Chapter 1: Inductive and Deductive Reasoning
1.1
Making Conjectures: Inductive Reasoning
Inductive reasoning involves looking at specific examples. By observing patterns and
identifying properties in these examples, you may be able to make a general conclusion,
which you can state as a conjecture.
-
A conjecture is based on evidence you have gathered.
More support for a conjecture strengthens the conjecture, but does not prove it.
Conjecture: a testable expression that is based on available evidence but is not yet proved.
Inductive Reasoning: drawing a general conclusion by observing patterns and identifying
properties in specific examples.
1.2
Exploring the Validity of Conjectures
Some conjectures initially seem to be valid, but are shown not to be valid after more
evidence is gathered.
-
The best way we can say about a conjecture reached through inductive reasoning is that
there is evidence either to support or deny it.
A conjecture may be revised, based on new evidence.
1.3
Using Reasoning to Find a Counterexample
Once you have found a counterexample to a conjecture, you have disproved the conjecture.
This means that the conjecture in invalid.
You may be able to use a counterexample to help you revise a conjecture.
-
A single counterexample is enough to disprove a conjecture.
Even if you cannot find a counterexample, you cannot be certain that there is not one. Any
supporting evidence you develop while searching for a counterexample, however, does
increase the likelihood that the conjecture is true.
Counterexample: an example that invalidates a conjecture.
1.4
Proving Conjectures: Deductive Reasoning
Deductive reasoning involves starting with general assumptions that are known to be true
and, through logical reasoning, arriving at a specific conclusion.
-
A conjecture has been proved only when it has been shown to be true for every possible
case or example. This is accomplished by creating a proof that involves general cases.
When you apply the principles of deductive reasoning correctly, you can be sure that the
conclusion you draw is valid.
A demonstration using an example is not a proof.
1.5
Proof: a mathematical argument showing that a statement is valid in all cases, or that no
counterexample exists.
Generalisation: drawing a specific conclusion through logical reasoning by starting with
general assumptions that are known to be valid.
Transitive Property: if two quantities are equal to the same quantity, then they are equal to
each other. If a = b and b = c, then a = c.
Proofs That Are Not valid
A single error in reasoning will break down the logical argument of a deductive proof.
This will result in an invalid conclusion, or a conclusion that is not supported by the proof.
1.6
Division by zero always creates an error in a proof, leading to an invalid conclusion.
The reason you are writing a proof is so that others can read and understand it.
Invalid Proof: a proof that contains an error in reasoning or that contains invalid
assumptions.
Premise: a statement assumed to be true.
Circular Reasoning: an argument that is incorrect because it makes use of the conclusion
to be proved.
Reasoning to Solve Problems
Inductive and deductive reasoning are useful in problem solving.
-
Inductive reasoning involves solving a simpler problem, observing patterns, and drawing a
logical conclusion from your observation to solve the original problem.
Deductive reasoning involves using known facts or assumptions to develop an argument,
which is then used to draw a logical conclusion and solve the problem.
1.7
Analysing Puzzles and Games
Both inductive and deductive reasoning are useful determining a strategy to solve a puzzle
or win a game.
-
Inductive reasoning is useful when analysing games and puzzles that require recognising
patterns or creating a particular order.
Deductive reasoning is useful when analysing games and puzzles that require inquiry and
discovery to complete.
Chapter 2: Properties of Angles and Triangles
2.1
Exploring Parallel Lines
When a transversal intersects a pair of parallel lines, the corresponding angles that are
formed by each parallel line and transversal are equal.
When a transversal intersects a pair of lines creating equal corresponding angles, the pair of
lines is parallel.
- When a transversal intersects a pair of non-parallel lines, the corresponding angles are not
equal.
- There are also other relationships among the measures of the eight angles formed when a
transversal intersects two parallel lines.
- Transversal: a line that intersects two or more other lines at distinct points.
- Interior Angles: any angles formed by a transversal and two parallel lines that lie inside the
parallel lines.
- Exterior Angles: any angles formed by a transversal and two parallel lines that lie outside
the parallel lines.
