Math Mammoth Division 2 Contents Introduction... 4 Review of Division... 8 Division Terms and Division with Zero... 11 Dividing with Whole Tens and Hundreds... 13 Finding Fractional Parts with Division... 15 Order of Operations and Division... 18 The Remainder, Part 1... 20 The Remainder, Part 2... 23 Long Division 1... 26 Long Division 2... 30 Long Division 3... 33 Long Division with 4-Digit Numbers... 37 More Long Division... 41 Remainder Problems... 44 Long Division with Money... 48 Long Division Crossword Puzzle... 50 Average... 51 Problems with Fractional Parts... 54 Problems to Solve... 56 Divisibility... 59 Prime Numbers... 63 Finding Factors... 66 Review... 68 Answers... 71 More from Math Mammoth... 85 3
Introduction Math Mammoth Division 2 is a continuation from the Math Mammoth Division 1 book. It includes lessons on division, long division, remainder, problem solving, average, divisibility, and factors. The book is most suitable for fourth grade. We start out by reviewing basic division with single-digit numbers. Then students study division terms and dividing by whole tens and hundreds. The lesson Finding Fractional Parts with Division shows an important relationship between fractions and division. For example, we can find 3/4 of a number by first finding 1/4 (divide by 4), then multiplying that result by 3. Next we briefly study the order of operations, this time including divisions in the problems. In the lesson The Remainder, Part 1, we study the concept of remainder, first using pictures and small numbers. In the second lesson on remainder, we still use small numbers, but students work the problems using the long division symbol or corner, as I like to call it. That is of course preparing them for long division. Next, long division is taught in several small steps over many lessons. We start with the situation where each of the thousands, hundreds, tens, and ones can be divided evenly by the divisor. Then is introduced the remainder in the ones. Next comes the situation where we have a remainder in the tens. Finally, we have a remainder in the hundreds, and so on. We also have lots of word problems to solve. After long division is mastered, we study the concept of average and problem solving involving a fractional part of a whole. I have included many bar diagrams and pictorial representations of these problems to help the students. The last section deals with elementary number theory topics. We study some basic divisibility rules (though not all of them), prime numbers, and find all factors of a given two-digit number. Answers are at the end of the book. I wish you success in teaching math! Maria Miller, the author 4
Helpful Resources on the Internet DIVISION CONCEPT AND DIVISION FACTS The Forty Frogs Game Learn to find fractions of sets by dividing a set of baby frogs into equal groups. http://www.mathactivities.net/frogs.htm Patty's Paints Division Help Patty paint cars by solving basic division questions. Lastly you can drive your newly painted car in a fun race! http://www.multiplication.com/games/play/pattys-paints-division Flying High Division Fly your plane safely through the storm clouds by answering the division facts correctly. http://www.multiplication.com/games/play/flying-high-division Times or Divide Bingo A useful class teaching resource on division and multiplication by 10 and 100. It includes decimals and is suitable for use on an interactive whiteboard. http://www.topmarks.co.uk/flash.aspx?f=bingotimesordivide Multiply & Divide Whole Numbers by 10, 100, 1000 Practice multiplying and dividing by 10, 100, and 1,000 in this interactive timed quiz that includes word problems. Choose Level 3. https://www.mangahigh.com/en/maths_games/number/calculations_with_whole_numbers_(integers)/multiply_and_divide_whole_numbers_by_10,_100_and_1000 Fraction of a Number Practice finding a fraction of a given number. http://www.mathplayground.com/fractions_fractionof.html Order of Ops Save seven members of a Royal Family from prison by using your order of operation skills. Choose the expression to be solved in each step, and solve it. The program uses a visual representation of a stairway to show how the expression gets shorter at each step. http://mrnussbaum.com/orderops/ Free worksheets for the order of operations Generate printable and customizable worksheets for the order of operations. Choose from five operations and parentheses. You can choose the number range used, number of problems, and more. http://www.homeschoolmath.net/worksheets/order_of_operations.php ITP Remainders This ITP sets up an empty grid into which you can place counters. Removing or highlighting extra counters will change the calculation displayed. http://mathsframe.co.uk/en/resources/resource/67/itp_remainders Leftovers game with beads Practice division with this fun dice game! http://www.learn-with-math-games.com/long-division-games-for-the-classroom.html Moving Remainders Division Game Practice your division skills with this printable board game for two or more players. http://www.lauracandler.com/filecabinet/math/pdf/movingremaindersgame.pdf 5
