July 20, 2009

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- Mathematics > General

- Grade 1
- Grade 2
- Grade 3
- Grade 4
- Grade 5
- K
- Grade 6
- Grade 7
- Grade 8
- Grade 9
- Grade 10
- Grade 11
- Grade 12

Multiply and divide whole numbers. Understand and use standard algorithms for multiplication and division.

Compare fractions, decimals and common percents.

Understand and perform addition and subtraction with fractions, including fractions with different denominators and mixed numbers. Add and subtract decimals, including money in decimal notation.

Measure angles and describe angles in degrees, and identify, classify and draw polygons and triangles.

Find and use the perimeter and area of triangles, parallelograms and trapezoids, and the surface area and volume of rectangular prisms.

Evaluate simple algebraic expressions.

Use two-dimensional coordinate grids to represent points in the first quadrant that fit linear equations. Draw the line determined by the points.

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Table of Contents

- Evaluating Algebraic Expressions Lesson
- Area of a trapezoid (trapezium)
- Measuring With Inches
- Fabulous Fractions
- Algebra 1A
- math level5
- rags to riches
- Polynomial Test
- Systems of Equations Baseball
- Math Games Lessons
- Flips, Slides and Turns
- Pythagorean Theorem Battleship
- Averages & Variation
- Choices Beginning of Year Math Activity
- Bridges of Oregon
- Rational-ity
- Numbers 1 to 99: Activities
- Grade 8 mathematics
- Number Race 0 to 12
- HiLoTarget Game
- Algebra Lesson Plans and Creative Lessons
- Arithmetic quiz spreadsheet
- Blank Multiplication Chart
- Linear Equation Card Sort
- Trigonometry
- Factor Monster
- Algebra 1A
- Lesson: Serving up Numerators and Denominators
- Team Challenge
- I See The Sine
- Investigation Backpack 03: A One-day Hike
- Investigation Backpack 02: WordsWorth Plus
- Investigation Backpack 02: WordsWorth Plus 1 to 26
- Activity Content
- Problem Solving with Subtraction
- Comparing Fractions
- Comparing Fractions Math Start Power Point Game
- Comparing Fractions with Unlike Denominators
- Modeling Fractions
- Solving two-step equations
- Comparing Fractions with Like Denominators
- NASA Kids' Club
- Popcorn Math Kit
- Intermediate Level Measurement Math Kit
- Geometry Analysis
- Pythagorean theorem and the distance formula
- Paper Crease
- Pick a Shape
- Pie Free Circles
- The Square Problem
- The Goat Problem
- Two Triangles
- The Right Plot
- The Truncated Square
- Square Peg, Round Peg
- Security Cameras
- Cool math Lessons - Calculus - What's a limit
- Optimization Problems
- Cotton Candy
- Target Practice
- How Many Triangles
- Inscribed Rectangle
- edHelper com - Math Review Worksheet
- Parabolas and transformations
- Rigid Transformations
- Parabolas from Points
- Interactive Math Websites for students
- Integrated Algebra sample New York Regents 2007
- Factoring Worksheet
- Exponent Worksheets
- Two Problems
- Mixture Problems
- The Escalator Problem
- The Gears Problem
- Algebra Analysis
- Introduction to exponents
- Venn Diagram Project
- regents strategies
- Balance Problems
- Interactive Math Websites
- Grading Policy Percentage Lesson
- Rates of Growth
- Writing and Comparing Whole Numbers and Decimals Math Stars Power Point Game
- Properties of Real Numbers
- Assessment
- Subtracting Signed Numbers
- Calculating Sample Space and SImple Probabilities
- Theoretical Probability
- Evaluating Theoretical Probability
- Probability of Multiple Events
- Binomial Distribution
- Adopt a Famiily
- Math
- Math Maven's Mysteries
- Figure This! Math Challenge
- King's List of Math Resources
- An Interactive Math Dictionary for Kids
- Cool Math
- Evaluating Variable Expressions
- Comparing Reals
- Math T.V./ Math Playground
- Finishing the Personal Analysis
- Class Party!!!
- Generating a Final Report Spreadsheet
- Gathering and Compiling Sales Data
- Topic 6: Evaluating and presenting results
- Running the Class Business
- Topic 5: Running the business
- Preparing a loan presentation
- Using TI-82/83+ Calculators to Solve Systems
- Solving Systems with the Substitution Method
- Topic 4: Solving systems of linear equations
- 6th Grade Math Vocabulary Game
- Understanding and Applying Slope
- Topic 3: Linear Regression
- Creating and Executing a Market Research Survey
- Linear Regression Analysis
- Writing 2-variables equations
- Graphing 2-variable equations using a table
- Smartboard Activities
- Teaching linear equations through running a small business
- Notes on pacing
- Introduction to the project
- Introduction to Economics
- Topic 2: Graphing 2-variable linear equations
- Evaluating Algebraic Expressions Classwork
- Evaluating Algebraic Expressions Lesson
- HW-order of operations
- Distributive Property
- Property Match Game
- Buy One get X Free
- Circum Circle
- Exploring Rational Functions
- Finding an equation from a graph
- Finding the area of a trapezoid
- Multiplying and Dividing Signed Numbers
- Converting Fractions to Decimals to Percents
- Graphing Quadratics Investigation
- Introduction to 2-variable expressions
- Calculating Sample Space
- Number Detective
- Steps to Solving Word Problems-Primary
- Fun with Fractions
- REVERSE
- Subtraction Poem
- Money - Naming Coins
- Model 6E + S Lesson Plan from Nortel LearniT: Ancient Observatories & Timeless Knowledge
- Patterns, Variables and Functions: Integrated, Project-Based, Science and Mathematics Inquiries for Diverse Fifth and Sixth Grade Learners
- Decimal Basketball
- Pre-Algebra
- Integrated Algebra sample New York Regents 2007 Solutions
- Rendezvous with a Comet Teacher's Guide
- Combinatorics - The Fine Art of Counting
- Coaster Creator
- Comparing Reals
- Area of basic shapes powerpoint
- Geometry Bingo
- Problem Solving with Addition
- National Library of Virtual Manipulatives
- Integers
- Applying Integers
- Integers
- Rounding
- Rounding and Estimation Math Stars
- Rounding and Estimation
- Computation with Addition
- Computation Math Stars
- Teacher Tips
- Problem Solving Math Stars
- Problem Solving
- Factors
- Multiples
- Algebraic Thinking: A Basic Skill
- Math Focal Points - Grade 5
- 2.1 Introduction to Proportions and Percents--Sample Lesson
- Math Focal Points - Grade 6
- Math Focal Points - Grade 7
- 2.1 - 2.6: Unit on Proportional Reasoning, Rates, Ratios and Percents
- Probability
- 2.2 Proportions (continued) and Intro to Unit Rate
- Math Focal Points - Grade 8
- 2-2 Balance with Multiplication
- Classroom Architect
- Math - Place Value
- 1-1 The Linear Pattern
- 1-2 Situations of Constant Change
- 2-1 Balance with Addition
- 2-4 Balancing with Parentheses
- 2-5 Balancing with Parts
- 3-2 Tables, Graphs, and Equations, Oh My!
- 3-3 You have got to start somewhere
- 3-5 Starting from over there
- 4-2 Two Constant Changes
- 4-3 Respect the General
- 4-4 Substituting in the System
- Mathematics, Grades 10-12, Basics
- Mathematics, Chapter 17, Exponents
- Mathematics, Grade 12, Chapters 35-45
- Mathematics
- 2-6 Dealing with Imbalance
- 3-4 Partials in your patterns
- 3.6 Integers: Inquiry and Multiple Representations
- 3.6b Integer Subtraction--Pt. 1
- 3.6b Integer Subtraction Game--Pt. 2
- Algebra I course
- Squares of Numbers
- Tangrams
- Discovering Pi
- Water to the Max
- Significant Figures
- 21 Game
- Block It
- Fact Families
- Facts, Just Gimme the Facts
- Fractions in Everyday Life
- Fractions with Oranges
- Eccentricity of Conic Sections
- Let Us Go Shopping
- Learning Fractions with Pictures
- Learning Fractions with M&M's
- Arithmetic
- Free Standing Structure
- Roman Numerals
- Cube Coloring Problem
- Number Prefixes
- Discovering Pi
- Smile Metric Style
- Probability - The Study of Chance

Reviewed Resources with a score of 2.

This is a lesson for teaching evaluating algebraic expressions. This file does not include the classwork.

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Apply area of trapezoid formula to solve problems in real-life contexts

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Students will measure various objects using an 'inchworm ruler' by lining up the ruler correctly and counting the inches along the given object.

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Students will develop an understanding of ratio, proportion and percent using models. Students will analyze models for data. Students will use software to design and create models representing fractions, decimals, and percents.

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Algebra 1A, of the National Repository of Online Courses.

Unit 1 of 4.

Basic algebra principles

Lesson 1: Real numbers & algebraic expressions Lesson 2: Simplifying expressions Lesson 3: Solving equations Lesson 4: More on solving equations

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level 5 math

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A game to solve equations and build up money up a money ladder

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This test covers rules of exponents, identifying types of polynomials, degrees of polynomials, ascending and descending orders, adding and subracting polynomials and multiplying polynomials.

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Review game for systems of equations

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This collection of lesson plans is created by MIT students taking an Intro to K-12 Math/Science Teaching course. Each lesson consists of a lesson plan for the teacher, and activity sheets for the students.

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This Power Point Presentation is a math practice assessment on Flips, Slides and Turns geared for 3rd grade students.

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It's a battle of wits against the computer. Remember that a²+b²=c²

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A high school probability and statistics unit on averages and variation.

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Blueprints and activities to do with Oregon Bridges

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This collection contains chapters of a textbook about rational functions in high school.

The original open-source educational resource can be found here: http://cnx.org/content/col10350/latest/ .

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This is a unit for teachers about the set of natural numbers 1 to 99. More than 100 questions, activities, and conjectures about natural numbers 1 to 99. Odd numbers, even numbers, prime numbers, composite numbers, factors, sums of factors, and so on, Plus uphill numbers, flat numbers, downhill numbers, palprimes, and emirps.

This is the third unit in a set of three units. The other two units are the reference units:

Natural Numbers 1 to 99

Natural Numbers 1 to 99 -Tables

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Lessons for grade 8 mathematics

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This is the first of several number race games played on a set of "racetracks." In this beginner's game, each player has five racetracks, and puts a "racehorse" on the zero (0) end of each track. Each turn, a player rolls 2D6 (two 6-sided dice) and uses the numbers on the dice to move one or two horses along the track. The object of the game is to get all five horses to twelve (12).

Number Race 0 to 12 describes three game variations.

