SO YOU HAVE THE DIVIDEND, THE QUOTIENT, THE DIVISOR, AND THE REMAINDER. STOP THE MADNESS WE'RE TURNING INTO MATH ZOMBIES.

Transcription

1 SO YOU HAVE THE DIVIDEND, THE QUOTIENT, THE DIVISOR, AND THE REMAINDER. STOP THE MADNESS WE'RE TURNING INTO MATH ZOMBIES.

2 HELLO. MY NAME IS MAX, AND THIS IS POE. WE'RE YOUR GUIDES THROUGH WHAT WE CALL, MATH CRUSH. IN THIS BOOK, WE'RE GOING TO STUDY LONG DIVISION. ALSO KNOWN AS THE QUOTIENT, GROUPING, DIVISIBLE, OR MULTIPLICATION BACKWARDS. MATHCRUSH.COM LICENSE AND REGISTRATION PLEASE. DON'T MAKE ME GO KAMIKAZE ON YOU. Math Crush grants you a limited and revocable license to use Mathcrush.com for the purpose of personal use. Permission to use downloaded materials is valid for the duration of the subscription term. Continued personal or classroom use requires a current Math Crush License. You may not distribute, display, create derivative works from, translate, modify, reverse-engineer or otherwise exploit Mathcrush.com or any portion of it. You may not make any commercial use of any information provided on Mathcrush.com or make any use of Mathcrush.com for any commercial benefit. Math Crush reserves the right to refuse service or terminate accounts in its discretion, including, without limitation, if Math Crush determines that any customer conduct violates applicable law or is harmful to the interests of Mathcrush.com.

3 MULTIPLICATION FACTS BELOW IS A TABLE OF THE BASIC MULTIPLICATION FACTS. ON PAGE 3 IS ANOTHER VERSION. YOU CAN CHOOSE WHICH ONE YOU WANT TO USE. SINCE DIVISION IS THE OPPOSITE OF MULTIPLICATION YOU NEED TO KNOW YOUR BASIC MULTIPLICATION FACTS BEFORE YOU CAN DIVIDE. X

4 HOW TO USE TABLE X HERE'S A PIECE OF THE MULTIPLICATION TABLE. THE DARKER TOP AND SIDE NUMBERS ARE WHAT YOU ARE MULTIPLYING. SO IF I ASKED WHAT IS 6 X 5? YOU WOULD FIND WHERE 6 AND 5 CROSS, WHICH EQUALS 30. IF YOU LOOK CLOSELY YOU WILL SEE THEY CROSS SOMEWHERE ELSE TOO NO PROBLEM. SO IF I WAS LOOKING FOR 4 X 3 I JUST LOOK FOR WHERE THEY CROSS, WHICH IS AT 12. X THAT'S NOT SCARY AT ALL. 2

5 MULTIPLICATION FACTS - CHART 2 1 X 1 = 1 1 X 2 = 2 1 X 3 = 3 1 X 4 = 4 2 X 1 = 2 2 X 2 = 4 2 X 3 = 6 2 X 4 = 8 3 X 1 = 3 3 X 2 = 6 3 X 3 = 9 3 X 4 = 12 4 X 1 = 4 4 X 2 = 8 4 X 3 = 12 4 X 4 = 16 5 X 1 = 5 5 X 2 = 10 5 X 3 = 15 5 X 4 = 20 6 X 1 = 6 6 X 2 = 12 6 X 3 = 18 6 X 4 = 24 7 X 1 = 7 7 X 2 = 14 7 X 3 = 21 7 X 4 = 28 8 X 1 = 8 8 X 2 = 16 8 X 3 = 24 8 X 4 = 32 9 X 1 = 9 9 X 2 = 18 9 X 3 = 27 9 X 4 = X 1 = X 2 = X 3 = X 4 = X 1 = X 2 = X 3 = X 4 = X 1 = X 2 = X 3 = X 4 = X 1 = X 2 = X 3 = X 4 = 52 1 X 5 = 5 1 X 6 = 6 1 X 7 = 7 1 X 8 = 8 2 X 5 = 10 2 X 6 = 12 2 X 7 = 14 2 X 8 = 16 3 X 5 = 15 3 X 6 = 18 3 X 7 = 21 3 X 8 = 24 4 X 5 = 20 4 X 6 = 24 4 X 7 = 28 4 X 8 = 32 5 X 5 = 25 5 X 6 = 30 5 X 7 = 35 5 X 8 = 40 6 X 5 = 30 6 X 6 = 36 6 X 7 = 42 6 X 8 = 48 7 X 5 = 35 7 X 6 = 42 7 X 7 = 49 7 X 8 = 56 8 X 5 = 40 8 X 6 = 48 8 X 7 = 56 8 X 8 = 64 9 X 5 = 45 9 X 6 = 54 9 X 7 = 63 9 X 8 = X 5 = X 6 = X 7 = X 8 = X 5 = X 6 = X 7 = X 8 = X 5 = X 6 = X 7 = X 8 = X 5 = X 6 = X 7 = X 8 = X 9 = 9 1 X 10 = 10 1 X 11 = 11 1 X 12 = 12 2 X 9 = 18 2 X 10 = 20 2 X 11 = 22 2 X 12 = 24 3 X 9 = 27 3 X 10 = 30 3 X 11 = 33 3 X 12 = 36 4 X 9 = 36 4 X 10 = 40 4 X 11 = 44 4 X 12 = 48 5 X 9 = 45 5 X 10 = 50 5 X 11 = 55 5 X 12 = 60 6 X 9 = 54 6 X 10 = 60 6 X 11 = 66 6 X 12 = 72 7 X 9 = 63 7 X 10 = 70 7 X 11 = 77 7 X 12 = 84 8 X 9 = 72 8 X 10 = 80 8 X 11 = 88 8 X 12 = 96 9 X 9 = 81 9 X 10 = 90 9 X 11 = 99 9 X 12 = X 9 = X 10 = X 11 = X 12 = X 9 = X 10 = X 11 = X 12 = X 9 = X 10 = X 11 = X 12 = X 9 = X 10 = X 11 = X 12 = X 13 = 169 3

