Modeling
Word Problems:
Addition and
Subtraction
*** Note: Problems written in italics within the handout may not be solved as part of this session. The problems are in the handout as a reference for participants. An answer key with answers to all problems within the packet will be distributed at the end of the session. *** 1 Directions Step 1 Read each problem individually. Solve with a model, and then algebraically. Step 2 Compare your model with a partner. 1. Leonard spent of his money on a sandwich. He spent 2times as much on a gift for his brother as on some comic books. He had of his money left. What fraction of his money did he spend on comic books? (G5 M3 L7) 2. 94children are in a reading club. One‐third of the boys and three‐sevenths of the girls prefer fiction. If 36 students prefer fiction, how many girls prefer fiction? 2 Directions Step 1 Read and analyze the boxed problem types silently for 2 minutes. Step 2 Jot down a word problem on page 4 that corresponds to each of the 6 problem situations. Step 3 When signaled by the facilitator, stand and share one of your problems with another participant from a different table. Step 4 Analyze your partner’s problem situation. Step 5 At the signal, change partners and share a different problem, etc. 3 Word Problems for Each Problem Type: 4 Reflection  Compare Polya’s process and the RDW process.  What obstacles might students encounter using each approach?  Eureka Math has chosen to use the RDW process. Why do you think so? Instead of Polya’s Problem Solving Process 
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Understand the problem. Devise a plan Carry out the plan. Look back. We use the following process: RDW: Read, Draw, Write Read the problem, Draw and label a model as you reread.  Can I draw something?  What can I draw?  What conclusions can I make from my drawing? Write an equation or equations that help solve the problem. Write a statement of the answer to the question. Polya’s process culminates with “look back.” The RDW process, on the other hand, culminates with a statement and a labeled drawing, an illustration of the story. The statement puts the answer back into context. Does the statement make sense? Does it correspond correctly to the drawing? Does the drawing tell the story? This is MP.2 in action, “reasoning abstractly and quantitatively.” The drawing precipitates the reasoning. The student does not figure out the problem and then draw but rather decodes the relationships through the drawing. The abstract numbers are manipulated in the calculation and restored as quantities in the statement. 5 Notes on Pedagogy: The RDW process often involves moving back and forth between reading and drawing. Students might first read the problem entirely then reread the first sentence. Draw and label. Reread the second sentence. Draw and label, etc. Consider the following example: Leonard spent of his money on a sandwich. He spent 2times as much on a gift for his brother as on some comic books. He had of his money left. What fraction of his money did he spend on comic books? Read: Leonard spent of his money on a sandwich. Draw: Read: He spent 2 times as much on a gift for his brother as on some comic books. Draw: Read: He had of his money left. Draw: Read: What fraction of his money did he spend on comic books? Draw and write: 6 Directions Step 1
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Complete the first problem role‐playing a student as the facilitator models the RDW process. Read the Problem Solving Protocol. Apply the protocol to Set 1. Use both the addition/subtraction and multiplication/division situation charts (the latter when applicable) to analyze and classify the situations. Set 1 1. There are 4 snakes on a rock. 2 more snakes slither over. How many snakes are on the rock now? (GK M4 L16) 2. Lila is collecting honey from her beehives. From the first hive she collects gallon. Then from the other two hives she collects gallon each. How many gallons of honey does Lila collect in all? (G5 M3 Mid‐Module Assessment) 7 Directions: Analyze the students’ models below. Then notate the models from the most simple to the complex. 8 9 Set 2 1. Dominic has 6 yellow stickers and 2 blue stickers. How many stickers does he have in all? (GK M4 L17) 2. A bakery used 12,674 kg of flour. Of that, 1,802 kg was whole wheat and 888 kg was rice flour. The rest was all‐purpose flour. How much all‐purpose flour did they use? (G4 M1 L17) 10 Set 3 1. Five green frogs were sitting on a log. It was so hot that 2 of the frogs went for a swim! How many frogs were still sitting on the log? (GK M4 L21) 2. Tiffany spent 4 sevenths of her money on a teddy bear. If the teddy bear costs $24, how much money did she have left? (G5 M4 L7) 11 Directions Step 1 Read the different Notes on Application Problems and Modes of Instructional Delivery. Step 2 Determine which type of delivery is being modeled in the video. Step 3 Discuss what you would hear, see, and experience with the two other modes of delivery. Step 4 Discuss the strengths and weaknesses of each mode of instructional delivery and when each might be the right choice. Notes on Application Problems The Application Problems (AP) of a Story of Units are a distinct component of each lesson though also included within the Concept Development (CD), and its Lesson 7 Objective: Multiply any whole number by a fraction using tape diagrams. Problem Sets, and in Homework. They are presented, in general, after the Fluency and prior to the Concept Suggested Lesson Structure Development (above right) though at times they follow the CD or precede the Fluency (below right). Though at times the Application Problem segues Fluency Practice (12 minutes) directly into the CD, in general, the AP is the most moveable component of the lesson. You might use it as a “Do Now” when students enter the classroom in Read Tape Diagrams (4 minutes) the morning, after recess, or after lunch. Like the Fluency, it might be superficially customized to better serve the needs and interests of your students. Fluency Practice Application Problem Concept Development Student Debrief (12 minutes) (5 minutes) (33 minutes) (10 minutes) Total Time (60 minutes) Read Tape Diagrams 5.NF.4 (4 minutes) Half of Whole Numbers 5.NF.4 (4 minutes) Fractions as Whole Numbers 5.NF.3 (4 minutes) Materials: (S) Personal white boards Note: This fluency prepares students to multiply fractions by whole numbers during the Concept Development. T: S: T: S: 
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Change the names to those of your students Change the numbers to be more or less challenging Change the context to be more relevant When there is good reason to thoughtfully customize the problem more deeply, of course that might be the best path. 
