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Step-by-Step Model Drawing Solving Word Problems the Singapore Way

(Char Forsten, 2010) Crystal Springs Books, www.SDE.com/crystalsprings, 1-800-321-0401

• Provides a transition from abstract to concrete levels of processing, giving word problems a visual context • Basically a “graphic organizer” that shows the relationships among numbers • Since the process isn’t appropriate for every problem, other problem-solving strategies may still be needed • A “step-by-step approach” that provides explicit, guided instruction to help student organize and apply a problem-solving strategy The “Basic” 8-Step Singapore Approach

1. Read the entire problem 2. Determine who is involved in the problem 3. Determine what is involved in the problem 4. Draw a unit bar for each variable 5. Place the numbers 6. Place the question mark 7. Do the computation to the side or underneath the unit bars 8. Write the answer in a complete sentence

Forsten’s 7-Step Approach

1. Read the entire problem 2. Rewrite the question in a sentence form, leaving a space for the answer 3. Determine who and/or what is involved in the problem 4. Draw the unit bar(s) 5. Chunk the problem, adjust the unit bars, and fill-in the question mark 6. Correctly compute and solve the problem 7. Write the answer in the sentence, and make sure the answer makes sense

Step 2 promotes student understanding of the problem and helps guide their thinking and modeling

Student Goals: Learn how to use the model (process) Automatize the process Apply the process Contents

27 illustrated & explained model-drawing examples, covering addition, subtraction, multiplication and division of whole numbers, fractions and decimals, plus ratio, rate, percentage, and “bridge to algebra” problems, plus 43 practice problems with illustrated solutions

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Scope & Sequence

Whole Numbers

Addition Discrete Model (one-to-one, number-square correspondence) Continuous Model (numeral-in-bar representation) 3 Addends

Subtraction Continuous Model Comparison

Mixed Operations (Addition & Subtraction)

Multiplication Single-Digit Factors Comparison

Division “Partitive” “Quotive” or Measurement

Mixed Operations (Multiplication & Addition) Mixed Operations (Multiplication & Subtraction)

Fractions Addition

Subtraction Multiplication Division Mixed Operations

Decimals Addition Subtraction Multiplication Division Mixed Operations

Ratio

Rate

Percentage

Bridge to Algebra Teaching Tips “Reflectively” read the problems (without reading the numerals) before reading the entire problem

Easier for students to visualize the problem without the numbers interfering with their thoughts

Read the punctuation marks in the problems Model reading word problems carefully Have students place a checkmark over each numeral as they add it to the model

Model accuracy and attention to detail

Have students put a slash after each sentence to show they have finished adding the information to the model

Model accuracy and attention to detail

“Milk the problem for all it’s worth” – additional questions using the same model

Extend student thinking & computational practice

Use graph paper or a vertical “starting line” to help students with alignment of unit bars

Helps students line up unit bars in a model

Use individual objects, manipulatives or illustrations to represent numerals in “number-bond” stories before introducing continuous model drawing

Helps students with number (quantity) - numeral (symbol) correspondences, and move from the concept of “discrete” cardinality to “continuous” modeling

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Problem 1 Addition: Whole Numbers - Discrete Model (one-to-one, number-square correspondence)

Janet picked 3 daisies and 2 sunflowers from her garden. How many total flowers did Janet pick from her garden?

Step 1: Read the Problem “Let’s read the entire problem and picture what it’s about.” Optional: Read “reflectively” (without numerals)

Step 2: Re-Write the Question as a Sentence “We’ll come back and add the answer to our sentence.”

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

Step 6: Compute and Solve the Problem

Step 7: Write the Answer in the Sentence (above) Make Sure the Answer Makes Sense

“Milking the Problem” How many more daisies than sunflowers did Janet pick? If Janet wanted to pick the same number of sunflowers as daisies, how many more sunflowers would she need to pick?

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Problem 2 Addition: Whole Numbers - Continuous Model (numeral-in-bar representation)

One bag of lettuce weighed 14 ounces. Another bag weighted 12 ounces.

