Balanced Assessment Test Third Grade 2008

Balanced Assessment Test Third Grade 2008 Core Idea Task Score Number Operations The Pet Shop This task asks students to use addition, subtraction, multiplication, and division to solve problems about pets. Successful students could work finding half as much and twice and much and solve multi-step problems. Number Properties House Numbers The task asks students to solve problems using odd and even numbers. Students need to use multiple constraints to reason out solutions to problems and explain their thinking. The problem also allows students to use multiplication in context. Successful students could develop a logical reason using 2 constraints to justify their answer. Algebra Blob Bugs The task asks students to identify and work with a number sequence derived from diagrams. Students draw and extend patterns and diagrams. Successful students could work backward from a place in the pattern to the number in the sequence. Geometry Looking Glass Land The task asks students to identify shapes with line symmetry and draw in the line of symmetry. Students are also asked to design a shape with two lines of symmetry and mark in the lines of symmetry. Measurement Time to Get Clean The task asks students to work with a table of activities and times. Students need to reason about fractions of an hour, and add time together. Successful students could convert minutes to hours and calculate elapsed time. 1

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The Pet Shop This problem gives you the chance to: Use adding, subtracting, multiplying and dividing whole numbers in real contexts 1. Four baskets of puppies are on sale today. In each basket there are five puppies. In all, how many puppies are on sale? 2. There are 12 snakes in the pet shop. Each snake is about 2 feet long. If they are placed end to end how long would they be? feet Show how you figured this out. 3. In the window of the pet shop are some rabbits. Inside the shop there are 12 more rabbits. In all, there are 45 rabbits. How many rabbits are in the shop window? Show how you figured this out. 4. Three parrots eat 14 bags of parrot food each week. How many bags of parrot food do three parrots eat each day? Show how you figured this out. 5. In the pet shop fish tank there are 18 goldfish. There are twice as many angel fish as goldfish in the fish tank. And there are half as many guppies as goldfish in the fish tank. In all, how many fish are there in the pet shop fish tank? Show how you figured this out. Copyright 2008 by Mathematics Assessment Resource Service. 10 5 Third Grade - The Pet Shop

The Pet Shop Rubric The core elements of performance required by this task are: use adding, subtracting, multiplying and dividing whole numbers in real contexts Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Gives correct answer: 20 1 1 2. Gives correct answer: 24 feet Shows correct work such as: 2 x 12 Special case: Accept answer: 10 3. Gives correct answer: 33 Shows correct work such as: 45 12 4. Gives correct answer: 2 Shows correct work such as: 14 7 5. Gives correct answer: 63 fish Shows correct work such as: 18 goldfish + 36 angel fish + 9 guppies 1 1 1s.c. 1 2 1 2 1 1 2 Partial credit One error (1) 3 Total Points 10 1 2 Copyright 2008 by Mathematics Assessment Resource Service. 6 Third Grade - The Pet Shop

The Pet Shop Work the task and look at the rubric. What are the key mathematical ideas that a student needs to use to be successful on this task? One difficulty students had with the task was choosing on operation. As you analyze each part of the task, think about the role operation played in student errors. As you look through student work, notice which students are using multiplication for part 2 and how many still rely on repeated addition as their most comfortable strategy? In part 4 how many students are comfortable using multiplication or division as a strategy? How many are still drawing pictures to think about the relationships? While students may know how to do multiplication in the setting of practice problems that are set up for them, it is important to notice when and if they make the connection to using that operation when given choices. What kinds of models do students have to help them think about the meaning of operations? How often do students have opportunities to work with word problems and pick an operation? In part one, how many of your students thought there were 5 or 4 puppies (used a number in the problem)? How many of your students put 9 puppies (adding instead of multiplying)? How many of your students put 25 (multiplied the wrong numbers)? Look at student work for part two of the task, how long are 12 snakes. How many of your students put: 24 60 10 26/28 12/2/5 Other What kinds of misunderstandings led to each type of error? How are the implications for instruction different for the error types? Now look at work for part 3, subtraction. How many of your students used addition (57)? Why do you think your students struggled with this part of the task? Is there too much emphasis on key words? Do students have opportunities to draw or act out actions to word problems? Do students have exposure to all three types of subtraction situations: take away, missing addend, and comparison subtraction? Look at student work for part 4; find the bags of food per day. How many of your students put: 2 43 1 3 4 11 21 Other 7

