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# 8 Step Model Drawing

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1. 8 Step Model Drawing Singapore’s Best Problem-Solving MATH Strategies Presented by – Julie Martin Adapted from a presentation by Bob Hogan and Char Forsten

2. What is Singapore Math? • The term “Singapore Math” refers to the mathematics curriculum in the country of Singapore, developed by the Curriculum Planning & Development Institute of Singapore and approved by Singapore’s Ministry of Education. • In the U.S., the term generally refers to the Primary Mathematics Series, which is the textbook series for grades K-6.

3. What Are the Strengths of the Singapore Math Curriculum? • The curriculum is highly coherent; it is taught in a logical, step-by-step manner that builds on students’ prior knowledge and skills. It follows a concrete to pictorial to abstract approach. (What we call C-R-A.) • Fewer topics are taught in greater depth. The goal is mastery. • Problem solving is the heart of the Singapore curriculum. Place value, mental math, and computation are reinforced through problem solving, particularly through the model drawing approach.

4. What Are the Strengths of the Singapore Math Curriculum? • Key instructional strategies used in Singapore math, related to place value, computation, mental math, and model drawing are applicable and highly effective with used with U.S. math programs. • The term “Singapore Math” is sometimes used interchangeably with the key strategies used in the curriculum. THEY ARE NOT THE SAME!

5. The Role of Model Drawing in Singapore Math • It is the key strategy taught and used to help students understand and solve word problems. • It is the pictorial stage in the learning sequence of concrete-representational-abstract. • It develops students’ visual-thinking capabilities and algebraic thinking. • It integrates and reinforces higher-level thinking, computation, and mental math strategies in a meaningful working context. • It is a process used regularly, not intermittently, to help students spiral their understanding and use of mathematics.

6. 3 Types of Bar/Model Drawing • Part to Whole: • A single bar with 2 or more sections • Used for addition and subtraction problems Example: Daniel and Peter have 450 marbles. Daniel has 248 marbles. How many marbles does Peter have? 450 248 ?

7. 3 Types of Bar/Model Drawing • Comparison of Quantities • A. Two bars that each represent a quantity Example: Daniel has 248 marbles. Peter has 202 marbles. Who has more marbles? How many more does he have? 248 202 ?

8. 3 Types of Bar/Model Drawing • Comparison of Quantities • B. Sum of 2 bars that each represent a quantity. Example: Mary had 120 more beads than Jill. Jill had 68 beads. How many beads did Mary have? How many beads did the two girls have together? ? 68 120 ?

9. 3 Types of Bar/Model Drawing • Combination of Part to Whole and Comparison • Two bars, each representing a quantity that is cut into pieces and compared to each other. • Used for fractions, decimals, percents, and multi-step problems Example: Julie has ½ as much money as Greg. Paul has 3 times as much money as Greg. If Paul has \$30, how much does Julie have? Greg Julie Paul 30

10. Setting Up the Model • Read the entire problem. • Determine who is involved in the problem. List vertically as each appears in the problem. • Determine what is involved in the problem. List beside the “who” from the previous step. • Draw unit bars (equal length to begin with).

11. Solving the Problem • Reread the problem, one sentence at a time, plugging the information into the visual model. Stop at each comma and illustrate the information on the unit bar. • Determine the question and place the question mark in the appropriate place in the drawing. • Work all computation to the right side or underneath the drawing. • Answer the question in a complete sentence, or as a longer response if asked.

12. Kate read 2 books. She also read 3 magazines. How many books and magazines did she read altogether? Kate’s books Kate’s magazines 2 ? 1 3 2 + 3 = 5 Kate read 5 books and magazines.

13. Alicia had \$6 more than Bobby. If Bobby had \$10.00, how much did they have altogether? Alicia’s money Bobby’s money \$10 \$6 ? \$10 \$10 + \$10 = \$20 OR \$10 + \$6 = \$16 \$20 + \$6 = \$26 \$16 + \$10 = \$26 Alicia and Bobby had \$26 altogether.

14. Max had 2 trucks in his toy chest. He added 3 more. How many total trucks did Max have in his toy chest? Max’s trucks toy chest ? 2 3 2 + 3 = 5 Max had 5 trucks in his toy chest.

15. Emily had 6 stickers. She gave 2 to a friend. How many stickers did Emily have left? Emily’s stickers friend left 2 6 ____ 2 ? 6 2 + ____ = 6 Emily had 4 stickers left.