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Change the situation type. Change from a multi to a single step problem Change from a single to a multi‐step problem Change to a problem that is open ended Sometimes students need more practice of a particular situation type, especially comparison problems. That is a good reason to customize. However, keep in mind that it is not easy to write a simple, clear word problem. When sufficient time is (Project a tape diagram with 10 partitioned into 2 equal units.) Say the whole. 10. On your boards, write the division sentence. (Write 10 ÷ 2 = 5.) Continue with the following possible sequence: 6 ÷ 2, 9 ÷ 3, 12 ÷ 3, 8 ÷ 4, 12 ÷ 4, 25 ÷ 5, 40 ÷ 5, 42 ÷ 6, 63 ÷ 7, 64 ÷ 8, and 54 ÷ 9. Application Problem (5 minutes) Mr. Peterson bought a case (24 boxes) of fruit juice. One‐third of the drinks were grape and two‐thirds were cranberry. How many boxes of each flavor did Mr. Peterson buy? Show your work using a tape diagram or an array. Note: This Application Problem requires students to use skills explored in G5 M4 Lesson 6. Students are finding fractions of a set and showing their thinking with models. Lesson 11
Objective: Use math drawings to represent additions with up to two compositions and relate drawings to the addition algorithm. Suggested Lesson Structure Application Problem
Fluency Practice
Concept Development
Student Debrief
(5 minutes) (10 minutes) (35 minutes) (10 minutes) Total Time (60 minutes) Application Problem (5 minutes)
Mr. Arnold has a box of pencils. He passes out 27 pencils and has 45 left. How many pencils did Mr. Arnold have in the beginning? Note: This is an add to with start unknown problem type that reviews two‐digit addition with one composition. Ask students to think about whether they know the parts or the whole and one part. This will guide them towards the recognition that the situation equation ___ 27 = 45 can be written as a solution equation, 45 + 27 = ___. Fluency Practice (10 minutes)
Place Value 2.NBT.1, 2.NBT.3
(3 minutes) NOTES ON MULTIPLE MEANS OF REPRESENTATION: Since we do not expect students to work the algorithm without place value charts and manipulatives in Grade 2, allow students to use disks to calculate the solution and to explain their thinking. They can even use straws to represent the pencils in the Application Problem. 12 not available to be thoughtful, it might be best to stay very close to the suggestions of the curriculum. Look for a prior example that is along the lines of what students need, and model the new problem after it, e.g., “My students really need more change unknown problems. I remember there was one in Lesson 4. I’ll use that to create another but with different characters, numbers, and a new context.” Always be thoughtful about number choice. Modes of Instructional Delivery A decision needs to be made about the mode of delivery of instruction each day as it is intentionally not dictated in the curriculum so that a teacher can determine what is best for a particular class of students. 