6 REVIEW - A BEFORE WE BEGIN, LET'S DO A QUICK REVIEW. YOU SHOULD KNOW YOUR BASIC MULTIPLICATION AND DIVISION BY NOW. REVIEW - A 3 X 5 = = 15 MULTIPLICATION IS A SHORTCUT FOR ADDITION. 3 X 5 = 11 X 6 = 7 X 6 = 7 X 3 = 6 X 8 = 8 X 9 = 9 X 12 = 9 X 9 = 7 X 3 = 11 X 13 = 5 X 11 = 12 X 9 = 13 X 8 = 6 X 13 = 11 X 11 = 5 X 13 = 11 X 5 = 9 X 4 = 8 X 4 = 10 X 0 = 3 X 9 = 0 X 13 = 12 X 6 = 12 X 13 = 12 X 5 = 11 X 7 = 9 X 8 = 4 X 4 = 4 X 9 = 4 X 7 = 6 X 9 = 3 X 13 = 3 X 8 = 1 X 13 = 6 X 11 = 7 X 9 = 11 X 4 = 8 X 10 = 4 X 7 = 4 X 11 = 11 X 8 = 9 X 6 = 6 X 10 = 0 X 4 = 7 X 13 = 8 X 3 = 6 X 8 = 10 X 11 = 7 X 5 = 3 X 13 = 13 X 13 = 3 X 8 = 4 X 12 = 12 X 11 = 9 X 9 = 2 X 10 = 9 X 7 = 12 X 6 = 5 X 7 = 7 X 8 = 10 X 7 = 8 X 8 = 5 X 4 = 4

7 REVIEW - B 56 IS A MULTIPLE OF 7 AND 8, BECAUSE 7 X 8 = 56. THE WORD, MULTIPLE, JUST MEANS THE ANSWER YOU GET WHEN YOU MULTIPLY. 7 X 8 = 4 X 13 = 5 X 12 = 9 X 5 = 2 X 8 = 7 X 6 = 4 X 7 = 0 X 12 = 9 X 11 = 6 X 8 = 11 X 3 = 0 X 3 = 8 X 1 = 4 X 7 = 11 X 6 = 12 X 3 = 10 X 2 = 13 X 11 = 11 X 7 = 4 X 8 = 5 X 4 = 9 X 4 = 2 X 6 = 5 X 9 = 10 X 10 = 9 X 13 = 8 X 10 = 0 X 7 = 11 X 2 = 11 X 9 = 5 X 4 = 6 X 1 = 13 X 8 = 8 X 9 = 9 X 0 = 5 X 5 = 9 X 7 = 9 X 7 = 9 X 11 = 12 X 7 = 11 X 5 = 6 X 9 = 11 X 4 = 10 X 3 = 9 X 10 = 5 X 10 = 9 X 8 = 13 X 2 = 3 X 9 = 5 X 11 = 8 X 8 = 1 X 3 = 4 X 12 = 11 X 12 = 9 X 0 = 6 X 9 = 12 X 11 = 13 X 2 = 5 X 7 = 7 X 10 = 6 X 13 = 4 X 13 = 0 X 11 = 8 X 5 = 7 X 11 = 5 X 12 = 10 X 4 = 4 X 0 = 8 X 5 = 13 X 12 = 7 X 4 = 13 X 3 = 9 X 11 = 6 X 13 = 11 X 9 = 7 X 8 = 6 X 12 = 6 X 9 = 4 X 12 = 8 X 8 = 9 X 8 = 6 X 0 = 9 X 5 = 11 X 5 = 5

8 REVIEW - C 32 4 = 8 DIVISION IS MULTIPLICATION BACKWARDS = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 6

9 REVIEW - D X AND FAMILY 4 X 5 = = 4 5 X 4 = = 5 MULTIPLICATION AND DIVISION ARE RELATED = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 7

10 REVIEW - E FILL IN THE BLANKS TO MAKE THE EQUATIONS CORRECT = 2. 5 X = X 7 = = 4 5. X 8 = = 7. 4 X 11 = 8. 6 X = = = = 12. X 6 = = X = = X = X 6 = = = X 10 = = = X 13 = X 7 = 25. X 12 = = X = = X 5 = = = X = X 5 = 34. X 6 = = X = = = 39. X 10 = = X = = 8

11 LONG DIVISION WELL, I HOPE THE REVIEW WASN'T TOO DIFFICULT. DO YOU REMEMBER THE OTHER WAYS OF SHOWING DIVISION? THESE ALL MEAN 45 DIVIDED BY 5, WHICH EQUALS = = = 9 YOU SHOULD BE VERY FAMILIAR WITH THE FIRST ONE, BUT WE NEED TO START SPENDING MORE TIME ON THE OTHER TWO. THIS PROBLEM IS AN EXAMPLE OF LONG DIVISION. IT TELLS US THAT 81 X 5 = X DO YOU SEE HOW ALL THE WORK IS SHOWN ON THE PROBLEM. I KNOW IT LOOKS CONFUSING, BUT DON'T START PANICKING. WE'VE JUST STARTED, SO HAVE NO WORRIES. BEFORE WE START SHOWING YOU HOW TO SOLVE THESE THE LONG WAY, QUICKLY ANSWER THE PROBLEMS BELOW. MAKE SURE YOU START WRITING THE ANSWER ON TOP OF THE PROBLEM

12 LONG DIVISION - CONTINUED TODAY OUR TEACHER WAS TRYING TO TEACH US LONG DIVISION. SHE SAID IT'S THE SAME AS SHORT DIVISION BUT LONG. I THINK SHE WAS SAYING IF YOU SEPARATE A LONG DIVISION PROBLEM INTO PIECES IT CAN LOOK A LOT EASIER. I DON'T KNOW ABOUT YOU, BUT I'M ALREADY CONFUSSSED. I LIKE LONG DIVISION IT'S SSSSSS-LONG. LET'S TAKE A LOOK AT A VERY SIMPLE PROBLEM. I KNOW THIS ONE 8 DIVIDED BY 4 EQUALS VERY GOOD POE, BUT LET ME SHOW YOU HOW TO SOLVE THIS USING LONG DIVISION. THE NUMBER ON TOP IS THE ANSWER. 2 X = 8-8 WHAT IS 8 DIVIDED BY 4 OR HOW MANY 4'S GO INTO 8? ANSWER IS 2, SO YOU PUT IT RIGHT ABOVE THE DIVIDE 2. MULTIPLY THERE'S A PATTERN...DIVIDE, MULTIPLY, AND SUBTRACT. THEN YOU MULTIPLY 2 TIMES 4, WHICH EQUALS 8. LAST, YOU SUBTRACT 8 MINUS 8, WHICH EQUALS 0. SINCE THERE ARE NO MORE DIGITS (NUMBERS) INSIDE YOU STOP SUBTRACT 10

