Table of Contents Page Page 2 Chapter 1 – Absolute Value of Numbers and 5.4 Factoring a[f(x)] + b(f(x)) + c, a≠ 0 151 Intro to Radicals 1.1 Absolute Value of a Number and the 2 5.5 Factoring a [f(x)] – b [g(y)] ; a 153 2 2 2 2 Number Line ≠ 0, b ≠ 0 1.2 Equations and Inequalities Involving 7 5.6 Combination of Factoring 155 Absolute Value 1.3 Powers and Roots of Numbers 14 5.7 Factor Theorem 157 1.4 Ordering Radicals and Using a 18 Chapter 6 – Relations and Quadratic Functions Calculator to Approximate Values 1.5 Simplifying Radicals by Factoring 22 6.1 Review of Relations and Functions 161 1.6 Adding and Subtracting Radicals 25 6.2 Graphs of Quadratic Functions 172 1.7 Multiplication and Division of Square 30 6.3 Transformations of Quadratic 186 Root Radicals Functions Chapter 2 – Properties and Applications of Radicals 6.4 Reciprocal Functions 191 2.1 Writing Radicals in Simplest Form 42 6.5 Graphing the Absolute Value 197 Function 2.2 Product of a Binomial times a 46 6.6 Solving Absolute-Value Equations 205 Binomial Algebraically and Graphically 2.3 Conjugates of Binomials and 50 Chapter 7 – Applications with Quadratic Functions Rationalizing Denominators 2.4 Relationships between Roots, 53 7.1 Completing the Square 214 Absolute Values, and Signs 2.5 Solving Equations Involving Radicals 55 7.2 Maximum and Minimum Problems 219 2.6 Problems Involving Radical 59 7.3 Solving Quadratic Equations 224 Expressions 7.3.1 Solving by Graphing 224 Chapter 3 – Rational Expressions and Equations 7.3.2 Solving by Factoring 229 3.1 Rational Numbers (Review) 69 7.3.3 Solving by Completing the 233 Square 3.2 Addition and Subtraction of Fractions 73 7.3.4 The Quadratic Formula 235 (Review) 3.3 Multiplication and Division of 77 7.4 The Discriminant 239 Fractions (Review) 3.4 Rational Expressions 82 Chapter 8 – Sequences and Series 3.5 Adding and Subtracting Rational 85 8.1 Arithmetic Sequences 246 Expressions 3.6 Multiplying and Dividing Rational 87 8.2 Arithmetic Series 252 Expressions 3.7 Multiple Operations with Rational 90 8.3 Geometric Sequences 257 Expressions 3.8 Rational Equations 92 8.4 Geometric Series 261 3.9 Solving Problems Involving Rational 95 8.5 Sums of Infinite Geometric Series 265 Equations Chapter 4 – Trigonometry Chapter 9 – Inequalities 4.1 Definition of Trig Functions and 103 9.1 Graphing Inequalities in One Variable 276 Angles in Standard Position in Two Dimensions 4.2 Special Angles 114 9.2 Graphing Inequalities in Two 279 Variables 4.3 Law of Sines 120 9.3 Graphing Systems of Linear and 284 Quadratic Inequalities 4.4 Law of Cosines 125 9.4 Graphing Quadratic Inequalities in 290 One Variable 4.5 Solving General Triangles 129 9.5 Problems for Quadratic Inequalities 295 (Ambiguous Case) Chapter 5 – Factoring Polynomials Chapter 10 – Linear and Quadratic Systems 5.1 Review of Factoring in General 140 10.1 Linear-Quadratic Systems 301 5.2 Factoring ax + bx + c, a ≠ 0 146 10.2 Quadratic-Quadratic Systems 304 2 2 2 2 2 5.3 Factoring a x – b y , a ≠ 0, b ≠ 0 149 10.3 Problems for Systems 307 Answers to Exercises and Chapter Tests 311

Grade 11 Mathematics CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 1 CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 5 1.1 Absolute Value of a Number and the Number Line 1.2 Equations and Inequalities Involving Absolute Value 1.3 Powers and Roots of Numbers 1.4 Ordering Radicals and Using a Calculator to Find Approximate Values 1.5 Simplifying Radicals by Factoring 1.6 Adding and Subtracting Radicals 1.7 Multiplication and Division of Square Root Radicals x 5 → -5 x 5 index radical sign x 5 → x 5or x 5 n a radicand It is unlawful to photocopy or reproduce content without permission from Dynamic Classroom Inc. This guide is licensed to the end user for PERSONAL use only. All rights reserved.

