A Non-profit Organization to Help Students Study, Explore, and Engage

Sharp Book Series

# Topic Shortcuts

(In Alphabetical Order)

Algebraic Expressions

Conics

Complete the Square...

Divisibility

Equations

Functions (transformations, inverse, ...)

Function Catalogue

Geometry

Geometry, 3D

Inductive Reasoning and Making a Conjecture

Linear Functions (gr 10)

Polynomials (gr 10)

Polynomials, Factoring (gr 10)

Probability and Statistics

Sequences and Series

Three-dimensional Vectors

Imaginary and Complex Numbers

Specific Units:

# Online Lessons

BC Math: Math 10, Math 11, Math 12

# Stuck with Homework?

Submit a grade 10-12 homework question!

## If you are serious about Math...

Don't just follow the normal curriculum, aim higher, try for example competition material.

American Mathematics Contest 10/12 (AMC 10/12)

AMC Problems and Solutions:

AMC 10 Problems and Solutions by AoP (Text version).

Many of these problems also have solutions presented in video tutorials on the AoP YouTube channel. Search the channel to check in a solution is available, for example:

Video Solution: Art of Problem Solving: 2017 AMC 10 A #22

Text version: select the year here, test A or B, next pick the problem or answer key: for ex. problem 22 solution in text version is here.

## Digital Tools

Cayley Contest Mathematics Contests, for Grade 10

Fermat Contest Mathematics Contests, for Grade 11

Euclid Contest Mathematics Contests, for Grade 12

# Tests Online

Assessment: Chinook Learning Services: Mathematics 20-1 or 20-2 or 20-3 Placement Test

# In Person...

Math Nite will be running every Wednesday from 6pm-8pm in MS-431 at the University of Calgary, from the following dates:
September 20th, 2017 to December 6th,
2017 January 17th, 2018 to April 11th, 2018.
More detailed information about Math Nite can be found here: http://www.math.ucalgary.ca/community_outreach/math_nite

Articles

"...The excitement quickly fades as students brush up against the reality of what David E. Goldberg, an emeritus engineering professor, calls “the math-science death march.” Freshmen in college wade through a blizzard of calculus, physics and chemistry in lecture halls with hundreds of other students. And then many wash out."

# Specific Topics

(see Shortcuts)

## Conics

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Try the activities first.

Scroll further down for lesson.

Activity:
Circle

Experiment with the circle.

Activities:
Parabola

Play with the parameters that influence the parabola, find the roots.

Focus of the Parabola
- use the mouse to drag the Focus and see the effect on the parabola's curve.
Experiment with the effect of coefficient a on the parabola.
ax2+bx+c
("standard form")
ax2+bx+c

when messing with the b of this function, the Min point of the parabola moves along its own similar parabola.
Use Desmos and your own calculations to figure out what is the formula/function for the trajectory that the Min point incurs during changes to b.

Interactive Tool: Manipulate the Sections of a Cone https://illuminations.nctm.org/Activity.aspx?id=3506
Interactive Tool: Manipulate the Sections of a Cone http://www.shodor.org/interactivate/activities/CrossSectionFlyer/

Conics in Generals

or on .teacherspayteachers.com.
Conic Sections (animation)

Lessons on Conics

http://www.shelovesmath.com/precal/conics/
https://njctl.org/courses/math/pre-calculus/conics/
https://illuminations.nctm.org/Lesson.aspx?id=2907

Activities and Exploration

Project: Using Conics to build Baymax

## Polynomials

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Lessons http://tutorial.math.lamar.edu/Classes/Alg/Polynomials.aspx

Polynomial Functions http://carlisleschools.ss13.sharpschool.com/cms/One.aspx?portalId=95563&pageId=768777

## Complete the Square

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How to complete_the_square.html
Mathisfun, Examples how to solve https://www.mathsisfun.com/algebra/completing-square.html

## Polynomials, Factoring

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Before you begin working on this concept:
Part I
There should be a good understanding of how the simplest brackets work first. When this fails it becomes a huge obstacle for the entire p. factorization concept. Practice extracting simple factors out in front of brackets.
Example
First, a factor distributes to all elements in the brackets when expanding the expression.