- Corresponding Angles: one interior angle and one exterior angle that are non-adjacent and
on the same side of a transversal. (F-Pattern)
2.2
Angles Formed by Parallel Lines
When a transversal intersects two parallel lines,
i.
the corresponding angles are equal.
ii. the alternate interior angles are equal.
iii. the alternate exterior angles are equal.
iv.
the interior angles on the same side of the transversal are supplementary.
- Alternate Interior Angles: two non-adjacent interior angles on the opposite sides of a
transversal. (Z-Pattern)
- Alternate Exterior Angles: two exterior angles formed between two lines and a transversal,
on the opposite sides of the transversal,
2.3
Angle Properties in Triangles
You can prove properties of angles in triangles using other properties that have already
been proven.
- In any triangle, the sum of the measure of the interior angles is proven to be 180o.
- The measure of any exterior angle of a triangle is proven to be equal to the sum of the
measures of the two non-adjacent interior angles.
- Non-Adjacent Interior Angles: the two angles of a triangle that do not have the same
vertex as an exterior angle.
2.4
Angle Properties in Polygons
You can prove properties of angles in polygons using angle properties that have already
been proved.
- The sum of the measures of the interior angles of a convex polygon with n sides can be
expressed as (n – 2)180o.
(𝑛−2)180°
- The measure of each interior angle of a regular polygon is
𝑛
- The sum of the measure of the exterior angles of any convex polygon is 360o.
- Convex Polygon: a polygon in which each interior angle measures less than 180o.
Chapter 3: Acute Triangle Geometry
3.1 Exploring Side-Angle Relations in Acute Triangles
𝑎
𝑏
𝑐
The sine law in an acute triangle: sin 𝐴 = sin 𝐵 = sin 𝐵
3.2 Proving and Applying the Sine Law
The sine law can be used to determine unknown side lengths or angle measures in acute
triangles when you have:
- two sides and the angle opposite a known side; or
- two angles and any side.
𝑎
𝑏
𝑐
When determining side lengths, it is more convenient to use: sin 𝐴 = sin 𝐵 = sin 𝐵
When determining angles, it is more convenient to use:
sin 𝐴
𝑎
=
sin 𝐵
𝑏
=
sin 𝐵
𝑐
3.3 Proving and Applying the Cosine Law
The cosine law can be used to determine an unknown side length or angle measure in an
acute triangle.
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶
𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 cos 𝐵
𝑎2 + 𝑏 2 − 𝑐 2
𝐶 = cos −1 (
)
2𝑎𝑏
𝑏 2 + 𝑐 2 − 𝑎2
𝐵 = cos −1 (
)
2𝑏𝑐
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
𝐴 = cos
−1
𝑏 2 + 𝑐 2 − 𝑎2
(
)
2𝑏𝑐
3.4 Solving Problems Using Acute Triangles
Drawing a clearly labelled diagram makes it easier to select a strategy for solving a
problem.
Chapter 4: Oblique Triangle Trigonometry
4.1 Exploring the Primary Trigonometric Ratios of Obtuse Angles
There are relationships between the value of a primary trigonometric ratio for an acute
angle and the value of the same primary trigonometric ratio for the supplement of the
angle.
For any angle:
sin = sin (180o - )
cos = -cos (180o - )
tan  = - tan (180o - )
- Oblique Triangle: a triangle that does not contain a 90o angle.
4.2 Proving and Applying the Sine and Cosine Laws for Obtuse Triangles
The sine law and cosine law can be used to determine unknown side lengths and angle
measures in obtuse triangle.
Be careful when using the sine law to determine the measure of an angle. The inverse sine
of a ratio always gives an acute angle, but the supplementary angle has the same ratio. You
must decide whether the acute angle or the obtuse angle is the correct angle for your
triangle.
Because the cosine ratios for an acute and its supplement are not equal, the measures of the
angles determined using the cosine law are always correct.
4.3 The Ambiguous Case of the Sine Law
The ambiguous case of the sine law may occur when you are given two side lengths and the
measure od an angle that is opposite one of these sides. Depending on the measure of the
given angle and the lengths of the given sides, you may need to construct and solve zero,
one, or two triangles.
Chapter 5: Statistical Reasoning
5.1 Exploring Data
Measure of central tendency (mean, median, mode) are not always sufficient to represent or
compare sets of data.