Grade 11 Mathematics CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 2 1.1 Absolute Value of a Number and the Number Line ▪ The absolute value of a number can be thought of as the distance that the number is located from 0 on a number line. The direction doesn’t matter and as a result, the absolute value of a number is never negative. e.g. ▪ The absolute value of 5 is 5. ▪ The absolute value of 5 is 5. ▪ Note: 5 is 5 units to the right of 0. ▪ Note: -5 is 5 units to the left of 0. ▪ The symbol for absolute value consists of two vertical line segments that appear on either side of the number or expression. e.g. (i) 5 means the absolute value of 5, which is equal to 5. (ii) 5- means the absolute value of -5, which is equal to 5. 1 1 1 1 (iii) = and - = 2 2 2 2 (iv) .0 27 = . 0 27 and 0- . 27 = . 0 27 Keep in mind the following rules: ▪ The absolute value of a positive number or 0 is the number. ▪ The absolute value of a negative number is its opposite. OR If x is any real number x 7 ▪ x = if x 0 e.g. 7 = ▪ x = − x if x < 0 e.g. 7- = ) 7 - ( - = 7 ▪ The absolute value of zero is equal to \"0\". Why? Just ask yourself: how far zero is from 0. Answer: zero units. So |0| = 0. ▪ We can simplify single absolute value terms containing more than one numerical expression inside the symbol by working inside of it first. e.g. (i) 2 − 5 = 3 - = 3 (ii) 7 − 8 = 1 - = 1 (iii) 12 − 8 = 4 = 4 (iv) 2.1 − 3 = 8 . 1 - = 8 . 1 It is unlawful to photocopy or reproduce content without permission from Dynamic Classroom Inc. This guide is licensed to the end user for PERSONAL use only. All rights reserved.

Grade 11 Mathematics CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 3 Further Examples with Solutions Simplify each of the following. Solution 3 1. 9 − 9 − 3 = 6 = 6 5 5 2. .4 − 5 . 4 − 5 = - 5 . 0 = 5 . 0 3. 2 − 3 2 2 − 3 2 = - 2 2 = 2 2 1 3 1 3 1 1 4. − − = - = 2 4 2 4 4 4 5. 1− 5 . 5 - 1− - 5 . 5 = 1+ 5 . 5 = 6 . 5 = 6 . 5 Order of Operations (Review) ▪ Next, let’s recall the order of operations with numbers that we will use to simplify expressions with more than one term. ▪ When there are several operations needed to simplify an expression, a special order of operations needs to be done. Following are 4 levels to be done in order, beginning with level 1. Remember BEDMAS: 1. Level 1 B (brackets) If more than one set, 2. Level 2 E (exponents) innermost first 3. Level 3 D (divide) Same level, do left to M (multiply) right 4. Level 4 A (add) Same level order doesn’t S (subtract) matter Examples: ▪ If there are no brackets (parentheses), perform all multiplication and division in the order they appear from left to right, before any addition or subtraction. e.g. 1. 2 + 3 x 5 + 6 = 2 + 15 + 6 = 23 2. 15 + 9 3 + 2 x 4 = 15 + 3 + 8 = 26 ▪ If there are brackets (parentheses) perform all operations inside the brackets before any other operations. e.g. 1. (5 + 6) x 3 = 11 x 3 = 33 2. 7 + (8 – 2) x 4 = 7 + 6 x 4 = 7 + 24 = 31 It is unlawful to photocopy or reproduce content without permission from Dynamic Classroom Inc. This guide is licensed to the end user for PERSONAL use only. All rights reserved.

