GED Math Graphics: Illustrated Guide to Self-Confidence www.gedmathgraphics.net Unit 1 GED SKILLS Basic Concepts Decimals Fractions by Howard Myers, Ed.D. Copyright © 2009 by Howard Edward Myers. All rights reserved. 2 CONTENTS Basic Concepts To Solve Word Problems .................................................................................................... 3 To Multiply ………………………………………………………………………………………… 4 To Divide ………………………………………………………………………........................... 5 Powers and Roots ………………………………………………………................................... 6 To Multiply Powers – Add ................................................................................................... 7 To Divide Powers – Subtract .............................................................................................. 8 Place Values and Rounding #1 ………………………………………………………………… 9 Place Values and Rounding #2 ………………………………………………………………… 10 To Multiply Groups of Numbers .......................................................................................... 11 Order of Operations ……………………………………………………………………………… 12 Mean and Median***……………………………………………………................................. 13 Perfect Squares*** ........................................................................................................... 14 Decimals Line Up the Decimals ……………………………………………………………………………. 15 Place Values and Rounding #3 ………………………………………………………………… 16 Place Values and Rounding #4 ………………………………………………………………… 17 10, 100, 1000 ……………………………………………………………………………………...18 Scientific Notation*** ........................................................................................................ 19 Fractions Measurements ……………………………………………………………………………………. 20 Ruler Fractions …………………………………………………………………………………… 21 Pizza Fractions ………………………………………………………….................................... 22 To Add or Subtract Fractions …………………………………………………………………… 23 To Multiply Fractions …………………………………………................................................. 24 To Divide Fractions ............................................................................................................. 25 *** Typical GED Questions 3 To Solve Word Problems 1. The first key step is simply: Read The Problem. (All of it!) 2. The next key step is to find out: What’s The Question? 3. Then and only then you are ready to: Answer what is asked Ignore the rest 4. Don’t forget to check: Go back to be sure your answer makes sense. 4 To Multiply Do you know three ways to write “one times two”? Be ready when you see them! 1 x 2 1 2 (1)(2) 5 To Divide Here are three ways to write “one divided by two.” Look now at the line in the fraction “one half.” That line always means “divided by.” 1 ÷ 2 2 ) 1 1 2 6 Powers and Roots Look at this pattern: Power 3 2 Base The Power tells how many times to multiply the Base by itself. 32 = 3 x 3, not 3 x 2. √ means Square Root. √ 9 means “the square root of 9” It asks, “what number multiplied by itself equals 9?” And that answer is 3. √ 9 = 3 7 To Multiply Powers – Add 32 x 33 = 35 (3 3)(3 3 3) = (3 3 3 3 3) 3(2+3) = 35 n2 x n3 = n5 (n n)(n n n) = (n n n n n) n(2+3) = n5 8 To Divide Powers – Subtract 35 (3 3 3 3 3) 32 (3 3) 3(5 – 2) = 33 n5 (n n n n n) n2 (n n) n(5 – 2) = n3 = = 9 Place Values and Rounding #1 Example: Round 4,238 to the tens place. 4 , 2 3 8 Solution: 1. Underline your rounding place for tens. 4,238 2. Look one place to the right. Is the 8 equal to 5 or more? In this case, yes, 8 > 5. So: Add one to the rounding place, and Write all digits to the right of the rounding place as zero. 4,240 10 Place Values and Rounding #2 Example: Round 4,238 to the hundreds place. 4 , 2 3 8 Solution: 1. Underline your rounding place for hundreds. 4,238 2. Look one place to the right. Is the 3 equal to 5 or more? In this case, no, 3< 5. So: Do not change the rounding place, and Write all digits to the right of the rounding place as zero. 4,200 11 To Multiply Groups of Numbers Step 1 3 ( 5 + 4) (3 x 5) = 15 Step 2 3 ( 5 + 4) (3 x 4) = 12 Step 3 15 +12 27 Step 1 3 ( 5 – 4) (3 x 5) = 15 Step 2 3 ( 5 – 4) – (3 x 4) = –12 Step 3 15 –12 3 12 Order of Operations Follow these steps: For Example: (3 – 1) + 52(7 – 3) – (6 – 2) 2 1. Groups first. (3 – 1) + 52(7 – 3) – (6 – 2) 2 2. Powers, from left to right. 