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Will students be encouraged to use a specific model to reason about the relationships within the problem (e.g. an array or tape diagram) or will any math drawing that makes sense be encouraged? Will this be a better problem to use a step‐by‐step guided approach because of new complexities, or will the students work independently and then share out their strategies? Will the students work independently or in pairs? In cooperative groups with a protocol or solve solo and then share with a partner? The chart below lays out three modes of delivery of instruction. There are many gradations within and between each one. Modeling with Interactive Questioning The teacher models the whole process with interactive questioning, some choral response, and talk moves such as, Guided Practice
Independent Practice
Each student has a copy of the
question. Though guided by the teacher, they work independently at times and then come together again. Timing is important. Students might hear, The students are given a problem
to solve and possibly a designated amount of time to solve it. The teacher circulates, supports, and is thinking about which student work to show to support the mathematical objectives of the lesson. When sharing student work, students are encouraged to think about the work with questions such as, problem, students might reflect , with a partner on the steps they down. Time to work together used to solve the problem. ck on what we selecting different student work did to solve this problem. What to share. then be given the same or similar problem to solve for homework. 13 Directions Step 1 Complete the first problem role‐playing a student as the facilitator models the RDW process. Step 2 Apply the protocol to solve the following problems. Step 3 Use both the addition/subtraction and multiplication/division situation charts (the latter when applicable) to analyze and classify the situations. Set 4 1. The students were playing with 7 balls on the playground. They kicked some into a puddle and now some are muddy! What is one way the balls might look? (GK M4 L18) 2. Manny bought a loaf of bread and cut it into 7 equal slices. He wants butter on some slices. What fraction of the slices might have butter and what fraction not have butter? (G4 M5) 3. Meyer has 0.64 GB of space remaining on his iPod. He wants to download a pedometer app (0.24 GB), a photo app (0.403 GB), and a math app (0.3 GB). Which combination of apps can he download? Explain your thinking. (G5 M1 L9) 14 Set 5 1. Nine dogs were playing at the park. Some more dogs ran in. Then there were 12 dogs in all. How many dogs ran in? (G1 M4 L20) 2. Martha, George, and Elizabeth sprinted a combined distance of 10,000 meters. Martha sprinted 3,206 meters. George sprinted 2,094 meters. How far did Elizabeth sprint? Solve using a simplifying strategy or an algorithm. (G4 M2 L1) 3. Leonard spent of his money on a sandwich. He spent 2times as much on a gift for his brother as on some comic books. He had of his money left. What fraction of his money did he spend on comic books? (G5 M3 L7) 15 Set 6 1. Ben and Peter caught 17 tadpoles. They gave some to Anton. They have 4 tadpoles left. How many tadpoles did they give to Anton? (G1 M4 L21) 2. All 3,000 seats in a theater are being replaced. So far, 5 sections of 136 seats and a sixth section containing 348 seats have been replaced. How many more seats do they still need to replace? (G4 M3 L13) 16 Directions Follow Protocol D: Digging Into a Vignette: “Question the Author” for G1‐M4‐Lesson 20. Step 4 17 18 19 20 21 22 Directions Step 1 Look at the student work for each problem. Step 2 Analyze the student work to uncover the child’s thinking. Step 3 Record how you might encourage and support the student after examining the work. 1. Jenny has 8 flowers in a vase. The flowers come in two different colors. Draw a picture to show what the vase of flowers might look like. Write a number sentence and a number bond to match your picture. (G1 M1 L7) 2. Dora found 5 leaves that blew in through the window. Then she found 2 more leaves that blew in. Draw a picture and use numbers to show how many leaves Dora found in all. (G1 M1 L1) 3. Dylan has 4 cats and 2 dogs at home. Laura has 1 dog and 5 fish at home. Laura says she and Dylan have an equal number of pets. Dylan thinks he has more pets than Laura. Who is right? Draw a picture, write 2 number bonds, and use a number sentence to show if Dylan and Laura have an equal amount of pets. 23 5. Shanika saw 5 pigeons on the roof. Some more pigoens flew onto the roof. She then counted 8 pigeons. How many pigeons flew over? (G1 M1 L31) Student A Student B 6. There are 8 juice boxes in the cubbies. Some children drank their juice. Now there are only 5 juice boxes. How many juice boxes were taken from the cubbies? (G1 M1 L32) Student C Student D 24 Directions Step 4 Complete the first problem role‐playing a student as the facilitator models the RDW process. Step 5 Apply the protocol to solve the problems of Set 7. Step 6 Use both the addition/subtraction and multiplication/division situation charts (the latter when applicable) to analyze and classify the situations. Set 7 1. It snowed 14 days. Some snowy days, we stayed home. Nine snowy days we were in school. How many snowy days did we stay home? (G1 M5 L11) 2. Three boxes weighing 128 pounds each and one box weighing 254 pounds were loaded onto the back of an empty truck. A crate of apples was then loaded onto the same truck. If the total weight loaded onto the truck was 2,000 pounds, how much did the crate of apples weigh? (G4 M3 L13) 25 Directions Step 1 Examine the models below and identify all the information that is given in the model. Step 2 Individually, create at least one word problem of the specified situation type that can be solved using each model. Step 3 When signaled, compare your word problems with a partner. 