Pure Mathematics 20 Unit 1 Systems of Equations and Linear

Pure Mathematics 20 Unit 1
Systems of Equations and Linear Inequalities
EXERCISES
TOPIC
1. Exploring Ordered Pairs and Solutions.
Pages 4-5: ALL
2. Solving Systems of Linear Equations Graphically
Pages 11-12: 1, 2, 4, 7, 17, 19, 25,
27, 29, 32, 33, 35, 37, 44
3. Solving Systems of Linear Equations by Substitution.
Pages 25-27: 9, 10, 14, 19, 23,
26 abc, 28, 29 b, 30, 36, 37, 47
5. Solving Systems of Linear Equations by Elimination.
Pages 38-40: 1, 2, 7, 8, 10, 12, 19,
27, 31, 35, 44, 46, 48, 58, 60
6. Solving Systems of Linear Equations (3 Variables)
Pages 44-46: 7, 8, 11, 15, 22, 31, 44,
51
7. Exploring Non-Linear Systems with a Graphing
Calculator.
Page 47: All
8. Reviewing Linear Inequalities in One Variable.
Pages 63-64: 4, 8, 25, 26, 33, 42, 47,
50, 51, 56, 59
9. Graphing Inequalities in the Coordinate Plane
Pages 68-71: All
10. Graphing Linear Inequalities in Two Variables.
Page 75: 1, 5, 6, 11, 12, 23, 24, 29,
32, 37, 41, 42
11. Review.
Pages 52-54: All
Pages 94-95: 5 – 50 (multiples of 5)
***UNIT EXAM***
***SEE DATE ON COURSE
OUTLINE***
Evaluation Summary
1. Describe the steps you would use to solve a system by elimination and the steps for
substitution. Which method do you prefer? Explain why.
2. Naomi invested $1000, part at 8% per annum and the rest at 10% per annum. In one
year, the two parts earned equal amounts of interest. How much did Naomi invest at
each rate?
Pure Mathematics 20 Unit 2
Quadratic Functions
TOPIC
EXERCISES
1. Exploring Transformations
(Absolute Value Function)
Worksheet
2. Transformations of Quadratic Functions. (1) Pages 109-111: 1, 3, 10, 13-19, 32, 33, 45
3. Transformations of Quadratic Functions. (2) Pages 118-121: 1, 3, 10, 11, 14, , 12, 16, 20,
24, 25, 29, 38, 40
4. Writing Equations of Quadratic Functions.
Pages 109-111: 48, 51, 54, 56, 60-66, 69
Pages 118-121: 46, 47, 48, 52, 54, 60, 62, 66,
70, 71, 72, 77, 78, 80
5. Completing the square. (1) a = 1
Page 131: 1–43 odd and worksheet
6. Completing the square. (2) a ≠ 1
Page 131: 44–63 and worksheet.
7. Maximums and minimums; zeros using
calculators.
8. Problems. (1) (formula, number, geometric
and revenue)
9. Problems. (2)
Page 131: 64–74
10. Review.
Page 144: 1-61 odd, 62-64 all
***UNIT EXAM***
Page 119: 74
Pages 132-134: 73-78, 81-84
Worksheet
***SEE DATE ON COURSE OUTLINE***
Evaluation Summary
1. Given the quadratic function y = s ( x + t )2 – u, describe how you would find each of the following and then
state the:
a) direction of opening
b) coordinates of the vertex
c) equation of the axis of symmetry
d) domain and range
e) maximum or minimum value
2. For y = 3x2 – 12x + 15, describe and illustrate the steps to complete the square and obtain the vertex.
3. Vehicles Incorporated currently sells an average of 20 compact cars each week at a price of $6400 each. The
sales department wants to increase the price, but the marketing department predicts that for every $300 increase,
sales will fall by one car. If the dealer cost (cost to the dealer) for each car is $4000, what price will maximize
profits for Vehicles Incorporated?
Pure Mathematics 20 Unit 3
Quadratic and Polynomial Equations
EXERCISES
TOPIC
1. Solving quadratic equations by graphing.
Pages 155-156: 1, 3, 7, 8, 11, 14, 15, 17, 19, 26,
34, 35, 36, 41, 42, 45, 51, 54, 55
2. Solving quadratic equations by factoring.
Factoring Review… Worksheet
− b ± b 2 − 4ac
3. The quadratic formula x =
.