13 LONG DIVISION - CONTINUED HEY MAX, WHY ARE WE DOING THIS LONG DIVISION STUFF IF THE PROBLEMS ARE SO EASY? I KNOW YOU CAN DO THESE IN YOUR HEAD, BUT IT IS EASIER TO EXPLAIN THE PROCESS IF WE START SIMPLE. OK, LET'S PRACTICE SOME MORE SO WE CAN MOVE TO BIGGER AND BETTER NUMBERS OK, HOW ABOUT 6 DIVIDED BY 2? WE JUST DIVIDE, MULTIPLY, AND SUBTRACT. NOW YOU TRY DOING THE ONES BELOW. 6 DIVIDED BY 2 EQUALS 3. X = MULTIPLY 3 TIMES 2, WHICH EQUALS 6. SUBTRACT 6 FROM 6, WHICH EQUALS 0. NO MORE DIGITS INSIDE SO STOP. SHOW YOUR WORK LIKE MAX DID

14 PRACTICE I WANT TO MAKE SURE YOU UNDERSTAND THE BASICS. REMEMBER, SHOWING YOUR WORK IS VERY IMPORTANT IN MATH. WHEN I'M FACED WITH A HARD PROBLEM I TRY TO REMEMBER SIMILAR EASY ONES, AND USE MY BASICS TO SOLVE IT. IN CASE YOU FORGOT, DIGITS ARE THE NUMBERS INSIDE THE ACTUAL NUMBER. FOR EXAMPLE, 526 IS THE NUMBER. BUT 5, 2, AND 6 ARE THE DIGITS. HELPFUL EXAMPLE 1 DIVIDE 2 2 MULTIPLY - 2 SUBTRACT 0 ANSWER

15 THE CONFUSING ZERO HEY MAX, I KNOW WE'RE DOING LONG DIVISION AND ALL, BUT I WAS WONDERING WHY CAN'T WE DIVIDE BY 0? WOW, GREAT QUESTION, POE. YOU KNOW, MOST PEOPLE GET VERY CONFUSED WHEN IT COMES TO 0 AND DIVISION......BUT IF YOU THINK ABOUT IT, AND REMEMBER THAT DIVISION IS MULTIPLICATION BACKWARDS. THEN WE CAN SEE WHY 0 CAUSES US TROUBLE. SO IF WE WERE GOING TO TURN THIS INTO A MULTIPLICATION PROBLEM. IT WOULD LOOK LIKE THIS = 6 6 X 4 = 24 AND THIS MAKES PERFECT SENSE NOW LOOK THE NEXT EQUATION. 0 7 = 0 0 X 7 = 0 AND IF WE TURNED THIS INTO A MULTIPLICATION PROBLEM IT WOULD STILL BE CORRECT. BUT CHECK THIS OUT. WHAT HAPPENS IF WE DIVIDE BY 0, AND THEN TRY AND TURN IT INTO A MULTIPLICATION PROBLEM? 8 0 =?? X 0 = 8 WAIT A SECOND, NOTHING TIMES 0 WILL EQUAL 8 THAT'S IMPOSSIBLE. EXACTLY, SO IF YOU REMEMBER THAT DIVISION IS MULTIPLICATION BACKWARDS, AND THEY ARE RELATED. THEN YOU CAN SEE THAT YOU CAN NOT DIVIDE BY 0. SO IF I HAD 36 0 =?, AND I TURNED IT INTO? X 0 = 36. NOTHING TIMES 0 WILL EQUAL 36, WHICH MEANS THERE IS NO ANSWER. 13

16 MATH ART BEFORE WE START GETTING INTO THE BIGGER NUMBERS, I THOUGHT WE SHOULD PRACTICE OUR BASICS. GOOD IDEA. IF WE CAN'T DO THE SIMPLE PROBLEMS THEN HOW ARE WE GOING TO DO THE HARD STUFF. I KNOW MOST OF YOU HAVE PROBABLY DONE THESE MATH ART PAGES. IN CASE SOMEONE FORGOT OR IS NEW, HERE IS A HELPFUL EXAMPLE TO GUIDE YOU. FIRST, I ANSWERED THE PROBLEMS IN PATTERN #1. THEN I CONNECTED THE DOTS IN THE ORDER THEY WERE IN. NOW YOU TRY PATTERN #2. PATTERN #1 PATTERN # = = 16 2 = = 42 7 = = 81 9 = = = = 50 5 = = = = = = = = LINE ENDS LINE ENDS THE PATTERNS ARE NOT CONNECTED TOGETHER. SO IN PATTERN #2, 8 DIVIDED BY 8 EQUALS 1 AND 5 DIVIDED BY 1 EQUALS 5. THEN I WOULD CONNECT THOSE TWO DOTS TOGETHER. 14

17 DIRECTIONS: NAME: ANSWER THE PROBLEMS BELOW AND CONNECT THE DOTS IN THE ORDER YOU CREATED. I STARTED THE PATTERN FOR YOU NOW YOU DO THE REST. BE CAREFUL SOME NUMBERS REPEAT. NOTE: PATTERNS ARE NOT CONNECTED TOGETHER. PATTERN #1 PATTERN #2 PATTERN #3 PATTERN # = = = 14 7 = 50 5 = = 36 9 = = = 88 8 = 99 3 = 62 2 = = = 30 2 = = = 28 2 = = 36 2 = 63 9 = 80 4 = 99 9 = 56 8 = = = 36 6 = = = = = = 44 2 = = 34 2 = = = 0 17 = 91 7 = = = = 80 8 = = = 50 2 = = 38 2 = LINE ENDS LINE ENDS LINE ENDS LINE ENDS *