Division with Remainders Practice modeling division with remainders in the quotients in this interactive online activity. http://www.harcourtschool.com/activity/elab2004/gr4/5.html LONG DIVISION MathFrog Dividerama! Interactive long division practice. Guided help available. http://cemc2.math.uwaterloo.ca/mathfrog/english/kidz/div5.shtml Mr. Martini's Classroom: Long Division An interactive long division tool. http://www.thegreatmartinicompany.com/longarithmetic/longdivision.html Drag-and-Drop Math Practice division interactively. Choose Division, 2-digit dividend, and 1-digit divisor. http://mrnussbaum.com/drag-and-drop-math/ Long Division Millionaire Game Learn to divide large numbers up to thousands. Can you answer all 15 questions? http://www.kidsmathtv.com/free/long-division-game-for-6th-grade-millionaire-game/ Division Jump board game Practice division of one-digit numbers into two, three, and four-digit numbers. http://www.learn-with-math-games.com/division-activities.html Bike Racing Math Average Race your motorcycle against others while answering questions about average. Speed up with each correct answer! http://www.mathnook.com/math/bike-racing-math-average.html Long Division Quiz Practice dividing four-digit numbers by single-digit numbers in this online quiz. http://i4c.xyz/nmenbdv Double-Division.org Double-division is a form of the long division algorithm that takes away the guesswork of finding how many times the divisor goes into the number to be divided. Also called 1-2-4-8 division. http://www.doubledivision.org/ Short Division This is a web page that explains short division in detail. Short division is the same algorithm as long division, but some steps are only done in your head and not written down. http://www.themathpage.com/arith/divide-whole-numbers.htm FACTORS AND PRIMES Arrays and Factors Drag rectangles to show the factorizations of a given number on a grid. http://www.shodor.org/interactivate/activities/factorizetwo/ 6
Factor Game Choose a number from the game board, and your opponent gets all the numbers that are its proper factors. If a player chooses a number with no proper factors remaining, that player loses a turn. You can adjust the number of rows and columns on the game board to get a more challenging (and interesting) game. This game can easily be adapted to be played offline, with paper and colored pencils. http://illuminations.nctm.org/activity.aspx?id=4134 Sliding Tile Factorization Game Slide a number over another to capture it, but you can only do this if the number you slide is a factor of the other. Number 1 is only supposed to be used to capture prime numbers. http://www.visualmathlearning.com/games/sliding_factors.html Octopus Factors Move counters up the legs of an octopus but only when the number on the circle is a multiple of the number on the card. http://www.counton.org/games/map-numbers/octopus/ Not a Factor Choose a number that is not a factor of the given number. http://www.helpingwithmath.com/resources/games/target_factors01/not_factor.html Product Game The players choose factors and the product of those gets colored in on the game board. The player who gets four products in a row wins. You can play against the computer or with a friend. This game can easily be adapted to be played offline, with paper and colored pencils. http://illuminations.nctm.org/activity.aspx?id=4213 Primes, Factors and Divisibility Explorer at CountOn.org Lessons explaining divisibility tests, primes, and factors. http://www.counton.org/explorer/primes Factoring Calculator This tool lists all the factors of a given number and shows an interesting visual that pairs the various factors of the number. You can even find all the factors of very large numbers, and it's fun to experiment with! http://www.dadsworksheets.com/factoring-calculator.html 7
Review of Division Multiplication has to do with equal-size groups: 2 4 means 2 groups of 4. Division also has to do with equal-size groups: 8 4 can mean, How many groups of 4 are in 8? It can also mean, How many in each group, when 8 things are put into 4 groups? Division is the opposite operation of multiplication. Division has two meanings : Dividing to find how many are in a group. Dividing into groups of certain size. 2 6 = 12 12 divided into 2 groups; how many in each group? 12 2 = 6 OR How many sixes are in 12? 12 6 = 2 6 2 = 12 1. Write a multiplication sentence and two division sentences. a. 2. Fact families: write two division and two multiplication sentences. a. 21 7 and 3 b. b. 24 4 and 12 divided into 6 groups; how many in each group? 12 6 = 2 OR How many twos are in 12? 12 2 = 6 c. c. 36 4 and 8 Math Mammoth Division 2 (Blue Series)