2a
4. The discriminant b 2 − 4ac .
5. The remainder theorem.
6. The factor theorem.
7. Solving polynomial equations.
8. Review.
***UNIT EXAM***
Pages 160-162: 1, 3, 7, 21, 24, 27, 36, 43, 44, 47,
49, 53, 55 60, 61, 65, 66, 70, 75, 80, 86, 89
Pages 178-180: 1, 2, 3, 6, 15, 16, 17, 18, 21, 29,
32, 34, 37, 47, 51, 55, 62, 64, 70, 77, 82
Pages 172-173: 57, 63, 65, 66 a, c, 67 a, c, 69
Pages 189-190: 1, 3, 5, 11, 12, 21, 23, 27, 28, 33,
34, 35, 43, 47, 54 a
Pages 202-203: 17, 19, 29, 31, 40, 41, 44, 47, 48,
51, 52, 55, 60
Pages 209-211: 1, 2, 7, 8, 14, 15, 23, 31, 37, 52,
55, 57, 59, 63, 65, 66, 67, 69, 82a, 85
Pages 219-221: 1, 2, 3, 4, 5, 6, 27, 28, 31, 35, 38,
41, 42, 46, 49, 78, 84, 93, 100
Page 232: (omit 33 - 34 and 57 - 75 )
***SEE DATE ON COURSE OUTLINE***
Evaluation Summary
1
a. Given an example for each situation. Include the equation being solved, the function being
graphed, and draw a graph illustrating solution(s), vertex and y-intercept.
i) two solutions
ii) one solution
iii) no solution
b. Explain how the graph relates to the solution(s).
2.
Use the Remainder Theorem to find k if P(x)=4x3 + kx2 – 6x – 5 and P(x) ÷ (x + 2) has a
remainder of –13.
3
a. Use the Factor Theorem to find one factor of P(x) = 2x3 + x2 – 13x + 6.
b. Use your answer from part a) to algebraically find all solutions of 2x3 + x2 – 13x + 6 = 0. In
your solution, include the fully factored form of the given polynomial equation.
Pure Mathematics 20 Unit 4
Functions
EXERCISES
TOPIC
1. Operations with Functions
Pages 247-250: 4, 6, 10, 26, 34, 37, 44, 54, 55,
60, 63, 68, 70
2. Composition of Functions
Pages 256-259: 1, 3, 10, 12, 16, 23, 24, 33, 35,
36, 44, 54, 55, 60, 65, 72, 73, 78
Pages 268-270: 1, 11, 16, 17, 31, 32, 36, 42-47,
48, 67, 69, 74, 85, 89, 91, 96
3. Inverse of a Function
4. Graphing Polynomial Functions
Pages 272-273: All
5. Solving by Graphing … All Types
Worksheet
Questions with asterisks * should be solved algebraically
(and solutions confirmed graphically).
6. Polynomial Equations and Inequalities
7. Absolute Value Equations and Inequalities
8. Rational Equations and Inequalities
9. Radical Equations and Inequalities
10. Review
Pages 282-285: 1-8, 9, 11, 13-16, 29, 30, 35, 41,
43, 45*, 47*, 49, 52, 53*, 54*, 55*, 56*, 69
Pages 296-298: 2*, 4*, 6*, 12*, 26*, 30*, 33, 34,
41, 46, 52, 56, 65, 78, 80, 83, 96, 102, 104, 108
Pages 309-312: 1-12, 13, 20, 21, 32, 43, 49*,
50*, 51*, 65*, 70*, 77, 79, 89, 91, 110*, 115*,
129
Pages 323-326: 1, 2, 11, 16, 24*, 26*, 35*, 37,
44, 45, 48*, 51*, 53*, 68, 71, 72, 73, 82, 83, 118*,
124
Pages 330-331: 40*, 41*, 42*, 43*, 60*, 61*,
64*, 65*, 66*, 67*, and all other questions.
Page 332: 32*, 33*, 36*, 37*, 38*, 39*, and all
other questions.