Grade 11 Mathematics CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 4 3. 12 (2 + 4) + 7 = 12 6 + 7 = 2 + 7 = 9 4. 12 2 + (4 + 7) = 12 2 + 11 = 6 + 11 = 17 ▪ The same rules apply when we are working with terms containing absolute value. e.g. 1. 2- + 3• 5 = - 2 + 15 = 13 = 13 1 1 1 1 2. 7 – − 1 = 7 − - = 7 − = 6 2 2 2 2 Further Examples with Solutions Solution 1. Simplify. 1. Since there are no brackets, perform the 2 + 3 x 4 multiplication first. 2 + 3 x 4 = 2 + 12 2. Now add. 2 + 12 = 14 2. Simplify. 1. Perform operations inside of the brackets first. (2 + 3) x 4 (2 + 3 ) x 4 = 5 x 4 2. Now multiply. 5 x 4 = 20 3. Simplify. 1. Perform operations inside of brackets first. 32 4 – (3 x 2) 32 4 – (3 x 2) = 32 4 - 6 2. Now perform division before adding or subtracting. 32 4 – 6 = 8 – 6 3. Now subtract.t 8 – 6 = 2 4. Simplify. 1. Perform operations in brackets first. 1.3 + 2(5 + 0.25) – 4 1.3+2(5 + 0.25) – 4 = 1.3+2(5.25) – 4 2. Now perform multiplication before adding or subtracting. 1.3 + 10.5 – 4 3. Now add and subtract to get 7.8. It is unlawful to photocopy or reproduce content without permission from Dynamic Classroom Inc. This guide is licensed to the end user for PERSONAL use only. All rights reserved.

Grade 11 Mathematics CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 5 5. Simplify. 1. Work inside of the absolute value sign first to 2 − 5 . 3 + 2 • 5 ge:t 2 − 5 . 3 + 2• 5 = 5 . 1 - + 2• 5 2. Clear the absolute value sign to get: 1- 5 . + 2• 5 = 5.1 + 2• 5 3. Perform multiplication before addition. 5.1 + 2• 5 = 5.1 + 10 4. Now perform addition. 5.1 + 10 = 11 5 . 6. Simplify. 1. Work inside of the absolute value sign first to 5 − 5 . 3 - − 12 4 get: 5 − 5 . 3 - − 12 4= 5 + 5 . 3 − 12 4 2. Clear the absolute value sign to get: 5 + 5 . 3 − 12 4 = 5 . 8 − 12 4 3 . Perform division before subtraction. 5.8 − 12 4 = 5 . 8 − 3 4. Now perform subtraction. 5.8 − 3 = 5 . 5 7. Simplify. 1. Work inside of the absolute value sign first (perform multiplication before subtraction). 7 − 3• 4 − 5 - 7 − 3• 4 − 5 - = 7 − 12 − 5 - 2. Clear the absolute value sign to get: 7 − 12 − 5 - = 5 - − 5 - = 5 − 5 3. Now perform subtraction. 5 − 5 = 0 Exercises 1.1 1. Simplify each of the following terms. a. 12 − b. 15 − 11 5 . 3 5 . c. 11 − 15 d. 21 − 22 2 . 5 . e. 3 2 − 7 2 f. 27 − 5 3 It is unlawful to photocopy or reproduce content without permission from Dynamic Classroom Inc. This guide is licensed to the end user for PERSONAL use only. All rights reserved.

Grade 11 Mathematics CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 6 3 3 1 1 g. − h. 2 − 3 2 4 4 8 i. 1− 3 . 3 - j. 2 − 8 . 5 - 2. Use the order of operations to simplify each of the following expressions. a. 1− 5 . 4 + 3• 7 b. 2 − 5 + 4 8 2 c. .2 − 5 . 3 - − 5 . 1 • 4 d. .5 − - 10 − 5• 2 . 4 7 3 e. 8 − 2 • 4 − 3 - f. 2 − 3• 4 − 5 . 1 - g. 2- − 3• 5 h. 15- 2 . − 2• 5 . 1 1 2 i. 7 – − 1 j. 5 – − 2 3 3 3. Answer each of the following questions. a. Write a term involving absolute value that shows a distance of 25 m below sea level. b. Riley jumped from a plane that was at an altitude of 850 m. Show the distance he jumped from the plane using the absolute value sign. It is unlawful to photocopy or reproduce content without permission from Dynamic Classroom Inc. This guide is licensed to the end user for PERSONAL use only. All rights reserved.