You can draw inferences from numerical data by examining how the data is distributed
around the mean or the median.
- To compare sets of data, the data must be organised in a systematic way.
- When analysing two sets of data, it is important to look at both similarities and differences
in the data.
- Outlier: a value in a data set that is very different from other values in the set.
- Line Plot: a graph that records each data value in a data set as a point above a number line.
- Dispersion: a measure that varies by the spread among the data in a set; dispersion has a
value of zero if all the data in a set is identical, and it increases in value as the data
becomes more spread out.
5.2 Frequency Tables, Histograms, and Frequency Polygons
Large sets of data can be difficult to interpret. Organising the data into intervals and
tabulating the frequency of the data in each interval can make it easier to interpret the data
and draw conclusions about how the data is distributed.
A frequency distribution is a set of intervals and can be displayed as a table, a histogram, or
a frequency polygon.
- Frequency Distribution: a set of intervals (tables or graph), usually of equal width, into
which raw data is organised: each interval is associated with a frequency that indicates
the number of measurements in this interval.
- A frequency distribution should have a minimum of 5 intervals and a maximum of 12
intervals.
- The interval width can be determined by dividing the range of the data by the desired
number of intervals and then rounding to a suitable interval width.
- Histogram: the graph of a frequency distribution, in which equal intervals of values are
marked on a horizontal axis and the frequencies associated with these intervals are
indicated by the areas of the rectangles drawn for these intervals.
- The height of each bar in a histogram corresponds to the frequency of the interval it
represents.
- Frequency Polygon: the graph of a frequency distribution, produced by joining the
midpoints of the intervals using straight lines.
- Frequency polygons are useful when comparing multiple sets of data because they can be
graphed on top of each other.
5.3 Standard Deviation
To determine how scattered or clustered the data in a set is, determine the mean of the data
and compare each data value to the mean, 𝑥̅ .
𝑥̅ =
∑𝑥
𝑛
𝑥̅ =
∑(𝑓)(𝑥)
𝑛
The standard deviation, 𝜎, is a measure of dispersion of data about the mean.
𝜎=√
∑(𝑥 − 𝑥̅ )2
𝑛
The mean and standard deviation can be determined using technology for any set of
numerical data, whether or not the data is grouped.
- When data is concentrated close to the mean, the standard deviation, σ, is low. When data
is spread far from the mean, the standard deviation is high. As a result, standard
deviation is a useful statistic to compare the dispersion of two or more sets of data.
- Standard deviation is often used as a measure of consistency.
5.4 Normal Distribution
You can make reasonable estimates about data that approximates a normal distribution,
because data that is normally distributed has special characteristics.
Normal curves can vary in two main ways: the mean determines the location of the centre
of the curve on the horizontal axis, and the standard deviation determines the width and
height of curve.
The properties of normal distribution can be summarised as follows:
- The graph is symmetrical. The mean, median, and mode are equal (or close) and fall at the
line of symmetry.
- The normal curve is shaped like a bell, peaking in the middle, sloping down towards the
sides, and approaching zero at the extremes.
- About 68% of the data is within one standard deviation of the mean.
- About 95% of the data is within two standard deviations of the mean.
- About 99.7% of the data is within three standard deviations of the mean.
- The area under the curve can be considered as 1 unit, since it represents 100% of the data.
Generally, measurements of living things (such as mass, height, and length) have a normal
distribution.
5.5 Z-Scores
The standard normal distribution is a normal distribution with mean, 𝜇, of 0 and a standard
deviation, 𝜎, of 1. The area under the curve of a normal distribution is 1.
Z-scores can be used to compare data from different normally distributed sets by
converting their distributions to the standard normal distribution.
- A z-score indicates the number of standard deviations that a data value lies from the mean.
It is calculated using this formula:
𝑧=
𝑥−𝜇
𝜎
- A positive z-score indicates that the data value lies above the mean.
- A negative z-score indicates that the data value lies below the mean.
- The area under the standard normal curve, to the left of a particular z-score, can be found in
a z-score table. The area to the right is the area to the left subtracted from 1.
5.6 Confidence Intervals
It is often impractical, if not impossible, to obtain data for a complete population. Instead,
random samples of the population are taken, and the mean and standard deviation of the
data are determined. This information is then used to make predictions about the
population.