***UNIT EXAM***
***SEE DATE ON COURSE OUTLINE***
The students should be able to solve the following equations and inequalities algebraically and graphically:
absolute value equations (single absolute value only)
radical equations (up to 2 radicals with one being simple e.g.
rational equations
polynomial equations
polynomial inequalities (this includes quadratic inequalities)
x)
The students should be able to solve the following inequalities graphically only:
absolute value inequalities
radical inequalities
rational inequalities
Evaluation Summary
1. Summarize the features of linear, quadratic, cubic, quartic and quintic functions in a table
using the following headings. Name, degree, general form, end behavior if ‘a’ is positive, end
behavior if ‘a’ is negative and greatest number of turning points possible.
2. In the method used to solve a radical equation, both sides of the equation are squared to
remove the radicals. Explain why radicals are isolated before both sides are squared.
3. Make a list of the operations that can be applied to two functions to define new functions.
Make up functions of your own, and use these to demonstrate each point in your list.
4. In your own words, describe the difference between
a) f(g(x)) and g(f(x))
b) fg(x) and f(g(x))
Pure Mathematics 20 Unit 5
Reasoning
TOPIC
EXERCISES
1. Inductive Reasoning, Conjectures and
Counterexamples
Pages 340-342: 1-6, 12-15, 18, 19, 21, 22
Pages 345-346: 1-20
2. Connecting words
Page 354-356: 1-7, 11, 12, 16, 17, 18, 24, 25,
26, 33, 36- 39, 40, 41, 56, 57, 58, 60, 61
3. If-Then statements
Pages 361-362: 1-32
4. Deductive Reasoning (and Direct Proofs)
Page 349: 1-14, 16, 17
Page 372: 15-19
5. Congruent Triangles
Pages 364-365: All
Page 383: 1, 2, 5, 6
6. Geometric Proofs
Pages 389-392: 1-10, 16, 17, 19, 28
7. Review.
Pages 376-378: 4-30
Page 379: 1-19
***UNIT EXAM***
***SEE DATE ON COURSE OUTLINE***
Evaluation Summary
1. Explain what is meant by the converse of an “If . . . then” statement. Illustrate your
explanation with an example of a true statement that has a true converse, and a true statement
that has a false converse.
2. Explain what is meant by the contrapositive of an “If . . . then” statement. Illustrate your
explanation with an example. Use your example to explain why the contrapositive has the
same meaning as the original statement.
Pure Mathematics 20 Unit 6
The Circle
1. Introduction
Preview Worksheet (Theorems and
Vocabulary)
Page 382 All
2. Chord Properties
Pages 400-401: 1-20, 23-26, 30
3. Angles in a Circle.
Pages 412-414: 1-26, 28, 35
4. Cyclic Quadrilaterals.
Pages 419-421: 1-16, 18, 21, 25
5. Tangents to a Circle
Pages 431-432: 1-14, 16, 18-21
6. Angles and Polygons.
Pages 442-443: 1-4, 5, 8, 11, 14, 17, 20,
23, 26, 29, 32-36
9. Review
Pages 446-447: Omit 19, 21
Pages 448: Omit 14-16
***UNIT EXAM***
***SEE DATE ON COURSE
OUTLINE***
Formal Proof
Opening Statement: Consisting of what is given and what is to be proved.
Body of the Proof: Statements with corresponding reasons.
Closing Statement: Conclusion of proof. ie. Hence or therefore.
Required Proofs
1. Perpendicular bisector of a chord contains the centre of the circle.
2. Angle inscribed in a semicircle is a right angle.
3. Tangent segments to a circle from any external point are congruent.
NB: Trigonometric ratios can be used to justify proofs, in addition to circle
properties.
Pure Mathematics 20 Unit 7
Coordinate Geometry and Trigonometry
EXERCISES
TOPIC
1. Getting Started
Worksheet – Coordinate Geometry
Preview
2. Connecting Coordinate Geometry and Plane
Geometry
Pages 460-461: 1, 2, 3, 4, 8, 15, 16,
20, 21, 24
3. Distances Between Points and Lines
Pages 475-476: 1, 4, 7, 12, 15, 18,
20, 29, 31, 33, 36, 39, 42, 47, 51, 56
4. The Equation of a Circle.
Page 481-482: 1, 5, 7, 8, 13, 16, 20,
21, 24, 27-36, 38, 40, 44, 50, 52
5. Intersections of Lines and Circles
Pages 488-491: 1, 2, 8, 10, 19, 21,
24, 25, 27, 29, 33, 35, 39, 42, 45, 47,
52, 54, 55, 57, 61, 63, 67, 82, 90
6. Review
Pages 516-517: 1-51
Page 518: 1-33
***UNIT EXAM***
***SEE DATE ON COURSE
OUTLINE***
Evaluation Summary
1. Explain in detail the process that you would follow to determine the equation of the tangent
line to a circle at a given point on the circle. Use the circle define by ( x − 2) 2 + ( y + 3) 2 = 58
and the point ( 5, 4 ) to illustrate your explanation.
