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2324273341456169758389INTRODUCTION TO SINGAPORE MATHWelcome to Singapore Math! The math curriculum in Singapore has been recognized worldwidefor its excellence in producing students highly skilled in mathematics. Students in Singapore haveranked at the top in the world in mathematics on the Trends in International Mathematics andScience Study (TIMSS) in 1993, 1995, 2003, and 2008. Because of this, Singapore Math has gainedin interest and popularity in the United States.Singapore Math curriculum aims to help students develop the necessary math concepts andprocess skills for everyday life and to provide students with the ability to formulate, apply, and solveproblems. Mathematics in the Singapore Primary (Elementary) Curriculum cover fewer topics butin greater depth. Key math concepts are introduced and built on to reinforce various mathematicalideas and thinking. Students in Singapore are typically one grade level ahead of students in theUnited States.The following pages provide examples of the various math problem types and skill sets taught inSingapore.1319252750Used for subtraction of two numbers. Crossout the number of items to be taken away.Count the remaining ones to find the answer.20 – 12 81. The Counting-On MethodUsed for addition of two numbers. Count onin 1s with the help of a picture or numberline.A number bond shows the relationship in asimple addition or subtraction problem. Thenumber bond is based on the concept “partpart-whole.” This concept is useful in teachingsimple addition and subtraction to youngchildren.7 4 1199033. The Crossing-Out MethodAt an elementary level, some simplemathematical skills can help studentsunderstand mathematical principles. Theseskills are the counting-on, counting-back, andcrossing-out methods. Note that these methodsare most useful when the numbers are small. 17 18 19 110112. The Counting-Back MethodUsed for subtraction of two numbers. Countback in 1s with the help of a picture ornumber line.16 – 3 1313–114–115–1wholepartpartTo find a whole, students must add the twoparts.To find a part, students must subtract the otherpart from the whole.16The different types of number bonds areillustrated on the next page.Singapore Math Level 2A & 2BCD-704680 SINGAPORE MATH BINDUP 2AB TXT.indd 5510/21/14 2:24 PM
1. Number Bond (single digits)Students should understand that multiplicationis repeated addition and that division is thegrouping of all items into equal sets.9361. Repeated Addition (Multiplication)Mackenzie eats 2 rolls a day. How many rollsdoes she eat in 5 days?3 (part) 6 (part) 9 (whole)9 (whole) 3 (part) 6 (part)9 (whole) 6 (part) 3 (part)2 2 2 2 2 105 2 102. Addition Number Bond (single digits)9 She eats 10 rolls in 5 days.512. The Grouping Method (Division)Mrs. Lee makes 14 sandwiches. She givesall the sandwiches equally to 7 friends. Howmany sandwiches does each friend receive?4 9 1 4 10 4 14Make a ten first.3. Addition Number Bond (double andsingle digits)2 15Each friend receives 2 sandwiches.514 7 210 2 5 10 7 10 17One of the basic but essential math skillsstudents should acquire is to perform the 4operations of whole numbers and fractions.Each of these methods is illustrated below.Regroup 15 into 5 and 10.1. The Adding-Without-RegroupingMethodH T OO:Ones4. Subtraction Number Bond (doubleand single digits)122–738 8 9H: Hundreds10Since no regrouping is required, add thedigits in each place value accordingly.10 7 33 2 52. The Adding-by-Regrouping MethodO:OnesH T O5. Subtraction Number Bond (doubledigits)20–151010104 9 2T: Tens1 15 3H: Hundreds6 4 55In this example, regroup 14 tens into 1hundred 4 tens.10 5 510 10 05 0 56CD-704680 SINGAPORE MATH BINDUP 2AB TXT.indd 62 1T : Tens 5 6 8Singapore Math Level 2A & 2B10/21/14 2:24 PM
7. The Multiplying-Without-RegroupingMethod3. The Adding-by-Regrouping-TwiceMethodH T OO:Ones11T OO:Ones2 42 8 6T : Tens 3 6 56 5 1H: Hundreds 2T : Tens4Regroup twice in this example.First, regroup 11 ones into 1 ten 1 one.Second, regroup 15 tens into 1 hundred 5tens.Since no regrouping is required, multiply thedigit in each place value by the multiplieraccordingly.8. The Multiplying-With-RegroupingMethodH T OO:Ones4. The Subtracting-Without-RegroupingMethodH T OO:Ones311, 0 4 7H: HundredsIn this example, regroup 27 ones into 2 tens7 ones, and 14 tens into 1 hundred 4 tens.Since no regrouping is required, subtract thedigits in each place value accordingly.9. The Dividing-Without-RegroupingMethod2 4 15. The Subtracting-by-RegroupingMethodH T OO:Ones72 4 8 2–48–82– 208 111T : Tens– 2 4 73 3 4H: HundredsIn this example, students cannot subtract 