1. (GK M4 L16) Put together with result unknown. 2. (G2 M5 L6) Put together with change unknown. Choose one set of units given (discreet, measurement, fractional) to write at least one word problem. 26 3. (G5 M3 L11) Take from with change unknown 4. Draw a model to represent a “change unknown” situation. Have your partner write a corresponding word problem. 27 Directions Step 1 Complete the first problem role‐playing a student as the facilitator models the RDW process. Step 2 Apply the protocol to solve the problems of Set 8. Step 3 Use both the addition/subtraction and multiplication/division situation charts (the latter when applicable) to analyze and classify the situations. Set 8 1. Mr. Arnold has a box of pencils. He passes out 27 pencils and has 45 left. How many pencils did Mr. Arnold have in the beginning? (G2 M5 L11) 2. Madame Curie made some radium in her lab. She used kg of the radium in an experiment and had 1 kg left. How much radium did she have at first? (G5 M3 L9) (Extension: If she performed the experiment twice, how much radium would she have left?) 3. Max had x brownies. He ate 4 brownies and shared the remaining brownies among his 6 friends equally. How many brownies did each friend receive? Express your answer in terms of x. 28 Directions Step 1 Examine the models below and identify all the information that is given in the model. Step 2 Individually, create at least one “start unknown” word problem that can be solved using each model. Step 3 When signaled, compare your word problem with a partner. 1. (G2 M4 L8) 2. (G2 M5 L2) 29 Directions Step 1 Read and analyze the highlighted problem types silently for 2 minutes. Step 2 Jot down 3 word problems on the next page, one that corresponds to each of the bottom three columns. Step 3 When signaled by the facilitator, stand and share one of your problems with another participant from a different table. Step 4 Analyze your partner’s problem situation. Step 5 At the signal, change partners and share a different problem, etc. 30 Word Problems for Each Problem Type: 31 Modeling Comparison Problems: When modeling comparison problems with second grade students, it is probably best to begin by using the double bar. The ability to recognize that a problem involves comparison is an important insight. Students start by reading the entire problem: Vincent counts 30 dimes and 87 pennies in a bowl. How many more pennies than dimes are in the bowl? Read: Vincent counts 30 dimes. Draw: Read: and 87 pennies in a bowl Draw: Read: How many more pennies than dimes are in the bowl? Draw: In Grade 2 Module 2 Lessons 1‐9 of the 15‐16 Eureka Math Revision, students are introduced to the full range of comparison word problem situations. (See Appendix F for that specific set of problems and their models as they are to appear in the revised G2 M2.) Lesson 10 guides students to solve two‐step problems involving comparison. For example, if Vincent’s Coins were to be two‐step, it would instead ask for the total number of pennies and dimes in the bowl. 32 Directions Step 1 Complete the first problem role‐playing a student as the facilitator models the RDW process. Step 2 Apply the protocol to solve the following problems. Step 3 Use both the addition/subtraction and multiplication/division situation charts (the latter when applicable) to analyze and classify the situations. Set 9 1. Rose wrote 8 letters. Nikii wrote 12 letters. How many more letters did Nikii write than Rose? (G1 M6 L11) 2. Two thirds of Maria’s wire is equal to half of Robert’s wire. The total length of their wires is 4 ft 1in. How much longer is Robert’s wire than Maria’s? (G5 M3 L16) 33 Set 10 1. Ben played 9 songs on his banjo. Lee played 3 more songs than Ben. How many songs did Lee play? (G1 M6 L1) 2. Emi planted 12 flowers. Rose planted 3 fewer flowers than Emi. How many flowers did Rose plant? (G1 M6 L2) 3. Jayden has 325 marbles. Elvis has 4 times as many as Jayden. Presley has 599 fewer than Elvis. How many marbles does Presley have? (G4 M3 L12) 34 Directions Step 1 Choose one problem in either Set 9 or Set 10 that you found difficult to model. Step 2 Share the model you drew with a partner and explain why the problem was difficult. Step 3 Follow the Deliberate Practice Protocol with the difficult problem. Step 4 Repeat these steps with the other partner. 