TABLE OF CONTENTS. Introduction to Finish Line Indiana Math 10. UNIT 1: Number Sense, Expressions, and Computation. Real Numbers
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1 TABLE OF CONTENTS Introduction to Finish Line Indiana Math 10 UNIT 1: Number Sense, Expressions, and Computation LESSON 1 8.NS.1, 8.NS.2, A1.RNE.1, A1.RNE.2 LESSON 2 8.NS.3, 8.NS.4 LESSON 3 A1.RNE.3 LESSON 4 A1.RNE.4 LESSON 5 8.C.1 LESSON 6 8.C.2 LESSON 7 A1.RNE.5 LESSON 8 A1.RNE.7 LESSON 9 A1.RNE.6 LESSON 10 A1.RNE.6 Real Numbers Exponents and Square Roots Rational Exponents Radical Expressions Solving Problems with Rational Numbers Scientific Notation Simplifying Rational Expressions Polynomial Expressions Factoring Polynomials Factoring Trinomials UNIT 1 REVIEW UNIT 2: Geometry and Measurement LESSON 1 8.GM.1 LESSON 2 8.GM.2 LESSON 3 8.GM.3, 8.GM.6 LESSON 4 8.GM.4, 8.GM.5 LESSON 5 8.GM.7, 8.GM.8. 8.GM.9 Three-Dimensional Objects and Cross Sections Volume and Surface Area Transformations Congruence and Similarity Pythagorean Theorem UNIT 2 REVIEW UNIT 3: Linear Equations, Inequalities, and Functions LESSON 1 8.AF.1, 8.AF.2, A1.L.1 LESSON 2 A1.L.11 LESSON 3 A1.L.1 LESSON 4 A1.L.8 LESSON 5 8.AF.1, A1.L.2 LESSON 6 A1.L.3 LESSON 7 8.AF.3, A1.F.1, A1.F.3, A1.F.4 LESSON 8 8.AF.4, A1.F.2 Solving Linear Equations in One Variable Equations and Formulas Solving Linear Inequalities in One Variable Compound Linear Inequalities Representing Linear Relationships in One Variable Algebraic Proportions Understanding Functions Graphs of Functions
2 LESSON 9 8.AF.5, 8.AF.6, 8.AF.7, Writing Linear Equations A1.L4, A1.L5, A1.L.6 LESSON 10 8.AF.7,A1.L.4, A1.L.5 LESSON 11 A1.L.7 LESSON 12 A1.L.9. A1.L.10 Representing Linear Functions Representing Linear Inequalities Absolute Value Equations UNIT 3 REVIEW UNIT 4: Systems of Equations and Inequalities LESSON 1 8.AF.8, A1.SEI.1 LESSON 2 A1.SEI.2 LESSON 3 A1.SEI.3 LESSON 4 A1.SEI.4 LESSON 5 A1.SEI.4 Understanding Systems of Equations Solving Systems of Linear Equations Writing Systems of Linear Equations Systems of Linear Inequalities Graphing Systems of Linear Inequalities UNIT 4 REVIEW UNIT 5: Quadratic and Exponential Equations and Functions LESSON 1 A1.QE.1 LESSON 2 A1.QE.2 LESSON 3 A1.QE.3 LESSON 4 A1.QE.4, A1.QE.6 LESSON 5 A1.QE.4 LESSON 6 A1.QE.5 LESSON 7 A1.QE.6, A1.QE.7 Comparing Linear and Exponential Functions Representing Exponential Functions Graphing Exponential and Quadratic Functions Solving Quadratic Equations The Quadratic Formula Writing Quadratic Equations Interpreting Quadratic Functions UNIT 5 REVIEW UNIT 6: Data Analysis, Statistics, and Probability LESSON 1 A1.DS.1 LESSON 2 8.DSP.1, A1.DS.2 Sampling and Data Bivariate Numerical Data LESSON 3 8.DSP.1, 8.DSP.3, Lines of Best Fit A1.DS.3, A1.DS.4 LESSON 4 A1.DS.5 Bivariate Categorical Data LESSON 5 8.DSP.4, 8.DSP.5, Compound Probability 8.DSP.6 LESSON 6 A1.DS.6 Misleading Statistics UNIT 6 REVIEW Glossary Formula Reference Sheet