18 LONG DIVISION - BIG NUMBERS OK MAX, I'M READY TO LEARN HARDER LONG DIVISION. GREAT LET'S START WITH TWO DIGITS INSIDE. CHECK OUT THE NEXT PROBLEM. 378 FIRST I COVER UP ALL NOW I ASK MYSELF THEN I MULTIPLY THE NUMBERS INSIDE HOW MANY 3'S THE TOP NUMBER EXCEPT THE FIRST ONE. CAN GO INTO 7? 2. BY 3 (2 X 3 = 6). 2 X x 3 = 3 2 x 3 = 6 * x 3 = 9 6 NEXT I SUBTRACT THE NOW I UNCOVER SINCE NOTHING IS HOLDING UP THE 8, 7 BY 6 (7-6 = 1). THE NEXT NUMBER. HE FALLS DOWN NEXT TO THE HELP I'M FALLING NOW I START OVER AGAIN. I THEN I MULTIPLY THE NEED TO FIGURE OUT HOW NEW TOP NUMBER LAST, I SUBTRACT THE MANY 3'S GO INTO 18? 6. BY 3 (6 X 3 = 18). 18 BY 18 (18-18 = 0). 2 6 X x 3 = x 3 = x 3 = 18 * 7 x 3 = I NEED ANOTHER EXAMPLE. AND YOU WANTED HARDER ONE'SSS. ANSWER IS 26 OR 3 X 26 = 78. THERE ARE NO MORE DIGITS TO BRING DOWN SO I CAN STOP. 16

19 DIVISION - MORE BIG NUMBERS WHEN YOU DO LONG DIVISION YOU BASICALLY: DIVIDE MULTIPLY START SUBTRACT OVER BRING DOWN YES, AND YOU KEEP DOING THAT UNTIL THERE IS NO MORE DIGITS TO BRING DOWN. IT'S LIKE A CONTINUOUS PATTERN. I'M GOING TO TRY THIS ONE ON MY OWN. WHY DON'T YOU FOLLOW ALONG COVER NUMBERS EXCEPT FIRST ONE 96 DIVIDED BY 4 I NEED TO FIGURE OUT WHAT TIMES 4 COMES CLOSE TO 9 WITHOUT GOING OVER X 4 = 4 2 X 4 = 8 * 3 X 4 = 12 HOW MANY 4'S GO INTO 9 WITHOUT GOING OVER? ANSWER: 2 X 2 X 2 X 2 THEN MULTIPLY THE TOP SUBTRACT NUMBER 2 BY 4. 9 MINUS UNCOVER THE NEXT NUMBER AND BRING IT DOWN. X 2 4 HOW MANY 4'S X 2 4 X GO INTO 16? THEN MULTIPLY ANSWER: 4. THE TOP NUMBER 4 BY x 4 = 16 * 5 x 4 = DIVIDED BY 4 = 24 OR 24 X 4 = 96 SUBTRACT 16 MINUS 16. THERE ARE NO MORE DIGITS TO BRING DOWN, SO YOU ARE DONE. THE ANSWER IS

20 PRACTICE - A GIVE THESE A TRY AND REMEMBER THE PATTERN: 1. DIVIDE 2. MULTIPLY 3. SUBTRACT 4. BRING DOWN I JUST NOTICED SOMETHING ALL THE DIGITS (NUMBERS) INSIDE HAVE A DIGIT ABOVE IT. HELPFUL EXAMPLE THE CLOSES YOU CAN COME TO 7 WITHOUT GOING OVER IS 3 X 2 = 6. X 3 7 X ANOTHER WAY TO LOOK AT IT X 2 = GOES PERFECTLY INTO 14, BECAUSE 7 X 2 = GOES INTO 9, BUT THE 0 IS TOO SMALL. WHEN THIS HAPPENS PUT A 0 ON TOP. 3 X 3 = 9, BUT 3 X 30 = 90 SEE WHY YOU NEED A 0? 18

21 PRACTICE - B SO HOW'S IT GOING? DO YOU THINK YOU'RE GETTING IT? REMEMBER, PRACTICE MAKES PERFECT. HEY, LOOK ON THE BRIGHT SIDE AT LEAST WE'RE NOT DOING LONG DIVISION. HELPFUL EXAMPLE THE CLOSES YOU CAN COME TO 8 WITHOUT GOING OVER IS 2 X 3 = 6. X ANOTHER WAY TO LOOK AT IT X 3 = GOES PERFECTLY INTO 24, BECAUSE 8 X 3 = 24. IT'S VERY IMPORTANT THAT YOU KEEP ALL THE NUMBERS LINED UP CORRECTLY SINCE 2 CAN NOT GO INTO 0, WHAT DO YOU PUT ON TOP? 19

22 PRACTICE - C

23 LONG DIVISION PUZZLE SSSO DO YOU GET IT NOW? SSSURE, BUT LETSSS DO A FEW MORE. I HAVE AN IDEA LET'SS PLAY A GAME. I'LL GIVE YOU A PROBLEM AND THE ANSWERSS, BUT YOU NEED TO PUT THE ANSWERSS IN THE CORRECT BLANKSSS. I GET IT. WE USE THE NUMBER BOXES ON THE RIGHT TO FILL IN THE ANSWERS TO THE PROBLEM...SINCE 2 GOES INTO 6 ONLY 3 TIMES, WE NEED TO PUT THE 3 IN THE BOX. THEN CROSS OUT THE 3, BECAUSE WE CAN ONLY USE EACH ANSWER BOX ONCE. ALRIGHTY SMIGHTY. ON THIS ONE WE WOULD PUT THE 1 IN THE FIRST BOX, BECAUSE 4 GOES INTO 6 ONLY 1 TIME. THEN WE NEED TO CROSS OUT THE 1 IN THE ANSWER BOXES

24 LONG DIVISION PUZZLE - CONTINUED FINISH THESE PUZZLES AND THEN TAKE A MATH ART BREAK. YOU CAN DO IT. I LIKE THOSE MATH ART PAGES. DON'T FORGET YOU CAN ONLY USE EACH NUMBER BOX ONCE, SO CROSS THEM OUT AFTER YOU USE THEM. THIS LONG DIVISION ISS NOT THAT BAD YAH IT'SS LONG, BUT ONCE YOU GET MOVING IT'SS PRETTY EASY

25 DIRECTIONS: NAME: ANSWER THE PROBLEMS BELOW AND CONNECT THE DOTS IN THE ORDER YOU CREATED. I STARTED THE PATTERN FOR YOU NOW YOU DO THE REST. BE CAREFUL SOME NUMBERS REPEAT. NOTE: PATTERNS ARE NOT CONNECTED TOGETHER. PATTERN #1 PATTERN #2 PATTERN #3 PATTERN # = = = 0 19 = 36 6 = = = = = 44 2 = = = 80 2 = = 48 4 = = = 34 2 = 32 2 = = = 90 9 = = 66 2 = = = 50 2 = 45 5 = 36 9 = 14 1 = = 77 7 = = = 38 2 = = = 56 7 = = = = = 36 2 = = LINE ENDS LINE ENDS LINE ENDS LINE ENDS *