35 Directions Step 1 Complete the first problem role‐playing a student as the facilitator models the RDW process. Step 2 Apply the protocol to solve the following problems. Step 3 Use both the addition/subtraction and multiplication/division situation charts (the latter when applicable) to analyze and classify the situations. Set 11 1. Moses sold 24 raffle tickets on Monday and 4 fewer on Tuesday. How many tickets did he sell in all on both days? (G2 M4 L10) 2. Tiffany bought 1 kg of cherries. Linda bought kg of cherries less than Tiffany. How many kilograms of cherries did they buy altogether? (G5 M3 L7) 3. Terry weighs 40 kg. Janice weighs 2¾ kg less than Terry. What is their combined weight? (G7 M3 L6) 36 Set 12 1. Eduardo has 14 stamps which is 8 more than the stamps that Paula has. How many stamps do they have altogether? (G2 M3 L31) 2. Julia, Keller, and Israel are fund raising. Keller collected $42.50 more than Julia. Israel collected $10 less than Keller. Altogether, the three collected$184.50. How much did each person collect? (G7 M3 L8) 37 Directions Step 1 Examine the models below and identify all the information that is given in the model. Step 2 Individually, create at least one “comparison” word problem of the specified situation type that can be solved using each model. Step 3 When signaled, compare your word problems with a partner. 1. (G2 M5 L8) 2. (G5 M3 L13) 38 Directions Step 1 Attempt the opening challenge problem again using your new knowledge of models. Step 2 Compare your model with a partner. 1. Leonard spent of his money on a sandwich. He spent 2times as much on a gift for his brother as on some comic books. He had of his money left. What fraction of his money did he spend on comic books? (G5 M3 L7) 2. 94children are in a reading club. One‐third of the boys and three‐sevenths of the girls prefer fiction. If 36 students prefer fiction, how many girls prefer fiction? 39 Appendix A: Protocols and Deliberate Practice Stems Protocol A for Problem Solving: “Draw and Re‐Draw” Pencils are required for this protocol. 1. Independently model and solve the set’s problems on the left side of the space provided below each one as student would do. 2. Share your models and solutions with a partner or within a triad. 3. On the right side of the space provided, redo the models as a tape diagram to clarify the relationships within each and the coherence from one problem to the other. Protocol B for Deliberate Practice: “Teach and Repeat Using Feedback” Time frames may be adjusted. 1. (90 seconds) The “teacher” and “students” plan for the instruction and prepare needed materials. 2. (60 seconds) The “teacher” stands and delivers instruction to other team members. 3. (90 seconds) The “students” make a positive statement about A’s teaching strategy and then suggest an improvement. Partner A “calls the shot” by restating the feedback that will be practiced. 4. (60 seconds) Partner A reteaches precisely the same segment of instruction, using the feedback. 5. (60 seconds) Partners give feedback again. 6. (60 seconds) Partner A uses the feedback again. Protocol C for SHARE IT: Whole Group Participation 1. Write “Share It” on four brightly colored 3 x 5 or 4 x 6 cards. 2. As you, the teacher/facilitator, circulate during small group talk, listen for an interesting insight, thought, discussion that you think would be helpful for the whole group to hear. 3. Ask the students/participant(s) if they would be willing to share their thoughts with the whole group. If willing, hand over a “Share It” card. 4. Once the small group sharing has concluded, call on those to whom you passed the “Share It” cards. Protocol D for Digging Into a Vignette: “Question the Author” Designate roles as teacher, student, and parentheticals reader. Start reading through the designated component of the transcript, saying and doing what is indicated each step of the way. Each member of the triad is committed to holding each other accountable for understanding what is being said and done by asking at least once, “What’s the author’s intent?” (Meaning, what is the point of saying/doing that?”) When done, deliberately practice one chunk while waiting for other groups to conclude their role play. 40 Possible Deliberate Practice Stems: “Push Me to My Best Practice!” As adult learners, we communicate to each other and to our students that anyone’s teaching practice or learning can be improved with effective effort. on’t give up on colleagues, on ourselves or on students! Stems for Positive Feedback during Deliberate Practice: 
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You motivated the students to focus when you_____. The connection between the pictorial and the abstract was clarified when you _____. It helped when you gestured to show the relationship between _____ and _____. Stems Suggesting Improvements during Deliberate Practice: 
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One “push” I would give would be to_____. Next time, try _____. You might clarify the relationship between the pictorial representation and the number sentence by ____. You might remain silent and demonstrate (gesture) as you_____. You might say _____ instead of _____. You might shorten/lengthen the wait time after _____. _______ might be good to tell where _______ might be better to elicit. When you check for understanding you might _____. After you asked ________, your cue for a response could be crisper. When you checked our personal boards, you might _____. Do it again. Do it better by_____. You mention a jump between Problem ___ and Problem ___. What is a bridge problem you could inset between the two? Reflection on Practice: “How are we doing at practicing deliberately?” 
