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6th Grade Study Guide for Chapter 2: Decimals Games you will play for decimals: Doggone Decimal (Everyday Math) Students form two 2-digit numbers with decimal points whose product is closest to the target number for each round. Smallest to Greatest (Investigations) Students strategize through placing decimal cards on a 3 by 3 grid such that they are able to play 9 decimal numbers in order before their opponent. Lessons 2-1+2-2: Reading and Writing Very Large and Very Small Numbers In standard notation digits take on the value of the place value they occupy. For example: In 8,654.15 the 6 represents 6 hundred, while the 5 represents 5 tens or fifty. In expanded notation numbers are written as the sum of the values of its digits. For example: 8,654.15 would be written as (8*1000) + (6*100) + (5*10) + (4*1) + (1*1/10) + (5* 1/100). In number-and-word notation numbers are written as numerals and words. For example: 8,654.15 would be written as 8 thousand, 6 hundred, fifty-four, and fifteen hundredths. Practice: Standard notation Expanded notation Number-and-word notation 754 (3* 1/10) + (5* 1/100) Seventy-five hundredths (9*100) + (5* 10) Sixty-six thousandths Answer Key: Standard notation Expanded notation Number-and-word notation 754 (7*100) + (5*10) + (4*1) 7 hundred, fifty-four 0.35 (3* 1/10) + (5* 1/100) Thirty-five hundredths 0.75 (7* 1/10) + (5* 1/100) Seventy-five hundredths 950 (9*100) + (5* 10) 9 hundred, fifty 0.065 (6* 1/100) + (5* 1/1000) Sixty-six thousandths Lessons 2-3 to 2-6: Addition, Subtraction, and Multiplication of Decimals When adding or subtracting decimals, you have to line the decimals carefully so that you are adding ones to ones, tenths to tenths, etc. For example, 0.50 + 1.50 would be written as follows: 0.50 + 1.50 The answer is: 2.00 When multiplying decimals, move the decimal to the right the same number of places as the exponent when the exponent is positive. For example, $0.50 * 100 students = $50.00 because we moved the decimal two places to the right. Move the decimal to the left the same number of places as the exponent when the exponent is negative. For example, $0.50 * 0.1 (or 1/10) = $0.05. Practice: Player One: You’re playing Doggone Decimal. The target number is 100. You get the digits 2, 5, 3, and 8. How will you build two decimal numbers whose product is closest to the target number? Player Two: Your opponent gets the digits 2, 1, 5, and 9. How should he/ she build two decimal numbers? Who wins? How much did they win by? Answer Key: Player One: Possible Products: 23. * 5.8= 133.4 25. * 3.8= 95.0 28. * 3.5= 98.0 32. * 5.8= 185.6 35. * 2.8= 98.0 38. * 2.5= 95.0 52. * 3.8= 197.6 53. * 2.8= 148.4 58. * 2.3= 133.4 82. * 3.5= 287.0 83. * 2.5= 207.5 85. * 2.3= 195.5 Player Two: Possible Products: 21. * 5.9= 123.9 25. * 1.9= 47.5 29. * 5.1= 147.9 12. * 5.9= 70.8 15. * 9.2= 138.0 19. * 5.2= 98.8 52. * 1.9= 98.8 51. * 2.9= 147.9 59. * 2.1= 123.9 92. * 1.5= 138.0 91. * 2.9= 263.9 95. * 114.0 Player Two will win IF he/she does 52. * 1.9= 98.8 or 19. 8 5.2= 98.8, because the best that player one can do is 35. * 2.8= 98.0 or 28. * 3.5= 98.0. Player Two wins by 0.8 points. This was a close game! *Play Doggone Decimal for more practice with multiplying decimals. Lessons 2-8: Dividing Decimals and Interpreting Remainders Use estimation to help you to place your decimal point in your answer as you did with the multiplication of decimals. Make sure that your answer is reasonable. For example, 80 brownies divided by 100 people will result in less than one whole brownie each, since 80 / 100 = 0.8000. 08.000 would not make sense, since this is more than one. Another way to think of dividing is multiplying by a decimal; so the problem above could be represented as: 80 * .01= 0.80, which is less than one brownie each. You can add as many zeroes as you’d like to whole numbers to the right of the decimal point. For example, if you wanted to find your average score for quizzes: 87, 63, 95, and 68, you would compute the sum (313) then divide this by 4. You’ll find that 4 * 78 is 312, so you have a remainder of 1. If you add a decimal point and 2 zeroes, you can continue to divide until you get 78.25 with no remainder. 78.25 4/ 313.00 28 33 32 10 8 20 20 0 Practice: You go out to eat with 3 friends. The bill comes to $51. How much does each person owe? Answer Key: 12.75 4l 51.00 4 11 8 30 28 20 20 0 Lessons 2-9: Scientific Notation for Very Large and Very Small Numbers 102 has a positive exponent and means 10 * 10 or 100. 