26 REMAINDERS SO MAX, WHAT DO WE DO IF THE NUMBER DOES NOT DIVIDE PERFECTLY INTO THE OTHER NUMBER? GOOD QUESTION, POE. WHEN THAT HAPPENS WE ACTUALLY HAVE A FEW CHOICES, BUT LET'S LEARN THE EASY WAY FOR NOW. THE EASY WAY IS CALLED A REMAINDER. A REMAINDER? I WILL ASSUME THAT IS THE SAME THING A SOMETHING THAT REMAINS OR IS LEFT BEHIND. EXACTLY. A REMAINDER IS WHAT IS LEFT OVER WHEN YOU DIVIDE. THINK OF IT LIKE CHANGE WHEN YOU BUY FOOD. THAT'S SIMPLE ENOUGH, BUT WHY DON'T YOU SHOW US AN EXAMPLE. NO PROBLEM. JUST REMEMBER YOU DIVIDE THE SAME WAY AS BEFORE NOW DO YOU SEE THE ONE? THERE ARE NO MORE DIGITS TO BRING DOWN SO WE CAN STOP. WE CALL THE ONE A REMAINDER. ANSWER 32 R1 "R" MEANS REMAINDER. 24

27 PRACTICE - A THE CLOSES YOU CAN COME TO 7 WITHOUT GOING OVER IS 1 X 4 = 4. AFTER YOU SUBTRACT YOU GET 3. THEN BRING DOWN THE 5 TO MAKE 35. HELPFUL EXAMPLE THIS MEANS: 18 X 4 = 72 THEN ADD REMAINDER 3: = R THE CLOSES YOU CAN COME TO 35 WITHOUT GOING OVER IS 8 X 4 = 32. THE REMAINDER HAS TO BE SMALLER THAN X 4 = 72, WHICH MEANS YOU HAVE A REMAINDER OF SINCE 7 CAN NOT GO INTO 4, WHAT DO YOU PUT ON TOP? 25

28 PRACTICE - B ON THIS PROBLEM, EVERY TIME YOU SUBTRACT THE ANSWER HAS TO BE SMALLER THAN THE 2. DO YOU KNOW WHY? HELPFUL EXAMPLE 4 8 R * * 1 THIS DIVISION PROBLEM IS ASKING HOW MANY 2'S CAN FIT INTO

29 PRACTICE - C THESE ARE ALL MIXED UP. SOME HAVE REMAINDERS SOME DON'T. 1 9 R * * 2 ON THIS PROBLEM, EVERY TIME YOU SUBTRACT THE ANSWER HAS TO BE SMALLER THAN THE PRACTICE YOUR BASICS WITH A MATH ART PAGE. 27

30 DIRECTIONS: NAME: ANSWER THE PROBLEMS BELOW AND CONNECT THE DOTS IN THE ORDER YOU CREATED. I STARTED THE PATTERN FOR YOU NOW YOU DO THE REST. BE CAREFUL SOME NUMBERS REPEAT. NOTE: PATTERNS ARE NOT CONNECTED TOGETHER. PATTERN #1 PATTERN #2 PATTERN #3 PATTERN #4 8 8 = = 38 2 = = = = 30 3 = 48 8 = = = 28 2 = = = = = = = 12 6 = = 35 5 = = 34 2 = = = 56 8 = = 27 9 = = = 88 4 = 55 5 = 24 8 = 54 6 = = = 77 7 = 36 2 = 72 3 = 42 7 = = 20 4 = = = 78 6 = 80 4 = 60 6 = 45 3 = 56 7 = 78 6 = = 88 2 = = = = 93 3 = = = = = 84 4 = LINE ENDS LINE ENDS LINE ENDS LINE ENDS

31 REWRITING DIVISION HEY MAX, THIS LONG DIVISION STUFF IS PRETTY EASY, BUT WHAT DO WE DO IF THE PROBLEM IS WRITTEN ONE OF THE OTHER WAYS? = = 7 DO YOU MEAN LIKE THIS? YES, BUT HOW WOULD WE SOLVE THEM IF THEY WERE BIGGER? REMEMBER, POE, THEY ALL MEAN THE SAME THING 42 DIVIDED BY 6 EQUALS 7. SO WE CAN WRITE THE PROBLEM EITHER OF THE THREE WAYS. I KIND OF SEE WHAT YOU MEAN, BUT CAN YOU SHOW US AN EXAMPLE? LET'S START WITH THE FIRST WAY. IF YOU HAVE A PROBLEM LIKE THIS YOU CAN REWRITE IT SO IT LOOKS LIKE A LONG DIVISION PROBLEM. THEN WE CAN SOLVE IT LIKE BEFORE = X X = = YOU HAVE TO BE CAREFUL REWRITING THESE. I MEAN LOOK HOW THE 72 AND 4 SWITCH. I GUESS PRACTICE WILL HELP US ALONG. 29

32 PRACTICE HELPFUL EXAMPLE 85 3 = COME BACK! REWRITE THE PROBLEMS AND SOLVE = = = = = = = = = = 30

33 REWRITING DIVISION - CONTINUED STRANGE HOW THE NUMBERS SWITCH ORDER, BUT WE STILL READ THE PROBLEM THE SAME WAY. SURE. TOO TRUE SO YOU READY FOR THE OTHER WAY? THIS IS CALLED A FRACTION, WHICH IS ALSO USED AS DIVISION. A MATTER OF FACT, LATER ON IN MATH YOU WILL SEE THIS A LOT = I DO NOT WANT TO CONFUSE YOU BY GIVING YOU TOO MUCH INFORMATION, SO LET'S JUST STICK WITH THE LONG DIVISION PART. THIS IS GOING TO GET CONFUSING. I MEAN HOW ARE WE SUPPOSE TO REMEMBER WHICH NUMBER GOES IN THE INSIDE. I JUST THINK OF AN UGLY MONSTER WHO HATES FRACTIONS. THIS MONSTER HATES THEM SO MUCH THAT HE LOVES PUSHING THEM OVER. PLEASE DON'T CALL ME A UGLY. I HAVE FEELINGS TOO BUT YOU'RE RIGHT I DO LOVE PUSHING FRACTIONS OVER. UH OH! I THINK I'LL BUILD A HOUSE FOR PROTECTION. OUCH! WE'RE OK. I REMEMBER DOING THESE IN THE FIRST BOOK, BUT NOW IT MAKES MORE SENSE. 31

