Chapter 1: Inductive and Deductive Reasoningmskmoon.weebly.com/uploads/1/7/1/8/17186300/final_review.pdf · 2019-11-10 · Chapter 1: Inductive and Deductive Reasoning a) Analyze

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.


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.


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


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)


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. .


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:


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


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.


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.


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?


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


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.


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 ( ______, ______)


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


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


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


3. Determine the equation of a parabola with vertex (3, -3) and point (7, -19).

4. Solve using the quadratic formula.

a. b.


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?


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.

Documents

Chapter 1: Inductive and Deductive Reasoningmskmoon.weebly.com/uploads/1/7/1/8/17186300/final_review.pdf · 2019-11-10 · Chapter 1: Inductive and Deductive Reasoning a) Analyze