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Singapore Mathematics: Problem Solving Linda King Singapore Math Coach lking@windsor.k12.mo.us By Linda King, Windsor C-1 Singapore Math & Solving Word Problems Singapore Math moves students through 3 stages of learning: – Concrete – Pictorial – Abstract By Linda King, Windsor C-1 Singapore Math & Solving Word Problems 8-Step Model Drawing is a pictorial way to represent word problems. This method helps students to SEE the problem and can dramatically increase student understanding. 8-Step Model Drawing provides students with a framework that can be used to attack MOST word problems. By Linda King, Windsor C-1 8 Steps for Model-Drawing 1. Read the entire problem. 2. Decide who is involved in the problem. 3. Decide what is involved in the problem. 4. Draw unit bars of equal length. 5. Read each sentence, one at a time. 6. Put the question mark in place. 7. Work computation to the side or underneath. 8. Answer the question in a complete sentence. By Linda King, Windsor C-1 Problem 1: 1. Adam read 8 books. Haley read 6 books. How many books did they read altogether? (1st grade problem) By Linda King, Windsor C-1 Step 1: Read the entire problem. 1. Adam read 8 books. Haley read 6 books. How many books did they read altogether? By Linda King, Windsor C-1 Step 2: Who is the problem about? 1. Adam read 8 books. Haley read 6 books. How many books did they read altogether? By Linda King, Windsor C-1 Step 2: Who is the problem about? 1. Adam read 8 books. Haley read 6 books. How many books did they read altogether? Adam By Linda King, Windsor C-1 Step 2: Who else is the problem about? 1. Adam read 8 books. Haley read 6 books. How many books did they read altogether? By Linda King, Windsor C-1 Step 2: Who else is the problem about? 1. Adam read 8 books. Haley read 6 books. How many books did they read altogether? Adam Haley By Linda King, Windsor C-1 Step 3: What is the problem about? 1. Adam read 8 books. Haley read 6 books. How many books did they read altogether? Adam Haley By Linda King, Windsor C-1 Step 3: What is the problem about? 1. Adam read 8 books. Haley read 6 books. How many books did they read altogether? Adam’s books Haley’s books By Linda King, Windsor C-1 Step 4: Draw unit bars of equal length. Adam’s books Haley’s books By Linda King, Windsor C-1 Step 4: Draw unit bars of equal length. Adam’s books Haley’s books By Linda King, Windsor C-1 Step 5: Read each sentence, one at a time. Adam read 8 books. Adam’s books Haley’s books By Linda King, Windsor C-1 Add the information to your drawing. Adam read 8 books. Adam’s books Haley’s books By Linda King, Windsor C-1 Add the information to your drawing. Adam read 8 books. Adam’s 1 1 1 1 1 1 1 1 books Haley’s At the beginning of books 1st grade, one unit bar represents 1 unit. By Linda King, Windsor C-1 Step 5: Read each sentence, one at a time. Haley read 6 books. Adam’s 1 1 1 1 1 1 1 1 books Haley’s 1 books By Linda King, Windsor C-1 Add the information to your drawing. Haley read 6 books. Adam’s 1 1 1 1 1 1 1 1 books Haley’s 1 books By Linda King, Windsor C-1 Add the information to your drawing. Haley read 6 books. Adam’s 1 1 1 1 1 1 1 1 books Haley’s 1 1 1 1 1 1 books By Linda King, Windsor C-1 Step 5: Read each sentence, one at a time. How many books did they read altogether? Adam’s 1 1 1 1 1 1 1 1 books Haley’s 1 1 1 1 1 1 books By Linda King, Windsor C-1 No new information. Move to step 6. Adam’s 1 1 1 1 1 1 1 1 books Haley’s 1 1 1 1 1 1 books By Linda King, Windsor C-1 Step 6: Put the question mark in place. How many books did they read altogether? Adam’s 1 1 1 1 1 1 1 1 books Haley’s 1 1 1 1 1 1 books By Linda King, Windsor C-1 Step 6: Put the question mark in your drawing. Adam’s 1 1 1 1 1 1 1 1 books Haley’s 1 1 1 1 1 1 ? books By Linda King, Windsor C-1 Before working computation, add information to the drawing as needed. Adam’s 1 1 1 1 1 1 1 1 