16. There are 4 fishbowls in the science classroom. Each bowl contains 2 fish. How many fish are there in all 4 bowls? Fish in bowls 1 2 3 4 ? 2 2 2 2 ( 4 groups of 2) 4 x 2 = 8 There are a total of 8 fish in all 4 bowls.

17. Mr. Carter had 12 cookies. He wanted to divide them evenly among 3 students. How many cookies will each student receive? Students’ cookies child child child 3 units = 12 1 unit = 4 12 ? • ÷ 3 = 4 • Each student will receive 4 cookies.

18. Anna and Raul caught fireflies one hot summer night. Anna caught 4 more fireflies than Raul. Raul caught 5 fireflies. How many fireflies did they catch altogether? Anna’s fireflies Raul’s fireflies 5 4 9 ? 5 5 5 + 4 = 9 9 + 5 = 14 They caught 14 fireflies altogether.

19. Mr. Carter had 12 cookies. He wanted to put them into bags, so that each bag would have just 3 cookies. How many bags will he need? Cookies in bags 1 bag 12 3 3 3 3 12 - 9 6 - 3 3 - 3 0 3 + 3 + 3 + 3 = 12 (3, 6, 9, 12) Mr. Carter will need 4 bags.

20. Becca and Sari strung beads on a necklace. They each began with 34 beads, but Becca took off 14 beads, while Sari added another 43 beads. How many more beads does Sari’s necklace have than Becca’s? Becca’s beads Sari’s beads took off 14 20 added on 34 43 20 14 34 57 Sari has 57 more beads than Becca.

21. Amy had 5 baseball cards. Jeff had 3 times as many cards as Amy. How many baseball cards did they have altogether? Amy’s baseball cards Jeff’s baseball cards 5 ? 5 5 5 5 + 5 + 5 + 5 = 20 4 x 5 = 20 They have 20 baseball cards altogether.

22. Eddie had 3 times as much money as Velma. Tina had 2 times as much money as Velma. If Tina had \$60, how much money did Eddie have? Eddie’s \$ Velma’s \$ Tina’s \$ \$30 \$30 \$30 \$30 ? \$60 \$30 \$30 Two units = \$60 \$60 ÷ 2 = \$30 \$30 + \$30 + \$30 = \$90 Eddie has \$90.

23. Two-thirds of a number is 8. What is the number? A number 2 units = 8 1 unit = 4 ? 12 4 4 4 8 4 + 4 = 8 8 + 4 = 12 Two-thirds of 12 is 8.

24. In the fourth grade, 3/7 of the students were boys. If there were 28 girls in the grade, how many boys were in the grade? girls boys 28 4 units = 28 1 unit = 7 7 7 7 7 ? 21 7 7 7 There were 21 boys in the fourth grade.

25. Mrs. Owen bought some eggs. She used ½ of them to make cookies and ¼ of the remainder to make a cake. She had 9 eggs left. How many eggs did she buy? Mrs. Owen’s eggs cookies 12 12 cake 3 units = 9 1 unit = 3 3 3 3 3 9 4 x 3 = 12 12 + 12 = 24 Mrs. Owen bought 24 eggs.

26. The ratio of the number of boys to the number of girls is 3:4. If there are 88 girls, how many children are there altogether? # of boys # of girls 222222 66 ? 22 22 22 22 88 4 units = 44 1 unit = 22 88 + 66 = 154 or 22 x 7 = 154 There are 154 students altogether.

27. Mr. Frank N. Stein correctly answered 80% of the questions on his science test. If there were 30 questions on the test, how many questions did he answer correctly? Mr. Stein’s questions 3 3 3 3 3 3 3 3 3 3 30 10 units = 30 1 unit = 3 correct ? 8 x 3 = 24 Mr. Stein got 24 correct answers on the test.

28. Mutt and Jeff collected a total of 52 aluminum cans for a recycling project. If Mutt collected 12 more cans than Jeff, how many cans did each boy collect? Mutt’s cans Jeff’s cans 52 – 12 = 40 40 ÷ 2 = 20 ? 20 12 52 equal ? 20 Pre-Algebra: x + x + 12 = 52 2x + 12 = 52 2x = 40 x = 20 32 + 20 = 52 Mutt collected 32 cans and Jeff collected 20 cans.

29. One number is one fourth of another number. If the difference between the numbers is 39, find the two numbers. one # another # 13 3 units = 39 1 unit = 13 13 ? ? 13 13 13 13 4 x 13 = 52 difference is 39 One number is 12 and the other number is 52.