Algorithms in Everyday Mathematics

Copyright © Wright Group/McGraw-Hill , children first learn to understand the mathematics , children first learn includes a variety of standard computational algorithms includes a variety of standard computational Everyday Mathematics Everyday

Home Connection Handbook

Everyday Mathematics become skilled at representing ideas with objects, words, pictures, and symbols. become skilled at representing ideas with objects, words, pictures, problems. develop persistence and confidence in dealing with challenging learn that their intuitive methods are valid and that mathematics makes sense. learn that their intuitive methods are valid and that mathematics become more proficient with mental arithmetic. opposed to learning are motivated because they understand their own methods, as by rote. ᭜ ᭜ ᭜ ᭜ ᭜ Algorithm Invention an operation, children are encouraged to invent their In the early phases of learning This approach requires children to focus on the own methods for solving problems. learn to think and use their common sense, as well meaning of the operation. They as new skills and knowledge. Children who invent their own procedures: How do children learn to use algorithms for computation? How do children learn to use a variety of computational methods and the Ideally, children should develop that is most appropriate in a given situation. flexibility to choose the procedure Everyday Mathematics The program leads children through as well as children’s invented procedures. mathematical operation (addition, subtraction, three phases as they learn each multiplication, and division). behind the problems they solve. Then, quite often, they come up with their own they solve. Then, quite often, behind the problems prove that they “get it.” Through this process, they unique working algorithms that one algorithm for computing answers to addition, discover that there is more than division problems. Having children become subtraction, multiplication, and essential to their growth and development as comfortable with algorithms is problem solvers. In In mathematics, an algorithm is a specific series of steps that will give you the an algorithm is a specific series In mathematics, school, you and your classmates time. For example, in grade correct answer every are, for multiplying. Chances and memorized a certain algorithm probably learned it worked, but it did! no one knew why An algorithm is a well-defined procedure or set of rules guaranteed to achieve a guaranteed to or set of rules procedure is a well-defined An algorithm to follow the directions every time you an algorithm You use certain objective. or defrost something in toy, use a recipe to make cookies, put together a new the microwave. What is an algorithm? What is 30 Algorithms in Algorithms Copyright © Wright Group/McGraw-Hill proficient withatleastone. Children areencouragedtoexperimentwithvariousalgorithmsandbecome technological world. others areentirelynewalgorithmsthathavesignificantadvantagesintoday’s commonly beentaughtintheU.S.orsimplificationsofthosealgorithms.And children haveinventedontheirown.Othersaretraditionalalgorithmswhich operation. Someofthesealgorithmsmaybethesameorsimilartomethods computational strategies,theyareintroducedtoseveralalgorithmsforeach After childrenhavehadmanyopportunitiestoexperimentwiththeirown Alternative Algorithms method foreachoperation. promote flexibilitywhileensuringthatallchildrenknowatleastonereliable any alternativetheypreferwhensolvingproblems.Theaimofthisapproachisto algorithm. Oncetheycanreliablyusethefocusalgorithm,childrenmayitor All childrenareexpectedtolearnanddemonstrateproficiencywiththefocus terminology, andsupportacrossgradelevelsofthecurriculum. understand andlearn.Theyalsoprovidecommonconsistentlanguage, algorithm. Focusalgorithmsarepowerful,relativelyefficient,andeasyto For eachoperation,theprogramdesignatesonealternativealgorithmasa“focus” Demonstrating Proficiency rd-is utato ihclmstrade-firstsubtractionwith trade-first subtractionwithcolumns Teacher MastersforFamily Communication an unnecessarytrade 31 Copyright © Wright Group/McGraw-Hill 11 268 483 268 483 61411 268 483 7411 268 483 751 11). 