Grade 11 Mathematics CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 7 c. A point is graphed on the number line below. Graph another point on the line that is the same distance from 0. d. A point is graphed on the number line below. Graph another point on the line that is the same distance from 0. 1.2 Equations and Inequalities Involving Absolute Value When an equation includes a variable, not equal to zero, under the absolute value sign, it is equal to two different values. e.g. If x = 3, then we know that 3 = and that 3- = 3 3 x = 3 or x = - 3 is called the translation of the equation x = 3 The graph of the solution to x = 3 is shown below. Each point in the graph above is 3 units from zero. Several other examples follow. 1. Equation Translation x = x = 8 or x = -8 (This is also the 8 solution.) Graph 2. Equation Translation and Solution x + 2 = 7 x + 2 = 7 or x + 2 = -7 → x = 5 or x = -9 Graph It is unlawful to photocopy or reproduce content without permission from Dynamic Classroom Inc. This guide is licensed to the end user for PERSONAL use only. All rights reserved.

Grade 11 Mathematics CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 8 Additional Examples with Solutions 1. Write the translation and find the solution to each of the following equations. Equation Translation Solution 1 1 1 1 1 a. x = x = or x = - x = or x = - 3 3 3 3 3 b. x − 1 = 5 x – 1 = 5 or x – 1 = -5 x = 6 or x = -4 c. x2 − 1 = 7 2x – 1 = 7 or 2x – 1 = -7 2x = 8 or 2 x = -6, so x = 4 or x = -3 2. The graph of the solution for an absolute value equation is shown below. Write an absolute value equation for each solution. a. Solution: x = 10 or x = -10, so x = 10 b. 3 Solution: x = 3 or x = -3, so x = It is unlawful to photocopy or reproduce content without permission from Dynamic Classroom Inc. This guide is licensed to the end user for PERSONAL use only. All rights reserved.

Grade 11 Mathematics CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 9 Absolute Value Inequalities Less Than or Less Than or Equal to 5 ▪ We know that x = means that x = 5 or x = -5. 5 ▪ However, if x , then x lies between -5 and 5, this translates to x > -5 AND x < 5. It can be written as -5 < x < 5. ▪ The graph of the solution would look like the following: ▪ Notice that the endpoints are not part of the solution. ▪ If x then x lies between -5 and 5 inclusive, this translates to x -5 AND 5 x 5. It can be written as -5 x 5. ▪ The graph of this solution would look like the following: Greater Than or Greater Than or Equal to ▪ If the absolute value inequality were x , then x is greater than 5 or it is less 5 than -5, this translates into x < - 5 OR x >5 ▪ The graph would look like the following: ▪ As shown, the endpoints for each ray are not part of the solution. ▪ If the absolute value inequality were x 5, then x is greater than or equal to 5 or it is less than or equal to -5, this translates into x 5 OR x -5 ▪ The graph would look like the following: It is unlawful to photocopy or reproduce content without permission from Dynamic Classroom Inc. This guide is licensed to the end user for PERSONAL use only. All rights reserved.

Grade 11 Mathematics CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 10 Additional Examples with Solutions 1. Write the translation for each of the following inequalities. Inequality Translation a. x 32 -32 x 32 1 1 1 b. x < - < x < 7 7 7 c. x > 3.2 x < -3.2 or x > 3.2 d. x 9 x - 9 or x 9 2. Write the translation and inequality for each of the following graphs. a. Translation: x 6 and x - 6 → -6 x 6 Inequality: x 6 b. Translation: x < 2 and x > -2 → -2 x 2 Inequality: x < 2 c. Translation: x 4 or x - 4 Inequality: x 4 d. Translation: x < - 1 or x > 1 Inequality: x > 1 It is unlawful to photocopy or reproduce content without permission from Dynamic Classroom Inc. This guide is licensed to the end user for PERSONAL use only. All rights reserved.