What did students need to understand in order to solve this problem? What strategies did students use to make sense of this problem? In part 5, students needed to work with several sets of constraints in order to find the number of fish. As you look through student work, could your students make sense of the number of angel fish (36)? Were they able to find the number of guppies (9)? Now look at their answers, how many of your students put: 63 72 53 45 54 60 20 36 27 Other What kinds of experiences do students need to help them learn to organize their information in word problems? Recognize the operations needed? Think through longer chains of reasoning than that required for a one-step problem? How often do students work with reasoning chains that are longer than two steps? 8

Looking at Student Thinking on The Pet Shop Student A is comfortable with multiplication and division. The student is starting to reason about rates in the attention to labels in part 4. Notice that in part 5 the student labels the intermediary steps to keep track of information before doing the final addition. Student A 9

Student B is also comfortable with division facts in part 4 and organizes the information needed to solve part 5 by making a table. Notice that for this student the halving and doubling are just known. The student could do the computations in his/her head. Student B Student C is may or may not be as comfortable with division. The student uses an alternative strategy of portioning out bags per day of the week to find the solution to part 4. Student C 10

Student D uses repeated addition to find the solution for part 4 and then checks the work with multiplication. In part 5 the student uses a counting by 3 s strategy to think about halving the 18. Student D The cognitive demands for part 5 were challenging for students. Student E shows a good grasp of labeling and understanding what is being calculated in parts 3 and 4. In part 5 the student is able to double and halve, but forgets about including the original goldfish in finding the final solution. How often do students in your class have the opportunity to solve problems involving more than 2 steps? How do you help students develop the habit of mind of labeling their work or making sense of what is produced from a calculation? Student E 11

Now contrast this thinking with the work of Student F. This student adds the goldfish to the guppies and calculates the number of angel fish. Where does the student go wrong? How might labels have helped this student? What labels would you add? Student F Student G was able to find that there were 2 bags of parrot foot per day and even say it in words. However, the student felt the need to then take the unneeded information of 3 parrots to do further calculations. What would be a good question to pose to this child? Notice that in part 5 the student finds the angel fish and guppies, but forgets to add in the original goldfish. Student G 12

Student H is starting to struggle with operations. In part 2 the student uses a counting strategy to multiply 2 times 12. In part 3 the student has trouble identifying the correct operation and so tries to find a multiplication problem to get the total of 45. In part 4 the student has the correct answer of 2 bags, however there are two sets of diagrams. What question would you like to pose to the child to probe their understanding? In part 5 the student again has difficulty with choosing the operation. Instead of doubling to find the goldfish, the student added two. The student then correctly takes half of the goldfish to get 10 guppies. What types of experience does this student need to move forward? Student H 13

Student I does a considerable amount of thinking. In part 4 how do you think the student arrived at the numbers in his problem? In part 5 the student adds together three quantities. Which numbers do you think the student used for goldfish? For angel fish? For guppies? In each case, what misconception did the student have? Student I Now look at the work for Student J. In part 4 the student tries two strategies for finding the solution. Are you convinced the student could have gotten the correct answer if the number of bags was divisible by 3? Can you explain the thinking of the student in part 5? Which quantities did the student calculate? What misconceptions did the student have? Student J 14

Student K makes some of the most common mistakes for this task. Notice that in part 3 and part 4 the student chooses the incorrect operation. In part 5 the student shows a calculation for goldfish and angel fish. However the student lacks a basic understanding of working with problems in context. Does it make sense to add two kinds of fish to find out the amount of a completely different fish? What types of experiences or questions would help push this student s thinking? Why? Student K 15