751 hundreds tens ones hundreds tens ones hundreds tens ones 140 400). 600 80). 140 3). Partial-Sums Column Addition Add the tens (60 Add the ones (8 Add the hundreds (200 Add the hundreds (200 Example: Add the partial sums (600 Add the partial sums Since 14 tens is 1 hundred plus 4 tens, add 1 to the hundreds column, and change the number in the tens column to 4. Since 11 ones is 1 ten plus 1 one, add 1 to the tens column, and change the number in the ones column to 1. Example: Add the digits in each column. Home Connection Handbook “200 plus 400 is 600. But 7. 60 and 80 is 140. (looking at the next column) I need to adjust that, so I write more to do, I can just But that needs adjusting, so I write 5. 8 and 3 is 11. With no write 1.” For some children, the above process becomes so automatic that they start at the left and write the answer column by column, adjusting as they go without writing any of the intermediate steps. If asked to explain, they might say something like this: Column Addition To add using the column- addition algorithm, draw vertical lines to separate the ones, tens, hundreds, and so on. Add the digits in each column, and then adjust the results. You can add two numbers by You can add two sums, working calculating partial column at a time, one place-value all the sums to and then adding find the total. Focus Algorithm: Partial-SumsFocus Algorithm: Addition This section presents just a few of the possible algorithms for adding whole numbers. for adding whole algorithms a few of the possible presents just This section Addition Algorithms Addition 32 Copyright © Wright Group/McGraw-Hill For example,consider: other part,theresultremainssame. and subtractthesamenumberfrom If youaddanumbertoonepartofsum Opposite-Change Rule at atime. concentrate ononething allowing childrento done beforethesubtraction, except thatallthetradingis the traditionalU.S.algorithm This algorithmissimilarto Trade-First Subtraction Focus Algorithm: subtraction algorithms. This sectionpresentsseveral subtraction ismoredifficult. for addition,probablybecause algorithms forsubtractionthan There areevenmore Subtraction Algorithms Now add2tothe8,andsubtractfrom7: ends inzeros. numbers beingaddedsothatoneofthem This ideacanbeusedtorenamethe (8 2) (7 8 7 2) 15 10 5 any order. Now subtractcolumnbyin hundreds columnbecomes8. 12, andthetopnumberin number inthetenscolumnbecomes number, borrow1hundred.Thetop tens columnlargerthanthebottom To makethetopnumberin column becomes2. and thetopnumberintens ber intheonescolumnbecomes12, number, borrow1ten.Thetopnum- ones columnlargerthanthebottom To makethetopnumberin larger thanthebottomnumber. in eachcolumnisaslargeor make tradessothatthetopnumber Examine thecolumns.Youwantto Example: 15 Teacher MastersforFamily Communication Trade-First Subtraction the first. Rename thesecondnumberandthen the second. Rename thefirstnumberandthen Example: 268 483 268 483 utat2 Subtract30. Subtract 2. utat7 Subtract10. Subtract 7. d .Add10. Add 7. d .Add30. Add 2. Opposite-Change Rule udestn ones tens hundreds udestn ones tens hundreds ones tens hundreds udestn ones tens hundreds 490 261 481 270 932 932 576 356 356 356 932 356 932 8 8 12 12 212 212 212 251 751 500 300 751 451 33 Counting Up Example: Counting Up To subtract using the counting-up algorithm, To find 932 – 356, start with 356 and count up to 932. start with the number you 356 are subtracting (the Add 4 to count up to the nearest ten. subtrahend), and “count up” to the number you are 360 subtracting from (the Add 40 to count up to the nearest hundred. minuend) in stages. Keep 400 track of the amounts Add 500 to count up to the largest possible 100. you count up at each 900 stage. When you are Add 32 to count up to 932. finished, find the sum 932 of the amounts. Now find the sum of the numbers you added. 4 40 500 32 576 So, 932 356 576.