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The distractions from the deliberate practice were _____. I could best label the emotions accompanying giving and getting “pushes” as _____. 41 Appendix B: Multiplication and Division Situations 42 Appendix C: Word Problems from Models GK M4 L16 There were 6 girls playing soccer. A boy came to play. How many children were playing soccer then? G2 M5 L6 Maria made 60 cupcakes for the school bake sale. She sold 28 cupcakes on the first day. How many cupcakes did she have left? G5 M3 L11 Bill reads of a book on Monday. He reads of the book on Tuesday. If he finishes reading the book on Wednesday, what fraction of the book did he read on Wednesday? G2 M4 L8 At the school fair, 29 cupcakes were sold and 19 were left over. How many cupcakes were brought to the fair? G2 M5 L2 Max has 42 marbles in his marble bag after he added 20 marbles at noon. How many marbles did he have before noon? G2 M5 L8 Susan have 37 pennies. M.J. has 55 more pennies than Susan. How many pennies do they have all together? G5 M3 L13 Mark jogged 3 km. His sister jogged 2 km. How much farther did Mark jog than his sister? 43 Appendix D: Stages of Model Drawing The foundation of the tape diagram begins in Kindergarten. As students advance through the years, there are different ways to model and to scaffold the labeling, as you can see in the sequence on the next page. The primary concern is that the model supports the student in decoding the situation and makes sense in relationship to the problem. To differentiate you might label pictorially, with the number inside the tape, or the number outside the tape. In deciding how to model for a whole class the question is, “What is the most sophisticated method I can use that students can readily access?” You will notice that in Stages 1‐5 the tapes are labeled with the whole on the bottom of the tape. It has advantages in that it readily transitions to the number line. Does this mean it is always correct? Not necessarily. Consider a situation such as: Jordan uses 3 lemons to make 1 pitcher of lemonade. He makes 4 pitchers. The natural way might be to draw one unit first, label the unit as having a value of 3 and then draw 3 more like units. It would make sense to label the ‘3’ on the bottom rather than the top if labeling outside the tape just because the student is seeing the unit while labeling rather than covering it with his hand. At the end of drawing, the student might then label the unknown at the top of the tape. That said, consistent approaches can be helpful as students are learning to model, e.g. labeling the total on the top or the bottom, inside or outside the tape, but take care that initial restrictions and consistencies don’t turn into rigid rules that become meaningless procedures rather than tools for reasoning about relationships. 1 A rule of thumb might be to let the situation guide the drawing as you work chunk by chunk with an eye towards labeling so that students will understand. As you use the tape diagram be aware of the choices you are making. Over time, expect students to label and model in their own ways when working independently. 1
The progression outlined is derived from and inspired by an excellent resource, “A Handbook for Mathematics Teachers in Primary School” by Julianan Ng Chye Huat and Lim Kian Huat 44 6 Stages of Model Drawing Directions: Analyze and notate the new complexities at each stage. Addition and Subtraction Equal Groups Stage 1 Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6 45 Appendix E: Analysis of Evidence of the Standards for Mathematical Practice within a Grade 2 Classroom Directions: Watch the short video clip of the Grade 2 student’s thinking about his work. Discuss with your partner: What Standards for Mathematical Practice are evidenced? How are they evidenced? MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.6 Attend to precision. MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning. 46 Modeling Word Problems K‐2 Answer Key 1. Leonard spent of his money on a sandwich. He spent 2times as much on a gift for his brother as on some comic books. He had of his money left. What fraction of his money did he spend on comic books? (G5 M3 L7) 2. 94children are in a reading club. One‐third of the boys and three‐sevenths of the girls prefer fiction. If 36 students prefer fiction, how many girls prefer fiction? Set 1 1. There are 4 snakes on a rock. 2 more snakes slither over. How many snakes are on the rock now? (GK M4 L16) 2. Lila is collecting honey from her beehives. From the first hive she collects gallon. Then from the other two hives she collects gallon each. How many gallons of honey does Lila collect in all? (G5 M3 Mid‐Module Assessment) Set 2 1. Dominic has 6 yellow stickers and 2 blue stickers. How many stickers does he have in all? (GK M4 L17) 2. A bakery used 12,674 kg of flour. Of that, 1,802 kg was whole wheat and 888 kg was rice flour. The rest was all‐purpose flour. How much all‐purpose flour did they use? (G4 M1 L17) Set 3 1. Five green frogs were sitting on a log. It was so hot that 2 of the frogs went for a swim! How many frogs were still sitting on the log? (GK M4 L21) 2. Tiffany spent 4 sevenths of her money on a teddy bear. If the teddy bear costs $24, how much money did she have left? (G5 M4 L7) Set 4 1. The students were playing with 7 balls on the playground. They kicked some into a puddle and now some are muddy! What is one way the balls might look? (GK M4 L18) 2. Manny bought a loaf of bread and cut it into 7 equal