GED Prep Live: Number Sense & Basic Algebra

Learning Objectives By the end of this lesson, you will be able to: Apply number sense concepts, including ordering rational numbers, absolute value, multiples, factors, and exponents Add, subtract, multiply, divide, and use exponents and roots of rational, fraction, and decimal numbers Write, evaluate, and compute with expressions Write, manipulate, and solve linear equations Write, manipulate, and solve linear inequalities

The GED Math Test Total Time: 1 hour, 15 minutes Roughly 40 45 questions First section: 5 7 questions, no calculator allowed Second section: calculator allowed (Texas Instruments TI-30XS MultiView calculator) Slightly less than half of the content in the math test focuses on numerical problem solving Slightly more than half focuses on algebraic problem solving Questions test procedural skill, fluency, and problem solving The problem solving skills tested are taken from both academic and workforce contexts You are provided with an on-screen calculator (the Texas Instruments TI-30XS Multiview scientific calculator) for use on most questions You are also given reference sheets for formulas and calculator instructions

The GED Math Test The GED Math Test uses a variety of question types to assess your skills, including: Multiple choice Fill-in-the-blank Drop-down Drag and drop Hot-spot

Number Lines A number line represents numbers in order from least to greatest: As you move to the left along the number line, values decrease As you move to the right along the number line, values increase A number line can also represent fractions and mixed numbers: p. 214-215

Number Lines 1. What is the value of the point on the number line below? A. 0 B. C. 1 2 2 3 D. 1 p. 253

Powers and Roots Powers are a way to show repeated multiplication 5 5 can be expressed as 5 to the second power The number 5 appears in the multiplication problem two times, so it is written as 5 2 5 2 = 25 Roots are a way to show the number that produces a specific quantity when multiplied by itself The second (or square ) root of 49 is the number that, when squared, results in 49 49 = 7 p. 320

Powers and Roots Powers can also expand beyond exponents of 2 5 5 5 can be expressed as 5 to the third power The number 5 appears in the multiplication problem three times, so it is written as 5 3 5 3 = 125 You can also take more than the square root of a number The third (or cubic ) root of 125 is the number that, when cubed (raised to the third power), results in 125 3 125 = 5 p. 320

Powers and Roots A number raised to the first power equals itself 1 3 = 3 14 1 = 14 1,325 1 = 1,325 A number other than zero raised to the power of zero equals 1 0 0 0 3 = 1 14 = 1 1,325 = 1 A number raised to a negative exponent is equal to a fraction with a numerator of 1 = 1 2 3 3 2 = 1 5 14 14 5 = 3 1,325 1,325 3 1 p. 320

Powers and Roots 2. Determine the value of 10-3. You may NOT use a calculator. A. 1,000 B. C. 1 1000 1 1000 D. 1,000 p. 321

Order of Operations GED Math questions may require multiple operations to solve. To solve these questions, you need to tackle the operations in the correct order: from L to R from L to R P = parentheses (and brackets) E = exponents (and roots) M = multiplication D = division A = addition S = subtraction 2 + 3 (5 2) 2 6 = 2 + 3 (3) 2 6 = 2 + 3 (9) 6 = 2 + 27 6 = 2 + 162 = 164 p. 324

Order of Operations 3. In the expression below, what is the last operation you should perform to find the value of the expression? 2 10 5+ [ 7( ) + ( 6 2)( 3)] 10 A. Subtract 2 from 6. B. Add 5. C. Multiply by 2. D. Find the square of 10. p. 325

Absolute Value The absolute value of a number is its distance from zero on the number line The absolute value of 2 is 2, since 2 is two spaces away from zero on the number line The absolute value of 2 is also 2, since 2 is two spaces away from zero on the number line Absolute value is written using two straight lines around the number, variable, or expression 2 x 2d+6 p. 326

Absolute Value 4. Determine the value of 110 201. A. 311 B. 91 C. 91 D. 311 p. 327

Mixed Practice 5. Which of the following expressions has the least value? A. 3-3 B. 4 0 C. 4 1 D. 2-4 p. 321