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The HiLoTarget dice game is a game for two or more players. There are three versions of the game called Try for High, Go for Low, and Hit the Target. You use three polyhedra dice:

A tetrahedron with four sides used to roll 1, 2, 3, or 4 An octahedron with eight sides used to roll 1, 2, 3, 4, 5, 6, 7, or 8. A dodecahedron with 12 sides used to roll 1, 2 ,3, 4, 5, 6, 7, 8, 9, 10, 11, or 12.

A turn consists of three dice rolls. Each player does the following:

Roll #1: Roll all three dice, choose one of the dice, and record the score on a score sheet.

Roll #2: Roll two of the dice, excluding the die chosen on the first roll, choose one of the two dice, and record the score on the score sheet.

Roll #3: Roll the die not choosen on the first two rolls, and record the score on the score sheet.

We usually play four rounds.

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Lesson Plans and Creative Lessons for any Integrated Algebra Course

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Spreadsheet with two tabs

Tab 1 - 40 question quiz in 4 parts (addition, subtraction, multiplication and division w/fractions). Generates questions and answers automatically. Fold under answers on paper.

Tab 2 - Long division calculator carries out calculations for whole numbers.

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This is a blank Multiplication Chart 1-12

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A card sort activity that covers the basic Algebra 1 standards on linear equations. Students are required to sort different representations of linear equations. They also have to decide what goes on additional blank cards.

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Lesson Plans and Creative Lessons for Trigonometry Classroom.

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Factor Monster is our name for a classic game about natural numbers, factors, proper factors, prime numbers, composite numbers, and related algebraic alakazams. A computer version called Taxman appeared in People's Computer Company in 1972. It was also known as The Factor Game and described in the article "The Factor Game" by J.B. Harkin and D.S. Martin in the Arithmetic Teacher, volume 20, pages 580-582 (1973).

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This curriculum emphasizes a multi-representational approach to algebra, with concepts, results, and problems being expressed graphically, analytically, and verbally. It develops algebraic fluency by providing students with the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. In addition, the course develops proficiency with operations involving monomial and polynomial expressions. The main unifying themes of the course include understanding, writing, solving, and graphing linear equations, systems of linear equations and inequalities, quadratic equations, and rational equations. Upon completion of this course you will:

- Perform operations with real numbers
- Simplify and evaluate algebraic expressions
- Use equations to solve word problems
- Graph and solve problems involving inequalities and absolute value
- Graph and solve linear equations
- Solve systems of equations
- Solve many types of real-world problems
- Factor polynomial equations
- Understand relations and functions
- Solve quadratic equations
- Work with radical expressions and rational equations

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Fractions are introduced through the use of fiction, rich media, and hands-on examples. Colored candies are used in this lesson to highlight the concepts of numerator and denominator. The lesson culminates in students creating fractions on their own.

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Ratio, proportion and percent review game suitable for Prealgebra or Algebra I.

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This is an introduction to basic y=sinx, y=cosx graphs and critical values. Using the unit circle special angles and values for sinx we construct the sine curve from the 4 quadrants of the unit circle. "Unravel the mystery of the sine graph"

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Prepare to go on a one-day hike. Shop in camping catalogs or online for the right stuff to wear and carry. Include essential stuff recommended by hiking/camping pros.

**This resource is part of the Investigation Backpack collection. **

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Grab your favorite dictionary and play WordsWorth Plus. We'll show you how to calculate the WordsWorth Plus of a word and suggest a bunch of activities and investigations. As you play, you'll learn about reverses, palindromes, semordnilaps, permutations, and anagrams.

**This resource is part of the Investigation Backpack collection. **

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Grab your favorite dictionary and play WordsWorth. As you play, you'll learn about permutations of words called reverses, palindromes, semordnilaps, and anagrams.

Assign a letter score to each letter in the alphabet, a through z, as follows: a = 1, b = 2, c = 3, and so on, up to z = 26. The WordsWorth of a word is the sum of the letter scores of the word's letters.

Most of the play in WordsWorth is thinking: knowing words, learning more words, and devising strategies for finding answers. People are well equipped to do this type of play. Some of the play is more mundane: looking up letter scores and adding them to get the WordsWorth of a word. First grade students might start with base-10 blocks as their WordsWorth calculator, and then move on to mental math and paper and pencil math as their addition skills improve by playing WordsWorth.

**This resource is part of the Investigation Backpack collection. **

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To help kids understand the difference between tenths and hundredths -- pick ten kids to take ten shots each with a Nerf ball and hoop.

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This centers-based lesson on problem solving with subtraction is designed for two-days of 45-minute lessons or one-day of a 90-minute lesson block. The lesson provides differentiated instruction using the H.O.P. Centers Framework. H.O.P. stands for Hands-On, Open-Ended, and Practice which are the themes for each of the three centers in the framework. Lesson structure is designed using brain research principles. In this lesson students will learn how to solve problems using subtraction with a short engaging lesson, hands-on activity, practice writing about a word problem using multiplication, and engage in tiered practice of the concept.

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This unit provides centers-based differentiated lesson plans that have been structured using brain research principles. In this unit you will find teacher tips to help with the transition to centers-based teaching. Each lesson within the unit comes with: a complete lesson Plan, a lesson presentation for Smart Boards (pdf provided as well), a student packet for recording and assessing student work, files with all teacher-made materials for centers activities, and answer key. This unit should be completed within four 90-minute block classes or eight 45-minute classes.

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This jeopardy-style power point game is designed to review the unit on Comparing Fractions.

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This centers-based lesson on comparing unlike denominators is designed for two-days of 45-minute lessons or one-day of a 90-minute lesson block. The lesson provides differentiated instruction using the H.O.P. Centers Framework. H.O.P. stands for Hands-On, Open-Ended, and Practice which are the themes for each of the three centers in the framework. Lesson structure is designed using brain research principles. In this lesson students will learn how to compare fractions with unlike denominators with a short engaging lesson, hands-on activity in which they explore fractions using sliced lemons, practice writing about a word problem using fractions with unlike denominators, and engage in tiered practice of the concept.

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This centers-based lesson on modeling fractions is designed for two-days of 45-minute lessons or one-day of a 90-minute lesson block. The lesson provides differentiated instruction using the H.O.P. Centers Framework. H.O.P. stands for Hands-On, Open-Ended, and Practice which are the themes for each of the three centers in the framework. Lesson structure is designed using brain research principles. In this lesson students will learn how to model fractions to convert between improper fractions and mixed numbers with a short engaging lesson, hands-on activity in which students use play doh to create different fraction circles, practice writing about a word problem using fractions, and engage in tiered practice of the concept.

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Algebra lesson using other web sites to solve two-step equations.

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This centers-based lesson on comparing fractions with like denominators is designed for two-days of 45-minute lessons or one-day of a 90-minute lesson block. The lesson provides differentiated instruction using the H.O.P. Centers Framework. H.O.P. stands for Hands-On, Open-Ended, and Practice which are the themes for each of the three centers in the framework. Lesson structure is designed using brain research principles. In this lesson students will learn how to compare fractions with like denominators with a short engaging lesson, hands-on activity utilizing play doh, practice writing about a word problem using fractions with like denominators, and engage in tiered practice of the concept.

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NASA Kids' Club has educational games, engaging multimedia and visuals, and educational activities to cover K-4 students' developmental and learning abilities as addressed in national education standards in math, science and technology. The skills levels provide a natural progression through the site that allows users to find games that are best suited to their varying abilities. Developmentally appropriate content is based on national education standards and benchmarks per grade level. Content is written within the K-4 reading levels as determined by the Flesch-Kincaid Grade Level Score, a tool available in Microsoft Word.

To visit the Kids' Club interactive website, click here.

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This unit provides students with the opportunity to conduct mathematical investigations using popcorn. Students will measure length and volume of popped and unpopped kernels along with the mass. Students will compare and contrast their data from three types of popcorn, and calculate the average, ratio, percent, and probability. Temperature readings will be taken and graphed, and scatter and stem and leaf plots will be created to represent student results. The mathematical skills emphasized in Popcorn Math are measuring, collecting data, calculating for average, ratio, percent and probability, and graphing.

This collection is available as downloadable PDFs, but portions of the content are found in wiki format (see WIKI Learning Experiences and Resources) which can be edited or built up with other materials on the same topic by members of the Curriki community.

More About CABOCES

Welcome to the Math, Science, and Technology Department (MST) of Cattaraugus Allegany Board of Cooperative Educational Services (CA BOCES). As a department of Instructional Support Services (ISS), we provide Math, Science, and Technology kits along with support services to meet the needs of the individual districts in our area. The BOCES Teaching and Learning Center at Allegany provides hands-on math, science, and technology kits to over twenty area school districts in Cattaraugus and Allegany counties. Our department provides forty-three different varieties of kit topics. The kits cover topics in math, life science, physical science, chemistry, and earth science. Each kit contains a teacher's manual, student activity book, materials inventory list, instructional video, and materials needed to complete each learning experience. Each kit aligns with the New York State standards and prepares students for the New York State assessments.

Contact Information

BOCES Teaching & Learning Center @ Allegany Math, Science, Technology & Health Dept. 80 N Fourth St, Third Floor Allegany, NY 14706 Phone: (716)-376-8272; Fax: (716)-372-0621

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This unit provides students with the opportunity to measure various objects and use their data to determine area, volume, radius, diameter, and density. They will use Math rulers, protractors, and Vernier Calipers to make their measurements rounding to four different scales. They will measure the radius and diameter of a cylinder and use Pi to calculate the area of various cylindrical objects. Students will measure the circumference and determine the surface area of a sphere and using the thickness of the wall of a hollow sphere, determine the volume inside the sphere as well. The mathematical skills emphasized in Intermediate Level Measurement are measuring, collecting data, and calculating for volume, area, and density.

This collection is available as downloadable PDFs, but portions of the content are found in wiki format (see WIKI Learning Experiences and Resources) which can be edited or built up with other materials on the same topic by members of the Curriki community.

More About CABOCES

Welcome to the Math, Science, and Technology Department (MST) of Cattaraugus Allegany Board of Cooperative Educational Services (CA BOCES). As a department of Instructional Support Services (ISS), we provide Math, Science, and Technology kits along with support services to meet the needs of the individual districts in our area. The BOCES Teaching and Learning Center at Allegany provides hands-on math, science, and technology kits to over twenty area school districts in Cattaraugus and Allegany counties. Our department provides forty-three different varieties of kit topics. The kits cover topics in math, life science, physical science, chemistry, and earth science. Each kit contains a teacher's manual, student activity book, materials inventory list, instructional video, and materials needed to complete each learning experience. Each kit aligns with the New York State standards and prepares students for the New York State assessments.