books Haley’s 1 1 1 1 1 1 ? books By Linda King, Windsor C-1 Before working computation, add information to the drawing as needed. Adam’s 1 1 1 1 1 1 1 1 8 books 6 Haley’s 1 1 1 1 1 1 ? books By Linda King, Windsor C-1 Step 7: Work the computation. Adam’s 1 1 1 1 1 1 1 1 8 books 6 Haley’s 1 1 1 1 1 1 ? books By Linda King, Windsor C-1 Step 7: Work the computation. Adam’s 1 8 1 1 1 1 1 1 1 books Haley’s 1 1 1 1 1 1 6 ? books 8 + 6 = 14 or count the unit bars By Linda King, Windsor C-1 Place the answer next to the question mark in your drawing. Adam’s 1 8 1 1 1 1 1 1 1 books 14 Haley’s 1 1 1 1 1 1 6 ? books 8 + 6 = 14 By Linda King, Windsor C-1 Step 8: Answer in a complete sentence. Adam’s 1 8 1 1 1 1 1 1 1 books 14 Haley’s 1 1 1 1 1 1 6 ? books 8 + 6 = 14 1st graders fill in the blank in a sentence. By Linda King, Windsor C-1 Step 8: Answer in a complete sentence. Adam’s 1 8 1 1 1 1 1 1 1 books 14 Haley’s 6 ? 1 1 1 1 1 1 books 8 + 6 = 14 Adam and Haley read 14 books altogether. By Linda King, Windsor C-1 Problem 2: 40¢ each 2. Sara bought 2 pears. She paid _____¢ . (1st grade problem) By Linda King, Windsor C-1 Step 1: Read the entire problem. 40¢ each 2. Sara bought 2 pears. She paid _____¢ . By Linda King, Windsor C-1 Step 2: Who is the problem about? 40¢ each 2. Sara bought 2 pears. She paid _____¢ . By Linda King, Windsor C-1 Step 2: Who is the problem about? 40¢ each 2. Sara bought 2 pears. She paid _____¢ . Sara By Linda King, Windsor C-1 Step 3: What is the problem about? 40¢ each 2. Sara bought 2 pears. She paid _____¢ . By Linda King, Windsor C-1 Step 3: What is the problem about? 2. Sara bought 2 pears. 40¢ each She paid _____¢ . Sara’s pears By Linda King, Windsor C-1 Step 4: Draw unit bars of equal length. 2. Sara bought 2 pears. 40¢ each She paid _____¢ . Sara’s pears By Linda King, Windsor C-1 Step 4: Draw unit bars of equal length. 2. Sara bought 2 pears. 40¢ each She paid _____¢ . Sara’s pears By Linda King, Windsor C-1 Step 5: Read each sentence, one at a time. 2. Sara bought 2 pears. 40¢ each She paid _____¢ . Sara’s pears By Linda King, Windsor C-1 Add the information to your drawing. 2. Sara bought 2 pears. She paid _____¢ . 40¢ each Sara’s pears By Linda King, Windsor C-1 Add the information to your drawing. 2. Sara bought 2 pears. She paid _____¢ . 40¢ each Sara’s pears 1 1 By Linda King, Windsor C-1 Step 5: Read each sentence, one at a time. 2. Sara bought 2 pears. She paid _____¢ . 40¢ each Sara’s pears 1 1 By Linda King, Windsor C-1 Add new information to your drawing. 2. Sara bought 2 pears. 40¢ each She paid _____¢ . Sara’s pears 1 1 By Linda King, Windsor C-1 No new information in this sentence. 2. Sara bought 2 pears. 40¢ each She paid _____¢ . Sara’s pears 1 1 By Linda King, Windsor C-1 Is there any other information that I could add to the drawing? 2. Sara bought 2 pears. She paid _____¢ . 40¢ each Sara’s pears 1 1 By Linda King, Windsor C-1 Add this information to your drawing. 2. Sara bought 2 pears. She paid _____¢ . 40¢ each Sara’s pears 1 1 By Linda King, Windsor C-1 Add this information to your drawing. 2. Sara bought 2 pears. She paid _____¢ . 40¢ each Sara’s 40¢ pears 1 1 By Linda King, Windsor C-1 Step 6: Put the question mark in place. 2. Sara bought 2 pears. She paid _____¢ . 40¢ each Sara’s 40¢ pears 1 1 By Linda King, Windsor C-1 Step 6: Put the question mark in place. 2. Sara bought 2 pears. She paid _____¢ . 40¢ each Sara’s 40¢ pears 1 1 ?¢ By Linda King, Windsor C-1 Is there any more information that we could add to the drawing? 2. Sara bought 2 pears. She paid _____¢ . 40¢ each 40¢ Sara’s pears 1 1 ?¢ By Linda King, Windsor C-1 Is there any more information that we could add to the drawing? 2. Sara bought 2 pears. She paid _____¢ . 40¢ each Sara’s 40¢ (40¢) pears 1 1 ?¢ By Linda King, Windsor C-1 Step 7: Work computation. 2. Sara bought 2 pears. She paid _____¢ . Sara’s 40¢ (40¢) pears ?