When data approximates a normal distribution, a confidence interval indicates the range in
which the mean of any sample of data of a given size would be expected to lie, with a
stated level of confidence. This confidence interval can then be used to estimate the range
of the mean for the population.
Sample size, confidence level, and population size determine the size of the confidence
interval for a given confidence level.
- Margin of Error: the possible difference between the estimate, as determined from the
random sample, and the true value for the population; usually expressed as  a %.
Chapter 6: Systems of Linear Inequalities
6.1 Graphing Linear Inequalities in Two Variables
Graph linear inequality boundaries in the same manner that you would graph a line.
𝑦 = 𝑚𝑥 + 𝑏, the slope intercept form
𝑚=
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛
=
𝑦2 −𝑦1
𝑥2 −𝑥1
, b is the y-value of the y-intercept
≥, is read as greater than or equal to and is graphed with a solid boundary.
≤, is read as less than or equal to and is graphed with a solid boundary.
>, is read as greater than and is graphed with a dashed boundary.
<, is read as less than and is graphed with a dashed boundary.
If you arrange the inequality into the slope-intercept form, the solution region can be
represented by shading the area defined by y. e.g. if y < mx + b, then shade below the
boundary; if y > mx + b, then shade above the boundary.
Alternately, you can graph the boundary and use a test point to determine the region.
6.2 Exploring Graphs of Systems of Linear Inequalities
A system of linear inequalities involves two or more boundaries. These boundaries are
graphed as described in section 6.1. The solution region for the system is the area where
the solutions regions for each inequality overlaps.
Rather than shading each individual inequality region, it may be easier to use small arrows
to represent the solution regions, then shade the final overlapping solutions region.
6.3 Graphing to Solve Systems of Linear Inequalities
Many real-world problems are found in the first quadrant, since they involve positive
numbers, and often involve whole number because they involve counting items. Make sure
you consider the context when answering questions.
6.4 Optimisation Problem – Creating the Model
When creating a model, the first step is to represent the situation algebraically. An
algebraic model includes the following:
- a defining statement of the variables (let x represent…);
- a statement describing the restrictions;
- a system of linear inequalities that describes the constraints; and
- an objective function that shows how the variables are related to the quantity to be
optimised.
Represent the model graphically.
- Optimisation Problem: a problem where a quantity must be maximised or minimised
following a set of guidelines or conditions.
- Constraint: a limiting condition of the optimisation problem being modelled, represented
by a linear inequality.
- Objective Function: the equation that represents the relationship between the two variables
in the system of linear inequalities and the quantity to be optimised.
- Feasible Region: The solution region for a system of linear inequalities that is modelling
an optimisation problem.
6.5 Optimisation Problem – Exploring Solutions
The value of the objective functions for a system of inequalities varies throughout the
feasible region, but in a predictable way
The optimal solutions (minimum or maximum) to the objective function are represented by
points at the intersections of the boundaries of the feasible region.
- You can verify each optimal solution to make sure it satisfies each constraint by
substituting the values of its coordinates into each linear inequality in the system.
- The intersection points of the boundaries are called vertices, or corners, of the feasible
region.
6.6 Optimisation Problem – Linear Programming
To solve an optimisation problem using linear programming, begin by creating algebraic
and graphical models of the problem (as shown in 6.4). Use the objective function to
determine which vertex of the feasible region results in the optimal solution.
Chapter 7: Quadratic Functions and Equations
7.1 Exploring Quadratic Relations
7.2 Properties of Graphs of Quadratic Functions
7.3 Solving Quadratic Equations by Graphing
7.4 Factored Form of a Quadratic Function
7.5 Solving Quadratic Equations by Factoring
7.6 Vertex Form of a Quadratic Function
7.7 Solving Quadratic Equations Using the Quadratic Formula
7.8 Solving Problems Using Quadratic Models
Chapter 8: Proportional Reasoning
8.1 Comparing and Interpreting Rates
8.2 Solving Problems That Involve Rates
8.3 Scale Diagrams
8.4 Scale Factors and Areas of 2-D Shapes
8.5 Similar Objects: Scale Models and Scale Diagrams
8.6 Scale Factors and 3-D Objects