Now learn to recognize this in reverse:

“a” is multiplied in both expressions. Recognize it as a factor that can play its role in front of brackets.
Practice creating brackets from simple expressions.
Before you begin working on this concept:
Part II
Recognizing the major algebraic formulas in reverse is very useful in factorizing polynomials as it gives immediate results.
For example:
a2 - b2
should be immediately recognized in its factorized form simply based on the formula
(a+b)(a-b)
Or for example

should always be recognized as

Factoring Methods

What is Factorizing Polynomials?
More on Factorizing Polynomials here.

Divisibility of numbers is a useful concept for factoring polynomials: Quick ways to determine divisibility:

A number is divisible by 2 if it terminates with 2, 4, 6, 8, or 0.
A number is divisible by 3 if the sum of its digits is divisible by 3.
A number is divisible by 4 if the last two digits are divisible by 4.
A number is divisible by 5 if it terminates with 5 or 0.
A number is divisible by 6 if it is divisible by both 2 and 3.
A number is divisible by 8 if the last three digits are divisible by 8.
A number is divisible by 9 if the sum of digits is divisible by 9.
A number is divisible by 10 if it terminates with 0.

Worksheet 2.6 Factorizing Algebraic Expressions

Lectures and Practice

Factoring Trinomials Tips and Tricks
Practice and solutions http://tutorial.math.lamar.edu/Classes/Alg/Factoring.aspx

Art of Problem Solving: Simon's Favorite Factoring Trick
Criss Cross Method

(why does it work?)

Factoring Trinomials

Various Calculators on Wolfram

https://www.wolframalpha.com/examples/mathematics/algebra/

Factoring Calculator on Wolfram

https://www.wolframalpha.com/input/?i=6x%5E2+-+x+-+2

## Linear Functions

(see first: Functions)

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What does "Linear" mean?
"Linear" is anything that represents the formula of a line. This can be linear equations, linear graphs, linear progressions, etc.

The typical formula for a line is given as:
y = mx + b, where m is the slope of the line and b is its y-intercept.

Lessons and other resources

Graphing Linear Equations https://njctl.org/courses/math/algebra-i/graphing-linear-equations/
Solving and Graphing Linear Inequalities https://njctl.org/courses/math/algebra-i/solving-and-graphing-linear-inequalities/
Parallel and Perpendicular Lines http://map.mathshell.org/lessons.php?unit=9220&collection=8

Practice: Parallel Lines
Practice Exercises
Domain and range
Plug in values
Table of values
Finding points on the graph
Intercepts and points on the graph (p.17)
Graphing volume (p.18)
Interpreting Graphs
http://www.math10.ca/lessons//UnitBooks/Math10C_Workbook_Unit4.pdf
Practice: Shortest Distance between a Point and Line - Equations of Lines
Practice:
Given a line y = mx + b, find the formula of a parallel line at a distance d.
Practice:
Shortest distance
between two parallel lines
(based on the solution above):

Problems that require finding the shortest distance between two parallel lines:
Using the above formula, we can isolate the distance d. This is assuming that the two line formulas are given. In other words:
- m & b1 are known,
- and b2 the parallel line's y-intercept is also given (or can be calculated).

Since the distance is a length, taking the positive value is what we want.

Practice EXAM http://www.math10.ca/lessons/relationsAndFunctions/practiceExam/Math10C_PracticeExam_RelationsAndFunctions.pdf

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Lessons and Notes

Scroll down to: Unit 3: Chapters 3 and 4 (these are Video presentations).