Student L again has problems working in context. Notice in part one the student puts one of the numbers from the problem, rather than doing a computation. In part 2 the student only thinks about the snakes in the picture, rather than the amount of snakes mentioned in the problem. Notice that the student uses a counting on strategy for subtraction in part 3. While this works, it is very long and can lead to errors. How do we help students make the connection between adding on and subtraction? How do we give them a reason to let go of the comfortable to learn new, more efficient strategies? In part 5 the student shows a fairly good understanding, but makes a calculation or transcription error in finding the amount of angel fish. Student L 16

Student M also struggles with context. In part 1 and 2, it is unclear what the student is thinking because there is no work. The answers are not typical errors. In part 4 the student uses the wrong information and operation. In part 5 the student may or may not have a good grasp of the situation. The student may have taken half of the angel fish instead of half the goldfish. Paying attention to the referent (number I need to find half of) is critical to reading in mathematics. This is perhaps a different skill from reading in other subject areas. What opportunities can we provide to help students to learn reading skills that are particular to mathematics? Student M 17

Student N struggles with operation. In part 2 the student uses a diagram but doesn t include enough snakes. In part 4 the student s explanation doesn t give us enough information to understand what is being counted or how it relates to days in a week or 14. In part 5 the student chooses incorrect operations. The student finds the correct amount of angel fish, and then subtracts the goldfish to get 20. Then the student finds half that amount (10) and subtracts from the previous amount to get 8. Then there is another subtraction. How do we help students connect meaning to calculations? The student is making sense of labels for things that he/she understands. For problems that are more challenging the labels are left off. Where would you go next with this student? Student N 18

Student O is able to make sense of multiplication as forming equal groups in parts one and two. The student is not able to identify both the number of groups and the quantity of each group. In part 3 the student can accurately compute, but picks the incorrect operation. Can you think of a reason for putting the string of +5 s? In part 5 the student, who seems to be thinking that everything on the page should be multiplication, uses the same size groups for every type of fish. What does this student need to move forward? Student O 19

3 rd Grade Task 1 The Pet Shop Student Task Core Idea 2 Number Operations Use adding, subtracting, multiplying and dividing whole numbers in real contexts. Understand the meaning of operations and how they relate to each other, make reasonable estimates, and compute fluently. Understand different meanings of addition and subtraction of whole numbers and the relationship between the two operations. Develop fluency in adding and subtracting whole numbers. Understand multiplication as repeated addition, an area model, an array, and an operation on scale. Mathematics in this task: Identifying operations in word problems Separating out unneeded information Missing addend subtraction Knowing the number of days in a week and applying that information in a problem situation Doubling and halving in context Organizing information in a multi-step problem, tracking what has been calculated and what still needs to be found Multiplication and multiplication fact families Based on teacher observations, this is what third graders know and are able to do: Use multiplication to find the numbers of puppies in four baskets and to find the total length of snakes Understood key words in multiplicative thinking: twice and half Counting by 2 s Areas of difficulty for third graders: Recognizing subtraction in context Choosing operations, recognizing division in context Organizing multi-step problems Labeling their work Identifying unnecessary information Measurement in context Strategies used by successful students: Used more than one operation to check their work Drawing pictures Labeling their calculations or writing sentences to describe each answer Counting on, repeated addition, modeling were often strategies of students with lower scores, students with higher scores were more comfortable with multiplication and division 20

The maximum score available is 10 points. The minimum score for a level 3 response is 5 points. Most students, 87%, could multiply in context to find the number of puppies and the length of the snakes. Many students, 80%, could also show strategies for the multiplication. More than half the students could use multiplication in context, show their work, and use subtraction to find a missing addend with work. Some students, almost 15%, could meet all the demands of the task, including recognizing unnecessary information, supplying a number to help solve the problem, and use doubling and halving to solve a problem involving multiple solution steps. Almost 6% of the students scored no points on the task. All the students in the sample with this score attempted the task. 21