10-2 has a negative exponent and means 1/10 * 1/10 or 1/100. This can be written as decimals: 0.1 * 0.1 = 0.01. Zero exponents are always equal to one, while an exponent of one means the number times one. For example, 100 = 1, while 101 = 10 * 1 or 10; likewise 20 = 1, while 21 = 2. We use powers of ten to write very large numbers or very small numbers using scientific notation. Numbers written in scientific notation are written as products of a factor more than one and less than 10 and a power of ten. For example, the width of a hair is about 2.5 * 10-7( or 0.0000025), while the number of hairs on average will be about 5 * 106 (or 5,000,000). Practice: Standard notation Scientific notation Expanded notation 356,900,000 9.8 * 105 (3*0.01) + (5* 0.0001) -3 7.45 * 10 (5* 1000)+(4* 100)+(3*1) Answer Key: Standard notation Scientific notation Expanded notation 356,900,000 3.569 * 108 (3* 100,000,000)+ (5*10,000,000) + (6*1,000,000)+(9*100,000) 980,000 9.8 * 105 (9*100,000)+(8*10,000) 0.0305 3.05 * 10-2 (3*0.01) + (5* 0.0001) 0.0745 7.45 * 10-3 (7*0.01)+(4*0.001)+(5*0.0001) 5,403 5.403 * 103 (5* 1000)+(4* 100)+(3*1) Expectations Goals: Key Understandings: Students will understand that… Decimals have a distinct position on a number line and in an ordered list, which may seem counter-intuitive at first, such that the longer numeral is not always the larger. [Example: 0.099999 is less than 0.199.] There are equivalent forms for any real number, including standard, number-and- word, expanded and scientific notations. The results of an operation depend on the types of numbers involved: multiplying and dividing decimals might seem counter-intuitive as multiplying by a decimal gives you a smaller value, while dividing by a decimal gives you a larger value. Essential Questions: Where would decimals be positioned on a number on the number line and how can you compare the values of various decimals? What is the relationship between place value and digits to the left and the right of the decimal? What do you know about decimals as compared to whole #s and how does this help you to determine the reasonableness of your answer? How can estimation be used to determine the reasonableness of an answer and when and how should you round numbers to various place values? Lesson by lesson expectations: Students will know… Students will be able to… There are equivalent forms for any real Read and write very large numbers. number, including standard, number- Interpret and translate between and-word, and expanded notations. expanded, number-and-word, and standard notations for very large numbers. (2.1) Order very large numbers via the __________________________________ game Number Top-It (p 463+464 MM) ___________________________________________________ Decimals represent specific points on Read and write very small numbers. a number line and can be organized in (2.2) an ordered list. Interpret and translate between standard, number-and-words, and You must line up the decimals when expanded, notations as well as adding or subtracting so that you are fractions, and decimals. adding or subtracting similar place values (ones with ones and tenths with Order decimal numbers via the game tenths). High-Number Toss (p455 MM). _ Add and subtract decimals. (2.3) Explore precision (Ex: 3.0) and ________________________________________________ rounding decimals (Ex: 2.9 to 3). We can use powers of 10 to facilitate Multiply by powers of ten while mental math. exploring exponential notation, both Positive exponents involve multiplying positive and negative. (2.4) tens, while negative exponents involve Play Doggone Decimal Game (p 310 multiplying tenths. SRB). ____________________________________ ______________________________________ The relationship between the powers of Estimate products of decimals. (2.5) 10 and place values. Practice decimal multiplication. Play Multiplication Bull’s Eye (p328 Multiplying and dividing decimals might SRB). seem counter-intuitive as multiplying by a decimal gives you a smaller _______________________________ value, while dividing by a decimal Locate decimal points in products. gives you a larger value. (2.6) Multiply decimals using the lattice method. *Only for those students who already prefer to use the lattice method. __________________________________ Divide whole numbers. (2.7) Partial quotients method reviewed. _________________________________ ___________________________________ Divide decimals. (2.8) Remainders may be represented in a number of ways depending upon the Write remainders as decimals. situation. Interpret remainders. ____________________________________ _____________________________________ A positive power of 10 is a number Explore positive and negative powers that can be represented as a product of ten. (2.9) whose only factors are 10’s. Translate between scientific and A negative power of 10 is a number standard notations. that can be represented as a product Play Scientific Notation Toss (p 331 whose only factors are 0.1’s or 1/10’s. SRB and p472 MM). _________________________________ ________________________________ Use the power key on a calculator. (2.10) Review exponential notation. Play Exponent Ball (p 311 SRB and pp 404 + 436 MM) Multiply and divide decimals. ____________________________________ Use scientific notation on a calculator. (2.11) Explore patterns with powers of 10.

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