Art of Problem Solving: Vieta for Quadratics Part 1
Art of Problem Solving: Vieta for Quadratics Part 2

Parabola Interactive Activity on Desmos https://teacher.desmos.com/activitybuilder/custom/56fade8fcd70f60d1ca0ca05
Review: Quadratic Theory - Warm-up Questionnaire

## Functions

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Explaining y = f(x)

Functions Catalog

Functions vs. Relations http://www.learnalberta.ca/content/memg/Division04/Relation/index.html

http://carlisleschools.ss13.sharpschool.com/cms/One.aspx?portalId=95563&pageId=768692

Piecewise Functions https://www.mathsisfun.com/sets/functions-piecewise.html

Function Transformations
Vertical and Horizontal Transformations

Stretch and Compression

Transforming the square-root function

The Inverse of a Function The Inverse of a Square Root Function

https://www.mathsisfun.com/sets/function-inverse.html

Rational Functions

http://carlisleschools.ss13.sharpschool.com/cms/One.aspx?portalId=95563&pageId=768791

Function Operation and Composition http://blumalg2.blogspot.com/2018/08/august-22-2018-function-operations-and.html

http://www.purplemath.com/modules/fcnops.htm

https://www.mathsisfun.com/sets/functions-composition.html

Reconstructing a function from its roots and y-intercept into factored form y = a(x - x1)(x - x2)

http://www.analyzemath.com/Graphing/graphing_square_root_func.html

https://www.mathway.com/popular-problems/Algebra/279714

## Equations

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Linear Equations with Two Variables http://map.mathshell.org/lessons.php?unit=9235&collection=8
Systems of Linear Equations https://njctl.org/courses/math/algebra-i/systems-of-linear-equations/

## Exponent Laws

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The rules:
What is an "Integral exponent"? An "Integral exponent" is an exponent that is a whole number
(opposite to the powers of radicals for example which are 1/n, i.e. fractional)  .

## Sequences and Series

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Lessons

http://carlisleschools.ss13.sharpschool.com/cms/One.aspx?portalId=95563&pageId=768844

## Three-dimensional Vectors

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Activity Use Tinkercad to model 3D vectors from the video below.

## Imaginary and Complex Numbers

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Complex Numbers: Introduction http://www.purplemath.com/modules/complex.htm

Lesson - Part I
(see the entire series here)
Complex Numbers http://www.mathsisfun.com/numbers/complex-numbers.html

## Geometry

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Interactive: Circle & Line Geometry
Interactive: Incenter-Incircle

Interactive: Angle Bisector
Mathematics Vision Project:

Angles
Parallel Lines
Triangles
Similar Triangles and Trigonometry
Congruent Triangles
Circles
Transformations
Area of Figures
https://njctl.org/courses/math/geometry/

## 3D Geometry

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Activity Use Tinkercad to identify the cross section of 3D objects.
Build a hypercube

Cavalieri's Principle (Volume)

The Volume of a Cone

## Analytic Geometry (= Coordinate Geometry = Cartesian Geometry)

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Origin of Analytic Geometry
Distance Formula
Midpoint Formula
Partitions of a Line Segment
Slopes of Parallel and Perpendicular Lines
Eq of Parallel and Perpendicular Lines
Triangle Coordinate Proofs
Equation of a Circle and Completing the Square
https://njctl.org/courses/math/geometry/analytic-geometry/

 Inductive Reasoning and Making a Conjecture Khan Academy https://www.khanacademy.org/math/algebra-home/alg-series-and-induction#alg-deductive-and-inductive-reasoning Activity: Leap Frog Numbers Patterns (Needs Flash) https://brocku.ca/mathematics/resources/learningtools/learningobjects/frogs/fullscreendemo.html

# Articles

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# Books

 Free online version available.. This book is also available in Kindle format. Chapters have Self Grading Quizzes Topics include: Sierpinski Triangles, Menger Sponge, Fractals Fibonacci and The Golden Ratio. Currently (Dec 2017),  free on Kindle. A Book of Abstract Algebra Charles C. Pinter Elementary Linear Algebra Howard Anton, Chris Rorres Make It Stick offers techniques for becoming more productive learners, and cautions against study habits and practice routines that turn out to be counterproductive By road testing many of the counterintuitive techniques described in this book, Carey shows how we can flex the neural muscles that make deep learning possible.

# Extras for Fun

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