34 PRACTICE HELPFUL EXAMPLE I COULD DO THIS ALL DAY LONG. THEY JUST FALL OVER LIKE DOMINOS. NOW YOU TRY JUST REMEMBER TO KNOCK THEM OVER AND THEN DIVIDE = 2. = = 4. = = = 7. = = = 10. = = 32

35 BIGGER NUMBER SO WHAT DO YOU THINK? IF YOU KEEP WITH THE PATTERN OF: 1. DIVIDE 2. MULTIPLY START 3. SUBTRACT OVER 4. BRING DOWN LONG DIVISION IS ALL THE SAME NO MATTER HOW BIG THE NUMBERS GET. TRUE MAX, BUT I LIKE WHEN YOU SHOW US AN EXAMPLE. ANOTHER EXAMPLE COMING UP. LET'S MAKE THE NUMBER WE'RE DIVIDING, THE NUMBER INSIDE, BIGGER ANSWER X NOW YOU GIVE THESE THREE A TRY. OR

36 PRACTICE - A HELPFUL EXAMPLE BEFORE YOU START, TAKE A LOOK AT THIS EXAMPLE. DO YOU SEE THE 0 ON TOP? SINCE THE 4 IS SMALLER THAN THE 5 WE PUT A 0. 0 X 5 = R ANSWER: 91 R1 DO YOU KNOW WHY WE DON'T HAVE TO WRITE THE 0 IN THE FINAL ANSWER?

37 PRACTICE - B HELPFUL EXAMPLE WHEN YOU BRING DOWN THE 1 IT IS SMALLER THAN THE 4, SO YOU HAVE TO PUT A 0 ABOVE THE 1. 0 X 4 = ANSWER: 203 DO YOU SEE HOW THE 8 HAS THE 2 ABOVE IT, THE 1 HAS A 0, AND THE 2 A 3? ALL THE DIGITS INSIDE HAVE A DIGIT ABOVE IT

38 PRACTICE - C BEFORE WE MOVE ON, LET'S DO ONE PAGE WITH ALL THREE TYPES OF DIVISION AND A MATH ART PAGE. HERE ARE A COUPLE OF EXAMPLES TO HELP YOU ALONG = = = = = = = = = 8. = = = 36

39 DIRECTIONS: NAME: ANSWER THE PROBLEMS BELOW AND CONNECT THE DOTS IN THE ORDER YOU CREATED. I STARTED THE PATTERN FOR YOU NOW YOU DO THE REST. BE CAREFUL SOME NUMBERS REPEAT. NOTE: PATTERNS ARE NOT CONNECTED TOGETHER. PATTERN #1 PATTERN #2 PATTERN #3 PATTERN # = = = = 60 5 = = 91 7 = = = 80 4 = = 32 8 = = 93 3 = 66 3 = = 99 3 = = = = = 88 2 = = 69 3 = = = = 0 13 = 80 8 = = 54 9 = 96 8 = = = 84 3 = = 60 4 = 63 3 = = 34 2 = = 63 9 = 27 9 = 90 5 = = 30 6 = = = 48 3 = = = 78 3 = 36 9 = 96 4 = = = LINE ENDS LINE ENDS LINE ENDS LINE ENDS *

40 DIVIDING BY TWO DIGITS WELL POE, YOU SHOULD BE FEELING PRETTY COMFORTABLE WITH DIVIDING, BUT WE NEED TO START DIVIDING BY TWO DIGITS. THAT MEANS THE OUTSIDE NUMBER WILL BE BIGGER. WOW, THAT'S GOING TO TAKE MORE TIME TO SOLVE. YES AND NO. FIRST I WANT TO TEACH YOU THE LONG WAY, SO YOU CAN SOLVE A PROBLEM NO MATTER WHAT. THEN I WANT TO SHOW YOU A SHORTCUT. I SEE WHAT YOU'RE SAYING, MAX. I MEAN IT'S A GOOD IDEA TO KNOW HOW TO SOLVE A PROBLEM BEFORE LEARNING SHORTCUTS. TAKE A LOOK AT THIS PROBLEM. JUST REMEMBER, THIS IS THE LONG WAY. IT WILL GET EASIER IN A BIT JUST LIKE BEFORE, WE NEED TO LOOK AT ONE NUMBER AT A TIME. AS YOU CAN SEE I COVERED UP THE 5 AND SINCE THE 9 IS WAY TOO SMALL I PUT A 0 ABOVE THE 9. 0 X 34 = 0. THEN I BRING DOWN THE 5 AND START OVER NOW WE NEED TO FIND OUT WHAT TIMES 34 COMES CLOSE TO 95 WITHOUT GOING OVER. LET'S LIST SOME MULTIPLES OF 34. TOO BIG X X X 2 68 WE BRING DOWN THE 2 AND FIND OUT WHAT TIMES 34 COMES CLOSE TO 272 WITHOUT GOING OVER. LET'S LIST SOME MORE MULTIPLES OF X X X X X

41 PRACTICE - A HELPFUL EXAMPLE R X X 3 X 4 X X 6 X 7 X DO YOU SEE HOW THIS CAN TAKE A LONG TIME? HAVE YOU NOTICED ANY WAYS OF MAKING THIS EASIER? 1 58 X TOO BIG

42 PRACTICE - B DID SOME PROBLEMS TAKE LESS TIME? DO YOU KNOW WHY? 40

43 ESTIMATING HEY POE, LET'S TRY AND MAKE THESE BIGGER PROBLEMS A LITTLE LESS TIME CONSUMING. WE NEED TO USE WHAT IS CALLED ESTIMATING. IT SHOULD CREATE LESS WORK FOR US. ESTIMATING HUH. IS THAT WHEN WE MAKE THE NUMBER LOOK EASIER? YES, THAT'S A PRETTY GOOD WAY OF LOOKING AT IT. MOST MATH PEOPLE LIKE TO SAY IT'S AN APPROXIMATION. ANOTHER BIG WORD. SO IF YOU TAKE AN ESTIMATION OF A NUMBER YOU ARE JUST TELLING WHAT IT ABOUT EQUALS, BUT REALLY YOU ARE JUST TRYING TO MAKE THE NUMBER EASIER. POE IS RIGHT LET'S TAKE A LOOK AT SOME PROBLEMS TO BETTER SHOW WHAT HE MEANS. 62 IS CLOSE TO 60 AND 38 IS CLOSE TO 40, AND 60 AND 40 ARE MUCH EASIER NUMBERS. ACTUAL ESTIMATE YOU KNOW I USE ESTIMATING ALL THE TIME, ESPECIALLY WHEN I GO SHOPPING. I MEAN IF SOMETHING COSTS 99 CENTS I ESTIMATE IT TO 1 DOLLAR. IT MAKES THE NUMBER EASIER. SO DO I. THAT WAY IT'S LESS DIFFICULT TO FIGURE OUT HOW MUCH MONEY I NEED TO BUY STUFF. I DO NOT HAVE TO THINK SO HARD. TAKE A LOOK AT THESE PROBLEMS. DO YOU SEE HOW THERE COULD BE MORE THAN ONE ANSWER? IT JUST MATTERS HOW EASY DO YOU WANT THE NUMBER TO BE. ACTUAL EASY EASIER