What was the total weight of the 2 bags of lettuce? Step 1: Read the Problem “Let’s read the entire problem and picture what it’s about.” Optional: Read “reflectively” (without numerals)

Step 2: Re-Write the Question as a Sentence “We’ll come back and add the answer to our sentence.”

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

Step 6: Compute and Solve the Problem

Step 7: Write the Answer in the Sentence (above) Make Sure the Answer Makes Sense

“Milking the Problem” How many more pounds of lettuce would be needed to make 2 pounds? What fraction of a pound is the 12-ounce bag? What fraction of a pound is the 14-ounce bag?

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Problem 3 Addition: Whole Numbers – 3 Addends

254 people attended a concert on Friday night. 357 people attended a concert on Saturday night.

137 people attended a concert on Sunday afternoon. How many total people attended the concerts?

Step 1: Read the Problem “Let’s read the entire problem and picture what it’s about.” Optional: Read “reflectively” (without numerals)

Step 2: Re-Write the Question as a Sentence “We’ll come back and add the answer to our sentence.”

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

Step 6: Compute and Solve the Problem

Step 7: Write the Answer in the Sentence (above) Make Sure the Answer Makes Sense

“Milking the Problem” How many people attended the concert on Friday and Saturday nights? How many more people attended the concert on Saturday night than on Friday night? How many more people attended the concert on Saturday night than on Sunday afternoon?

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Problem 4 Subtraction: Whole Numbers - Discrete Model (one-to-one, number-square correspondence)

There were 6 birds sitting on a tree branch. Two of the birds flew away. How many birds were left?

Step 1: Read the Problem “Let’s read the entire problem and picture what it’s about.” Optional: Read “reflectively” (without numerals)

Step 2: Re-Write the Question as a Sentence “We’ll come back and add the answer to our sentence.”

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

Step 6: Compute and Solve the Problem

Step 7: Write the Answer in the Sentence (above) Make Sure the Answer Makes Sense

“Milking the Problem” If 2 more flew away, how many birds would be left sitting on the tree branch? If still 2 more flew away, how many birds would be left sitting on the tree branch?

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Problem 5 Subtraction: Whole Numbers - Continuous Model (numeral-in-bar representation)

A total of 438 people were at a concert. There were 213 children and the rest were adults.

How many adults were at the concert? Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

Problem 6 Subtraction: Whole Numbers – Comparison

Rico has $45. Tom has $28.

How much more money does Rico have than Tom? Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

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Problem 7 Multiplication: Whole Numbers – Single-Digit Factor

Ling put 3 photos on each page of her album. If there were 6 pages, how many photos did Ling put in her album?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

Problem 8 Multiplication: Whole Numbers – Two-Digit Factor

Luis ran 2 miles each day during the month of September. How many total miles did Luis run in September?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

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Problem 9 Multiplication: Whole Numbers – Comparison

Itty and Bitty each did jumping jacks. Itty did 8 jumping jacks.

Bitty did 4 times as many jumping jacks as Itty. How many jumping jacks did Itty and Bitty do altogether?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

Problem 10 Division: Whole Numbers – Partitive

Simon arranged chairs in the gym for an assembly. He put 42 chairs into 6 equal rows.

How many chairs were in each row?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

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Problem 11 Division: Whole Numbers with Remainder – Quotitive or Measured

Mr. Rodriguez is finding drivers to take 17 of his students on a field trip. If only 4 students can ride in each car, how many cars will Mr. Rodriguez need?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

Problem 12 Mixed Operations: Whole Numbers (addition & subtraction)

Jerome’s goal is to read 345 pages of his book over a 3-day period. If he reads 142 pages on Monday, and 115 pages on Tuesday,

how many pages will Jerome read on Wednesday to reach his goal?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

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Problem 13 Mixed Operations: Whole Numbers (multiplication & division)

Sasha picked 6 baskets of apples, putting 10 apples in each basket. She then divided the baskets of apples evenly among 3 people.