slices. He wants butter on some slices. What fraction of the slices might have butter and what fraction not have butter? (G4 M5) 3. Meyer has 0.64 GB of space remaining on his iPod. He wants to download a pedometer app (0.24 GB), a photo app (0.403 GB), and a math app (0.3 GB). Which combination of apps can he download? Explain your thinking. (G5 M1 L9) Set 5 1. Nine dogs were playing at the park. Some more dogs ran in. Then there were 12 dogs in all. How many dogs ran in? (G1 M4 L20) 2. Martha, George, and Elizabeth sprinted a combined distance of 10,000 meters. Martha sprinted 3,206 meters. George sprinted 2,094 meters. How far did Elizabeth sprint? Solve using a simplifying strategy or an algorithm. (G4 M2 L1) 3. Leonard spent of his money on a sandwich. He spent 2times as much on a gift for his brother as on some comic books. He had of his money left. What fraction of his money did he spend on comic books? (G5 M3 L7) Set 6 1. Ben and Peter caught 17 tadpoles. They gave some to Anton. They have 4 tadpoles left. How many tadpoles did they give to Anton? (G1 M4 L21) 2. All 3,000 seats in a theater are being replaced. So far, 5 sections of 136 seats and a sixth section containing 348 seats have been replaced. How many more seats do they still need to replace? (G4 M3 L13) Set 7 1. It snowed 14 days. Some snowy days, we stayed home. Nine snowy days we were in school. How many snowy days did we stay home? (G1 M5 L11) 2. Three boxes weighing 128 pounds each and one box weighing 254 pounds were loaded onto the back of an empty truck. A crate of apples was then loaded onto the same truck. If the total weight loaded onto the truck was 2,000 pounds, how much did the crate of apples weigh? (G4 M3 L13) 8 Set 8 1. Mr. Arnold has a box of pencils. He passes out 27 pencils and has 45 left. How many pencils did Mr. Arnold have in the beginning? (G2 M5 L11) 2. Madame Curie made some radium in her lab. She used kg of the radium in an experiment and had 1 kg left. How much radium did she have at first? (G5 M3 L9) (Extension: If she performed the experiment twice, how much radium would she have left?) 3. Max had x brownies. He ate 4 brownies and shared the remaining brownies among his 6 friends equally. How many brownies did each friend receive? Express your answer in terms of x. 9 Set 9 1. Rose wrote 8 letters. Nikii wrote 12 letters. How many more letters did Nikii write than Rose? (G1 M6 L11) 2. Two thirds of Maria’s wire is equal to half of Robert’s wire. The total length of their wires is 4 ft 1in. How much longer is Robert’s wire than Maria’s? (G5 M3 L16) 10 Set 10 1. Ben played 9 songs on his banjo. Lee played 3 more songs than Ben. How many songs did Lee play? (G1 M6 L1) 2. Emi planted 12 flowers. Rose planted 3 fewer flowers than Emi. How many flowers did Rose plant? (G1 M6 L2) 3. Jayden has 325 marbles. Elvis has 4 times as many as Jayden. Presley has 599 fewer than Elvis. How many marbles does Presley have? (G4 M3 L12) 11 Set 11 1. Moses sold 24 raffle tickets on Monday and 4 fewer on Tuesday. How many tickets did he sell in all on both days? (G2 M4 L10) 2. Tiffany bought 1 kg of cherries. Linda bought kg of cherries less than Tiffany. How many kilograms of cherries did they buy altogether? (G5 M3 L7) 3. Terry weighs 40 kg. Janice weighs 2 kg less than Terry. What is their combined weight? (G7 M3 L6)
12 Set 12 1. Eduardo has 14 stamps which is 8 more than the stamps that Paula has. How many stamps do they have altogether? (G2 M3 L31) 2. Julia, Keller, and Israel are fund raising. Keller collected $42.50 more than Julia. Israel collected $10 less than Keller. Altogether, the three collected $184.50. How much did each person collect? (G7 M3 L8) 13 Word Problems from 15‐16 Edition of G2 M2 Eureka Math Lesson 1: Vincent counts 30 dimes and 87 pennies in a bowl. How many more pennies than dimes are in the bowl? NOTES ON MULTIPLE MEANS OF EXPRESSION:
To avoid inhibiting children’s natural drawings during the RDW process, be careful not to communicate that the tape diagram is the best or “right “ way. If a drawing makes sense, it is right. Regularly, you might guide students through the modeling of a problem with the tape so that this important model gradually enters into their tool kit Note: This compare with difference unknown problem presents an opportunity to work through the common misconception that “more” means add. After drawing the two tapes, ask guiding questions such as, “Does Vincent have more dimes or pennies?” “Does Vincent have 30 pennies?” (Yes!) “Tell me where to draw a line to show 30 pennies.” “This part of the tape represents 30 pennies. What does this other part of the pennies tape represent?” (The part that is more than the dimes.) This will help students recognize that they are comparing, not combining, the quantities. This problem has an interesting complexity because, though there are more of them, the pennies are worth less. Ask students, “Could you buy more with Vincent’s pennies or with his dimes? How do you know?” Lesson 2: With one push, Brian’s toy car traveled 40 centimeters across the rug. When pushed across a hardwood floor, it traveled 95 centimeters. How many more centimeters did the car travel on the hardwood floor than across the rug? Note: This compare with difference unknown problem gives students further practice with comparing quantities. A new complexity is to compare length measurements rather than numbers of discreet objects. NOTESON
DIFFERENTIATING
THEAPPLICATION
PROBLEM: The9ApplicationProblemsof
Module2areallcomparison
situations.