Contact Information

BOCES Teaching & Learning Center @ Allegany Math, Science, Technology & Health Dept. 80 N Fourth St, Third Floor Allegany, NY 14706 Phone: (716)-376-8272; Fax: (716)-372-0621

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Template for Analysis of Mathematics Programs/Series
Mathematics Core Curriculum
Mathematics Toolkit
Curriculum Guidance Materials & Resources
Template for Analysis of Mathematics Programs/Series

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Students use the pythagorean theorem in order to derive the distance formula. Includes pythagorean theorem practice problems

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A fun geometry problem that takes a little creativity to visualize. Students investigate the length of a crease that is formed when you fold a piece of paper corner to corner.

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Students investigate Picks Theorem, involving the area of an irregular polygon on a lattice point grid. This problem has students look for patterns, try to establish a formula, and test the formula with a more difficult case. The area of a lattice point Pterodactyl.

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A problem involving the area of a crescent. Many students are surprised to discover that their answers do not involve pi.

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A great problem that combines practice in graphing linear functions with a challenging problem involving the area of a square.

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This problem has students investigate the area of a pasture in which a goat would be able to graze.

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Students look at two triangles and must prove whether one line is longer than another. The results are surprising, and there are many different ways to prove the answer to be true.

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A math problem involving the area of an irregular quadrilateral and the Pythagorean Theorem

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Students use the Pythagorean theorem to solve a tricky puzzle involving a square with four isosceles triangles cut from the corners.

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This activity has students investigate which fits better, a square peg in a round hole or a round peg in a square hole.

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Students use the relationship between central angles and inscribed angles in a circle to solve a problem involving the installation of security cameras in an art gallery.

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This lesson will explain the concept of a limit from various points of view.

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A worksheet with five common optimization problems.

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Students use the idea of volume of a cylinder to figure out the length of a single fiber of cotton candy.

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Students find the area of an annulus and find that the result is unexpectedly intuitive.

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This is a fun problem that has students investigate how many triangles they can create out of rods of various lengths. Students use the triangle inequality, and look for a pattern in their solutions.

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Students will explore some classic optimization problems using The Geometers Sketchpad. This lesson has students begin by exploring a pattern, then has them extrapolate their findings into a generalization about inscribed shapes. Extension questions are included.

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Worksheet: Finding the vertices, factoring, solving equations by factoring, midpoint, and system of equations (requires email registration).

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Students will graph various parabolas and compare the graphs to the parabola y = x^2 . They will determine what type of transformation each parabola undergoes to better understand parabolic equations.

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Students investigate horizontal and vertical shifts in quadratic, cubic, square root, and absolute value functions

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Students will find equations of parabolas from three points on the parabola, and then study what happens to the parabola when one of the points is changed slightly.

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This is a list of webpages that are sorted by the lesson for teachers use for review in class.

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This is a practice test for the New York State regents for Integrated Algebra. Solutions are posted under Intergrated Algebra sample New York Regents 2007 Solutions

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Workshhet on Factoring Quadratic equations and difference of squares.

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Exponent Worksheet that includes Scientific Notation

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Two problems that are good for developing number sense and problem solving skills.

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An assortment of good mixture problems to encourage good problem solving skills.

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A puzzle of a problem that challenges students intuition about how an escalator should work. Tricky.

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Students examine a set of cogs and make guesses as to which direction they will turn in and how many times they will rotate. Students use a specific case to make a general statement about any number of rotations.

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Template for Analysis of Mathematics Programs/Series for the state of New York
Mathematics Core Curriculum
Mathematics Toolkit
Curriculum Guidance Materials & Resources
Template for Analysis of Mathematics Programs/Series

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This lesson explores the famous question of whether it would be better to recive one lump sum of money or a small amount that is doubled many times in order to introduce exponents.

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Students will make both qualitative and Quantitative Venn Diagrams

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Introduces backsolving and pick a number strategies to solve regents problems

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Students are given a worksheet to build skills that will help with solving equations and systems of equations. This sheet involves no numbers, just symbols, so that students are not distracted by the nomenclature of algebraic expressions, and can focus on building equation solving intuition. The columns of problems get increasingly more challenging as they go on. It is suggested that the students work on the more difficult problems in pairs or small groups.

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This resource is a list sorted by subject and individual lessons that include web pages for students to obtain extra practice.

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This is just a worksheet. For the rest of the lesson I do a review of basic percentages, then go over the grading policy, and finally give this worksheet out for the end of class and the homework.

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A collection of problems to encourage students to explore the differences between types of growth, including linear and exponential growth.

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This jeopardy-style review game is designed for review of the unit on writing and comparing whole numbers and decimals.

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Lesson 3

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Quiz and Test options for the Numbers, Operations, and Properties Unit.

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This is a lesson to teach students to subtract signed numbers. See other file that contains worksheet for students to practice.

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This worksheet will allow students to practice calculating the sample space of independent events. Simple probabilities follow the calculations.

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This folder contains lesson plans that explore the concept of theoretical probability.

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This worksheet is a good way to initiate a conversation about what probability actually means. It will help students conceptualize when things are more or less likely in relation to each other, and give them the opportunity to think about what they would do in certain situations.

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This folder includes lessons concerning probability of independent events.

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Uses a visual and hands-on approach to understanding what a Bernoulli trial is and what you would use a binomial distribution for.

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The theme of this activity is decimals, research, and collaboration.

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Resources for teaching math.

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Each Math Maven Mystery focuses on a particular area of math reasoning and computation that requires students to use creative thinking and sharp math skills to "crack the case" and help Math Maven find the solution and solve the mystery.

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Real World Math Challenges

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List of math resources to use in classroom.

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An interactive math dictionary for kids.

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Great resource for math games, and a lot more.

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Lesson 2

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Lesson 4.

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Math TV - interactive video word problems, Math Playground - online math games, Thinking Blocks - interactive math tool developed by classroom teachers to help students learn how to solve multistep word problems.

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In this lesson, students engage in reflection on the project itself and on their role in the project.

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In this lesson, students receive the fruits of their labors! :)

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This lesson focuses on helping students learn some of the more dynamic functionalities of Microsoft Excel through analyzing their own data. It may require 2 or 3 45-minute class periods to complete.

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In this lesson, students compile all of the data generated by the actual running of their business. This data will be used in the next two lessons to analyze their actual effectiveness and to prepare an Annual Report.

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In this final sub-unit, students gather and analyze the data generated by this project (number of units sold, revenue, profit, etc.). This sub-unit also includes a heavy focus on technology, as students generate Excel spreadsheets to report on their results and use some of the dynamic functionalities of the program. Similarly, they employ Microsoft Word to create an "Annual Report" for their business. The unit ends with a class party, where they use their proceeds to provide food!

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This is just a "place holder" lesson plan describing some of the key elements to watch for while executing the business. The actual business should run for 2 weeks. This is actually an ideal time to begin another brief unit or do review for a different subject.

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This is less a sub-unit than just a place-holder in the scope and sequence to allow for the actual execution of the project. It is a good time to introduce a new unit or to review information/prepare for tests, etc.

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In this lesson, students work collaboratively to prepare a loan proposal presentation for the school principal, using the data they have gathered for their project. This lesson will take several 45-minute class periods, depending upon how much outside work the students are assigned. It also requires some previous communication with a member of the administration to schedule their appearance in class.

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This lesson exposes students to a new functionality of the TI-82/83+ calculator: solving systems of equations. Students practice solving systems using provided examples and then verifying their own data.

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This lesson addresses the substitution method of solving systems of equations. The students are introduced to the method through the use of a "real-world" problem.

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In this sub-unit, students will be introduced to the concept of systems of linear equations and their solutions. They will learn to solve these systems using accurate graphing, substitution, and linear combination. The concept will be introduced in the context of a business model (cost and revenue curves) and the need to determine how many units of a product will need to be sold before the business becomes profitable. Students will then conduct their own research into cost and revenue and solve systems of equations that they themselves have derived.

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This game is similar to Jeopardy, but it is dealing with 6th grade math vocabulary.

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This lesson introduces the notion of slope through intuitive, geometric, and Algebraic means. This lesson may take 2 (or even 3) class periods to allow for full student comprehension, as slope is traditionally one of the most difficult-to-acquire concepts in the Algebra curriculum.

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In this sub-unit, students are introduced to the concept of linear regression and determining the equation of a function in 2-variables from data, using a linear model. They will develop and conduct a "market survey", in order to use the data to perform their own regression. There is a strong emphasis on the use of technology in this sub-unit.

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In this lesson, students will identify key questions that they will need answers to in order to have the necessary information to plan their business. They will design an appropriate survey and administer this survey to other students at the school. Finally, they will compile the results into data sets in order to analyze them.

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This lesson exposes students to the method of linear regression for determining the equation for a linear model of a data set. It progresses from a more intuitive understanding of the method to using technology to generate an equation using the least-squares method (students will not be exposed to the actual least squares method).

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In this lesson, students will learn how to translate situations in real-life into 2-variable equations and generate different ordered pair solutions.

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This lesson introduces the notion of graphing 2-variable equations by making use of a function table. It uses examples from the format of the project in order to solidify the concepts of cost and revenue.

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A list of Smart board (links) activities for all sujects.

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This is an 8-week unit of instruction for an Algebra I course that covers all the generally required material for the topics of graphing linear equations, solving systems of linear equations, simple linear regression, and applying systems of linear equations to real-life situations. The unit is constructed using a Project-Based Learning approach, where the students work collaboratively to run a class business during the 8 weeks. As they move through the various stages of planning and executing the business, they learn various concepts related to linear equations and how they directly apply to predicting demand or cost/revenue information. Finally, the project incoporates technology education by having the students use Microsoft Excel and Powerpoint to report and analyze the sales results of their business.

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These are some general notes on pacing for this project.

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This lesson is designed to introduce students to the project that they will be performing over the next 8 weeks. In it, they will be given a checklist for the portfolio that they will create, as well as the rubric by which that portfolio will be graded.

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In this lesson, the teacher introduces fundamental concepts of economics to the class. This should create a "working vocabulary" for the class for the remainder of the project.

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In this sub-unit, students are introduced to 2-variable linear equations. They learn to graph these equations usinig a variety of methods and to determine the equation of a line from its graph. The bulk of the concepts in this unit will be introduced through the vocabulary of the economics study in Topic 1 and practiced in the context of the students' business.

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This file contains a worksheet for student to practice evaluating algebraic expressions.

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This is a lesson for teaching evaluating algebraic expressions. This file does not include the classwork.

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This is a homework assignment for students to practice order of operations.

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This activity helps students to discover the distributive property. It also has students explore the uses and usefulness of the distributive property.

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This activity helps students match definitions with the property they describe.