44 PRACTICE ON THESE PRACTICE PROBLEMS TRY AND MAKE THE NUMBERS AS EASY AS POSSIBLE. CHECK OUT THE EXAMPLES HELPFUL EXAMPLES ACTUAL IS BETWEEN 20 AND 30, BUT IT'S CLOSER TO IS BETWEEN 500 AND 600, BUT IT IS CLOSER TO ESTIMATE SOMETIMES THE NUMBER IS RIGHT IN THE MIDDLE OF TWO NUMBERS. FOR EXAMPLE, 35 IS EXACTLY IN THE MIDDLE OF 30 AND 40. IF THIS HAPPENS JUST MAKE THE NUMBER BIGGER

45 MULTIPLES OF 10 NOW THAT YOU KNOW HOW TO ESTIMATE, LET'S REVIEW OUR MULTIPLES OF 10. IN OTHER WORDS, LET'S MAKE SURE YOUR KNOW HOW TO MULTIPLY YOUR ESTIMATED NUMBERS. HELPFUL EXAMPLE 6 X 20=120 6 X 2 = 12. THEN JUST CARRY OVER THE 0. ANSWER: X 500= X 80= 3. 3 X 80= 4. 9 X 90= 5. 1 X 400= 6. 4 X 40= 7. 9 X 30= 8. 5 X 500= 9. 8 X 20= 10. 6X 70= X 600= 12. 7X 90= 8 X 80= 14. 5X 30= 3 X 20= 16. 4X 700= 7 X 100= 18. 6X 40= 5 X 70= 20. 8X 800= 43

46 DIVIDING BY TWO DIGITS NOW LET'S USE OUR ESTIMATING SKILLS TO HELP US SOLVE THE SAME KIND OF DIVISION PROBLEMS. WE'RE GOING TO DIVIDE BY TWO DIGITS, BUT THIS TIME USE ESTIMATING TO SAVE US SOME TIME. EXACTLY, SO LET'S GET STARTED. IT HELPS TO LOOK AT AN EASIER PROBLEM FIRST. WHEN WE SOLVED A PROBLEM LIKE THIS EARLIER, WE LISTED THE MULTIPLES OF X X X X X BUT INSTEAD OF WASTING TIME, ESTIMATE 21 TO AN EASIER NUMBER LIKE 20. NOW WE CAN DO THE MULTIPLES OF 20 IN OUR HEAD. IN OTHER WORDS, WE DO NOT HAVE TO LIST ALL THE MULTIPLES OF 21. SO WE USE 20 TO HELP US FIND THE ANSWER FOR R CAN 20 GO INTO 9? NO, BECAUSE 9 IS TOO SMALL. WE PUT A 0 ABOVE THE 9 AND 0 X 21 = VERY IMPORTANT. DO YOU SEE THAT WE'RE MULTIPLYING 21 NOT 20. WE ONLY USE 20 AS A HELPER. AFTER BRINGING DOWN THE 1, WE NOW HAVE 91. ASK YOURSELF HOW MANY 20'S CAN GO INTO 91. WELL 4 X 20 = 80 AND THAT'S AS CLOSE AS WE CAN GET. SO USE 4. 4 X 21 = 84. NOW DON'T PANIC WE'RE JUST STARTING. CHECK OUT THE NEXT PAGE FOR ANOTHER EXAMPLE. 44

47 DIVIDING BY TWO DIGITS - CONTINUED OK, LET'S LOOK AT THIS PROBLEM. FIRST, WE NEED TO MAKE THE 43 EASIER. IF I ESTIMATE IT TO 40 THAT WILL HELP NOW USE THE 40 TO HELP US FIND AN ANSWER. FIRST, 40 DOES NOT GO INTO 6. IT IS TOO SMALL, WHICH MEANS WE NEED TO PUT A 0 ON TOP. 0 X 43 = THE 6 IS ONLY AN ESTIMATE. IT IS NOT ALWAYS THE CORRECT ANSWER, SO BE CAREFUL R AFTER I SUBTRACT AND BRING DOWN THE 8, WE GET 68. SO HOW MANY 40'S GO INTO 68? 1 IS THE CLOSES WE CAN GET, SO I WILL USE 1. 1 X 43 = 43. NOW LET'S STOP FOR A SECOND. DON'T FORGET THAT WHEN YOU SUBTRACT THE ANSWER HAS TO BE SMALLER THAN 43. IF IT IS BIGGER THAT MEANS ANOTHER 43 CAN FIT INTO THE NUMBER. IF YOU DON'T KNOW WHAT I MEAN, HAVE NO WORRIES, WE WILL SHOW YOU A PROBLEM LIKE THAT IN A BIT X 16 NOW I HAVE NO IDEA HOW MANY 43'S GO INTO 259, BUT IF WE USE 40 THAT IS A LOT EASIER. 6 X 40 = 240, SO I WILL USE 6. 6 X 43 = X SMALLER THAN 43. OK ANSWER 16 R1 CHECKING MY ANSWER R *

48 PRACTICE - A YOU THINK THIS MIGHT HELP SAVE YOU SOME TIME? IF YOU ARE TOTALLY CONFUSED JUST SOLVE THESE PROBLEMS USING THE LONG WAY. REMEMBER, THERE ARE MANY DIFFERENT WAYS TO SOLVE A PROBLEM. USE THE ONE THAT YOU FEEL MOST COMFORTABLE WITH. YOU ARE DIVIDING BY MULTIPLES OF 10. DO YOU SEE HOW ALL THE OUTSIDE NUMBERS END IN A 0? THAT MEANS YOU DO NOT HAVE TO ESTIMATE. THEY ARE ALREADY IN THE SIMPLEST FORM