Grade 11 Mathematics CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 11 Exercises 1.2 1. Write the solution for each equation. a. x = 2.3 b. x = 21.5 3 13 c. x = d. x = 2 5 1 e. x − = 2 f. x + = 7 3 1 2 g. x − 1 = 2 h. x + 1 = 2 2 3 3 1 i. x2 − = 5 j. x3 − = 8 2. Translate each of the following into an equivalent sentence without the absolute value sign. 4 b. x < 2.7 a. x < 5 1 1 c. x d. x 1 2 2 e. x2 < 6 1 1 f. x < 2 2 3 g. x2 + < 2.5 h. x − < 3 1 i. x 3 j. x 5.5 It is unlawful to photocopy or reproduce content without permission from Dynamic Classroom Inc. This guide is licensed to the end user for PERSONAL use only. All rights reserved.

Grade 11 Mathematics CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 12 k. x > 1 3 l. x > 4 m. x2 > 6 1 n. x > 2.1 3 1 o. x + > 9 p. x2 − > 10 1 3. Graph the solution of each of the following on the number line. a. x = 3 b. x3 = 1 1 c. x2 + = 7 1 d. x < 2 e. x > 3 f. x − 1 2 1 g. x + 1 2 2 It is unlawful to photocopy or reproduce content without permission from Dynamic Classroom Inc. This guide is licensed to the end user for PERSONAL use only. All rights reserved.

Grade 11 Mathematics CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 13 4. What is the solution for each of the following absolute value equations? a. The absolute value of x is equal to 9. b. The absolute value of two times x is equal to 11. c. The absolute value of one half x is equal to 9. d. The absolute value of (x plus two) is equal to 8.6. e. The absolute value of (two times x minus one) is equal to 9. 5. Translate each of the following into an equivalent sentence without an absolute value sign. a. The absolute value of x is less than 5. b. The absolute value of x is greater than one half. c. The absolute value of two times x is less than 12. d. The absolute value of (x plus two) is greater than 9.5. e. The absolute value of (two times x minus one) is less than 1. 6 . Graph the solution for each of the following absolute value inequalities on the number line below. a. The absolute value of (two times x plus one) is equal to seven. b. The absolute value of one third times x is less than 1. It is unlawful to photocopy or reproduce content without permission from Dynamic Classroom Inc. This guide is licensed to the end user for PERSONAL use only. All rights reserved.

Grade 11 Mathematics CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 14 1.3 Powers and Roots of Numbers ▪ Recall from last year the difference between powers and roots of numbers. At that time we worked with the following ideas. ▪ If we raise a number to a positive integer power we multiply it by itself that 3 number of times. For example, 5 to the power 3 is 5 which is equal to 2 5 x 5 x 5 = 125 or 7 to the power 2 is 7 which is equal to 7 x 7 = 49 ▪ On the other hand, when we go in the opposite direction, instead of raising a number to a power, we find the root of a number. For example the second or square root of 16 is 4 since 4 is one of its two equal factors. Or the third or cube root of 27 is 3 because it is one of its three equal factors. ▪ We use the radical sign to represent the root of a number. The number under the radical sign is called the radicand and the root is called the index. An example is shown below. index radical sign n a radicand e.g. 2 64 ▪ the radicand is 64 - the square (or second) root of 64 ▪ the index is 2 ▪ (since square roots are very common we usually leave the index out so 2 that 64 = 64 3 1000 ▪ the radicand is 1000 - the cube (or third) root of 1000 ▪ the index is 3 Raising a number to a Power ▪ The process of squaring a number (raising it to the power 2) involves multiplying it by itself for a total of 2 factors. 3 3 3 9 2 2 e.g. 5 = 5(5) = 25, or ( ) = ( )( ) = 7 7 7 49 ▪ The process of raising a number to the power 3 involves multiplying it by itself twice for a total of 3 factors. 3 3 e.g. 2 = 2(2)(2) = 8, or 7 = 7(7)(7) = 343 Finding the Root of a number ▪ The process of finding the square root of a number is to determine one of its two equal factors. e.g. 25 = 5 ( )( ) 5 = 5(it should be noted that 25 = 5 - ( )( ) 5 - = − 5 as well). However, in most cases, when finding the square root of a number we will take only the positive square root of a number which is called the principal square root. It is unlawful to photocopy or reproduce content without permission from Dynamic Classroom Inc. This guide is licensed to the end user for PERSONAL use only. All rights reserved.