3 LESSON 8 Polynomial Expressions A1.RNE.7 A polynomial is an algebraic expression of one or more terms, which are parts of an expression separated by the operations of addition or subtraction. A monomial is a polynomial with only one term, like 8, 8y, or 8y 2. A binomial is a polynomial with two terms, like 6x 2 7. A trinomial has three terms, like x Terms of a polynomial can be variables that are raised to whole-number powers, such as x 2 or 6y 3 z. The exponents, however, must be nonnegative for the expression to be a polynomial. For example, 2x 2 is not a polynomial. A polynomial has no variable in the denominator. 5 x is not a polynomial. A variable written without an exponent has a power of 1. x 5 x 1 Adding and Subtracting Polynomials To add polynomials, simply combine like terms. See how it works in this example: (x 2 2 9x 1 3) 1 (2x 2 1 x 2 4) x 2 1 2x 2 2 9x 1 x x 2 2 8x 2 1 To subtract polynomials, change the sign of each term in the second polynomial and then add. For example, (2x 3 1 4x 2 2 6x) 2 (4x 3 2 3x 2 1 5x) Subtracting is adding the opposite. a 2 b 5 a 1 _ b+ is written as (2x 3 1 4x 2 2 6x) 1 ( 4x 3 1 3x 2 2 5x) which equals 2x 3 1 7x x. Try this sample question. SAMPLE 1 What is (4x 2 2 6) 2 (x 2 1 2) in simplest form? A 3x B 3x C 5x D 5x Changing the sign of each term in the second polynomial gives the addition problem (4x 2 2 6) 1 ( x 2 2 2), which simplifies to 3x Choice A is correct. 44 UNIT 1 Number Sense, Expressions, and Computation
4 Try this sample question. SAMPLE 2 Write the simplified form of this expression. (6x 2 1 4y 2 ) 1 (5x 2 2 3y 2 ) To add these binomials, simply combine like terms: 6x 2 1 5x x 2 and 4y 2 2 3y 2 5 y 2. So the simplified form is 11x 2 1 y 2. Multiplying Polynomials To multiply polynomial expressions, multiply each term in the first polynomial by each term in the second polynomial. Then combine like terms, if needed, to simplify the resulting polynomial expression. For example: (4x 1 3)(x 2 2) 5 4x(x 2 2) 1 3(x 2 2) 5 4x 2 2 8x 1 3x x 2 2 5x 2 6 When multiplying variable terms with like bases, add the exponents. x 5 x 2 5 x x 7 Try this sample question. SAMPLE 3 The length of a rectangle is 5x 1 1 units long. The width of the rectangle is 2x 1 5 units long. What is the area, in square units, of the rectangle? A 7x C 10x 2 1 7x 1 5 B 10x D 10x x 1 5 The area of a rectangle is length 3 width. So, the area of this rectangle is (5x 1 1)(2x 1 5). To find the area, multiply each term in 5x 1 1 by each term in 2x 1 5: (5x 1 1)(2x 1 5) 5 5x(2x 1 5) 1 1(2x 1 5) 5 10x x 1 2x 1 5 Combine like terms. This gives 10x x 1 5. Choice D is correct. UNIT 1 Number Sense, Expressions, and Computation 45
5 Dividing Polynomials by Monomials Sometimes you may be asked to divide a polynomial expression by a monomial expression. Each term of the polynomial expression must be divided by the monomial. In the example below, the monomial 4p divides each term of the polynomial, creating separate fractions. Then common terms are canceled within each fraction to simplify the expression. When dividing variable terms with like bases, subtract the exponents. x 5 4 x 2 5 x x 3 16p 5 2 4p p 2 16p 5 5 4p 4p 2 4p 3 4p 2 12p 2 4p 5 4p 4 2 p 2 2 3p Try this sample question. SAMPLE 4 What is the simplified form of the expression below? _9x 4 1 6x x x 2 A 3x 2 1 2x 2 5 C 3x 2 1 6x x 2 B 6x 2 1 3x 2 12 D 6x 2 1 6x x 2 This item can be rewritten as the sum and difference of separate fractions: 9x 4 1 6x x 2. 