2 + 52 (4) – 4 2 3. x and ÷, from left to right. 2 + 25 (4) – 4 2 4. + and –, from left to right. 2 + 100 – 2 100 13 Typical GED Questions – Mean and Median The question gives a group of numbers, pictured like this: 3, 7, 2, 9, 5, 4 Find the mean (“average”) Add, then divide the total by how many numbers there are. 3 + 7 + 2 + 9 + 5 + 4 = 30 ÷ 6 = 5 Find the median (middle number) Arrange in order by size, find the number in the middle. 2 + 3 + 4 + 5 + 7 + 9 (4 + 5) ÷ 2 = 4.5 For two middle numbers, find the average. 14 Perfect Squares Typical GED Question n n x n = n2 √ n2 = n √ ?2 = ??? 1 1 x 1 = 1 √ 1 = 1 √100 = 10 2 2 x 2 = 4 √ 4 = 2 √400 = 20 3 3 x 3 = 9 √ 9 = 3 √900 = 30 4 4 x 4 = 16 √16 = 4 √1,600 = 40 5 5 x 5 = 25 √25 = 5 √2,500 = 50 6 6 x 6 = 36 √36 = 6 √3,600 = 60 7 7 x 7 = 49 √49 = 7 √4,900 = 70 8 8 x 8 = 64 √64 = 8 √6,400 = 80 9 9 x 9 = 81 √81 = 9 √8,100 = 90 10 10 x 10 = 100 √100 = 10 √10,000 = 100 11 11 x 11 = 121 √121 = 11 √1.21 = 1.1 12 12 x 12 = 144 √144 = 12 √1.44 = 1.2*** 13 13 x 13 = 169 √169 = 13 √1.69 = 1.3 14 14 x 14 = 196 √196 = 14 √1.96 = 1.4 15 15 x 15 = 225 √225 = 15 √ 2.25 = 1.5*** Look for one of these (especially THESE ***!) on your GED. 15 Line Up the Decimals Example: $20 + $200 + $20.20 + $2.02 + $0.22 2 0 . 2 0 0 . 2 0 . 2 0 2 . 0 2 . 2 2 TOTAL 2 4 2 . 4 4 And know your PLACE VALUES. Here is a worksheet for practice. _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ 16 Place Values and Rounding #3 Example: Round 4,238.9347 to the thousandths place. _________________________________________________________________________ 4 , 2 3 8 9 3 4 7 Solution: 1. Underline your rounding place for thousandths. 4,238.9347 2. Look one place to the right. Is the 7 equal to 5 or more? In this case, yes, 7 > 5. So: Add one to the rounding place, and Drop all digits to the right of the rounding place. 4,238.935 17 Place Values and Rounding #4 Example: Round 4,238.9347 to the hundredths place. _________________________________________________________________________ 4 , 2 3 8 9 3 4 7 Solution /Solución: 1. Underline your rounding place for hundredths. 4,238.9347 2. Look one place to the right. Is the 4 equal to 5 or more? In this case, no, 4 < 5. So: Do not change the rounding place, and Drop all digits to the right of the rounding place. 4,238.93 18 10, 100, 1000 . . . Move point right, one place per zero. To Multiply 1 2 3 4 x 1 0 = 1 2 3 4 1 2 3 4 x 1 0 0 = 1 2 3 4 1 2 3 4 x 1 0 0 0 = 1 2 3 4 0 Zero “place holder.” Move point left, one place per zero. To Divide 1 2 3 4 ÷ 1 0 = 1 2 3 4 1 2 3 4 ÷ 1 0 0 = 0 1 2 3 4 1 2 3 4 ÷ 1 0 0 0 = 0 0 1 2 3 4 Zero “place holder.” These easy shortcuts save precious time on the test! Do you use them? 19 Typical GED Question – Scientific Notation Do it like this 1 0 = 1.0 x 101 1 0 0 = 1.0 x 102 1 0 0 0 = 1.0 x 103 1 0 0 0 0 = 1.0 x 104 2 3 5 0 0 = 2.35 x 104 How to do it 1. Write a decimal place after the first digit. 2. To the right of the point: -keep only the first zero (if any), and all non-zero digits. 3. Drop the other zeros. 4. Count how many places the point moved left. 5. That is the power of your 10. ??? 20 Measurements 1 Yard 1 Foot 1 Foot 1 Foot 1 Foot 12 Inches ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 1 Pound = 16 Ounces 21 Ruler Fractions 1 2 1 3 4 4 1 3 5 7 8 8 8 8 0 1 0/8 1/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8 1/8 = 1/8 3/8 = 3/8 5/8 = 5/8 6/8 = 3/4 7/8 = 7/8 8/8 = 1 4/8 = 1/2 2/8 = 1/4 22 Pizza Fractions = 1/8 + 1/8 = 2/8 = 1/4 1 1 8 8 1 1 8 8 1 1 8 8 1 1 8 8 = 1/4 + 1/4 = 2/4 = 1/2 Two eighths make one fourth, and two fourths make one half. Study the picture until you are sure you understand it. It’s really important! 23 To Add or Subtract Fractions 1. Work only with the top numbers. 2. Bottom numbers must be equal. 3. If not—find a common denominator. 1 2 3 5 5 5 4 1 3 5 5 5 1 1 4 2 1 2 3 4 4 4 + = = + + = NO 24 To Multiply Fractions 1. Top times top, bottom times bottom. 2. Cancel where possible. 1 1 1 2 3 6 2 x 2 1 4 2 2 5 5 3 x 3 2 9 3 3 10 5 5 x 2 x = x = x = 25 To Divide Fractions 1. Invert the divisor, and multiply. 2. Cancel where possible. 1 1 3 2 2 1 1 2 2 3 1 3 2 2 5 3 3 2 2 3 3 5 2 5 ÷ = x = ÷ = x =