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This activity deals with the fundamental notion of a limit. The activity has students analyze a real world business scenario and to explore limits by looking at trends in data.

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A nice problem that encourages students to think backwards. Instead of inscribing a triangle in a circle, the triangle is given, and the student must discover the diameter of the circle.

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This lesson has students compare and contrast differences between rational functions to understand vertical asymptotes, horizontal asymptotes, and removable and non removable discontinuities.

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This lesson focuses on "working backwards" from a given graph to derive its equation. This is a prerequisite skill for linear regression, which will be taught in the next sub-unit.

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Students cut a parallellogram and then tape it to form a trapezoid. They then answer a series of questions which enable them to discover the area of a trapezoid.

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This lesson teaches students to multiply and divide signed numbers. See other file that contains class work that goes with lesson.

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This activity will help students remember how to convert a fraction into a decimal and then into a percent.

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This activity starts with students graphing y equals x squared on the calculator and then moving on to shifting, etc.

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In this lesson, students are introduced to the concept of a 2-variable linear expression. They learn to create these expressions by defining two variables representing different kinds of quantities, and they learn to evaluate these expressions.

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This file includes the following methods of calculating sample space: tree diagrams, multiplication counting principle, permutations, and combinations.

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The students have to figure the missing multipication number to figure out the number riddle

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Great "recipe" for primary students to assist with remembering the steps to solve addition/subtraction word problems.

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This unit will introduce the concept of fractions with activities and lessons that cater to visual, auditory, and tactile learning styles. Students will create fractions with a variety of manipulatives, solve problems with fractions, play games with fractions, and explore fractions in their everyday lives. The unit will also integrate language arts, as students write fraction stories and read literature related to fractions.

Students will gain an understanding of basic fractions, including 1/2, 1/3, 1/4, and whole. Students will learn key vocabulary words: whole, fraction, numerator, and denominator. Students will understand how fractions relate to their everyday lives.

Lessons include:

- Lesson #1: Introduce Whole and One-Half
- Lesson #2: More about One-Half
- Lesson #3: Practice with One-Half
- Lesson #4: Introduce Numerator and Denominator
- Lesson #5: Going Beyond One-Half
- Lesson #6: Making Fractions I
- Lesson #7: Making Fractions II
- Lesson #8: Comparing Fractions I
- Lesson #9: Comparing Fractions II
- Lesson #10: Fraction Bingo
- Lesson #11: Fraction Matching Game
- Lesson #12: Writing a Fraction Story I
- Lesson #13: Writing a Fraction Story II

Unit Resources include:

- Vocabulary Cards
- Problems of the Day
- Pizza Cut-Out Sheet
- One-Half Practice Sheet
- Candy Activity Sheet
- Candy Fractions Activity Sheet
- Clay Fractions Activity Sheet
- Comparing Fractions Activity Sheet
- Fraction Circles
- Bingo Caller Cars
- Bingo Game Board
- Fraction Matching Game
- Pre-Writing Graphic Organizer

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REVERSE was invented by Peter Lynn Sessions and published in People's Computer Company, May 1973. Begin with a list of numbers and try to put them in order with the smallest number on the left and the largest number on the right, according to the rules of the game.

REVERSE describes the standard game and suggests several variations

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This poem will assist students in remembering the rules for regrouping with double - multi digit numbers. It is colorful and lends itself to wall decor.

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A cute poem for each coin - quarter, dime, nickel, and penny accompanied by large easy to see images of each coin.

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This resource was created using the Nortel LearniT 6E + S lesson plan template. It addresses timekeeping using angles and the sun.

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Project-based learning stations that help students bridge observations to algebraic concepts through semester-long inquiries (each inquiry takes about a week). Once the class has completed the eight inquiries, the teacher may guide them through the inquiries again, exploring new patterns, variables and functions at a more complex level.

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To help kids understand the difference between tenths and hundredths -- pick ten kids to take ten shots each with a Nerf ball and hoop.

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Lesson Plans and Creative Lessons for Pre-Algebra Course.

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These are the solutions to the previously posted Integrated Algebra sample New York Regents 2007.

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The Rendezvous with a Comet Teacher's Guide has curriculum and instructional materials for implementing a 4-6 week unit associated with the Challenger Learning Center mission by the same name. This unit provides for 3-4 weeks of pre-mission activities, and for several days of post-mission activities. Whole chapters of the Teacher's Guide may be downloaded as PDF's by clicking chapter titles; sections of each chapter may be downloaded individually by clicking section headings.

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Love math but bored in math class? This is the course for you! Combinatorics is a fascinating branch of mathematics that applies to problems ranging from card games to quantum physics to the internet. The only pre-requisite is basic algebra; however we will be covering a lot of material. A mathematically agile mind will be helpful.

This is an open-source educational resource found online. Citation as follows:

This is an open-source educational resource found online. Citation as follows:

Andrew Sutherland, Combinatorics - The Fine Art of Counting, Summer 2007. (Massachusetts Institute of Technology: MIT OpenCouseWare), http://ocw.mit.edu/OcwWeb/hs/com/com/index.htm (Accessed 2 October 2008). License: Creative Commons BY-NC-SA

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Coaster Creator is an interactive game that explores the science behind roller coasters. Use your knowledge of potential energy and kinetic energy to design your own coaster. Rapid energy transfer is the key to roller coaster thrills but be careful — too much kinetic energy and you’ll crash, too little and you'll stall. Play this beta version of the game and find out if your coaster will be a smash or a snore!

http://www.jason.org/PublicPage/Curriculum/CoasterCreator.aspx

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Lesson 4 (including rational vs irrational)

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This powerpoint presentation gives a graphical description of the area formulas for squares, rectangles, triangles, and trapezoids.

For use with middle years or high school grades.

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This is a bingo game involving visuals to help students learn and identify key vocabulary for geometry on the 5th and 6th grade levels.

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This centers-based lesson on problem solving with addition is designed for two-days of 45-minute lessons or one-day of a 90-minute lesson block. The lesson provides differentiated instruction using the H.O.P. Centers Framework. H.O.P. stands for Hands-On, Open-Ended, and Practice which are the themes for each of the three centers in the framework. Lesson structure is designed using brain research principles. In this lesson students will learn how to solve problems using addition with a short engaging lesson, hands-on activity, practice writing about a word problem using addition, and engage in tiered practice of the concept.

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Interactive and online virtual manipulatives for mathematics.

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This centers-based lesson on integers is designed for two-days of 45-minute lessons or one-day of a 90-minute lesson block. The lesson provides differentiated instruction using the H.O.P. Centers Framework. H.O.P. stands for Hands-On, Open-Ended, and Practice which are the themes for each of the three centers in the framework. Lesson structure is designed using brain research principles. In this lesson students will learn how to locate integers on a number line with a short engaging lesson, hands-on activity in which students use their sense of smell to sort numbers on a number line, practice writing about a word problem using integers, and engage in tiered practice of the concept.

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This centers-based lesson on applying integers is designed for two-days of 45-minute lessons or one-day of a 90-minute lesson block. The lesson provides differentiated instruction using the H.O.P. Centers Framework. H.O.P. stands for Hands-On, Open-Ended, and Practice which are the themes for each of the three centers in the framework. Lesson structure is designed using brain research principles. In this lesson students will learn how to use integers to describe real work situations with a short engaging lesson, hands-on activity in which they experiment with temperature, practice writing about a word problem using integers, and engage in tiered practice of the concept.

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This unit provides centers-based differentiated lesson plans that have been structured using brain research principles. In this unit you will find teacher tips to help with the transition to centers-based teaching. Each lesson within the unit comes with: a complete lesson Plan, a lesson presentation for Smart Boards (pdf provided as well), a student packet for recording and assessing student work, files with all teacher-made materials for centers activities, and answer key. This unit should be completed within three 90-minute block classes or six 45-minute classes.

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This centers-based lesson on rounding is designed for two-days of 45-minute lessons or one-day of a 90-minute lesson block. The lesson provides differentiated instruction using the H.O.P. Centers Framework. H.O.P. stands for Hands-On, Open-Ended, and Practice which are the themes for each of the three centers in the framework. Lesson structure is designed using brain research principles. In this lesson students will learn how to round with a short engaging lesson, hands-on kinesthetic activity, practice writing about a word problem using rounding, and engage in tiered practice of the concept.

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This Power Point game is designed for the review and informal assessment of the Rounding and Estimation Unit.

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This unit provides centers-based differentiated lesson plans that have been structured using brain research principles. In this unit you will find teacher tips to help with the transition to centers-based teaching. Each lesson within the unit comes with: a complete lesson plan, a lesson presentation for Smart Boards (pdf provided as well), a student packet for recording and assessing student work, files with all teacher-made materials for centers activities, and answer key. This unit should be completed within three 90-minute block classes or six 45-minute classes.

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This centers-based lesson on computation with addition is designed for two-days of 45-minute lessons or one-day of a 90-minute lesson block. The lesson provides differentiated instruction using the H.O.P. Centers Framework. H.O.P. stands for Hands-On, Open-Ended, and Practice which are the themes for each of the three centers in the framework. Lesson structure is designed using brain research principles. In this lesson students will learn how to compute using addition with a short engaging lesson, hands-on sensory activity, practice writing about a word problem using computation, and engage in tiered practice of the concept.

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This Power Point Game is designed for review and informal assessment of the Computation Unit.

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This folder contains tips for teachers implementing lessons designed with the H.O.P. Centers Framework.

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This Power Point game is designed for the review and informal assessment of the Problem Solving Unit.

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This unit provides centers-based differentiated lesson plans that have been structured using brain research principles. In this unit you will find teacher tips to help with the transition to centers-based teaching. Each lesson within the unit comes with: a complete lesson plan, a lesson presentation for Smart Boards (pdf provided as well), a student packet for recording and assessing student work, files with all teacher-made materials for centers activities, and answer key. This unit should be completed within five 90-minute block classes or ten 45-minute classes.

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This centers-based lesson on factors is designed for two-days of 45-minute lessons or one-day of a 90-minute lesson block. The lesson provides differentiated instruction using the H.O.P. Centers Framework. H.O.P. stands for Hands-On, Open-Ended, and Practice which are the themes for each of the three centers in the framework. Lesson structure is designed using brain research principles. In this lesson students will learn how to find the factors of a number with a short engaging lesson,in which they regroup counters to find different factors of a number, practice writing about a word problem using factors, and engage in tiered practice of the concept.

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This centers-based lesson on multiples is designed for two-days of 45-minute lessons or one-day of a 90-minute lesson block. The lesson provides differentiated instruction using the H.O.P. Centers Framework. H.O.P. stands for Hands-On, Open-Ended, and Practice which are the themes for each of the three centers in the framework. Lesson structure is designed using brain research principles. In this lesson students will learn how to find the multiples of a number with a short engaging lesson, hands-on activity in which students group counters to create their own multiples, practicing writing about a word problem using multiples, and engage in tiered practice of the concept.