Left-to-Right Subtraction Example: Left-to-Right Subtraction To use this algorithm, think of the number you are subtracting as a To find 932 356, think of 356 as the sum 300 50 6. Then subtract the parts of the sum sum of ones, tens, hundreds, and one at a time, starting from the hundreds. so on. Then subtract one part of 932 the sum at a time. Subtract the hundreds. 300 632

34 Home Connection Handbook Same-Change Rule Example: Same-Change Rule If you add or subtract the same number from both parts of a subtraction problem, the results Add the same number. remain the same. Consider, for example: Add 4. Add 40. 15 8 7 932 936 976 Now add 4 to both the 15 and the 8: 356 360 400 Subtract. 576 (15 4) (8 4) 19 12 7 Or subtract 6 from both the 15 and the 8: (15 6) (8 6) 9 2 7 Example: Same-Change Rule The same-change rule algorithm uses this idea Subtract the same number. to rename both numbers so the number being subtracted ends in zeros. Subtract 6. Subtract 50. 932 926 876 356 350 300 Subtract. 576

Partial-Differences Subtraction Example: Partial-Differences The partial-differences subtraction Subtraction algorithm is a fairly unusual method, 932 356 but one that appeals to some children. Subtract 100s. 900 – 300 600 The procedure is fairly simple: Write Subtract 10s. 30 – 50 20 partial differences for each place, record them, and then add them to find the total Subtract 1s. 2 – 6 4 difference. A complication is that some of Add the partial differences. 576 the partial differences may be negative. Copyright © Wright Group/McGraw-Hill

Teacher Masters for Family Communication 35 Multiplication Algorithms Children’s experiences with addition and subtraction algorithms can help them invent multiplication algorithms. For example, when estimating a product mentally, many children begin to compute partial products: “Ten of these would be. . ., so 30 of them would be. . ., and we need 5 more, so. . .” Beginning in Third Grade Everyday Mathematics, this approach is formalized as the partial-products multiplication algorithm. This algorithm and others are discussed in this section.

Focus Algorithm: Partial Products Example: Partial Products To use the partial-products algorithm, To find 67 53, think of 67 as 60 7 and think of each factor as the sum of ones, 53 as 50 3. Then multiply each part of tens, hundreds, and so on. Then multiply one sum by each part of the other, and add each part of one sum by each part of the the results. other, and add the results. 67 53 Rectangular arrays can be used to Calculate 50 60. 3,000 demonstrate visually how the partial- Calculate 50 7. 350 Calculate 3 60. 180 products algorithm works. The product Calculate 3 7. 21 14 23 is the number of dots in a Add the results. 3,551 14-by-23 array. The diagram below shows how each of the partial products is represented in the array.

20 3 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 14 23 (10 4) (20 3) 10 • • • • • • • •10 • • •20 • • • • • • • • • •10 • • 3 • • • • • • • • • • • • • • • • • • • • • • • (10 20) (10 3) (4 20) (4 3) • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 200 30 80 12 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 322 • • • • • • • •4 • • 20• • • • • • • • • • •4 • •3 4 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Copyright © Wright Group/McGraw-Hill

Modified Repeated Addition Example: Modified Repeated Addition Many children are taught to think of Think of 53 67 as 67 whole-number multiplication as fifty 67s plus three 67s. 53 repeated addition. However, using Since ten 67s is 670, 670 repeated addition as a computation fifty 67s is five 670s. 670 method is inefficient for anything but 670 So, 53 67 is five 670s 670 small numbers. For example, it would } plus three 67s. 670 be extremely tedious to add fifty-three 67 67s in order to compute 67 53. 67 } Using a modified repeated addition 67 3,551 algorithm, in which multiples of 10, 100, and so on, are grouped together, can simplify the process.

