Pre-Algebra Unit 1: Number Sense Unit 1 Review Packet Target 1: Writing Repeating Decimals in Rational Form Remember the goal is to get rid of the repeating decimal so we can write the number in rational form. To get rid of the repeating decimal, set up an equation where x equals the repeating decimals. Example 1: 0.7 Let x = 0.777 Since 1 digit repeats, multiply both sides by 10 Remember to multiply by ten, move decimal one place to right to make the number 10 times bigger. 10x = 7.777 Now subtract x from both sides (this gets rid of repeating decimal) 10x = 7.777-1x = -.777 9x = 7 Now solve one step equation for x: 9x 9 = 7 9 so the rational form is 7 9 Example 2: 0.6 Let x = 0.666 Since 2 digits repeats, multiply both sides by 100 Remember to multiply by a hundred, move decimal two places to right to make the number 100 times bigger. 100x = 6.666 Now subtract x from both sides (this gets rid of repeating decimal) 100x = 6.666-1x = -.666 99x = 6 Now solve one step equation for x: 99x = 6 99 99 so the rational form is 6 99 = 4 11
Example 2: 0.58 Let x = 0.58 Since 1 digit repeats, multiply both sides by 10 Remember to multiply by ten, move decimal one place to right to make the number 10 times bigger. 10x = 5.8 Now subtract x from both sides (this gets rid of repeating decimal)cancel out repeating parts that are on top of each other and subtract the rest (you ll get rid of decimal later) 10x = 5.8-1x = -.58 9x = 5.25 Now solve one step equation for x: 9x 9 = 5.25 9 To get rid of decimal, multiply top and bottom by 100 because that will move decimal to end to make numerator a whole number: 5.25(100) = 525 = 7 9(100) 900 12 Practice Problems: Write each of the following in rational form (as a fraction) Show All Work On a Separate Sheet to Receive Credit. 1. 0.2 2. 0.54. 0.26 Target 2: Perfect Square and Cube Roots Remember the square root of a number is finding the number that multiplied by itself to give you the square root. Example: 49 = 7 because 7 7 = 49 Remember that when working with area of a square, to find the length of each side of the square, you take the square root of the area.
Remember the cube root of a number is finding the number multiplied by itself times to give you the cube root. Example: 729 = 9 because 9 9 9 = 729 Remember that when working with volume of a square, to find the length of each side of the cube, you take the cube root of the volume. Remember if you are working with fractions you do the numerator and denominator separately and make sure to simplify your fraction if necessary. Example: 9 144 = 9 = and 144 = 12 so 9 144 = 12 = 1 4 Practice Problems: Simplify Each of the Following. Show All Work On a Separate Sheet to Receive Credit. 1. 169 2. 512. 64 121 4. 8 64 5. Find the length of a square that has an area of 100 units 2. 6. Find the length of a cube that has a volume of 27 units. Target : Rational and Irrational Numbers Remember: Whole Numbers: numbers that are positive and do not have fractions or decimals in them. These include positive perfect square and cube roots. Examples: 0, 5, 16, 64
Integers: positive and negative numbers that do not have fractions or decimals in them. These include both positive and negative perfect square and cube roots. These include ALL whole numbers. Examples: -8, - 6, - 8, 0, 12, 144, 216 Rational Numbers: numbers that can be written as fractions. These include ALL whole numbers and ALL integers (including all perfect square and cube roots) as well as ALL fractions (both regular, improper and mixed numbers) and ALL decimals that stop or repeat. Examples: -1, - 49, - 27, 0, 1,287, 121, 216,, 19, 6, 0.467, -1.8 5 7 11 Irrational Numbers: numbers that CAN NOT be written as fractions. These include decimals that do not stop and do not repeat, any numbers that have Pi (π) and any NON-PERFECT square and cube roots. Examples: -0.24, - 8, - 12, 1.010010001, 2, 9, 2π, π Practice Problems: Next to each number, write ALL categories of numbers as listed above that the number belongs to. 1. 0 2. π. 0.212212221 4. -20 5. 12,849 6. 10 7. 729 8. 4.81 9. 1 2
Target 4: Rational Approximation To find the rational approximation of an irrational square root, first find the two perfect square roots it falls between. Then set up a fraction to find the decimal part. If you are working with Pi, estimate Pi as.14 and perform the given operation. If a square root has a number in front of it, this means to multiply your rational approximation by that front number. Example: Find the rational approximation of 2 6 Step 1: 6 falls between 4 and 9 so Since 6 is closer to 4 than to 9, it will come before the halfway point. This means the answer to the nearest then has to be one of the following: 2.1 2.2 2. 2.4 2.5 To find which answer it is, we set up a fraction of our number. The distance from the beginning of the number line to our square root is the distance from 4 to 6 which is 2. This is the numerator. The whole distance across the number line us from 4 to 9 which is 5. This is the denominator so out fraction is 2 5. We then use long division to change the fraction to a decimal. We only need to divide until we get to the hundredths place value so we can use that to round to the nearest tenth. 0.40 2 5 = 2.0 so my answer is 2.4. Now I finish by multiply by 2 5 4 6 9 2 So 2.4 * 2 = 4.8. Example: 2π (multiply 2 and π) so 2 *.14 = 6.28 which rounds to 6. to nearest tenth. Practice Problems: Find the rational approximation of each of the given numbers to the nearest tenth: 1. 12 2. π 2