¢ 1 1 By Linda King, Windsor C-1 Step 7: Work computation. 2. Sara bought 2 pears. She paid _____¢ . Sara’s 40¢ (40¢) pears ?¢ 1 1 40 + 40 = 80 By Linda King, Windsor C-1 Place the answer in the drawing next to the question mark. 2. Sara bought 2 pears. She paid _____¢ . Sara’s 40¢ (40¢) pears ?¢ 1 1 40 + 40 = 80 By Linda King, Windsor C-1 Place the answer in the drawing next to the question mark. 2. Sara bought 2 pears. She paid _____¢ . Sara’s 40¢ (40¢) pears ?¢ 80 1 1 40 + 40 = 80 By Linda King, Windsor C-1 Step 8: Answer in a complete sentence. 2. Sara bought 2 pears. She paid _____¢ . Sara’s 40¢ (40¢) pears ?¢ 80 1 1 40 + 40 = 80 By Linda King, Windsor C-1 Step 8: Answer in a complete sentence. 2. Sara bought 2 pears. She paid _____¢ . Sara’s 40¢ (40¢) pears ?¢ 80 1 1 40 + 40 = 80 She paid 80¢. By Linda King, Windsor C-1 Problem 3: 70¢ 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. (1st grade problem) By Linda King, Windsor C-1 Step 1: Read the entire problem. 70¢ 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. By Linda King, Windsor C-1 Step 2: Who is the problem about? 70¢ 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. By Linda King, Windsor C-1 Step 2: Who is the problem about? 70¢ 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. Devi By Linda King, Windsor C-1 Step 3: What is the problem about? 70¢ 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. Devi By Linda King, Windsor C-1 Step 3: What is the problem about? 70¢ 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. Devi’s money By Linda King, Windsor C-1 Step 4: Draw unit bars of equal length. 70¢ 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. Devi’s money By Linda King, Windsor C-1 Step 4: Draw unit bars of equal length. 70¢ 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. Devi’s money By Linda King, Windsor C-1 Step 5: Read each sentence, one at a time. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ Devi’s money By Linda King, Windsor C-1 Add the information to the problem. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ Devi’s money By Linda King, Windsor C-1 Add the information to the problem. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ Devi’s 55¢ money This unit bar represents Devi’s money. Transition at end of 1st grade: one bar = 55¢. By Linda King, Windsor C-1 Step 5: Read each sentence, one at a time. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ Devi’s 55¢ money By Linda King, Windsor C-1 Add the information to the problem. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ Devi’s 55¢ money By Linda King, Windsor C-1 There is no new information to add to the drawing in this sentence. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ Devi’s 55¢ money By Linda King, Windsor C-1 Step 5: Read each sentence, one at a time. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ Devi’s 55¢ money By Linda King, Windsor C-1 Add information to the problem. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ Devi’s 55¢ money By Linda King, Windsor C-1 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ Devi’s 55¢ more money By Linda King, Windsor C-1 How much money does she need in all? Add the information to the problem. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ Devi’s 55¢ more money By Linda King, Windsor C-1 Add this information to your drawing. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ 70¢ Devi’s 55¢ more money By Linda King, Windsor C-1 Step 6: Put the question mark in place. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ 70¢ Devi’s 55¢ more money By Linda King, Windsor C-1 Step 6: Put the question mark in place. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ 70¢ Devi’s 55¢ more money ? By Linda King, Windsor C-1 Is there any other information that I could add to the drawing? 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ 70¢ Devi’s 55¢ more money ? By Linda King, Windsor C-1 No other information. