Math Notes for Rounding: I sometimes require notations and techniques that are not covered completely in the text. To get full credit, these should always be used in your homework, tests and quizzes. Showing work when ROUNDING a number Problems: 45. Round 4973 to Hundreds 45. 4 9 7 3 Show work: 1 5 0 00 51. Round 253,678 to Ten-thousands 51. 2 5 3, 6 7 8 0 2 5 0, 0 0 0 Summary: Write down the number to be rounded (leave space between digits) Underline the digit in the rounding position and all digits to its left Circle the digits immediately following the ones you underlined (the first circled digit is called the “check digit”). Rounding Up: If the check digit is 5 or greater, Write +1 directly under the right-most underlined digit, circle the digit 7 Since 7 ≥ 5, Add 1 to the underlined digits above (49 + 1 = 50 in the first example) Replace the check digit and all digits to its right with 0’s (73 becomes 00) Rounding Off: If the check digit is 4 or less, Write +0 under the right-most underlined digit, circle the check digit 3 Since 3 ≤ 4, Add 0 to the underlines digits above it (25 + 0 = 25 in the second example) Replace the check digit and all digits to its right with 0’s (3,678 becomes 0,000) Some students question this: “MrV, I already know how to do rounding in my head. Why do I have to use this technique?” My reply: “First, I need to grade you accurately. Showing work this way proves you properly identified the rounding position and the check digit before determining the answer. For 4973 in the 1st example, you might have looked at the 9 in the 100ths position as the check digit, so you would wind up with 5000 as the same answer.” “Second, you get the opportunity to improve your math habits. People who don’t like math have a tendency to write down as little intermediate work as possible. As math gets more complex, this often leads to lower grades.” “Third, it reduces the temptation to scam homework. Some students have actually written down the answers from the back of the book … and then expected magic to occur on tests!” More Examples: Round 389,702 to Hundreds 3 8 9, 7 0 3 Show work: 0 3 8 9, 7 0 0 Round 236,753,678 to Hundred-thousands 2 3 6, 7 5 3, 6 7 8 1 2 3 6, 8 0 0, 0 0 0 Math Notes for Whole Number Subtraction: MrV sometimes requires notations and techniques that are not covered completely in the text. These should always be used in your homework, tests and quizzes. Subtraction always involves two numbers, with the procedure of borrowing The text does a good with how to do subtraction, including how to do borrowing one digit at a time. Borrowing into 0 could use a better step-by-step explanation. Step-by-step Examples of borrowing into a 0 digit: 46. Original Borrow 10 problem from the 4 Borrow 10 from the 8 No borrow for 7 – 4 Borrow 10 from the 2 1 10 Borrow 10 from the 4 3 10 7 13 10 3 1110 840 8 4/ 0/ 8/ 4/ 0/ 76. 4 2 0 7 4 2/ 0/ 7 4/ 2/ 0/ 7 7 43 7 43 7 43 16 2 4 16 2 4 16 2 4 7 97 3 83 2 583 The text doesn’t give you a nice shortcut when you have to borrow from the digit 0. What is one less than 40? 39, of course. What is one less than 700? 699, of course. What is one less than 6000? 5999, of course. Trickier: What is one less than 100? Use 099 when placing the borrowing digits. Examples, showing multiple steps of the process. Original Borrow 10 problem from the 70 Borrrow 10 from the 3 6 9 16 37 0 6 3---7 0 6/ 2 16 9 16 2957 2957 3/---7 0 6/ 2957 58. 49 7 49 Original problem Borrow 10 from the 8000 7 9 9 9 14 80. 