3x 2 3x 2 3x 2 Simplifying each fraction gives 3x 2 1 2x 2 5. Choice A is correct. Polynomials and Closure Polynomials share a very important property with real numbers. When you add, subtract, or multiply any two real numbers, the result is always a real number. Likewise, when you add, subtract, or multiply any two polynomials, the result is always a polynomial. This property is called closure, and the set of all polynomials is closed under the operations of addition, subtraction, and multiplication. For example, the sum of the polynomials x and 2x 1 5 is the polynomial x 3 1 2x 1 4, their difference is x 3 2 2x 2 6, and their product is 2x 4 1 5x 3 2 2x 2 5. All three combinations yield a polynomial. However, closure fails under division. Although sometimes dividing one polynomial by a second yields a third polynomial, this is not always the case. 46 UNIT 1 Number Sense, Expressions, and Computation
6 For example, dividing the polynomial x 2 1 3x 1 2 by the polynomial x 1 2 yields the polynomial x 1 1. However, dividing the polynomial x 2 1 3x 1 2 by the polynomial x yields the expression x x 1. The term 2x 1 has a negative exponent, so this expression is not a polynomial. Try this sample question. SAMPLE 5 For the polynomials 2x 1 3 and 6x 2 1 x 2 15, verify that the sum, difference, and product are also polynomials. The sum (2x 1 3) 1 (6x 2 1 x 2 15) simplifies to 6x 2 1 3x 2 12, the difference (2x 1 3) 2 (6x 2 1 x 2 15) simplifies to 6x 2 1 x 1 18, and the product (2x 1 3)(6x 2 1 x 2 15) simplifies to 12x x x All three resulting expressions have terms with only non-negative integer exponents, and are therefore also polynomials. INDEPENDENT PRACTICE Read and solve each problem. 1 What is the simplified form of this expression? (2w 3 1 4w 2 1 5) 1 (3w 3 1 w 2 1 4w) A 5w 3 1 5w B 5w 3 1 5w 2 1 9w C 5w 3 1 5w 2 1 4w 1 5 D 5w 3 1 4w 2 1 5w 1 4 UNIT 1 Number Sense, Expressions, and Computation 47
7 2 Write each expression in simplest form. Show your work. A 5y 2 (8y 3 2 2y 2 1 7y) C (5x 1 3)(x 2 2 x 2 4) B 3x(2x 1 y) 2 y(x 1 2y 2 1) D (9y 2 2 4y 1 3) 2 (5y 2 1 2y 2 7) 3 Write an expression equivalent to 2s(4s 2 t 1 3) 2 t( 3s 1 5t) 1 6(s 2 2t 2 3) by entering the coefficients in each empty box. s 2 1 t 2 1 st 1 s 1 t 1 48 UNIT 1 Number Sense, Expressions, and Computation
8 20x 4 Simplify: x 7 2 4x 6. 4x 6 A 5x 2 1 4x 2 1 B 5x x 2 4 C 16x x 2 1 D 16x x The diagram shows a square and the length of one side. x 4 A Write an expression in simplest form that represents the perimeter of the square. B Write an expression in simplest form that represents the area of the square. 6 Divide 15y y 6 2 9y 5 by 3y 3. UNIT 1 Number Sense, Expressions, and Computation 49
9 7 The perimeter of the triangle shown is 19z z 9 8z 3 A Write an expression in simplest terms to represent the length of the remaining side. Show all steps in your work. B The height of the triangle is represented by 2y 1 3z. Write an expression that models the area of the triangle. Show your work. 50 UNIT 1 Number Sense, Expressions, and Computation
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