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This resource guide provides links to exemplary resources and insight on how to teach mathematics and science concepts at the middle school level. Concepts supported by this particular guide include decimals, fractions, division of whole numbers, and geometry.

The guides provide information on the needed content knowledge, science and mathematical pedagogical knowledge, exemplary lessons and activities, career information, and correlations to national mathematics and science standards.

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This resource guide provides links to exemplary resources and insight on how to teach mathematics and science concepts at the middle school level. The guides provide information on the needed content knowledge, science and mathematical pedagogical knowledge, exemplary lessons and activities, career information, and correlations to national mathematics and science standards.

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Introduces proportions and percents through a pictorial model, building up to the Equivalent Fractions method of solving proportions. Comes with a companion PowerPoint presentation to help implement the lesson. Fully editable.

A mix of direct instruction and small-group inquiry.

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This resource guide provides links to exemplary resources and insight on how to teach mathematics and science concepts at the middle school level. Concepts supported by this particular guide include decimals, fractions, division of whole numbers, and geometry.

The guides provide information on the needed content knowledge, science and mathematical pedagogical knowledge, exemplary lessons and activities, career information, and correlations to national mathematics and science standards.

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This resource guide provides links to exemplary resources and insight on how to teach mathematics and science concepts at the middle school level. Concepts supported by this particular guide include decimals, fractions, division of whole numbers, and geometry.

The guides provide information on the needed content knowledge, science and mathematical pedagogical knowledge, exemplary lessons and activities, career information, and correlations to national mathematics and science standards.

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An entire unit of activities for proportional reasoning (including percents). Designed for classes that need to re-teach a lot of middle school content while also meeting high school Algebra I objectives. Also a great resource for middle school teachers. The approach mixes direct instruction with guided inquiry and real-life applications. Students won't get lost, but they will construct some of the concepts themselves and see how the material applies to their lives.

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This folder contains all of the subtopics encompassed in the third unit of our high school Algebra curriculum.

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In teaching proportions, this lesson shifts students from 2.1's pictorial approach to 2.2's computational approach of equivalent fractions. With deeper comprehension, students move from the concrete (pictures) to the abstract (an algorithm).

After some differentiated drill practice, the lesson introduces the concept of Unit Rates. A mix of direct instruction and small-group inquiry.

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This lesson develops an intuitive and reason-based approach to the method of solving one-step linear equations using the multiplicative property of equality.

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This tool provides an opportunity for experimentation with the layout of your classroom without any heavy lifting. It can be used by teachers to plan their classroom, or it could be used for studying area and perimeter or writing a story.

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Resources for teaching place value.

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This algebra 1 course begins with a unit of linear equations and applications, with the first section being an examination of linear patterns. There are no equations, graphs, or tables… just real scenarios presented in picture form that require experimentation with linear patterns.

The course is meant to be investigative and for students to construct strategies that will be expanded upon and refined throughout the coursework and this section is exactly that.

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An examination of a trip and various linear relationships within it, including distance, speed, fuel efficiency, and budgeting.

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This lesson develops an intuitive and reason-based approach to the method of solving one-step linear equations using the additive property of equality.

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This lesson develops an intuitive and reason-based approach to the method of solving linear equations that contain parentheses by examining the distributive property.

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This lesson develops an intuitive and reason-based approach to the method of solving multi-step linear equations that contain parentheses, percentages, and decimals.

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We have studied numeric and analytic representations, now we are going to expand to graphical representations and the changing of one representation to another.

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We will explicitly de?ne slope-intercept form. We have already examined slope, y-intercepts, and graphing from tables, now we are putting all of that together.

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This lesson we will be examining the point-slope form in algebraic and application situations.

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Section 2 examines linear equations with x’s on both sides of the equation and introduces
the intersection of two lines.

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Section 3 examines general form of a line and solving for a variable in a formula. This
lesson will give us another form of a line and also the skills needed to solve linear system
applications.

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Section 4 examines linear system applications and solving by substitution.

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This folder contains 1 chapter from the FHSST (Free High School Science Texts) Mathematics textbook. FHSST is a project that aims to provide free science and mathematics textbooks for Grades 10 to 12 science learners. The project was initiated by young South African scientists, and now brings together scientists from around the world who are willing to contribute to the writing of the books.

This basic chapter is to be used in grades 10 through 12.

**This resource is part of the FHSST Mathematics collection.**

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**Click HERE to download the chapter as a PDF.**

Chapter 17, Exponents comes from the FHSST (Free High School Science Texts) Mathematics textbook. FHSST is a project that aims to provide free science and mathematics textbooks for Grades 10 to 12 science learners. The project was initiated by young South African scientists, and now brings together scientists from around the world who are willing to contribute to the writing of the books.

Below is a complete Table of Contents for this chapter.

**This resource is part of the FHSST Mathematics collection.**

**CONTENTS**

**III GRADE 11**

**Chapter 17: Exponents - Grade 11**

17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

17.2 Laws of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

17.2.1 Exponential Law 7 . . . . . . . . . . . . . . . . . . . . . . 251

17.3 Exponentials in the Real-World . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

17.4 End of chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

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This folder contains 11 chapters from the FHSST (Free High School Science Texts) Mathematics textbook. FHSST is a project that aims to provide free science and mathematics textbooks for Grades 10 to 12 science learners. The project was initiated by young South African scientists, and now brings together scientists from around the world who are willing to contribute to the writing of the books.

Below is a complete Table of Contents for this section.

