step graphs
739
Break-even analysis
740
Graphing a linear inequation
741
Linear programming problems 1
742
Linear programming problems 2