7ones from 1 one. So, regroup the tens andones. Regroup 8 tens 1 one into 7 tens 11ones.Since no regrouping is required, dividethe digit in each place value by the divisoraccordingly.10. The Dividing-With-RegroupingMethod6. The Subtracting-by-Regrouping-Twice MethodH T OO:Ones1 6 65 8 3 0–53 3–3 03 0–3 008 90 100– 5 9 3H: Hundreds2 0 77T : TensIn this example, students cannot subtract3 ones from 0 ones and 9 tens from 0 tens.So, regroup the hundreds, tens, and ones.Regroup 8 hundreds into 7 hundreds 9 tens10 ones.Singapore Math Level 2A & 2BCD-704680 SINGAPORE MATH BINDUP 2AB TXT.indd 74 92T : Tens 37 3 9T : Tens– 3 2 54 1 4H: Hundreds58In this example, regroup 3 hundreds into30 tens and add 3 tens to make 33 tens.Regroup 3 tens into 30 ones.710/21/14 2:24 PM
11. The Addition-of-Fractions Methodand division. It can also be applied to wordproblems related to fractions, decimals,percentage, and ratio. 1 2 1 3 5 2 3 6 24 3121212The use of models also trains students to thinkin an algebraic manner, which uses symbols forrepresentation.Always remember to make the denominatorscommon before adding the fractions.12. The Subtraction-of-FractionsMethodThe different types of bar models used to solveword problems are illustrated below. 1 5 – 1 2 3 5 – 2 2 55 21010101. The model that involves additionMelissa has 50 blue beads and 20 redbeads. How many beads does she havealtogether?Always remember to make the denominatorscommon before subtracting the fractions.13. The Multiplication-of-FractionsMethod1? 3 1 1 53950152050 20 70When the numerator and the denominatorhave a common multiple, reduce them totheir lowest fractions.2. The model that involves subtractionBen and Andy have 90 toy cars. Andy has 60toy cars. How many toy cars does Ben have?14. The Division-of-Fractions Method90 7 1 7 62 14 4 23 96 391360?90 – 60 30When dividing fractions, first change thedivision sign ( ) to the multiplication sign ( ).Then, switch the numerator and denominatorof the fraction on the right hand side. Multiplythe fractions in the usual way.3. The model that involves comparisonMr. Simons has 150 magazines and110 books in his study. How many moremagazines than books does he have?Model drawing is an effective strategy usedto solve math word problems. It is a visualrepresentation of the information in wordproblems using bar units. By drawing themodels, students will know of the variablesgiven in the problem, the variables to find, andeven the methods used to solve the problem.MagazinesBooks150110?150 – 110 40Drawing models is also a versatile strategy.It can be applied to simple word problemsinvolving addition, subtraction, multiplication,8CD-704680 SINGAPORE MATH BINDUP 2AB TXT.indd 8Singapore Math Level 2A & 2B10/21/14 2:24 PM
4. The model that involves two itemswith a differenceBrenda. Ellen bakes 9 fewer muffins thanGiselle. How many muffins does Ellen bake?A pair of shoes costs 109. A leather bagcosts 241 more than the pair of shoes. Howmuch is the leather bag?Shoes111 9Brenda(111 9) 5 24 241(2 24) – 9 39 1098. The model that involves sharing 109 241 350There are 183 tennis balls in Basket A and97 tennis balls in Basket B. How many tennisballs must be transferred from Basket A toBasket B so that both baskets contain thesame number of tennis balls?5. The model that involves multiplesMrs. Drew buys 12 apples. She buys 3times as many oranges as apples. She alsobuys 3 times as many cherries as oranges.How many pieces of fruit does she buyaltogether?Apples9Giselle?Bag?Ellen183Basket B?Oranges?Basket A1297Cherries183 – 97 8613 12 15686 2 436. The model that involves multiplesand difference9. The model that involves fractions1 ofGeorge had 355 marbles. He lost51the marbles and gave4 of the remainingmarbles to his brother. How many marblesdid he have left?There are 15 students in Class A. There are5 more students in Class B than in Class A.There are 3 times as many students in ClassC than in Class A. How many students arethere altogether in the three classes?Class AClass B355LBR15RR?5L: LostB: BrotherR: Remaining?Class C(5 15) 5 805 parts 355 marbles1 part 355 5 71 marbles3 parts 3 71 213 marbles7. The model that involves creating awholeEllen, Giselle, and Brenda bake 111 muffins.Giselle bakes twice as many muffins asSingapore Math Level 2A & 2BCD-704680 SINGAPORE MATH BINDUP 2AB TXT.indd 9910/21/14 2:24 PM