**This resource is part of the FHSST Mathematics collection.**

**CONTENTS**

**IV GRADE 12**

**Chapter 35: Logarithms - Grade 12**

35.1 Definition of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

35.2 Logarithm Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

35.3 Laws of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

35.4 Logarithm Law 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

35.5 Logarithm Law 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 448

35.6 Logarithm Law 3 . . . . . . . . . . . . . . . . . 448

35.7 Logarithm Law 4 . . . . . . . . . . . . . . . . . 449

35.8 Logarithm Law 5:. . . . . . . . . . . . . . . . . . . . . . . 450

35.9 Logarithm Law 6: . . . . . . . . . . . . . . . . . . . . . . 450

35.10Solving simple log equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

35.10.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

35.11 Logarithmic applications in the Real World . . . . . . . . . . . . . . . . . . . . . 454

35.11.1Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

35.12 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

**Chapter 36: Sequences and Series - Grade 12**

36.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

36.2 Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

36.2.1 General Equation for the nth-term of an Arithmetic Sequence . . . . . . 458

36.3 Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

36.3.1 Example - A Flu Epidemic . . . . . . . . . . . . . . . . . . . . . . . . . 459

36.3.2 General Equation for the nth-term of a Geometric Sequence . . . . . . . 461

36.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

36.4 Recursive Formulae for Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 462

36.5 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

36.5.1 Some Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

36.5.2 Sigma Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

36.6 Finite Arithmetic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

36.6.1 General Formula for a Finite Arithmetic Series . . . . . . . . . . . . . . . 466

36.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

36.7 Finite Squared Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

36.8 Finite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

36.8.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470

36.9 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

36.9.1 Infinite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . 471

36.9.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

36.10End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

**Chapter 37: Finance - Grade 12**

37.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

37.2 Finding the Length of the Investment or Loan . . . . . . . . . . . . . . . . . . . 477

37.3 A Series of Payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

37.3.1 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

37.3.2 Present Values of a series of Payments . . . . . . . . . . . . . . . . . . . 479

37.3.3 Future Value of a series of Payments . . . . . . . . . . . . . . . . . . . . 484

37.3.4 Exercises - Present and Future Values . . . . . . . . . . . . . . . . . . . 485

37.4 Investments and Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

37.4.1 Loan Schedules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

37.4.2 Exercises - Investments and Loans . . . . . . . . . . . . . . . . . . . . . 489

37.4.3 Calculating Capital Outstanding . . . . . . . . . . . . . . . . . . . . . . 489

37.5 Formulae Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

37.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

37.5.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

37.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

**Chapter 38: Factorising Cubic Polynomials - Grade 12**

38.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

38.2 The Factor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

38.3 Factorisation of Cubic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 494

38.4 Exercises - Using Factor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 496

38.5 Solving Cubic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

38.5.1 Exercises - Solving of Cubic Equations . . . . . . . . . . . . . . . . . . . 498

38.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

**Chapter 39: Functions and Graphs - Grade 12**

39.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

39.2 Definition of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

39.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

39.3 Notation used for Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

39.4 Graphs of Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

39.4.1 Inverse Function of y = ax + q . . . . . . . . . . . . . . . . . . . . . . . 503

39.4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

39.4.3 Inverse Function of y = ax2 . . . . . . . . . . . . . . . . . . . . . . . . 504

39.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

39.4.5 Inverse Function of y = ax . . . . . . . . . . . . . . . . . . . . . . . . . 506

39.4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

39.5 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

**Chapter 40: Differential Calculus - Grade 12**

40.1 Why do I have to learn this stuff? . . . . . . . . . . . . . . . . . . . . . . . . . 509

40.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

40.2.1 A Tale of Achilles and the Tortoise . . . . . . . . . . . . . . . . . . . . . 510

40.2.2 Sequences, Series and Functions . . . . . . . . . . . . . . . . . . . . . . 511

40.2.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

40.2.4 Average Gradient and Gradient at a Point . . . . . . . . . . . . . . . . . 516

40.3 Differentiation from First Principles . . . . . . . . . . . . . . . . . . . . . . . . . 519

40.4 Rules of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

40.4.1 Summary of Differentiation Rules . . . . . . . . . . . . . . . . . . . . . . 522

40.5 Applying Differentiation to Draw Graphs . . . . . . . . . . . . . . . . . . . . . . 523

40.5.1 Finding Equations of Tangents to Curves . . . . . . . . . . . . . . . . . 523

40.5.2 Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

40.5.3 Local minimum, Local maximum and Point of Inflextion . . . . . . . . . 529

40.6 Using Differential Calculus to Solve Problems . . . . . . . . . . . . . . . . . . . 530

40.6.1 Rate of Change problems . . . . . . . . . . . . . . . . . . . . . . . . . . 534

40.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

**Chapter 41: Linear Programming - Grade 12**

41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

41.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

41.2.1 Feasible Region and Points . . . . . . . . . . . . . . . . . . . . . . . . . 539

41.3 Linear Programming and the Feasible Region . . . . . . . . . . . . . . . . . . . 540

41.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546

**Chapter 42: Geometry - Grade 12**

42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

42.2 Circle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

42.2.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

42.2.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

42.2.3 Theorems of the Geometry of Circles . . . . . . . . . . . . . . . . . . . . 550

42.3 Co-ordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566

42.3.1 Equation of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566

42.3.2 Equation of a Tangent to a Circle at a Point on the Circle . . . . . . . . 569

42.4 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

42.4.1 Rotation of a Point about an angle ? . . . . . . . . . . . . . . . . . . . . 571

42.4.2 Characteristics of Transformations . . . . . . . . . . . . . . . . . . . . . 573

42.4.3 Characteristics of Transformations . . . . . . . . . . . . . . . . . . . . . 573

42.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574

**Chapter 43: Trigonometry - Grade 12**

43.1 Compound Angle Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577

43.1.1 Derivation of sin(? + ?) . . . . . . . . . . . . . . . . . . . . . . . . . . 577

43.1.2 Derivation of sin(? ? ?) . . . . . . . . . . . . . . . . . . . . . . . . . . 578

43.1.3 Derivation of cos(? + ?) . . . . . . . . . . . . . . . . . . . . . . . . . . 578

43.1.4 Derivation of cos(? ? ?) . . . . . . . . . . . . . . . . . . . . . . . . . . 579

43.1.5 Derivation of sin 2? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

43.1.6 Derivation of cos 2? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

43.1.7 Problem-solving Strategy for Identities . . . . . . . . . . . . . . . . . . . 580

43.2 Applications of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 582

43.2.1 Problems in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 582

43.2.2 Problems in 3 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 584

43.3 Other Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

43.3.1 Taxicab Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

43.3.2 Manhattan distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

43.3.3 Spherical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

43.3.4 Fractal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588

43.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

**Chapter 44: Statistics - Grade 12**

44.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

44.2 A Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

44.3 Extracting a Sample Population . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

44.4 Function Fitting and Regression Analysis . . . . . . . . . . . . . . . . . . . . . . 594

44.4.1 The Method of Least Squares . . . . . . . . . . . . . . . . . . . . . . . 596

44.4.2 Using a calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597

44.4.3 Correlation coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

44.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600

**Chapter 45: Combinations and Permutations - Grade 12**

45.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

45.2 Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

45.2.1 Making a List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

45.2.2 Tree Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

45.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

45.3.1 The Factorial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

45.4 The Fundamental Counting Principle . . . . . . . . . . . . . . . . . . . . . . . . 604

45.5 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

45.5.1 Counting Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

45.5.2 Combinatorics and Probability . . . . . . . . . . . . . . . . . . . . . . . 606

45.6 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

45.6.1 Counting Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

45.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

45.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

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This collection is a full course of material in the form of a textbook. The textbook is provided by FHSST (Free High School Science Texts). FHSST is a project that aims to provide free science and mathematics textbooks for Grades 10 to 12 science learners. The project was initiated by young South African scientists, and now brings together scientists from around the world who are willing to contribute to the writing of the books.

The FHSST Mathematics textbook contains a total of 45 chapters to be used in grades 10, 11, and 12. At the end of this description is a complete Table of Contents.

The textbook is broken into 5 sections: basics (Chapter 1), Grade 10 (Chapters 2-16), Grade 11 (Chapters 17-34), Grade 12 (Chapters 35-45), and Exercises. In this collection, you will find folders for each of these sections and chapters are found within.

**CONTENTS**

**I BASICS**

**Chapter 1: Introduction to Book**

1.1 The Language of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

**II GRADE 10**

**Chapter 2: Review of Past Work**

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 What is a number? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Letters and Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.6 Multiplication and Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.7 Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.8 Negative Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.8.1 What is a negative number? . . . . . . . . . . . . . . . . . . . . . . . . 10

2.8.2 Working with Negative Numbers . . . . . . . . . . . . . . . . . . . . . . 11

2.8.3 Living Without the Number Line . . . . . . . . . . . . . . . . . . . . . . 12

2.9 Rearranging Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.10 Fractions and Decimal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.11 Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.12 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.12.1 Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.12.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.12.3 Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.12.4 Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.13 Mathematical Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.14 Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.15 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

**Chapter 3: Rational Numbers - Grade 10**

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 The Big Picture of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Forms of Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Converting Terminating Decimals into Rational Numbers . . . . . . . . . . . . . 25

3.6 Converting Repeating Decimals into Rational Numbers . . . . . . . . . . . . . . 25

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.8 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

**Chapter 4: Exponentials - Grade 10**

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Laws of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3.1 Exponential Law 1 . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 30

4.3.2 Exponential Law 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3.3 Exponential Law 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.4 Exponential Law 4. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . 32

4.3.5 Exponential Law 5. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 32

4.3.6 Exponential Law 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

**Chapter 5: Estimating Surds - Grade 10**

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Drawing Surds on the Number Line (Optional) . . . . . . . . . . . . . . . . . . 38

5.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

**Chapter 6: Irrational Numbers and Rounding Off - Grade 10**

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.2 Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.3 Rounding Off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

**Chapter 7: Number Patterns - Grade 10**

7.1 Common Number Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7.1.1 Special Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7.2 Make your own Number Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.3.1 Patterns and Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

**Chapter 8: Finance - Grade 10**

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

8.2 Foreign Exchange Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

8.2.1 How much is R1 really worth? . . . . . . . . . . . . . . . . . . . . . . . 53

8.2.2 Cross Currency Exchange Rates . . . . . . . . . . . . . . . . . . . . . . 56

8.2.3 Enrichment: Fluctuating exchange rates . . . . . . . . . . . . . . . . . . 57

8.3 Being Interested in Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

8.4 Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

8.4.1 Other Applications of the Simple Interest Formula . . . . . . . . . . . . . 61

8.5 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

8.5.1 Fractions add up to the Whole . . . . . . . . . . . . . . . . . . . . . . . 65

8.5.2 The Power of Compound Interest . . . . . . . . . . . . . . . . . . . . . . 65

8.5.3 Other Applications of Compound Growth . . . . . . . . . . . . . . . . . 67

8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

8.6.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

8.6.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

8.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

**Chapter 9: Products and Factors - Grade 10**

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

9.2 Recap of Earlier Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

9.2.1 Parts of an Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

9.2.2 Product of Two Binomials . . . . . . . . . . . . . . . . . . . . . . . . . 71

9.2.3 Factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

9.3 More Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

9.4 Factorising a Quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

9.5 Factorisation by Grouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

9.6 Simplification of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

9.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

**Chapter 10 Equations and Inequalities - Grade 10**

10.1 Strategy for Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

10.2 Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

10.3 Solving Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

10.4 Exponential Equations of the form ka(x+p) = m . . . . . . . . . . . . . . . . . . 93

10.4.1 Algebraic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

10.5 Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

10.6 Linear Simultaneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

10.6.1 Finding solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

10.6.2 Graphical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

10.6.3 Solution by Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . 101

10.7 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

10.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

10.7.2 Problem Solving Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 104

10.7.3 Application of Mathematical Modelling . . . . . . . . . . . . . . . . . . 104

10.7.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 106

10.8 Introduction to Functions and Graphs . . . . . . . . . . . . . . . . . . . . . . . 107

10.9 Functions and Graphs in the Real-World . . . . . . . . . . . . . . . . . . . . . . 107

10.10Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

10.10.1Variables and Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

10.10.2Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

10.10.3The Cartesian Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

10.10.4Drawing Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

10.10.5Notation used for Functions . . . . . . . . . . . . . . . . . . . . . . . . 110

10.11Characteristics of Functions - All Grades . . . . . . . . . . . . . . . . . . . . . . 112

10.11.1Dependent and Independent Variables . . . . . . . . . . . . . . . . . . . 112

10.11.2Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

10.11.3 Intercepts with the Axes . . . . . . . . . . . . . . . . . . . . . . . . . . 113

10.11.4Turning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

10.11.5Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

10.11.6Lines of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

10.11.7 Intervals on which the Function Increases/Decreases . . . . . . . . . . . 114

10.11.8Discrete or Continuous Nature of the Graph . . . . . . . . . . . . . . . . 114

10.12Graphs of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

10.12.1Functions of the form y = ax + q . . . . . . . . . . . . . . . . . . . . . 116

10.12.2Functions of the Form y = ax2 + q . . . . . . . . . . . . . . . . . . . . . 120

10.12.3Functions of the Form . . . . . . . . . . . . . . . . . . . . . . 125

10.12.4Functions of the Form y = ab(x) + q . . . . . . . . . . . . . . . . . . . . 129