The Pet Shop Points Understandings Misunderstandings 0 All the students in the sample with this score attempted the task. They might be able to reason about equal groups or do some correct arithmetic, but could not sort through all the demands of any part. Students in this group often mistook a number in the problem for a solution, such as putting 5 puppies in part 1. (7% of all students put this answer). 5% of the students multiplied the wrong numbers in part one (5x5 instead of 5 x 4). These students were also most likely to add instead of subtract in part 3. Almost 20% of all 2 Students in this group could usually multiply to find the puppies in part 1 and find the length of snakes (either for 5 or 12). 8% of all students found the length for 5 snakes instead of twelve. 5 Students could use multiplication to find the number of puppies and length of the snakes and show their work. Students could also find the missing number of rabbits. 8 Students could solve one-step problems involving multiplication and subtraction. Students with this score generally missed all of part 4 or used only 2 of three correct amounts of fish in part 5. 10 Students could use multiplication, subtraction, and division in context. They had strategies for keeping track of several calculations needed to get a final answer. Many students at this level were good with labels and written descriptions for each calculation. students made this error in 3. More than 1/3 of the students, who used repeated addition as a strategy, did not get the correct answer. Too many students thought about rabbits in the window and more rabbits in the store making a total of 45 as an addition problem. Do students rely too heavily on key words? Students struggled with the operation in part 4 and recognizing unnecessary information. 18% of the students multiplied 3 parrots and 14 bags per week to get 42 bags per day. Almost 5% of the students thought it would be 1 bag per day or 21 bags per day. In part 5, 8% of had an answer of 45 (36 +9), 6% had 54 (36 +18), 6% had 72 (18 +18+36). Some students who had difficulty in part 5 added 2 or subtracted 2 for twice. 22

Implications for Instruction Students at this grade level need to transition from drawing and counting and repeated addition use of multiplication. More than one-third of the students who attempted repeated addition made errors. Many students struggled with choosing operations in this problem in almost all the parts. Working with models, like the bar model, help students to make sense of the action of the problem. More time and attention needs to be focused on understanding the purpose or usefulness of various operations. Students need to have many opportunities to solve problems in context and solve a variety of problem types for each operation. In this task, students had difficulty identifying a subtraction problem with a missing addend. Students also had difficulty identifying unnecessary information in a context. Sometimes in working problems in context, students need to use other information to solve a problem. In part 4 students needed to add in the knowledge that there are 7 days in a week. Students are too often only given problems with short reasoning chains, one-step problems. Students build their ability to think in multiple steps by being exposed to rich problems and discussing the logic of their solutions. In discussions about solutions, students pick up on strategies of other students for organizing information, labeling answers, and thinking through the steps: What am I looking for? What do I already know? What do I need to calculate to help me find the final answer? What happens to these labels if I add numbers together or if I multiply numbers? Ideas for Action Research Looking at the Meaning of Operations Students need to be exposed to a variety of contexts and types of addition, subtraction, multiplication, and division problems to help them make sense of operation. For example, students commonly understand the idea of take-away subtraction but have difficulties with comparison subtraction. The action of the story is different. Models are a good way for the student to record what is known, and think about what is needed. A good model can help the student think about the meaning of this operation in the context of the problem. See the types of models on the next page. A good source of model problems on the Singapore text books at www.singapore. Com or 8-Step Model Drawing by Bob Hogan and Char Forsten. 23

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House Numbers This problem gives you the chance to: use odd and even numbers Here is a street of 7 houses. House Number 1 2 3 4 5 6 7 1. Today, the mail man delivered two letters to each of the houses with odd numbers. How many letters, in all, did he deliver to these houses today? Show how you figured this out. 2. On each day of the week, a newspaper is delivered to each of the houses that has an even number. How many newspapers are delivered each week to these houses? Show how you figured this out. 3. There is a dog in the yard of each of the houses with an odd number between numbers 2 and 6. There is a cat in the yard of each of the first four houses. Which house has both a dog and a cat in its yard? Explain how you figured this out. 8 Copyright 2008 by Mathematics Assessment Resource Service 26