36 Home Connection Handbook Copyright © Wright Group/McGraw-Hill and historicalinterest(ithasbeenusedsince Everyday Mathematics Lattice Multiplication favorite ofmanychildrenin multiplication factsandaddingsingle-digitnumbers.Thismethodhasbecomea printed arithmeticbook,publishedin1478)andbecauseitprovidedpracticewith The followingexampleshowshowthemethodisusedtofind67 • Readtheproductfromupperlefttolowerright.Theis3,551. • Starting withthelower-rightdiagonal,addnumbersinsidelatticealong • Multiply eachdigitinonefactorbytheother.Writeproduct • Draw diagonalsfromtheupper-rightcornerofeachbox,extendingbeyond • Draw a2-by-2lattice,andwriteonefactoralongthetopoflattice Follow thesestepstofind67 Example: sum onthefourthdiagonalis3. column. Thesumalongthethirddiagonalisthen1 diagonal is5 The firstdiagonalcontainsonly1,sothesumis1.onsecond to thenextdiagonal. each diagonal.Ifthesumalongadiagonalisgreaterthan9,carrytensdigit diagonal andthe0below. example, since6 the productabovediagonalandonesdigitbelowdiagonal.For the cellwherecorrespondingrowandcolumnmeet.Writetensdigitof the lattice. other alongtheright.(Usealargerlatticetomultiplynumberswithmoredigits.) Lattice ϩ 2 ϫ ϩ 5 8 Multiplication ϭ ϭ 30, write30intheupper-leftboxwith3above 15. Writeonlythe5,andcarry1tonext initially includedthelatticemethodforitsrecreationalvalue ϫ 53. Everyday Mathematics Teacher MastersforFamily Communication ϩ 3 A ϩ . D 0 1100andappearedinthefirst . ϩ . 1, or5.The ϫ 53. 5 3 3 1 3 1 51 67 67 67 0 8 0 8 2 3 2 3 1 1 5 5 1 3 5 3 5 3 5 37 Division Algorithms One type of division situation involves making as many equal-size groups as possible from a collection of objects: How many dozens can you make with 746 eggs? How many 5-passenger cars are needed for 37 people? Such problems ask, “How many of these are in that?” More generally, a b can be interpreted as “How many bs are in a?” This idea forms the basis for the division algorithms presented in this section.

Focus Algorithm: Partial Quotients The partial-quotients algorithm uses a series of “at least, but less than” estimates of how many bs are in a.

Example: Partial Quotients

Estimate the number of 12s in 158. You might begin with multiples of 10 because they are simple to work with. There are at least ten 12s in 158 12ͤ1ෆ5ෆ8ෆ 10 first guess (10 12 120), but there are fewer than twenty 120 (20 12 240). Record 10 as a first estimate, and 38 3 second guess subtract ten 12s from 158, leaving 38. 36 2 13 sum of guesses

Now estimate the number of 12s in 38. 158 ، 12 13 R2 There are more than three (3 12 36) but fewer than four (4 12 48). Record 3 as the next estimate, and subtract three 12s from 38, leaving 2.

38 Home Connection Handbook Copyright © Wright Group/McGraw-Hill Example: at atime. vertical linesseparatingthedigitsofdivisorandworkoneplace-valuecolumn probably learnedinschool,butitiseasiertolearn.Tousethemethod,youdraw Column divisionisasimplificationofthetraditionallongalgorithmyou Column Division there are$3leftover.So,683 So, whenyoudivide$683among5people,eachpersongets$136,and 3 leftover. a totalof33one-dollarbills.Eachpersongets6,andthereare Trade the3leftoverten-dollarbillsfor30one-dollarbills.Younowhave 3 leftover. have atotalof18ten-dollarbills.Eachpersongets3,andthereare Trade theleftoverhundred-dollarbillfor10ten-dollarbills.Nowyou is oneleftover. First, divideupthehundred-dollarbills.Eachpersongetsone,andthere having 6hundred-dollarbills,8ten-dollarand3one-dollarbills. To find683 Column Division 5, imaginesharing$683among5people.Thinkabout 5 136 R3. Teacher MastersforFamily Communication 5 5 5 ͒ ͒ ͒ 1 1 5 3 68 13 1 518 3 68 36 13 1 518 3 68 33 15 15 3 33 30 39