10.13End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

**Chapter 11: Average Gradient - Grade 10 Extension**

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

11.2 Straight-Line Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

11.3 Parabolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

11.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

**Chapter 12: Geometry Basics**

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

12.2 Points and Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

12.3 Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

12.3.1 Measuring angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

12.3.2 Special Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

12.3.3 Special Angle Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

12.3.4 Parallel Lines intersected by Transversal Lines . . . . . . . . . . . . . . . 143

12.4 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

12.4.1 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

12.4.2 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

12.4.3 Other polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

12.4.4 Extra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

12.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

12.5.1 Challenge Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

**Chapter 13: Geometry - Grade 10**

13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

13.2 Right Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

13.2.1 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

13.2.2 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

13.3 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

13.3.1 Similarity of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

13.4 Co-ordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

13.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

13.4.2 Distance between Two Points . . . . . . . . . . . . . . . . . . . . . . . . 172

13.4.3 Calculation of the Gradient of a Line . . . . . . . . . . . . . . . . . . . . 173

13.4.4 Midpoint of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

13.5 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

13.5.1 Translation of a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

13.5.2 Reflection of a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

13.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

**Chapter 14: Trigonometry - Grade 10**

14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

14.2 Where Trigonometry is Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

14.3 Similarity of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

14.4 Definition of the Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 191

14.5 Simple Applications of Trigonometric Functions . . . . . . . . . . . . . . . . . . 195

14.5.1 Height and Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

14.5.2 Maps and Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

14.6 Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 199

14.6.1 Graph of sin ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

14.6.2 Functions of the form y = a sin(x) + q . . . . . . . . . . . . . . . . . . . 200

14.6.3 Graph of cos ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

14.6.4 Functions of the form y = a cos(x) + q . . . . . . . . . . . . . . . . . . 202

14.6.5 Comparison of Graphs of sin ? and cos ? . . . . . . . . . . . . . . . . . . 204

14.6.6 Graph of tan ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

14.6.7 Functions of the form y = a tan(x) + q . . . . . . . . . . . . . . . . . . 205

14.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

**Chapter 15: Statistics - Grade 10**

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

15.2 Recap of Earlier Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

15.2.1 Data and Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . 211

15.2.2 Methods of Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . 212

15.2.3 Samples and Populations . . . . . . . . . . . . . . . . . . . . . . . . . . 213

15.3 Example Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

15.3.1 Data Set 1: Tossing a Coin . . . . . . . . . . . . . . . . . . . . . . . . . 213

15.3.2 Data Set 2: Casting a die . . . . . . . . . . . . . . . . . . . . . . . . . . 213

15.3.3 Data Set 3: Mass of a Loaf of Bread . . . . . . . . . . . . . . . . . . . . 214

15.3.4 Data Set 4: Global Temperature . . . . . . . . . . . . . . . . . . . . . . 214

15.3.5 Data Set 5: Price of Petrol . . . . . . . . . . . . . . . . . . . . . . . . . 215

15.4 Grouping Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

15.4.1 Exercises - Grouping Data . . . . . . . . . . . . . . . . . . . . . . . . . 216

15.5 Graphical Representation of Data . . . . . . . . . . . . . . . . . . . . . . . . . . 217

15.5.1 Bar and Compound Bar Graphs . . . . . . . . . . . . . . . . . . . . . . . 217

15.5.2 Histograms and Frequency Polygons . . . . . . . . . . . . . . . . . . . . 217

15.5.3 Pie Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

15.5.4 Line and Broken Line Graphs . . . . . . . . . . . . . . . . . . . . . . . . 220

15.5.5 Exercises - Graphical Representation of Data . . . . . . . . . . . . . . . 221

15.6 Summarising Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

15.6.1 Measures of Central Tendency . . . . . . . . . . . . . . . . . . . . . . . 222

15.6.2 Measures of Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

15.6.3 Exercises - Summarising Data . . . . . . . . . . . . . . . . . . . . . . . 228

15.7 Misuse of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

15.7.1 Exercises - Misuse of Statistics . . . . . . . . . . . . . . . . . . . . . . . 230

15.8 Summary of Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

15.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

**Chapter 16: Probability - Grade 10**

16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

16.2 Random Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

16.2.1 Sample Space of a Random Experiment . . . . . . . . . . . . . . . . . . 235

16.3 Probability Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

16.3.1 Classical Theory of Probability . . . . . . . . . . . . . . . . . . . . . . . 239

16.4 Relative Frequency vs. Probability . . . . . . . . . . . . . . . . . . . . . . . . . 240

16.5 Project Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

16.6 Probability Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

16.7 Mutually Exclusive Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

16.8 Complementary Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

16.9 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

**III GRADE 11**

**Chapter 17: Exponents - Grade 11**

17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

17.2 Laws of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

17.2.1 Exponential Law 7 . . . . . . . . . . . . . . . . . . . . . . 251

17.3 Exponentials in the Real-World . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

17.4 End of chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

**Chapter 18: Surds - Grade 11**

18.1 Surd Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

18.1.1 Surd Law 1. . . . . . . . . . . . . . . . . . . . . . . . 255

18.1.2 Surd Law 2: . . . . . . . . . . . . . . . . . . . . . . . . . . 255

18.1.3 Surd Law 3: . . . . . . . . . . . . . . . . . . . . . . . . . . 256

18.1.4 Like and Unlike Surds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

18.1.5 Simplest Surd form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

18.1.6 Rationalising Denominators . . . . . . . . . . . . . . . . . . . . . . . . . 258

18.2 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

**Chapter 19: Error Margins - Grade 11**

**Chapter 20: Quadratic Sequences - Grade 11**

20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

20.2 What is a quadratic sequence? . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

20.3 End of chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

**Chapter 21: Finance - Grade 11**

21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

21.2 Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

21.3 Simple Depreciation (it really is simple!) . . . . . . . . . . . . . . . . . . . . . . 271

21.4 Compound Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

21.5 Present Values or Future Values of an Investment or Loan . . . . . . . . . . . . 276

21.5.1 Now or Later . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

21.6 Finding i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

21.7 Finding n - Trial and Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

21.8 Nominal and Effective Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . 280

21.8.1 The General Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

21.8.2 De-coding the Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 282

21.9 Formulae Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

21.9.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

21.9.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

21.10End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

**Chapter 22: Solving Quadratic Equations - Grade 11**

22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

22.2 Solution by Factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

22.3 Solution by Completing the Square . . . . . . . . . . . . . . . . . . . . . . . . . 290

22.4 Solution by the Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . 293

22.5 Finding an equation when you know its roots . . . . . . . . . . . . . . . . . . . 296

22.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

**Chapter 23: Solving Quadratic Inequalities - Grade 11**

23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

23.2 Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

23.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

**Chapter 24: Solving Simultaneous Equations - Grade 11**

24.1 Graphical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

24.2 Algebraic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

**Chapter 25: Mathematical Models - Grade 11**

25.1 Real-World Applications: Mathematical Models . . . . . . . . . . . . . . . . . . 313

25.2 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

**Chapter 26: Quadratic Functions and Graphs - Grade 11**

26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

26.2 Functions of the Form y = a(x + p)2 + q . . . . . . . . . . . . . . . . . . . . . 321

26.2.1 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

26.2.2 Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

26.2.3 Turning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

26.2.4 Axes of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

26.2.5 Sketching Graphs of the Form f(x) = a(x + p)2 + q . . . . . . . . . . . 325

26.2.6 Writing an equation of a shifted parabola . . . . . . . . . . . . . . . . . 327

26.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

**Chapter 27: Hyperbolic Functions and Graphs - Grade 11**

27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

27.2 Functions of the Form . . . . . . . . . . . . . . . . . . . . . . . . 329

27.2.1 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

27.2.2 Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

27.2.3 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

27.2.4 Sketching Graphs of the Form . . . . . . . . . . . . . . 333

27.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

**Chapter 28: Exponential Functions and Graphs - Grade 11**

28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

28.2 Functions of the Form y = ab(x+p) + q . . . . . . . . . . . . . . . . . . . . . . . 335

28.2.1 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

28.2.2 Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

28.2.3 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

28.2.4 Sketching Graphs of the Form f(x) = ab(x+p) + q . . . . . . . . . . . . . 338

28.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

**Chapter 29: Gradient at a Point - Grade 11**

29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

29.2 Average Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

29.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

**Chapter 30: Linear Programming - Grade 11**

30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

30.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

30.2.1 Decision Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

30.2.2 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

30.2.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

30.2.4 Feasible Region and Points . . . . . . . . . . . . . . . . . . . . . . . . . 346

30.2.5 The Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

30.3 Example of a Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

30.4 Method of Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

30.5 Skills you will need . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

30.5.1 Writing Constraint Equations . . . . . . . . . . . . . . . . . . . . . . . . 347

30.5.2 Writing the Objective Function . . . . . . . . . . . . . . . . . . . . . . . 348

30.5.3 Solving the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

30.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

**Chapter 31: Geometry - Grade 11**

31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

31.2 Right Pyramids, Right Cones and Spheres . . . . . . . . . . . . . . . . . . . . . 357

31.3 Similarity of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

31.4 Triangle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

31.4.1 Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

31.5 Co-ordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

31.5.1 Equation of a Line between Two Points . . . . . . . . . . . . . . . . . . 368

31.5.2 Equation of a Line through One Point and Parallel or Perpendicular to Another Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