House Numbers Rubric The core elements of performance required by this task are: use odd and even numbers. Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Gives correct answer: 8 Shows correct work such as: 4 x 2 Partial credit One error 2. Gives correct answer: 21 Shows correct work such as: 3 x 7 Special case: Accept answer 3 Accept repeated addition 3. Gives correct answer: House number 3 has both a cat and a dog. Gives correct explanation such as: the odd numbered houses between 2 and 6 are numbers 3 and 5. As only the first 4 houses have a cat, the only house 1 to have a cat and a dog is number 3. 3 Total Points 8 1 2 (1) 1 1 1s.c. 2 3 2 Copyright 2008 by Mathematics Assessment Resource Service 27

House Numbers Work the task and read the rubric. How do you think students might explain their thinking in part 3? What would you want in terms of a good response? Look at part one of the task. What is the mathematics students need to know to work this part of the task? How might a student show their thinking on this process? Now chart student answers. 8 14 4 6 3 Other The misconceptions are different for the different errors. Answers of 14 and 4 are failing to identify all the constraints or demands of the task. Six is choosing an incorrect operation. Students with this answer did not recognize a multiplication situation in context. Why might students have picked three as an answer? What other things did you notice when looking at student work? Now look at student work in part two. How many of your students put: 21 3 7 12 15 2 Other What are the implications for instruction? What types of learning activities do students need to help them recognize multiplicative situations? In part three, the task is assessing student skills in logic and justification. First look at their choice of house numbers. How many thought the house was: 3 4 2 5 2,3,4, 8 Other Now look at student reasoning, how many of your students gave: A convincing argument or a clear logical reason for their choice? Just restated the prompt? Were too vague (I figured it out by reading or I counted)? Tried to use the numbers for calculations Missed a constraint (forgot about dogs, forgot about evens) Nonmathematical reasons (house three is the biggest) 28

Misconceptions about between o Means in the middle ( 4 is between 2 and 6) o Means those numbers are excluded o Used only the end points (2 and 6) What opportunities do students have to make justifications or explain their logic? Are there class discussions that let students see the logic of others and begin building their own internal sense of what makes a convincing argument? How is information about what is valued in an explanation conveyed to students? Do they receive general information (use words, numbers and pictures) or specific information (I liked this explanation because it....)? 29

Looking at Student Work on House Numbers Student A has very clear, detailed explanations. In part 3 the student verifies their answer against each constraint. Student A 30

Student B also has a clear answer. Notice the use of editing to make the language more precise. This is a good mathematical habit of mind. Student B Student C struggles with the constraints. In part one the student does not use the idea of odd numbers, but recognized the multiplicative situation. In part two the student only attends to the constraint of even numbers, but does not think about days of the week. However the student has clear language to develop a logical, well-reasoned answer in part 3. Student C 31

Student D is able to think about the 3 even numbers, but continues with the idea of 2 per house rather than thinking about 1 newspaper for every day of the week. How do we help students identify constraints? How do we help them develop literacy for reading the details of labeling that is so important in mathematics for determining operation? Student D Student E overlooked the idea of even houses in working part 2. It is interesting to note that the student put 5 newspapers for each house in the diagram. Without being able to interview the student, it is hard to tell if there is a misunderstanding about days of the week (the week only includes school days) or if the student has life experience where newspapers are only delivered on workdays. 32

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Student F also only thinks about 5 days in a week. In part 5 the student attempts to use an elimination strategy, but fails to understand the constraint house with a dog and a cat. Student F 34

When students are learning an idea, it is fragile. Notice that Student G can use multiplication in part one, but then uses repeated addition in part 2. The student is not in disequilibrium that 7 groups of 2 had a different answer than the 2 x 7. In Part 3 the student confuses between with only considering the end points. Student G 35