Grade 11 Mathematics CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 15 ▪ The process of finding the cube or third root of a number is to determine one of its three equal factors. e.g. 125 = 3 5 ( )( 5 )( ) 5 = 5 3 th ▪ If we wanted to find the 4 root of a number, it would be one of its four equal factors. The fifth root would be one of its five equal factors, and so on. Examples with Solutions 1. What is the index and radicand of each of the following radicals? Solution a. 121 ▪ Index is 2. (Remember that if no number is shown as the nth root, it is understood to be 2.) ▪ Radicand is 121. 8 ▪ Index is 3. b. 3 8 27 ▪ Radicand is . 27 4 c. 0 . 008 ▪ Index is 4. ▪ Radicand is 0.008. 2. Find the value of each of the following numbers written to a power. Solution 4 4 a. 3 ▪ 3 = 3(3)(3)(3) = 81 1 3 1 3 1 1 1 1 b. ▪ = = 2 2 2 2 2 8 4 c. (0.5) 4 ▪ (0.5) = (0.5)(0.5)(0.5)(0.5) = 0.0625 3. Find the value of each of the following radicals. Solution a. 81 ▪ It is the square root, so look for one of two equal factors under the radical sign ▪ 81 = 9• 9 = 9 1 ▪ It is the third root, so look for one of b. 3 three equal factors under the radical. 8 ▪ 3 1 = 3 1 • 1 • 1 = 1 8 2 2 2 2 4 c. 16 ▪ It is the fourth root, look for one of four equal factors under the radical. 16 = 4 2• 2• 2• 2 = 2 4 It is unlawful to photocopy or reproduce content without permission from Dynamic Classroom Inc. This guide is licensed to the end user for PERSONAL use only. All rights reserved.

Grade 11 Mathematics CHAPTER 1 – ABSOLUTE VALUE OF NUMBERS AND INTRO TO RADICALS 16 Summary of Ideas Since Can be shown as We say We write nd 2 36 6 a 2 root of 36 is 6 36 = 6 • 6 = 6 3 rd 125 5 a 3 root of 125 is 5 125 = 3 5• 5• 5 = 5 3 5 th 32 2 a 5 root of 32 is 2 5 32 = 5 2• 2• 2• 2• 2 = 2 4 th 16 2 a 4 root of 16 is 2 16 = 4 2• 2• 2• 2 = 2 4 Exercises 1.3 1. What is the radicand and the index of each of the following radicals? 3 a. 27 4 b. 9 4 c. 625 4 d. 2 121 3 e. 3 . 375 f. 0 . 0032 5 2. Write a radical expression with each given index and radicand. a. index = 2, radicand = 81 b. index = 3, radicand = 216 c. index = 4, radicand = 625 d. index = 2, radicand = 0.09 e. index = 5, radicand = 32 27 f. index = 3, radicand = 125 3. Find the value of each of the following numbers written to a power. 3 5 a. 3 b. (0.03) 2 3 9 2 c. d. 3 2 It is unlawful to photocopy or reproduce content without permission from Dynamic Classroom Inc. This guide is licensed to the end user for PERSONAL use only. All rights reserved.

# BC grade 3 1 Chapter Sample w Aboriginal Applications for Parents

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**Description: ** BC grade 3 1 Chapter Sample w Aboriginal Applications for Parents

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