8 0,0 0 4 8,2 3 7 8 0, 0 0 4/ -------- 8, 2 3 7 7 1,7 6 7 Sometimes a subtraction results in a negative number (the bottom # is bigger that the top #). Example: 17 – 89 = ? Let’s look at how to set your work area up in a special MrV way. (not in the text) Normally, you would set it up as: 17 but switch the #’s, put – in the answer line, –89 then do the subtraction: can’t subtract 89 –17 –72 Math Notes for Multiplication: MrV sometimes requires notations and techniques that are not covered completely in the text. These should always be used in your homework, tests and quizzes. Proof of multiplication involves showing all carries and adding single digit products This chapter section only does a fair job showing how to do multiplication; it fails to give you any procedure for showing multiplication carries. Showing multiplication carries, step-by-step explanations. Single digit multiplication example: Original 4x9= problem 3 4 x 5 +3 = 2 3 6 3 28. 559 4 559 2 3 2 3 559 559 4 6 4 x 5 +2 = 22 4 36 (little # is a carry digit, big #’s go on product line) 4 2236 Two digit multiplication example; two rows of products will be added; and usually two little rows of carries: I recommend using – ‘s (dashes) rather than zeros to insure that the second and subsequent product rows are lined up correctly. Original problem 56. 9 5 7 7 35 5 x 9577 showing carries Cross off carries, 3 x 9577 Add 2 product rows, set up 2nd row show new carries show carries 2 3 3 -----2 3 3 1 2 2 2 3 3 ------ 957 7 957 7 957 7 35 35 35 1 2 2 2 3 3 ------ 957 7 35 1 1 1 4 7 885 4 7 885 4 7 885 4 7 885 – 2 8 7 31 – 2 8 7 31 – 0 0 3 3 5,19 5 The text doesn’t give you an Easy Way to multiply numbers with trailing 0’s. Let’s use 6,000 x 700 as an example. 1. Set up the multiplication as though the 0’s were not there. ( 6 x 7) 2. Do the multiplication. (6 x 7 = 42) 3. How many 0’s were ignored? Place them after the product. ( 4200000 ) 4. Insert commas from the right. ( 4,200,000 ) Examples, showing multiple steps of the process. 66. Original Set it up for problem the easy way 700 274 Do 7 x 274, but ignore 0’s 274 7 00 Count 0’s, place them after the product, add comma(s) 5 2 5 2 274 274 7 00 1 9 18 7 00 191, 8 0 0 The text shows an uncommon way of handling inside 0 multiplication. Use this instead: I recommend using –’s (dashes)rather than zeros to insure that the second and subsequent product rows are lined up correctly. You don’t need a product line for interior 0’s; just use an additional –. Cross off carries, Original 9 x 758 problem Show carries 5 7 74. se to skip one pos, 2 x 758 Add 2 product rows, Use a 2nd to skip x0 show new carries show carries, add comma 1 1 1 1 5 7 ---- 5 7 ----- 7 58 5 7 ---- 7 58 7 58 7 58 7 58 209 209 209 209 209 68 2 2 68 2 2 68 2 2 68 2 2 – – 15 1 6 – – 15 1 6 – – 1 15 8, 4 2 2 Math Notes for Long Division: Proof of division involves showing all work for each digit in the quotient. This chapter section does a good job showing how to do division, but it doesn’t show you how to simplify you division when both numbers have trailing 0’s. Simplifying divisions that have matching right-hand zeroes; step-by-step explanations. Some problems sometimes have lots of matching trailing zeros in both divisor and dividend: Note that the divisions below all have the same quotient. If you can remove the same number of trailing zeros from both divisor and dividend; it makes the division simpler. 9 9 9 9 6 54 60 540 600 5400 6000 54000 How would you simplify the divisions below?. 80 220 8 22 700 6480 70 648 Using MrV’s Times Table to do Division – selected pictures from the Powerpoint. Math Notes for Shorthand Division: Proof of division involves showing all work for each digit in the quotient. The textbook does not address shorthand division, which can save you a lot of time when you have to do Prime Factorization (the next page). Mostly, this technique is used for dividing by 2, 3, 5 or 7. Here are 2 examples that compare both ways. The notation is different: carries above, quotient below.