1. Comparing10. The model that involves ratioAaron buys a tie and a belt. The prices of thetie and belt are in the ratio 2 : 5. If both itemscost 539,Comparing is a form of thinking skill thatstudents can apply to identify similarities anddifferences.(a) what is the price of the tie?When comparing numbers, look carefullyat each digit before deciding if a number isgreater or less than the other. Students mightalso use a number line for comparison whenthere are more numbers.(b) what is the price of the belt?Tie 539BeltExample:? 539 7 77Tie (2 units) 2 x 77 154Belt (5 units) 5 x 77 3850123456783 is greater than 2 but smaller than 7.2. SequencingA sequence shows the order of a series ofnumbers. Sequencing is a form of thinkingskill that requires students to place numbersin a particular order. There are many terms ina sequence. The terms refer to the numbersin a sequence.11. The model that involvescomparison of fractions2 of Leslie’s height. Leslie’sJack’s height is33height is4 of Lindsay’s height. If Lindsay is160 cm tall, find Jack’s height and Leslie’sheight.?To place numbers in a correct order,students must first find a rule that generatesthe sequence. In a simple math sequence,students can either add or subtract to findthe unknown terms in the sequence.Jack?LeslieLindsay160 cmExample: Find the 7th term in the sequencebelow.1 unit 160 4 40 cmLeslie’s height (3 units) 3 40 120 cm1,4,7,10,13,1st 2nd 3rd4th5thterm term term term termJack’s height (2 units) 2 40 80 cmThinking skills and strategies are important inmathematical problem solving. These skillsare applied when students think through themath problems to solve them. The followingare some commonly used thinking skills andstrategies applied in mathematical problemsolving.Step 1: This sequence is in an increasingorder.Step 2: 4 – 1 37–4 3The difference between twoconsecutive terms is 3.Step 3: 16 3 19The 7th term is 19.10CD-704680 SINGAPORE MATH BINDUP 2AB TXT.indd 1016?6th7thterm termSingapore Math Level 2A & 2B10/21/14 2:24 PM
3. Visualizationadd the first and last numbers to geta result of 11. Then, add the secondand second last numbers to get thesame result. The pattern continuesuntil all the numbers from 1 to 10 areadded. There will be 5 pairs of suchresults. Since each addition equals11, the answer is then 5 11 55.Visualization is a problem solving strategythat can help students visualize a problemthrough the use of physical objects. Studentswill play a more active role in solving theproblem by manipulating these objects.The main advantage of using this strategy isthe mobility of information in the process ofsolving the problem. When students make awrong step in the process, they can retracethe step without erasing or canceling it.Step 4: Use the pattern to find the answer.The other advantage is that this strategyhelps develop a better understanding of theproblem or solution through visual objects orimages. In this way, students will be betterable to remember how to solve these typesof problems.Example: Sarah has a piece of ribbon.She cuts the ribbon into 4 equalparts. Each part is then cut into 3smaller equal parts. If the lengthof each small part is 35 cm, howlong is the piece of ribbon?Step 1: Simplify the problem.Find the sum of 1, 2, 3, 4, 5, 6, 7, 8,9, and 10.Step 2: Look for a pattern.2 9 114 7 113 35 105 cm4 105 420 cmStep 3: Describe the pattern.The piece of ribbon is 420 cm.When finding the sum of 1 to 10,Singapore Math Level 2A & 2BCD-704680 SINGAPORE MATH BINDUP 2AB TXT.indd 11The sum of all the numbers from 1 to100 is 5,050.The strategy of working backward appliesonly to a specific type of math word problem.These word problems state the end result,and students are required to find the totalnumber. In order to solve these wordproblems, students have to work backwardby thinking through the correct sequence ofevents. The strategy of working backwardallows students to use their logical reasoningand sequencing to find the answers.Example: Find the sum of all the numbersfrom 1 to 100.Note that the addition for each pairis not equal to 11 now. The additionfor each pair is now (1 100 101).5. Working BackwardThis strategy requires the use of observationaland analytical skills. Students have toobserve the given data to find a patternin order to solve the problem. Math wordproblems that involve the use of this strategyusually have repeated numbers or patterns.1 10 113 8 115 6 114. Look for a PatternSince there are 5 pairs in the sum of1 to 10, there should be (10 5 50pairs) in the sum of 1 to 100.50 101 5050Some of the commonly used objects forthis strategy are toothpicks, straws, cards,strings, water, sand, pencils, paper, and dice.1110/21/14 2:24 PM
