Foundations of Math 11 Ms Moon
Chapter 1: Inductive and Deductive Reasoning
a) Analyze and prove conjectures, using inductive and deductive reasoning, to solve
problems.
b) Analyze puzzles and games that involve spatial reasoning, using problem-solving
strategies.
Key Terms:
Conjecture
Inductive Reasoning
Counterexample
Proof
Generalization
Deductive Reasoning
Invalid Proof
Premise
Circular Reasoning
In Summary:
Key Idea:
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.
Once you have found a counterexample to a conjecture, you have disproved the
conjecture. This means that the conjecture is invalid.
You may be able to use a counterexample to help you disprove a conjecture.
Deductive reasoning involves starting with general assumptions that are known to be
true and, through logical reasoning, arriving at a specific conclusion.
A sing 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.
Inductive and deductive reasoning are useful in problem solving.
Need to Know
A conjecture is based on evidence you have gathered.
Foundations of Math 11 Ms Moon
More support for a conjecture strengthens the conjecture, but does not prove it.
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.
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.
The transitive property is often useful in deductive reasoning. It can be stated as
follows: Things that are equal to the same thing are equal to each other. If a=b and b=c
then a=c.
A demonstration using an example is not a proof.
Division by zero always creates an error in a proof, leading to an invalid conclusion.
Circular reasoning must be avoided. Be careful not to assume a result that follows from
what you are trying to prove.
The reason you are writing a proof is so that others can read and understand it. After
you write a proof, have someone else who has not seen your proof read it. If this
person gets confused, your proof needs to be clarified.
Inductive reasoning involves solving a simpler problem, observing patterns and
drawing a logical conclusion from your observations to solve the original problem.
Deductive reasoning involves using know facts or assumptions to develop an argument,
which is then used to draw a logical conclusion and solve the problem.
Foundations of Math 11 Ms Moon
Chapter 2: Properties of Angles and Triangles
a) Derive proofs and solve problems that involve the properties of angles and triangles.
1. Use the diagram to the right to complete the table.
Statement Justification
a = Corresponding angles are equal
c = Alternate interior angles are equal
b = Alternate exterior angles are equal
c + = 180 Interior angles on the same side of the transversal are supplementary
a + = 180 Exterior angles on the same side of the transversal are supplementary
f = Vertically opposite angles are equal
h = a
h + b = 180
d = h
d = e
d + f = 180
b = c
f + e = 180
Foundations of Math 11 Ms Moon
2. Complete the table.
The formula to find the sum of interior angles of a convex polygon with n-sides is:
The formula to find the measure of each interior angle is:
The sum of the measures of the exterior angles of a polygon is:
3. Consider the following Convex Polygons. Fill in the table with the appropriate values.
Convex
Polygon
The Sum of the
Measures of the
Exterior Angles
The sum of the
Measures of the
Interior Angles
The Measure of Each
Interior Angle of the
Regular Polygon
Square
(4 sides)
Heptagon
(7 sides)
Decagon
(10 sides)
Triskaidecagon
(13 sides)
Tetracontagon
(40 sides)
Foundations of Math 11 Ms Moon
Chapter 3: Acute Triangle Geometry
a) Solve problems about acute angle triangles that involve the cosine law and the sine law
sin A = cos A = tan A =
To call these trigonometric ratios quickly, remember the acronym:
___ ___ ___ ___ ___ ___ ___ ___ ___
1. Find the measure of angle X, to the nearest degree.
a. .
b. .
c. .
d. .
Foundations of Math 11 Ms Moon
e. Find side WX
f. Find side EF
2. Calculate the length of CD to the nearest tenth of a centimetre.
When you don’t have a right triangle, you may NOT use SOH CAH TOA. Why?
SINE LAW:
COSINE LAW 0 when you are looking for a side:
COSINE LAW – when you are looking for an angle:
Foundations of Math 11 Ms Moon
Chapter 4: Oblique Triangle Trigonometry
a) Solve problems about oblique angle triangles that involve the cosine law and the sin
law including the ambiguous case.
1. Determine the following:
sin 2°=
sin 10°=
sin 50°=
sin 60°=
sin 178°=
sin 170°=
sin 130°=
sin 120°=
cos 2°=
cos 10°=
cos 50°=
cos 60°=
cos 178°=
cos 170°=
cos 130°=
cos 120°=
In summary: For any angle Ɵ
sin Ɵ = _______________
cos Ɵ = _______________
2. Calculate the values for A (0° ≤ A ≤ 180°) that satisfy each of the equations listed. Give
your answer to the nearest degree.
a. sin A = 0.6428
b. cos A = 0.4226
c. sin A = 0.9659
Foundations of Math 11 Ms Moon
3. Given A = 35° and b = 20cm
a. Determine the height of the triangle to
the nearest tenth of a centimeter.
b. Determine and illustrate the
number of triangles that can be
drawn if a = 9cm.
c. Determine and illustrate the
number of triangles that can be
drawn if a = 25cm.
d. Determine and illustrate the number of triangles that can be drawn if a = 15cm.
4. Determine the measure of A to the nearest degree.
Foundations of Math 11 Ms Moon
5. Given A = 50° and b = 20cm, if a = 17,
determine the number triangles (zero, one, or
two) that are possible for these measurements.
Draw the triangle(s) to support your answer.
Determine side c and C.
Foundations of Math 11 Ms Moon
Chapter 5: Statistical Reasoning
a) Demonstrate an understanding of normal distribution, including: standard deviation
and z-scores
b) Interpret statistical data, using: confidence intervals, confidence levels, and margin of
error.