2. A circle centre O , is defined by x2 + y2 = r2. The point P(a,b) is outside the circle. T is the
point of contact on the circle of a tangent from P. Find the lengths of OT, OP and the length
of the tangent PT.
Pure Mathematics 20 Unit 8
Finance
EXERCISES
TOPIC
1. Unit Prices and Exchange Rates
Page 524: All
Page 526: All
2. Earning Income
Pages 530 – 531: 1 - 29
3. Net Income
Pages 536 – 537: 6, 7, 11, 12, 19, 20, 22,
30, 31, 32
4. Interest and Annuities
Page 542 - 543: 14 – 28 even, 29 –34
5. TVM Solver
Worksheet and Pages 544 – 545
6. Consumer Credit / Balancing a Budget
Pages 555 – 556: 1 – 20
Pages 565 - 566: 1 – 20, 23
7. Housing Costs
Pages 560 – 561: 2 – 32 even, 33, 35
8. Review
Page 572: 1 – 29
***UNIT EXAM***
***SEE DATE ON COURSE
OUTLINE***
Pure Math 20 AP
Course Syllabus
All material below is in addition to the regular Pure Math 20 Curriculum.
Unit 1
Systems of Equations (and Inequalities)
Matrices (3 x 4 … RREF on paper)
Matrices (2 x 3 … Solving using multiplicative inverse)
eg.
2 x − y = 12
3x + 5 y = 5
⎡2 − 1⎤ ⎡ x ⎤ ⎡12⎤
⎢3 5 ⎥ ⎢ y ⎥ = ⎢ 5 ⎥
⎣
⎦⎣ ⎦ ⎣ ⎦
⎡ 5 1 ⎤ ⎡2 − 1⎤ ⎡ x ⎤ ⎡ 5 1 ⎤ ⎡12⎤
⎢ − 3 2⎥ ⎢3 5 ⎥ ⎢ y ⎥ = ⎢ − 3 2⎥ ⎢ 5 ⎥
⎦⎣ ⎦
⎣
⎦⎣
⎦⎣ ⎦ ⎣
⎡13 0 ⎤ ⎡ x ⎤ ⎡ 65 ⎤
⎢ 0 13⎥ ⎢ y ⎥ = ⎢− 26⎥
⎦
⎣
⎦⎣ ⎦ ⎣
⎡1 0⎤ ⎡ x ⎤ ⎡ 5 ⎤
⎢0 1 ⎥ ⎢ y ⎥ = ⎢ − 2 ⎥
⎣
⎦⎣ ⎦ ⎣ ⎦
Cramer’s rule
Non-linear systems (algebraically)
eg.
1 3
− =2
x y
6 5
+ = −34
x y
or,
23 x + y = 0
53 x + 7 y = −5
Systems of Inequalities (2.5 in textbook)
Unit 2
Quadratic Functions
Transformations of all types of functions
(extensive preview of Pure math 30 transformations is appropriate here)
Challenge problems (old math contest questions)
Focus on Max / Min (optimization) questions.
Unit 3
Quadratic and Polynomial Equations
Tougher word problems (lots to choose from in textbook for this chapter)
Complex/Imaginary Numbers
Unit 4
Functions and Inequalities
Everything algebraic! (solve for cases rather than checking)
Equations… single and double (radical, rational and absolute value)
mixed type
Inequalities… single and double (radical, rational and absolute value)
Unit 5
Logic and Reasoning
Tougher Venn diagrams (requiring systems of equations)
Indirect Proofs
Lots of tougher proofs to assign from textbook in this chapter.
Unit 6
Circles
Tougher questions… material mainly the same.
Sector area and Arc Length
Complete the square to change circle from general to
Unit 7
Coordinate Geometry
Tougher proofs
Ambiguous case of Sine Law.( if not done in 10AP)
Unit 8
Finance
Business vs. Personal Financial Statements
Risk vs. Reward