49 PRACTICE - B I NEED A MATH ART BREAK

50 DIRECTIONS: NAME: ANSWER THE PROBLEMS BELOW AND CONNECT THE DOTS IN THE ORDER YOU CREATED. I STARTED THE PATTERN FOR YOU NOW YOU DO THE REST. BE CAREFUL SOME NUMBERS REPEAT. NOTE: PATTERNS ARE NOT CONNECTED TOGETHER. PATTERN #1 PATTERN #2 PATTERN #3 PATTERN # = = = = 42 7 = = 12 4 = = = = 27 3 = = 93 3 = = 75 5 = 82 2 = = 35 5 = = 20 4 = 80 4 = 0 23 = 44 2 = 32 2 = = 48 8 = = 72 8 = = = = = = 70 7 = = = 88 8 = = 45 9 = 63 3 = = = = = = = 90 2 = 27 9 = LINE ENDS LINE ENDS LINE ENDS LINE ENDS *

51 ESTIMATING IS NOT PERFECT WE'RE ALMOST DONE, BUT BEFORE YOU BECOME AN EXPERT AT DIVISION I WANTED TO SHOW YOU HOW ESTIMATING CAN SOMETIMES GIVE YOU THE WRONG ANSWER X YOU SHOULD ESTIMATE THE 16 TO 20 TO MAKE IT AN EASIER NUMBER TO FIND THE ANSWER. 6 IS TOO SMALL SO PUT A 0 ABOVE IT. 0 X 16 = BRING DOWN THE 2, WHICH MAKES 62. WHAT TIMES 20 COMES CLOSE TO 60? 3 X 20 = 60, SO USE 3. 3 X 16 = 48. SUBTRACT AND GET 14. NOW YOU MIGHT BE THINKING, "THIS IS WAY TOO CONFUSING." BUT IF YOU DON'T ESTIMATE A LITTLE YOU'RE GOING TO HAVE TO LIST A LOT OF MULTIPLES OF X R BRING DOWN THE 1 TO MAKE 141. HOW MANY 20'S GO INTO 141? 7 X 20 = 140, BUT WHAT HAPPENS IF WE USE 7 X 16? 8 X 16 = 128. THEN WE SUBTRACT AND GET A REMAINDER OF IS BIGGER THAN 16, WHICH MEANS WE CAN FIT ANOTHER 16 INTO 141. WE SHOULD'VE USED X MULTIPLE OF X X X X X X X X X

52 PRACTICE - A ESTIMATE 27 TO 30 (EASIER NUMBER). 30 DOES NOT GO INTO 6, SO PUT A 0 ABOVE THE 6. 0 X 27 = 0. PRACTICE 1 27 X X 3 81 HELPFUL EXAMPLE BRING DOWN THE 2, WHICH MAKES 62. WHAT TIMES 30 COMES CLOSE TO 62? 2 X 30 = 60, SO USE 2. 2 X 27 = 54. SUBTRACT AND GET 8. R 6 BRING DOWN THE 7 TO MAKE 87. HOW MANY 30'S GO INTO 87? 2 X 30 = 60 AND 3 X 30 = 90, SO USE 2. 2 X 27 = 54. **BUT WHEN YOU SUBTRACT: = IS BIGGER THAN 27, THIS TELLS US 2 IS TOO SMALL. WE SHOULD USE 3. 3 X 27 =

53 PRACTICE - B HELPFUL EXAMPLE ESTIMATE 42 TO 40 (EASIER NUMBER). 40 DOES NOT GO INTO 9, SO PUT A 0 ABOVE THE 9. 0 X 42 = 0. BRING DOWN THE 2, WHICH MAKES 92. WHAT TIMES 40 COMES CLOSE TO 92? 2 X 40 = 80, SO USE 2. 2 X 42 = 84. SUBTRACT AND GET 8. R 3 9 BRING DOWN THE 1 TO MAKE 81. HOW MANY 40'S GO INTO 81? 2 X 40 = 80, SO USE 2. 2 X 42 = 84. PRACTICE 42 X X 2 84 **BUT YOU CAN NOT SUBTRACT: IS TOO BIG, THIS TELLS US 2 IS TOO BIG. WE SHOULD USE 1. 1 X 42 =

54 PRACTICE - C

55 PRACTICE TEST YOU'RE ALMOST DONE. HERE'S A PRACTICE TEST AND THEN THE REAL MCCOY. MAKE SURE YOU HAVE SOMEONE CHECK YOUR ANSWERS TO SEE HOW YOU DID. DID HE SAY WE WERE DONE OR DUMB? YOUR SCORE: OUT OF 14 HOW YOU DID: = / = / 9 OR LESS =? 53

56 TEST YOUR SCORE: OUT OF 21 HOW YOU DID: = / = / 14 OR LESS =? 54

57 EXTRA PRACTICE - A HERE ARE A FEW EXTRA PAGES TO WORK ON. PRACTICE MAKES PERFECT = 2. 9 X = X 8 = = 4 5. X 6 = = 7. 5 X 10 = 8. 7 X = = = = 12. X 9 = = X = = X = X 5 = = = X 12 = = = X 11 = X 8 = 25. X 9 = = X = = X 5 = = = X = X 12 = 34. X 8 = = X = = = 39. X 7 = 7 55

58 DIRECTIONS: EXTRA PRACTICE - B NAME: ANSWER THE PROBLEMS BELOW AND CONNECT THE DOTS IN THE ORDER YOU CREATED. I STARTED THE PATTERN FOR YOU NOW YOU DO THE REST. BE CAREFUL SOME NUMBERS REPEAT. NOTE: PATTERNS ARE NOT CONNECTED TOGETHER. PATTERN #1 PATTERN #2 PATTERN #3 PATTERN # = = 60 3 = = = = = 38 2 = = = = 36 9 = 70 7 = = = = 4 2 = 72 8 = 36 2 = = = 3 3 = = 56 7 = = = 24 3 = 33 3 = = 34 2 = 81 9 = = = = 50 2 = = 25 5 = 84 7 = = 28 4 = 6 2 = = 7 7 = = 28 2 = = = = = 40 5 = 36 6 = = 44 4 = 9 3 = 12 3 = 26 2 = = = = 12 6 = LINE ENDS LINE ENDS LINE ENDS LINE ENDS * 56

59 EXTRA PRACTICE - C

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