3. Practice a little. Continue the patterns. a. 16 2 = 18 2 = 20 2 = 4. Fill in the tables. b. 45 5 = 40 5 = 35 5 = c. 90 10 = 100 10 = 110 10 = d. 56 7 = 49 7 = 42 7 = Eggs 6 12 24 36 54 78 Omelets 1 7 11 Thumbtacks 8 24 32 48 Pictures 1 8 10 12 13 5. Write a number sentence for each situation (It is not always division!). Explain what you find out from your calculation. a. Three children shared equally 18 marbles. 18 3 = 6. Each child got 6 marbles. c. A fruit store received a shipment of 400 apples in 4 boxes. e. Five boxes arrived at the bookstore, each containing 50 books. b. Jim has $34 and he wants a $45 book. d. Mrs. Davis shared 24 pieces of chocolate equally between 6 persons. f. Mom bought 2 books that cost $13 each. g. A herd of cows had a total of 20 legs. h. Sixty books were placed on 3 shelves. 9 Math Mammoth Division 2 (Blue Series)
6. Divide. a. 36 4 = 50 5 = 60 12 = 7. Find what number x stands for. a. 64 x = 8 x = 8 b. 54 9 = 42 7 = 48 6 = b. 35 x = 7 x = c. 32 8 = 64 8 = 72 9 = c. x 5 = 9 x = d. 24 3 = 27 9 = 35 7 = d. x 9 = 6 x = 8. For each division equation, write a multiplication sentence. Then find the value of the unknown. a. N 3 = 10 _ N = c. 60 T = 20 _ T = b. x 4 = 9 _ x = d. 81 y = 9 _ y = 9. Write a number sentence for each situation (It is not always division!). Explain what your answer tells you. a. How many $3 books can you buy with $21? b. A hundred apples were packed in 5 boxes. c. Five boxes of nails cost $30. d. A chocolate bar has 8 rows and 5 columns of squares. e. How many fives are in 45? f. Each of the 5 boxes weighs 12 pounds. 10 Math Mammoth Division 2 (Blue Series)
Division Terms and Division with Zero 1. What is missing from these divisions: the dividend, the divisor, or the quotient? Complete. a. 80 = 40 The is missing. b. 7 = 5 The is missing. c. 120 10 = The is missing. 2. Write the division problem. Solve for the unknown. 3. Make up: Study the terms in the picture. Notice: both the expression 56 7 and its answer are called the quotient! You can call 56 7 the quotient written, and the 8 as the quotient solved. a. The divisor is 7, the dividend is x, and the quotient is 3. ; x = b. The dividend is 140, the divisor is y, and the quotient is 7. ; y = c. The quotient is z, the divisor is 5, and the dividend is 150. ; z = a. three division problems with a quotient of 6 b. three division problems with a dividend of 24 4. Fill in the tables. Remember, the product of two numbers means they are multiplied. Numbers Product (written) Product (solved) Quotient (written) Quotient (solved) 12 and 3 12 3 36 10 and 5 20 and 4 100 and 10 11 Math Mammoth Division 2 (Blue Series)
Division with zero What about 0 0? We check a division problem by multiplication. We cannot really determine any single Is 0 3 = 0? Check if 0 3 = 0. Yes, it is. answer, because all of these could work: Is 0 11 = 0? Check if 0 11 = 0. Yes, it is. If 0 0 = 1, then check: 0 1 = 0 works. Is 3 0 = 0? Check if 0 0 = 3. It is not. If 0 0 = 7, then check: 0 7 = 0 works. If 0 0 = 0, then check: 0 0 = 0 works. Is 3 0 perhaps 3? Check if 0 3 = 3. It is not. So 0 0 is usually said to be In fact, dividing by zero is a real problem. an indeterminate form since we cannot No matter what number you suggest as determine an answer to it. an answer to the problem 3 0, the multiplication check will not work because you will end up multiplying by zero, and can never get the dividend as an answer. That is why division by zero is said to be undefined we cannot define a sensible answer. You can, however, divide zero by any number (except zero). The answer is always zero. 5. Divide. Mark off the problem if it is impossible to do. a. 64 8 = 0 8 = 32 32 = 6. Find what the unknown stands for. a. 64 x = 1 x = 7. Make up: Division by zero is undefined you cannot do it. b. 55 5 = 6 0 = 7 7 = b. 35 T = 35 T = a. two divisions with a quotient of 1 c. 50 1 = 0 10 = 0 0 = c. 0 x = 0 x = d. 0 1 = 1 1 = 9 0 = d. y 18 = 1 y = b. two divisions with a dividend of 0 Mark had two division problems with the same dividend and the same quotient, yet the divisors were different. How could that be? 12 Math Mammoth Division 2 (Blue Series)
Dividing with Whole Tens and Hundreds 1. Solve the multiplication and then write two matching divisions. 2. Divide. 4,800 60 =? Think backwards of multiplication: Because 60 80 = 4,800, then 4,800 60 = 80. a. 300 7 = 7 = 300 = a. 400 8 = 400 80 = 4,000 800 = 4,000 80 = Solving division problems always involves thinking backwards to multiplication, or how many times. 4,000 400 =? Or ask, How many times does 400 go into 4,000? Ten times. So, 4,000 400 = 10. b. 50 800 = = = b. 5,000 5 = 5,000 50 = 5,000 500 = 500 50 = c. 60 40 = = = c. 4,200 700 = 4,200 70 = 420 70 = 420 7 = 3. Solve the unknown factor problems and the division problems. Notice something special here! a. 6 = 540 540 6 = 4. Practice some more. a. 320 8 = 320 80 = b. 3 = 2,700 2,700 3 = c. 40 = 2,800 2,800 40 = b. 540 60 = c. 360 6 = 540 6 = 3,600 60 = 3,200 8 = 5,400 60 = 36,000 6 = 13 Math Mammoth Division 2 (Blue Series)