• Lessons1and2:Comparewith
DifferenceUnknown
• Lessons3and4:Comparewith
BiggerUnknown
• Lessons5and6:Comparewith
SmallerUnknown
• Lesson7:ComparewithSmaller
Unknownusing"morethan."
• Lesson8:ComparewithBigger
Unknownusing"lessthan."
• Lesson9:ComparewithBigger
Unknownusing"shorter
than.”
Thechallengingsituationtypesin
Lessons7,8,and9mightbe
frustratingifstudentshavenotbeen
successfulinLessons1‐6.Youmight
considereditingthesituationsin
Lessons7‐9toinsteadrepeatthoseof
1‐6,returningtothemorechallenging
problemtypesineitherModule3or4
Lesson 3: Jamie has 65 flash cards. Harry has 8 more cards than Jamie. How many flash cards does Harry have? Note: This problem type, compare bigger unknown, challenges students to make sense of the situation and determine the operation to solve. It follows the two previous compare difference unknown Application Problems to alert students to read and understand the situation instead of relying on key words that tell the operation. This problem exemplifies the error in using “more than” as a key word to subtract, since in this situation students solve by adding the parts. The problem could be represented using one tape, but since students are just beginning to do comparison problems at this level of sophistication with larger numbers, it may be wise to draw one tape to represent each boy’s cards emphasizing the fact of the comparison. Lesson 4: Caleb has 37 more pennies than Richard. Richard has 40 pennies. Joe has 25 pennies. How many pennies does Caleb have? Note: This problem has the added complexity of extraneous information, Joe’s pennies. You might ask, “Do I need to draw Joe’s pennies?” Depending on the needs of your students, this can be omitted in order to focus on the compare bigger unknown problem where “more than” is used to compare two quantities and addition is used to solve. Lesson 5: Ethan has 8 fewer playing cards than Tristan. Tristan has 50 playing cards. How many playing cards does Ethan have? Note: This compare smaller unknown problem uses the word “fewer”, which probably will suggest subtraction to students. The numbers were purposely chosen so students have the opportunity to use the take from ten strategy to solve. Lesson 6: Eve is 7 centimeters shorter than Joey. Joey is 91 centimeters tall. How tall is Eve? Note: In today’s lesson, students measure and compare lengths in centimeters and meters. This compare smaller unknown problem is similar to the problem in Lesson 5, but here measurement units are used with “shorter than” rather than “less than” or “fewer than”. Lesson 7: Luigi has 9 more books than Mario. Luigi has 52 books. How many books does Mario have? Note: This compare smaller unknown problem has the complexity that we subtract to find the number of books Mario has, though there is no action of taking away and the word “more” in the first sentence might suggest addition to students. “More” and “more than” are often mistakenly taught as “key words” signaling either to add or subtract. This approach distracts students from the more essential task of considering the part/whole relationships within a problem after representing it with a drawing. Lesson 8: Bill the frog jumped 7 centimeters less than Robin the frog. Bill jumped 55 centimeters. How far did Robin jump? Note: This compare bigger unknown problem uses the word “less”, which presents an opportunity for students to work through the easy mistake that “less” or “less than” means to subtract. Ask guiding questions such as, who jumped farther? This, along with a tape diagram, helps students recognize that Robin jumped farther and helps them determine the operation, addition. Lesson 9: Richard’s sunflower is 9 centimeters shorter than Oscar’s. Richard’s sunflower is 75 centimeters tall. How tall is Oscar’s sunflower? Note: This compare bigger unknown problem is similar to the problem in Lesson 8, but here the word “shorter” relates to measurement. This is in anticipation of today’s Concept Development, wherein students measure lengths of strings and use tape diagrams to represent and compare lengths. Name
Date
Read the word problem.
Draw a tape diagram or double tape diagram
and label.
Write a number sentence and a statement that
matches the story.
1. Nikil baked 5 pies for the contest. Peter baked 3 more pies than Nikil. How
many pies did Peter bake for the contest?
2. Emi planted 12 flowers. Rose planted 3 fewer flowers than Emi. How many
flowers did Rose plant?
3. Ben scored 15 goals in the soccer game. Anton scored 11 goals. How many more
goals did Ben score than Anton?
4. Kim grew 12 roses in a garden. Fran grew 6 fewer roses than Kim. How many
roses did Fran grow in the garden?
5. Maria has 4 more fish in her tank than Shanika. Shanika has 16 fish. How many
fish does Maria have in her tank?
6. Lee has 11 board games. Lee has 5 more board games than Darnel. How many
board games does Darnel have?