Mixed Practice 6. An expression is shown. ( ) 22 + 6 (14 5) 3 17 14 Simplify the expression completely. A. 2.73 B. 28 C. 76 D. 97 p. 325

Mixed Practice 7. In the number line below, what is the value of A minus B? A. 1 B. 1 1 2 C. 2 D. 1 3 2 p. 253

Mixed Practice 8. Milania has a score of -65 points, and Chris has a score of 55 points. By how many points is Milania losing to Chris? A. 55 B. 65 C. 120 D. 150 p. 327

Mixed Practice 9. One container of floor cleaner holds 3.79 liters. If Zachary bought 4 containers, how many liters of cleaner did he buy? Note: You may not use a calculator. A. 0.9475 B. 7.79 C. 12.83 D. 15.16 p. 243

Mixed Practice 10. Janice is creating a computer spreadsheet. A portion of her work is shown below. Using the information from the spreadsheet, what is the value of the expression A1 * C1 * A3 / (B3 * A3)? (Hint: In a spreadsheet, the symbol * means multiplication.) A. 21 B. 7 C. 1 7 D. 21 p. 319

Translating English into Math An algebraic expression uses numbers, operations, and variables to show number relationships Variables are letters that represent unknown numbers Each time a letter is used within the same expression, it represents the same number. Use different letters to represent different numbers 2x + 3x + x 8d 2a + 3b p. 328

Translating English into Math p. 328

Translating English into Math 11. Appliance City employees earn an hourly wage plus commission. Wage options are shown in the table below. Chandra is paid under Option B. If h represents the number of hours worked and s represents Chandra s total sales, which of the following expressions could be used to find her weekly pay? A. 6 + h + 0.03s B. 6h + 0.03s C. 6s + 0.03h D. 0.03hs p. 329

Simplifying and Evaluating Expressions Simplifying a linear expression means performing all the operations you can within an expression When working with variables, you must remember an important rule: you can add and subtract like terms only 5x + 2y 7z + 4x The only like terms are 5x and 4x They can be combined: 5x + 4x = 9x 9x + 2y 7z p. 330

Simplifying and Evaluating Expressions 12. Find the value of 6 (x + 2) + 7 when x = 2. A. 17 B. 21 C. 31 D. 168 p. 331

Factors and Multiples When you multiply two numbers to create a product, the numbers you multiply are factors 2 5 = 10 2 and 5 are factors All of the pairs of factors that can be multiplied to create a given product can be listed out in a T-chart 10 Factors of 10: 1 2 1 2 10 5 10 5 1 10 = 10 2 5 = 10 1 10 = 10 2 5 = 10 p. 332

Factors and Multiples Any non-prime number can be factored down to all prime numbers This is known as Prime Factorization 48 6 8 2 3 4 2 2 2 2 x 3 x 2 x 2 x 2 = 48 p. 332

Factors and Multiples The Greatest Common Factor (GCF) of two integers is the largest number that divides evenly into both integers To find the GCF: break down both integers into their prime factorizations find the prime factors they have in common multiply those prime factors together 48 6 8 2 3 4 2 2 2 72 6 12 2 3 6 2 3 2 2 x 3 x 2 x 2 = 24 p. 332

Factors and Multiples The Least Common Multiple (LCM) of two integers is the smallest number that both integers divide into evenly To find the LCM: break down both integers into their prime factorizations select each prime factor the most times it occurs multiply these factors 20 5 4 2 2 30 5 6 3 2 2 x 2 x 3 x 5 = 60 p. 332

Factors and Multiples 13. All numbers that are evenly divisible by both 6 and 14 are also divisible by which of the following numbers? A. 8 B. 12 C. 21 D. 28

Equations and Inequalities An equation is a mathematical statement that demonstrates the equality between two expressions 4 x 2 = 8 An inequality is a mathematical statement that connects two unequal expressions Unlike equations, you must flip the inequality sign when you multiply or divide by a negative value. Instead of an equal sign, an inequality will contain on of the following symbols: Greater than ( > ) Less than ( < ) Greater than or equal to ( ) 2x + 11 7 Less than or equal to ( ) 2x 18 11 11 2x 18 x 9 p. 358