How many apples did Sasha give to each of the 3 people?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

Problem 14 Fractions: Addition

Marissa spent 2/3 of her money at the mall and had $10 left. How much money did she spend at the mall?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

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Problem 15 Fractions: Subtraction

A pie was cut into 6 equal pieces. Luke ate 1/3 of the pie.

How many pieces were left?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

Problem 16 Fractions: Multiplication

Cal had 2 oranges. He wanted to share ¼ of his oranges with his sister, Anna.

What fraction of an orange should Cal give Anna?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

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Problem 17 Fractions: Division

There is ½ of a candy bar that needs to be divided evenly among 3 children. What fraction of the total candy bar will each child receive?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

Problem 18 Fractions: Mixed Operations

Terra’s monthly allowance is $48. She puts ½ of her allowance into savings, and gives ¾ of the remaining money to a local charity.

How much money does Terra have left for herself each month?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

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Problem 19 Decimals: Addition

Tiana spent $58.89 on clothes. She had $18.66 left.

How much money did she have at first?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

Problem 20 Decimals: Subtraction

Sam wants to buy a book that costs $15.99. He has $8.43 right now.

How much more money does Sam need to buy the book?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

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Problem 21 Decimals: Multiplication

An orange rope is 1.3 meters long. A green rope is 3 times as long as the orange rope.

What is the combined length of the orange and green ropes?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

Problem 22 Decimals: Division

Jiang needs to cut a ribbon into 3 equal pieces. If the ribbon is 2.4 meters long, what should be the length of each piece?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

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Problem 23 Ratio

The ratio of children to adults at the football game was 2:3. If there were 140 children at the football game, how many adults were there?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

Problem 24 Rate

If a photocopier can copy pages at the rate of 20 per minute, how many pages can it copy in ½ hour?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

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Problem 25 Percentage

Sixty percent of the animals at the neighborhood pet store were dogs. If there were a total of 40 animals at the pet store,

how many of the animals were dogs?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

Problem 26 Bridge to Algebra

Together, Jasmine and Laura earned a total of $64 babysitting. If Jasmine earned $14 more than Laura,

how much money did each girl earn?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

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Problem 27 Bridge to Algebra

Maria rented 2 video games and 3 DVDs for a total of $23. Each video game cost $1.50 more to rent than each DVD.

What was the cost of renting the DVDs?

Step 3: Determine Who and What is Involved “Write down who and/or what we are talking about.” Step 4: Draw the Unit Bar(s) “Draw the unit bar(s) to the right of the who and/or what terms.” Step 5 a: “Chunk” the Problem “Re-read the problem & add the information to the model.” b: Adjust the Unit Bar(s) “Label each unit bar.” c: Fill-In the Question Mark “Add a question mark to show the answer we need, and that we are ready to compute.”

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Next Steps Model Drawing for Challenging Word Problems

Finding Solutions the Singapore Way (Lorraine Walker, 2010)

Crystal Springs Books, www.SDE.com/crystalsprings, 1-800-321-0401

• Similar to problems found in US math textbooks for grades 6 through 9 • Goes beyond basic model drawing in a more complex way using slightly more advanced model-drawing techniques • Makes a good transition to solving word problems using algebraic equations

Contents Introduction Word Problems Whole Numbers Fractions Decimals Rate/Distance Ratio Percent

Two Unknowns Algebra

Working Backward & Forward from the Model

Examples Maria and Carlos each can earn $12 a day walking dogs. If they start with $25 and $14 respectively, and they both work every day, in how many days will their combined earnings equal $159? The sales team at Latti Motors sold 2 used cars on Monday. One car sold for $3,800, which resulted in a profit of 25% over the buying price. The other car sold for $2,480, which represented a 20% loss. Find the total loss or profit from these 2 sales. Terri won $250 playing bingo. She gave $b to each of her 3 friends. How much money did she have left? Express your answer in terms of b. Web Resources www.thesingaporemaths.com (help for teachers learning model drawing) www.teach-kids-math-by-model-method.com (help for parents learning model drawing) www.thinkingblocks.com (free, interactive website for learning & practicing model drawing)