6. The Before-After Concept“making an assumption.” Students can usethis strategy to solve certain types of mathword problems. Making assumptions willeliminate some possibilities and simplifiesthe word problems by providing a boundaryof values to work within.The Before-After concept lists all the relevantdata before and after an event. Students canthen compare the differences and eventuallysolve the problems. Usually, the Before-Afterconcept and the mathematical model gohand in hand to solve math word problems.Note that the Before-After concept can beapplied only to a certain type of math wordproblem, which trains students to thinksequentially.Example: Mrs. Jackson bought 100 piecesof candy for all the students inher class. How many pieces ofcandy would each student receiveif there were 25 students in herclass?Example: Kelly has 4 times as much moneyas Joey. After Kelly uses somemoney to buy a tennis racquet,and Joey uses 30 to buy a pairof pants, Kelly has twice as muchmoney as Joey. If Joey has 98 inthe beginning,(a) how much money does Kellyhave in the end?(b) how much money does Kellyspend on the tennis racquet?In the above word problem, assume thateach student received the same number ofpieces. This eliminates the possibilities thatsome students would receive more thanothers due to good behavior, better results,or any other reason.8. Representation of ProblemIn problem solving, students often userepresentations in the solutions to showtheir understanding of the problems. Usingrepresentations also allow students tounderstand the mathematical conceptsand relationships as well as to manipulatethe information presented in the problems.Examples of representations are diagramsand lists or tables.BeforeKellyJoeyAfter 98?KellyJoey(a) 98 - 30 682 68 136Kelly has 136 in the end. 30Diagrams allow students to consolidateor organize the information given in theproblems. By drawing a diagram, studentscan see the problem clearly and solve iteffectively.(b) 4 98 392 392 – 136 256Kelly spends 256 on the tennisracquet.A list or table can help students organizeinformation that is useful for analysis. Afteranalyzing, students can then see a pattern,which can be used to solve the problem.7. Making SuppositionMaking supposition is commonly known as12CD-704680 SINGAPORE MATH BINDUP 2AB TXT.indd 12Singapore Math Level 2A & 2B10/21/14 2:24 PM
9. Guess and CheckThe strategy of restating the problem is to“say” the problem in a different and clearerway. However, students have to ensure thatthe main idea of the problem is not altered.One of the most important and effectiveproblem-solving techniques is Guess andCheck. It is also known as Trial and Error.As the name suggests, students have toguess the answer to a problem and check ifthat guess is correct. If the guess is wrong,students will make another guess. This willcontinue until the guess is correct.How do students restate a math problem?First, read and understand the problem.Gather the given facts and unknowns. Noteany condition(s) that have to be satisfied.Next, restate the problem. Imagine narratingthis problem to a friend. Present the givenfacts, unknown(s), and condition(s). Studentsmay want to write the “revised” problem.Once the “revised” problem is analyzed,students should be able to think of anappropriate strategy to solve it.It is beneficial to keep a record of all theguesses and checks in a table. In addition,a Comments column can be included. Thiswill enable students to analyze their guess(if it is too high or too low) and improve onthe next guess. Be careful; this problemsolving technique can be tiresome withoutsystematic or logical guesses.11. Simplify the ProblemOne of the commonly used strategiesin mathematical problem solving issimplification of the problem. When aproblem is simplified, it can be “brokendown” into two or more smaller parts.Students can then solve the partssystematically to get to the final answer.Example: Jessica had 15 coins. Some ofthem were 10-cent coins and therest were 5-cent coins. The totalamount added up to 1.25. Howmany coins of each kind werethere?Use the guess-and-check method.Number of10 CoinsValueNumber of5 CoinsValueTotalNumber ofCoinsTotal Value77 10 70 88 5 40 7 8 1570 40 110 1.1088 10 80 77 5 35 8 7 1580 35 115 1.151010 10 100 55 5 25 10 5 15100 25 125 1.25There were ten 10-cent coins and five 5-centcoins.10. Restate the ProblemWhen solving challenging math problems,conventional methods may not be workable.Instead, restating the problem will enablestudents to see some challenging problemsin a different light so that they can betterunderstand them.Singapore Math Level 2A & 2BCD-704680 SINGAPORE MATH BINDUP 2AB TXT.indd 131310/21/14 2:24 PM
Singapore Math Level 2A & 2B Table of Contents . 1 5 3 6 4 5 CD-704680 SINGAPORE MATH BINDUP 2AB TXT.indd 6 10/21/14 2:24 PM. 7 Singapore Math Level 2A 2B 3. The Adding-by-Regrouping-Twice Method O