Equations and Inequalities 14. Solve: 3x 7 > 5. A. x < 4 B. x > 4 C. x < 2 3 D. x > 2 3 p. 359

Mixed Practice 16. The perimeter of a square is less than or equal to 64 inches. Which of the following represents the possible measures of the side of the square, in inches? From the formula sheet: Perimeter of a square: P = 4s A. s 16 B. s 16 C. s 8 D. s 64 p. 359

Mixed Practice Susan is in charge of planning Midvale Hospital s parent education classes. She uses the table below to determine the cost of each class. p. 325

Mixed Practice 18. A local foundation has offered to pay 75% of the cost of infant care classes, and the hospital will cover any remaining costs. There are 28 parents enrolled in the upcoming class. Which of the following expressions could be used to find the amount the hospital will pay? A. (100 75) (28) (30) B. (28) (30) (0.75) (30) C. (1 0.75) (28) (30) D. (1 0.75) (30) + 28 p. 325

Mixed Practice 11 2 19. What is the value of? 75 45 A. B. C. D. 23 225 43 225 3 10 3 5 p. 333

Mixed Practice 20. What is the value of the point on the number line below? A. 2 B. 2.5 C. 2.7 D. 2.8 p. 253

Mixed Practice 22. The cube shown below measures 6 inches on each side. Which of the following expressions represents the volume of the cube? From the formula sheet: Volume of a rectangular prism: V = lwh A. 6 1 B. 6 2 C. 6 3 D. 6 6 p. 321

Learning Objectives Now that you have completed this lesson, you should be able to: Apply number sense concepts, including ordering rational numbers, absolute value, multiples, factors, and exponents Add, subtract, multiply, divide, and use exponents and roots of rational, fraction, and decimal numbers Write, evaluate, and compute with expressions Write, manipulate, and solve linear equations Write, manipulate, and solve linear inequalities

Homework Don t stop now! Practice is important. To ensure you understand today s lessons, do the following for homework: Number Sense and Problem Solving Practice Questions: p. 234-236, #1-20 Decimals and Fractions Practice Questions: p. 256-258, #1-20 Algebra Basics, Expressions, and Polynomials Practice Questions: p. 348-350, #1-3, 6-8, 10, 12, 13, 18 Equations, Inequalities, and Functions Practice Questions: p. 384-386, #1, 2, 5, 8-12, 19, 21

Answer Key 1. C 2. C 3. B 4. B 5. A 6. B 7. B 8. C 9. D 10. B 11. B 12. C 13. C 14. B 15. A 16. A 17. A 18. C 19. A 20. D 21. B 22. C

Number Lines 1. What is the value of the point on the number line below? A. 0 B. C. 1 2 2 3 D. 1 1 1 From 0 to 3, the distance is 3. So, the distance between each 1 tick mark on the number line is. 3 1 1 2 + = 3 3 3 p. 253

Powers and Roots 2. Determine the value of 10-3. You may NOT use a calculator. A. 1,000 B. C. 1 1000 1 1000 D. 1,000 10 3 1 = 10 3 1 = 10 10 10 = 1 1000 p. 321

Order of Operations 3. In the expression below, what is the last operation you should perform to find the value of the expression? A. Subtract 2 from 6. B. Add 5. C. Multiply by 2. D. Find the square of 10. 2 10 5+ [ 7( ) + ( 6 2)( 3)] 10 100 5 + [ 7( ) + (6 2)(3)] 10 5 + [ 7(10) + (6 2)(3)] 5 + [ 7(10) + (4)(3)] 5 + [ 70 + 12] 5 + 82 p. 325

Absolute Value 4. Determine the value of 110 201. A. 311 B. 91 C. 91 D. 311 110 201 = 91 = (91) = 91 p. 327

Mixed Practice 5. Which of the following expressions has the least value? A. 3-3 B. 4 0 C. 4 1 D. 2-4 1 1 A. 3 3 = = 3 3 27 0 B. 4 = 1 1 C. 4 = 4 1 1 D. 2 4 = = 2 4 16 p. 321