31.5.3 Inclination of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

31.6 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

31.6.1 Rotation of a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

31.6.2 Enlargement of a Polygon 1 . . . . . . . . . . . . . . . . . . . . . . . . . 376

**Chapter 32: Trigonometry - Grade 11**

32.1 History of Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

32.2 Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 381

32.2.1 Functions of the form y = sin(k?) . . . . . . . . . . . . . . . . . . . . . 381

32.2.2 Functions of the form y = cos(k?) . . . . . . . . . . . . . . . . . . . . . 383

32.2.3 Functions of the form y = tan(k?) . . . . . . . . . . . . . . . . . . . . . 384

32.2.4 Functions of the form y = sin(? + p) . . . . . . . . . . . . . . . . . . . . 385

32.2.5 Functions of the form y = cos(? + p) . . . . . . . . . . . . . . . . . . . 386

32.2.6 Functions of the form y = tan(? + p) . . . . . . . . . . . . . . . . . . . 387

32.3 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

32.3.1 Deriving Values of Trigonometric Functions for 30, 45 and 60 . . . . . 389

32.3.2 Alternate Definition for tan ? . . . . . . . . . . . . . . . . . . . . . . . . 391

32.3.3 A Trigonometric Identity . . . . . . . . . . . . . . . . . . . . . . . . . . 392

32.3.4 Reduction Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

32.4 Solving Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 399

32.4.1 Graphical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

32.4.2 Algebraic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

32.4.3 Solution using CAST diagrams . . . . . . . . . . . . . . . . . . . . . . . 403

32.4.4 General Solution Using Periodicity . . . . . . . . . . . . . . . . . . . . . 405

32.4.5 Linear Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . 406

32.4.6 Quadratic and Higher Order Trigonometric Equations . . . . . . . . . . . 406

32.4.7 More Complex Trigonometric Equations . . . . . . . . . . . . . . . . . . 407

32.5 Sine and Cosine Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

32.5.1 The Sine Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

32.5.2 The Cosine Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

32.5.3 The Area Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

32.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

**Chapter 33: Statistics - Grade 11**

33.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

33.2 Standard Deviation and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . 419

33.2.1 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

33.2.2 Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

33.2.3 Interpretation and Application . . . . . . . . . . . . . . . . . . . . . . . 423

33.2.4 Relationship between Standard Deviation and the Mean . . . . . . . . . . 424

33.3 Graphical Representation of Measures of Central Tendency and Dispersion . . . . 424

33.3.1 Five Number Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

33.3.2 Box and Whisker Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 425

33.3.3 Cumulative Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

33.4 Distribution of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

33.4.1 Symmetric and Skewed Data . . . . . . . . . . . . . . . . . . . . . . . . 428

33.4.2 Relationship of the Mean, Median, and Mode . . . . . . . . . . . . . . . 428

33.5 Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

33.6 Misuse of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

33.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

**Chapter 34: Independent and Dependent Events - Grade 11**

34.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

34.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

34.2.1 Identification of Independent and Dependent Events . . . . . . . . . . . 438

34.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

**IV GRADE 12**

**Chapter 35: Logarithms - Grade 12**

35.1 Definition of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

35.2 Logarithm Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

35.3 Laws of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

35.4 Logarithm Law 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

35.5 Logarithm Law 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 448

35.6 Logarithm Law 3 . . . . . . . . . . . . . . . . . 448

35.7 Logarithm Law 4 . . . . . . . . . . . . . . . . . 449

35.8 Logarithm Law 5:. . . . . . . . . . . . . . . . . . . . . . . 450

35.9 Logarithm Law 6: . . . . . . . . . . . . . . . . . . . . . . 450

35.10Solving simple log equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

35.10.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

35.11 Logarithmic applications in the Real World . . . . . . . . . . . . . . . . . . . . . 454

35.11.1Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

35.12 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

**Chapter 36: Sequences and Series - Grade 12**

36.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

36.2 Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

36.2.1 General Equation for the nth-term of an Arithmetic Sequence . . . . . . 458

36.3 Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

36.3.1 Example - A Flu Epidemic . . . . . . . . . . . . . . . . . . . . . . . . . 459

36.3.2 General Equation for the nth-term of a Geometric Sequence . . . . . . . 461

36.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

36.4 Recursive Formulae for Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 462

36.5 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

36.5.1 Some Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

36.5.2 Sigma Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

36.6 Finite Arithmetic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

36.6.1 General Formula for a Finite Arithmetic Series . . . . . . . . . . . . . . . 466

36.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

36.7 Finite Squared Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

36.8 Finite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

36.8.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470

36.9 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

36.9.1 Infinite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . 471

36.9.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

36.10End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

**Chapter 37: Finance - Grade 12**

37.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

37.2 Finding the Length of the Investment or Loan . . . . . . . . . . . . . . . . . . . 477

37.3 A Series of Payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

37.3.1 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

37.3.2 Present Values of a series of Payments . . . . . . . . . . . . . . . . . . . 479

37.3.3 Future Value of a series of Payments . . . . . . . . . . . . . . . . . . . . 484

37.3.4 Exercises - Present and Future Values . . . . . . . . . . . . . . . . . . . 485

37.4 Investments and Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

37.4.1 Loan Schedules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

37.4.2 Exercises - Investments and Loans . . . . . . . . . . . . . . . . . . . . . 489

37.4.3 Calculating Capital Outstanding . . . . . . . . . . . . . . . . . . . . . . 489

37.5 Formulae Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

37.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

37.5.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

37.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

**Chapter 38: Factorising Cubic Polynomials - Grade 12**

38.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

38.2 The Factor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

38.3 Factorisation of Cubic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 494

38.4 Exercises - Using Factor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 496

38.5 Solving Cubic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

38.5.1 Exercises - Solving of Cubic Equations . . . . . . . . . . . . . . . . . . . 498

38.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

**Chapter 39: Functions and Graphs - Grade 12**

39.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

39.2 Definition of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

39.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

39.3 Notation used for Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

39.4 Graphs of Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

39.4.1 Inverse Function of y = ax + q . . . . . . . . . . . . . . . . . . . . . . . 503

39.4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

39.4.3 Inverse Function of y = ax2 . . . . . . . . . . . . . . . . . . . . . . . . 504

39.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

39.4.5 Inverse Function of y = ax . . . . . . . . . . . . . . . . . . . . . . . . . 506

39.4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

39.5 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

**Chapter 40: Differential Calculus - Grade 12**

40.1 Why do I have to learn this stuff? . . . . . . . . . . . . . . . . . . . . . . . . . 509

40.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

40.2.1 A Tale of Achilles and the Tortoise . . . . . . . . . . . . . . . . . . . . . 510

40.2.2 Sequences, Series and Functions . . . . . . . . . . . . . . . . . . . . . . 511

40.2.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

40.2.4 Average Gradient and Gradient at a Point . . . . . . . . . . . . . . . . . 516

40.3 Differentiation from First Principles . . . . . . . . . . . . . . . . . . . . . . . . . 519

40.4 Rules of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

40.4.1 Summary of Differentiation Rules . . . . . . . . . . . . . . . . . . . . . . 522

40.5 Applying Differentiation to Draw Graphs . . . . . . . . . . . . . . . . . . . . . . 523

40.5.1 Finding Equations of Tangents to Curves . . . . . . . . . . . . . . . . . 523

40.5.2 Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

40.5.3 Local minimum, Local maximum and Point of Inflextion . . . . . . . . . 529

40.6 Using Differential Calculus to Solve Problems . . . . . . . . . . . . . . . . . . . 530

40.6.1 Rate of Change problems . . . . . . . . . . . . . . . . . . . . . . . . . . 534

40.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

**Chapter 41: Linear Programming - Grade 12**

41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

41.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

41.2.1 Feasible Region and Points . . . . . . . . . . . . . . . . . . . . . . . . . 539

41.3 Linear Programming and the Feasible Region . . . . . . . . . . . . . . . . . . . 540

41.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546

**Chapter 42: Geometry - Grade 12**

42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

42.2 Circle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

42.2.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

42.2.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

42.2.3 Theorems of the Geometry of Circles . . . . . . . . . . . . . . . . . . . . 550

42.3 Co-ordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566

42.3.1 Equation of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566

42.3.2 Equation of a Tangent to a Circle at a Point on the Circle . . . . . . . . 569

42.4 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

42.4.1 Rotation of a Point about an angle ? . . . . . . . . . . . . . . . . . . . . 571

42.4.2 Characteristics of Transformations . . . . . . . . . . . . . . . . . . . . . 573

42.4.3 Characteristics of Transformations . . . . . . . . . . . . . . . . . . . . . 573

42.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574

**Chapter 43: Trigonometry - Grade 12**

43.1 Compound Angle Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577

43.1.1 Derivation of sin(? + ?) . . . . . . . . . . . . . . . . . . . . . . . . . . 577

43.1.2 Derivation of sin(? ? ?) . . . . . . . . . . . . . . . . . . . . . . . . . . 578

43.1.3 Derivation of cos(? + ?) . . . . . . . . . . . . . . . . . . . . . . . . . . 578

43.1.4 Derivation of cos(? ? ?) . . . . . . . . . . . . . . . . . . . . . . . . . . 579

43.1.5 Derivation of sin 2? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

43.1.6 Derivation of cos 2? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

43.1.7 Problem-solving Strategy for Identities . . . . . . . . . . . . . . . . . . . 580

43.2 Applications of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 582

43.2.1 Problems in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 582

43.2.2 Problems in 3 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 584

43.3 Other Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

43.3.1 Taxicab Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

43.3.2 Manhattan distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

43.3.3 Spherical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

43.3.4 Fractal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588

43.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

**Chapter 44: Statistics - Grade 12**

44.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

44.2 A Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

44.3 Extracting a Sample Population . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

44.4 Function Fitting and Regression Analysis . . . . . . . . . . . . . . . . . . . . . . 594

44.4.1 The Method of Least Squares . . . . . . . . . . . . . . . . . . . . . . . 596

44.4.2 Using a calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597

44.4.3 Correlation coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

44.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600

**Chapter 45: Combinations and Permutations - Grade 12**

45.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

45.2 Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

45.2.1 Making a List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

45.2.2 Tree Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

45.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

45.3.1 The Factorial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

45.4 The Fundamental Counting Principle . . . . . . . . . . . . . . . . . . . . . . . . 604

45.5 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

45.5.1 Counting Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

45.5.2 Combinatorics and Probability . . . . . . . . . . . . . . . . . . . . . . . 606

45.6 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

45.6.1 Counting Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

45.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

45.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

**V EXERCISES**

**Chapter 46: General Exercises**

**Chapter 47: Exercises - Not covered in Syllabus**

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Linear inequalities in one variable

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Slope-intercept form with fractions and decimal problems.

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Ever noticed how students mix up the rules for adding, subtracting, multiplying and dividing integers? Students need a unifying framework to help make sense of the diverse rules. This unit presents that unifying concept with a novel representation that combines the best aspects of two different representations: number lines, and algebra tiles.

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A concrete and conceptual approach to integer subtraction--helps students understand the meaning of the "adding the opposite" technique. Complete with fully-animated PowerPoint and student handout. The thought-provoking lesson culminates with an introduction to a game of integers, Attack & Defend, and how to play the game with subtraction.

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Students model integer subtraction with pictures in a creative and engaging game. Through pictures, they experience the patterns and meaning behind the skill. Only later, in the next lesson, will they formalize their understanding by generating the rule for subtracting with positives and negatives. Letting students discover the formal rule after they concretely experience a pattern is at the heart of inquiry-driven learning.

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This is a complete online Algebra I curriculum. There are 155 interactive web exercises, with printable randomly-generated worksheets and solutions. Lots of pre-algebra review, too. Mathematics is displayed correctly using MathML (Math Markup Language)---this is the future of math on the web. The first section, The Language of Mathematics, describes the philosophy of the course and explains the title.

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Allowing students to visualize the notion of the square of a number

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Often when students are introduced to tangrams,
they are asked to put the pieces together to form a square.
This is often a difficult and frustrating task because they
have no background as to how the pieces fit together.

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Many students tend to memorize, without
understanding, formulas that we use in geometry or other
mathematic areas. This particular activity allows students
to discover why pi works in solving problems dealing with
finding circumference.

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This lesson is one which I have used with the
class, however it will work equally as well as homework or
extra credit assignment. In addition to teaching, I am also
a track coach and this activity is of particular help to my
athletes in seeing what they should do to maximize their
performance. This activity is a great deal of fun in the
spring of the year when the students want to get outside to
enjoy the great outdoors and the water.

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Students traditionally have a very difficult time
understanding which digits are significant, especially
zeroes, in a number that represents something that has been
measured. This activity was designed so that students will
explore and truly understand which digits are significant
when dealing with numbers that represent measured values.

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As easy game to learn but difficult to master

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This problem solving game uses pattern blocks to reinforce computational skills in a challenging format rather than the drudgery of drill.

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Students are introduced to fact families and arrays. Students will complete a fact families worksheet and compose story problems using fact families.

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The teaching and learning of mathematics is facilitated by the development of a definite task analysis. Each skill builds upon and rests on the previous skills learned. Therefore, knowing all multiplication facts will help in the learning of subsequent skills in hard multiplication, division, fractions, etc.

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Many students can test well on multiplying and dividing fractions, but when it comes to real life situations do not understand how to use their knowledge.

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This lesson provides an introductory activity for fractions. Oranges are used to show the difference between one whole, one half, and one quarter.

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This article describes the concept of eccentricity of the conic sections as a unifying idea.

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This lesson will help students learn what total cost is. There will be a mock store set up in the classroom with objects that have price tags on them for the students to practice their addition skills in working with money.

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Through the use of pictures and manipulative materials, students learn the relationship between the numerator and denominator.

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This lesson is designed to improve students' understanding of fractions by using a fun manipulative (M & M candies). The students will sort their bowl of M & M's into colors and then, for each color, they will write out, both longhand and numerically, the fraction that represents how many of their M & M's are that color.

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This course covers the content of elementary school arithmetic, from addition and subtraction to fractions.

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This lesson has groups of students working together to plan, build, and calculate the costs of constructing a free standing structure

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This lesson provides a history of Roman Numerals as well as a primer on their use in arithmetic.

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This lesson introduces students to the geometry of cubes as a combinatorial problem.

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This lesson associates the various prefixes quantifying words with their numerical meanings.

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A lesson on computing pi from direct measurement

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A lesson on using the metric system

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An experiment with Rock Paper Scissors

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