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ 70¢ Devi’s 55¢ more money ? By Linda King, Windsor C-1 Step 7: Computation. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ 70¢ Devi’s 55¢ more money ? By Linda King, Windsor C-1 Step 7: Computation. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ Devi’s 55¢ more 70 – 55 = 15 money ? By Linda King, Windsor C-1 Place the answer by the question mark. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ Devi’s 55¢ more 70 – 55 = 15 money ? 15 By Linda King, Windsor C-1 Step 8: Answer in a complete sentence. 3. Devi has 55¢. She wants to buy the pen. She needs ________¢ more. 70¢ Devi’s 55¢ more 70 – 55 = 15 money ? 15 By Linda King, Windsor C-1 Step 8: Answer in a complete sentence. 3. Devi has 55¢. She wants to buy the pen. She needs 15¢ more. 70¢ Devi’s 55¢ more 70 – 55 = 15 money ? 15 By Linda King, Windsor C-1 Problem 4: 4. There were 287 people in a hall. 52 of them were children. How many adults were there? (2nd grade problem) Begin with Steps 1 through 4 of the 8-Steps for Model-Drawing. By Linda King, Windsor C-1 Problem 4: After Step 4. 4. There were 287 people in a hall. 52 of them were children. How many adults were there? people Now do Step 5, adding information, one sentence at a time. By Linda King, Windsor C-1 Problem 4: Step 5: After 1st sentence. 4. There were 287 people in a hall. 52 of them were children. How many adults were there? 287 people By Linda King, Windsor C-1 Problem 4: Step 5: After 2nd sentence. 4. There were 287 people in a hall. 52 of them were children. How many adults were there? 287 children people 52 By Linda King, Windsor C-1 Problem 4: Steps 5 & 6: Place the question mark, using the information in the third sentence. 4. There were 287 people in a hall. 52 of them were children. How many adults were there? 287 people children (adults)? 52 By Linda King, Windsor C-1 Problem 4: Step 7: Computation. 4. There were 287 people in a hall. 52 of them were children. How many adults were there? 287 people children (adults)? 52 By Linda King, Windsor C-1 Problem 4: Step 7: Completed. 4. There were 287 people in a hall. 52 of them were children. How many adults were there? 287 people children (adults)? 52 235 287 – 52 = 235 By Linda King, Windsor C-1 Problem 4 completed: 4. There were 287 people in a hall. 52 of them were children. How many adults were there? 287 people children (adults)? 52 235 287 – 52 = 235 There were 235 adults. By Linda King, Windsor C-1 Problem 5: 5. Tracy bought 6 bags of oranges. There were 10 oranges in each bag. How many oranges did she buy altogether? (2nd grade problem) Try this one on your own using the 8 Steps for Model Drawing. By Linda King, Windsor C-1 Problem 5 Drawing: 5. Tracy bought 6 bags of oranges. There were 10 oranges in each bag. How many oranges did she buy altogether? 1 2 3 4 5 6 Tracy’s 10 10 10 10 10 10 oranges ? By Linda King, Windsor C-1 Problem 5 Solution: 5. Tracy bought 6 bags of oranges. There were 10 oranges in each bag. How many oranges did she buy altogether? 1 2 3 4 5 6 Tracy’s 10 10 10 10 10 10 oranges ? 60 6 x 10 = 60 or 10 + 10 + 10 + 10 + 10 + 10 = 60 Tracy bought 60 oranges. By Linda King, Windsor C-1 Problem 6: 6. There are 304 girls in a school. There are 46 fewer boys than girls. a. How many boys are there in the school? b. How many children are there in the school? (2nd grade problem) Try this one on your own using the 8 Steps for Model Drawing. By Linda King, Windsor C-1 Problem 6 Drawing: 6. There are 304 girls in a school. There are 46 fewer boys than girls. a. How many boys are there in the school? b. How many children are there in the school? 304 girls b)? boys a)? fewer - 46 By Linda King, Windsor C-1 Problem 6 Solution: 6. There are 304 girls in a school. There are 46 fewer boys than girls. a. How many boys are there in the school? b. How many children are there in the school? 304 girls b)? 