Chapter 5 Test Percentages for a Math class
82.4 68.9 50 64.9 60.8 63.5 81.1 64.9 94.6 79.9
81.1 68.9 79.7 85.1 70.3 97.3 100 86.5 73 70.3
1. Use the set of data above to determine:
a) The Range of test scores:
b) The Median Score:
c) The Mean Score:
d) The Standard Deviation:
Chapter 2 Test Percentages for the same Math class (Note: There are 4 more test scores than Chapter 5 due to illnesses and/or students no longer enrolled in the class)
77.5 17.5 52.5 61.3 62.5 62.5 81.3 71.3
68.8 67.5 66.3 52.5 56.3 100 87.3 53.8
70.7 41.3 81.3 82.5 61.3 57.5 67 75
2. Use the set of data above to determine:
a) The Range of test scores:
b) The Median Score:
c) The Mean Score:
d) The Standard Deviation:
3. Which test has more consistent scores? Why?
4. If Karys scored μ + σ on the Chapter 2 test, what was her score?
5. If Oscar scored μ – 2σ on the Chapter 5 test, what was his score?
Foundations of Math 11 Ms Moon
6. Justin and Selena both wrote a provincial exam in mathematics. Justin wrote in
January, and Selena wrote in June. Their results are given below.
Name Mark (x) Provincial Mean (μ) Provincial Standard
Deviation (σ)
Justin 84% 71% 5.3%
Selena 82% 66% 6.2%
a) Determine which student’s result is better.
b) If the results of each exam are normally distributed, what percent of people who wrote
the exam in January scored better than Justin?
7. The flight between Vancouver and Winnipeg has a mean time of 156 min, with a
standard deviation of 3.5 min. Assuming that the flight times for this trip are normally
distributed, determine approximately what percent of the time you could expect the
flight time to be
a. less than 156 min
b. between 149 min and 156 min
Foundations of Math 11 Ms Moon
c. between 152.5 min and 163 min
d. over 163 min
8. A study of 500 Calgarian taxpayers revealed that 24.1% of these taxpayers make
charitable contributions. The study was considered accurate plus or minus 5%, 9 times
out of 10. In a particular year, there were 827 120 taxpayers in Calgary.
a. Determine: Margin of Error: ______________________
Confidence Level: ____________________
Confidence Interval: ___________________
b. Determine the projected range of the number of Calgary taxpayers who would
make a charitable donation that year.
Foundations of Math 11 Ms Moon
Chapter 6: Systems of Linear Inequalities
a) Model and solve problems that involve systems of linear inequalities in two variables.
1. Which of the following are solutions to the inequality ? Circle all that
apply.
a. (2, 5)
b. (4, 3)
c. (-2, 6)
d. (-6, -2)
e. (8, -3)
f. (0, 0)
2. Graph:
3. Graph:
4. You have just graphed the
system:
Where can the solutions to the system of inequalities be found? Give two valid
solutions.
Solutions: ( ______, ______) and ( ______, ______)
Foundations of Math 11 Ms Moon
5. A toy company manufactures two types of toy vehicles: racing cars and sport-
utility vehicles. No more than 40 racing cars and 60 sport-utility vehicles can be
made in a day. The company can make 70 or more vehicles, in total, each day. It
costs $8 to make a racing car and $12 to make a sport-utility vehicle. What
combinations will result in the minimum and maximum costs?
Step 1. Define your variables
Step 2. Determine constraints (i.e.
determine your inequalities –
not your objective function)
Step 3. Graph
Step 4. Write objective function
Step 5. Check all corners
Step 6. Answer in sentence
70
60
50
40
30
20
10
-20 20 40 60 80 100 120 140
Foundations of Math 11 Ms Moon
6. A vending machine sells pop and juice. The machine holds, at most, 240 cans of
drinks. Sales from the vending machine show that least 2 cans of juice are sold for
each can of pop. Each can of juice sells for $1.00 and each can of pop sells for
$1.25. Determine the maximum revenue from the vending machine.
Step 1. Define your variables
Step 2. Determine constraints (i.e.
determine your)
Step 3. Graph
Step 4. Write objective function
Step 5. Check all corners
Step 6. Answer in sentence
340
320
300
280
260
240
220
200
180
160
140
120
100
80
60
40
20
y
-50 50 100 150 200 250 300 350 400 450 500 550 600x
Foundations of Math 11 Ms Moon
Chapter 7: Quadratic Functions and Equations
a) Demonstrate an understanding of the characteristics of quadratic functions,
including: vertex, intercepts, domain and range, and axis of symmetry
Standard Form
Factored Form
Vertex Form
Quadratic Formula
1. Fill in the table for the relation
y-intercept
x-intercept(s)
Axis of symmetry
Vertex
Domain
Range
2. A quadratic function has an equation that can be written in the form
The graph of the function has x-intercepts at (1, 0) and (3, 0)
and passes through the point (-1, 16). Write the equation of the function.
10
8
6
4
2
-2
-4
-6
-8
-10
y
-15 -10 -5 5 10 15 20 25x
Foundations of Math 11 Ms Moon
3. Determine the equation of a parabola with vertex (3, -3) and point (7, -19).
4. Solve using the quadratic formula.
a. b.
Foundations of Math 11 Ms Moon
5. Suppose a pebble were to fall from a 200m cliff to the water below. The height of the
stone, h(t), in metres, after t seconds can be represented by the function
. How long would the stone take to reach the water?
6. Patricia dives from a platform that is 10m high. She reaches her maximum height of
0.5m above the platform after 0.32s. How long will Patricia take to reach the water?
Foundations of Math 11 Ms Moon
Chapter 8: Proportional Reasoning
a) Solve problems that involve the application of rates
b) Solve problems that involve scale diagrams, using proportional reasoning
c) Demonstrate an understanding of the relationships among scale factors, areas, surface
areas and volumes of similar 2-D shapes and 3-D objects.