Some students had difficulty identifying the constraints. Student H is able to identify the odd and even numbers, but uses them for addition rather than as defining the number of groups. The student doesn t recognize the multiplicative aspect of the problem. In part 3 the student is looking for a pattern, rather than working with the constraints in the task. Why types of questions might you pose for this student to help him see his misconceptions? An interesting aside, while the student received no marks on this task the student was able to meet standard on the overall exam. Does this change your thinking about the type of experiences the student needs? Student H 36

Student I is interesting because there are no arithmetic errors anywhere in the exam. All calculations are correct, yet the student s overall score in 10 points, below standard. Look carefully at the type of reasoning done by this student. What are the things that the student understands or doesn t understand about meaning of operations? About constraints? What types of experiences does this student need? Student I 37

Many students lack the ability to make a convincing argument or justify their solution in part 3. Student J confuses the idea of between with finding the middle. Also the student does not identify the need for odd numbers. Student J In using the drawing strategy, Student K is able to correctly use the between to separate the houses with his line. However the student needed to consider that clue a second time to pick between 1 and 3. Student K 38

3 rd Grade Task 2 House Numbers Student Task Core Idea 1 Number Properties Core Idea 2 Number Operations Use odd and even numbers in a problem situation. Use multiplication to solve problems. Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers. Develop understanding of the relative magnitude of whole numbers and the concepts of sequence, quantity, and relative positions of numbers Understand multiplication as repeated addition, an area model, an array, and an operation on scale. Mathematics of the task: Identifying constraints, like odd and even, between, range Supplying information, 7 days in a week Developing a logical justification with several steps Planning a solution strategy Choosing operations, recognizing multiplicative situations Based on teacher observations, this is what third graders know and are able to do: Find odd numbers and multiply by 2 or use repeated addition by 2 s Add or multiply by 3 s Use pictures as a problem-solving tool Areas of difficulty for third graders: Knowing there are 7 days in a week, not 5 days in a week Understanding between as a range, versus a midpoint Applying more than one constraint at a time Developing a justification Strategies used by successful students: Draw pictures Labels to make sense of models and calculations Checking their work: getting the answer in two ways 39

The maximum score available for this task is 8 points. The minimum score needed for a level 3 response, meeting standards, is 4 points. Many students, 85%, knew that there were 3 even houses. About 70% of the students could find the number of odd houses and use multiplication to find the number of letters delivered each day. More than half the students, 61%, could also find the number of even houses. Less than half the students, 42%, could use multiplication to find the number of letters and make a logical argument, using multiple constraints, to find the house with both a dog and a cat. Almost 18% of the students could meet all the demands of the task, including supplying the number of days of the week and use multiplication to find the number of newspapers delivered to even houses. 15% of the students scored no points on this task. All the students in the sample with this score attempted the task. 40

House Numbers Points Understandings Misunderstandings 0 70% of the students with this score attempted the task. Students had difficulty recognizing constraints and choosing operations. 10% of the students multiplied by 7 in part one ignoring the odd numbers. 8% of the students only wrote the number of odd houses in one, forgetting the 2 letters to each house. 5% added 4 odd numbers plus 2 letters instead of using 3 Students could find quantity of odd numbers and recognize a multiplicative situation to solve part 1. 4 Students with this score could find the solution to part 1 and identify the even houses in part 2 (special case). 5 Students could use constraints, supply needed extra information (days in a week), and use multiplication in context. multiplication. Students had difficulty with part 2. 13% picked the special case, 3 even numbers forgetting the days in a week. 13% chose 7, days in a week forgetting the even houses. 4% chose 15, thinking about 5 days in a week for 3 houses. Students had difficulty working with multiple constraints and organizing their information to develop a logical answer. 12% of the students thought the answer to part 3 was 4 (4 is between 2 and 6). 7% thought the answer was 2 (ignoring the idea of odd). 12% gave answers with more than one house. 4% gave answers larger than the house numbers available. 4% gave an answer of 5. 6 Students with this score generally missed all of part 2. 7 Students gave the special case in part 2. 8 Students could use constraints, supply needed extra information (days in a week), and use multiplication in context. Students could give a logical argument for their choice by making an organized list or using a diagram. 12% of the students who missed the explanation points in part 3 gave vague answers, e.g. I read it, I looked, I thought. 7% thought 4 was between 2 and 6. 4% drew incorrect pictures. 3% tried to use the numbers to do an arithmetic problem. 41