562 boys a)? 258 fewer - 46 304 – 46 = 258 258 + 304 = 562 There are 258 boys in the school. There are 562 children in the school. By Linda King, Windsor C-1 Problem 7: 7. Melissa has 6 postcards. Sally has 3 times as many postcards as Melissa. How many more postcards does Sally have than Melissa? (3rd grade problem) Try this one on your own using the 8 Steps for Model Drawing. By Linda King, Windsor C-1 Problem 7 Drawing: 7. Melissa has 6 postcards. Sally has 3 times as many postcards as Melissa. How many more postcards does Sally have than Melissa? Melissa’s 6 postcards 1x 2x 3x Sally’s postcards (6) (6) (6) ? By Linda King, Windsor C-1 Problem 7 Solution: 7. Melissa has 6 postcards. Sally has 3 times as many postcards as Melissa. How many more postcards does Sally have than Melissa? Melissa’s 6 postcards 6 + 6 = 12 1x 2x 3x Sally’s postcards (6) (6) (6) ? 12 Sally has 12 more postcards than Melissa. By Linda King, Windsor C-1 Problem 8 8. Brian has 6 goldfish. Work through He has 5 times as many Step 5: guppies as goldfish. sentence 2. STOP THEN, If he puts his guppies equally into 3 tanks, how many guppies are in each tank? (3rd grade problem) By Linda King, Windsor C-1 Problem 8: Step 5, sentence 3 8. Brian has 6 goldfish. He has 5 times as many guppies as goldfish. If he puts his guppies equally into 3 tanks, how many guppies are in each tank? Brian’s goldfish 6 x1 x2 x3 x4 x5 Brian’s guppies By Linda King, Windsor C-1 Problem 8: Step 5, sentence 3 8. Brian has 6 goldfish. He has 5 times as many guppies as goldfish. If he puts his guppies equally into 3 tanks, how many guppies are in each tank? Pull down the bar. Brian’s Too confusing to divide the 6 guppie bar again. goldfish Brian’s x1 x2 x3 x4 x5 guppies By Linda King, Windsor C-1 Problem 8: Step 5, sentence 3 8. Brian has 6 goldfish. He has 5 times as many guppies as goldfish. If he puts his guppies equally into 3 tanks, how many guppies are in each tank? NOW, divide the “pull-down” Brian’s 6 bar into 3 parts and label. goldfish Brian’s x1 x2 x3 x4 x5 guppies tank 1 tank 2 tank 3 By Linda King, Windsor C-1 Problem 8: Step 5, sentence 3 8. Brian has 6 goldfish. He has 5 times as many guppies as goldfish. If he puts his guppies equally into 3 tanks, how many guppies are in each tank? Brian’s Complete the problem. 6 goldfish Brian’s x1 x2 x3 x4 x5 guppies tank 1 tank 2 tank 3 By Linda King, Windsor C-1 Problem 8 Drawing: 8. Brian has 6 goldfish. He has 5 times as many guppies as goldfish. If he puts his guppies equally into 3 tanks, how many guppies are in each tank? Brian’s 6 goldfish Brian’s x1 x2 x3 x4 x5 guppies (6) (6) (6) (6) (6) (30) (30) tank 1 ? tank 2 tank 3 By Linda King, Windsor C-1 Problem 8 Solution: 8. Brian has 6 goldfish. He has 5 times as many guppies as goldfish. If he puts his guppies equally into 3 tanks, how many guppies are in each tank? Brian’s 6 goldfish Brian’s x1 x2 x3 x4 x5 guppies (6) (6) (6) (6) (6) (30) (30) 5 x 6 = 30 tank 1 ? tank 2 tank 3 30 ÷ 3 = 10 10 Brian has 10 guppies in each tank. By Linda King, Windsor C-1 Problem 9: 9. The capacity of a bucket is 6 liters. 5 buckets of water are needed to fill up a tank. What is the capacity of the tank? (3rd grade problem) Try this one on your own using the 8 Steps for Model Drawing. By Linda King, Windsor C-1 Problem 9 Drawing: 9. The capacity of a bucket is 6 liters. 5 buckets of water are needed to fill up a tank. What is the capacity of the tank? Bkt 1 Bkt 2 Bkt 3 Bkt 4 Bkt 5 Tank’s capacity 6 (6) (6) (6) (6) ? By Linda King, Windsor C-1 Problem 9 Solution: 9. The capacity of a bucket is 6 liters. 5 buckets of water are needed to fill up a tank. What is the capacity of the tank? Bkt 1 Bkt 2 Bkt 3 Bkt 4 Bkt 5 Tank’s capacity 6 (6) (6) (6) (6) ? 30 5 x 6 = 30 The capacity of the tank is 30 liters. By Linda King, Windsor C-1 Contact Linda King: lking@windsor.k12.mo.us (636) 464 – 4400, extension 4399 By Linda King, Windsor C-1