Mixed Practice 6. An expression is shown. ( ) 22 + 6 (14 5) 3 17 14 Simplify the expression completely. A. 2.73 B. 28 C. 76 D. 97 22 + 6 [ (14 5) 3 (17 14) ] = 22 + 6 [(9) 3(3)] = 22 + 6 (9 9) = 22 + 6 (1) = 22 + 6 = 28 p. 325

Mixed Practice 7. In the number line below, what is the value of A minus B? A. 1 1 3 1 = 1 1 2 2 B. 1 1 2 C. 2 D. 1 3 2 p. 253

Mixed Practice 8. Milania has a score of -65 points, and Chris has a score of 55 points. By how many points is Milania losing to Chris? A. 55 B. 65 C. 120 D. 150 M C 65 55 55 ( 65) = 120 p. 327

Simplifying and Evaluating Expressions 12. Find the value of 6 (x + 2) + 7 when x = 2. A. 17 B. 21 C. 31 D. 168 6 (2 + 2) + 7 6 (4) + 7 24 + 7 31 p. 331

Factors and Multiples 13. All numbers that are evenly divisible by both 6 and 14 are also divisible by which of the following numbers? A. 8 B. 12 C. 21 D. 28 Find a number that is evenly divisible by both 6 and 14, then determine which answer choice is also a factor of that number Multiples of 14: 14, 28, 42 Multiples of 6: 6, 12, 18, 24, 30, 36, 42 42 cannot be divided by 8, 12, or 28 without a remainder 42 21 = 2

Equations and Inequalities 14. Solve: 3x 7 > 5. A. x < 4 B. x > 4 C. x < 2 3 3x 7 > 5 + 7 + 7 3x > 12 3x 12 > 3 3 D. x > 2 3 x > 4 p. 359

Mixed Practice 15. The sum of 3 times a number and 4 times another number is divided by the sum of 2 and a third number. Which of the following expressions represents this series of operations? A. (3x + 4y) (2 + z) B. 3x + 4y (2 + z) C. 3x + 4y 2 + z D. (3x + 4y) 2z a number : x another number : y a third number : z 3 times a number : 3x 4 times another : 4y sum of : (3x + 4y) divide by : (2 + z) the sum of 2 and a third number : 2 + z p. 329

Mixed Practice 17. Which equation shows how to find the amount of money, c, the participants will pay based on the number of people, p, who attend both the Infant Care and the Teaching Your Child to Read workshops? A. c = 30p + 60p B. c = 30 + 60 + p C. c = 30p (60p) D. c = 35p + 60p c = total cost to participants p = number of participants Infant Care: $30 per person = 30p Teaching Your Child to Read: $60 per person = 60p c = 30p + 60p

Mixed Practice 11 2 19. What is the value of? 75 45 A. B. C. 23 225 43 225 3 10 Find LCM of 75 and 45: 75 = 3 x 25 = 3 x 5 x 5 45 = 9 x 5 = 3 x 3 x 5 11 33 = 75 225 2 10 = 45 225 D. 3 5 LCM is 3 x 3 x 5 x 5 = 225 33 10 23 = 225 225 225 p. 333

Mixed Practice 20. What is the value of the point on the number line below? A. 2 B. 2.5 C. 2.7 D. 2.8 Spaces between tick marks: 2.2 2 = 0.2 Distance from 2.6 to point on line: +0.2 2.6 + 0.2 = 2.8 p. 253

Mixed Practice 21. A school is selling t-shirts as a fund raiser. For each shirt that is sold, the school will earn 7 dollars. Which inequality shows how to find the minimum number of t-shirts, s, the school must sell in order to earn at least $2,500? A. 2,500 < 7s B. 2,500 7s C. 2,500 > 7s D. 2,500 7s s = number of t-shirts sold For each shirt... the school will earn 7 dollars : 7s in order to earn at least $2,500 = amount of money earned 2,500 7s 2,500