Implications for Instruction Students need to be able to recognize odd and even numbers and know common information, such as, the number of days in a week. Students need to work a variety of rich tasks with blank paper instead of scaffolded worksheets, so that they develop strategies for organizing their work and keeping track of what they have found out and what still needs to be calculated. Rich tasks allow students to think about multiple constraints. Encouraging students to share their thinking and having peers question their work helps students to start to internalize the qualities of showing their thinking and emphasize the need for labels as an organizing principal. Students also need to develop tools to help them understand operation. Many students struggled with choosing the correct operation. One such tool is to use model drawing to help visualize the action of the story problem. Ideas for Action Research Model Drawing Models are a powerful tool for making sense of information in a story problem by identifying what is known, what is needed, and seeing the picture helps students choose a correct operation. When learning the process, the student should have the necessary skills for solving the word problem. Here is one process for teaching model drawing: In section 12, the library has 127 books on dinosaurs and 78 books about reptiles. How many books in this section? 1. First read the question as a group. Ask students to answer what or who the question is about. In this case the problem is about dinosaur books and reptile books. So these labels should be listed on the left side of the paper. 2. Then reread the question for information or details. the library has 127 books on dinosaurs. The students can then draw a bar to represent the books next to the dinosaur label. Now ask students if this is a part or a whole. Because it is only a part of the number of books, the 127 should be placed inside the bar. 3. Continue reading and 78 books about reptiles. Now students should make a bar next to the reptile label. Ask if the bar will be larger or smaller than the bar for dinosaurs. These questions focus students on the quantities and relationship of the numbers. The questions help the students develop the language for thinking about problems on their own for developing their own internal self-talk. Again ask, is the 78 a part or a whole. Because it is a part, the 78 goes inside the bar. 4. Continue reading how many books in this section? Ask students what they need to find out. Responses should be the total number of books. So a bracket should connect the two bars on the right hand side with a question mark to indicate what is needed. See the completed model below. Dinosaurs Reptiles 78 127? 42

127 + 78 = 205 There are 205 books in this section of the library. 43

How does this model help students identify the operation needed? Now consider a different type of model problem. Jonah had 104 marbles to put in party bags for his 8 friends. How many marbles should he put in each bag? 1. Read the question with the class to get a feel of the context. Then ask who the task is about? What is the task about? Students should say its about Jonah s marbles. This label should be placed on the left side of the paper. 2. Now reread the task for details. Jonah had 104 marbles. Is this a part or the whole? In this case the 104 marbles is the whole amount. So students should draw a bar to represent the marbles and put the 104 to the right of the bar (location for wholes or totals). 3. Continue reading. to put into party bags for 8 friends. How many bags does he need? 8. So divide the bar into 8 equal groups. 4. Finally, read, How many marbles should he put in each bag? So what do we need to find out? The size of each part. So put a question mark inside one of the parts. Here is a sample model for this task. Jonah s marbles? 104 How does this model help students visualize the operation? Sample solution: 8 units = 104 1 unit = 104/8 = 13 1 unit equals 13 marbles. There are 13 marbles in each bag. Part of this process is to encourage students to write their solutions in complete sentences. After students have had some experience with the model drawing process. Have them try to work through the steps on their own or with a partner. (Try on a new strategy). Further problems are available in the Singapore math books from www.singapore.com or from 8 Step Model Drawing by Hagen and Forsten. 44