Balanced Assessment Test – Third Grade 2009 Core … Assessment Test – Third Grade 2009 Core Idea Task Number Operations Goldfish Bowls This task asks students to use addition,

Third Grade Copyright © 2009 by Noyce Foundation All rights reserved.

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Balanced Assessment Test – Third Grade 2009 Core Idea Task Number Operations Goldfish Bowls This task asks students to use addition, subtraction, multiplication, and division to solve problems about dividing goldfish into equal groups. Successful students could explain why 36 fish could not be divided into 5 bowls equally. Algebra Birthday Decorations The task asks students to identify and extend a geometric pattern in pictures and in a table. Successful students could extend the pattern beyond the table and could interpret their answers to determine if they had enough decorations. Geometry Making a Doll House The task asks students to identify and name several common geometric shapes. Students are asked about symmetry. Successful students could compare and contrast geometric attributes of two shapes, e.g. number of sides or angles, size of angles, parallel sides. Data The Math Test The task asks students to read and interpret data on a bar graph about test scores. Successful students to reason about the number of students with a score of “more than” a given number. Number Operation Valerie’s Puzzle The task asks students to work with a 3 by 3 square and fill in numbers from 1 to 9 to make given totals. Successful students were able to add accurately and follow the rule that each number could be used only once.


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Re-engagement Overview: The re-engagement lesson is designed to promote as much student discussion as possible. Students have already done the task on their own and now the important ideas need to be brought out and examined. In the process students need to have the opportunity to confront and see the error in the logic of their misconceptions. Often, as teachers, we try to prevent errors by giving frequent reminders, like “line up the decimal point”. But actually errors can provide great opportunities for learning. Students don’t let go of misconceptions until they understand why they don’t make sense. To the student there is some underlying logic to their misconceptions. Instead of going back to reteach a unit or lesson from scratch, we want to take advantage of the time students have already spent familiarizing themselves with a context. In this way we can maximize student learning from the work and build on their previous thinking. By using student work, students become very engaged in the process. They are genuinely interested in trying to figure out what someone else is thinking. This also ups the cognitive demand of the task and helps students learn to be more reflective about their thinking. During the re-engagement lesson students do not have their work in front of them. This is so they will be flexible in their thinking and not married to defending their previous ideas. We are hoping that discussion will allow them to change their minds. Teachers often give blank papers or “think sheets” to students to write on as the lesson progresses. The idea is to allow students to structure their own thinking and processes, and not confine their solution strategies to the teacher’s way of thinking. Teachers often begin the lesson by talking about wanting them to have a discussion like mathematicians. A teacher might say, “We want you to discuss why strategies make sense. We also want you to listen carefully to each other to see if you hear something that makes you change your mind. “ Classroom Norms: To maximize the amount of discussion in the classroom the following norms have proved helpful. Pose a question and then give everyone an opportunity for individual think time, before starting the discussion. In this way everyone has a better chance of contributing, not just the quick processors. Before going to a group discussion, do a pair/share to maximize the number of students getting to share their ideas and to give students the opportunity to use and develop mathematical language for a purpose of convincing others. Then open the discussion to the whole class to allow divergent ideas to be shared and examined. Continue probing to allow ideas to be “almost” over-clarified, so students really solidify their thinking. Try to avoid feedback about the “rightness” or “wrongness” of their ideas. Let the group try to reach consensus about what makes sense and why. Feedback stops the discussion. Some teachers like to also use the quiet thumbs up signal to let them know when the students are ready for the next part. So the flow of the lesson might look like this: Pose a question, allow think time, pair/share, and then class discussion of the question, and then pose a new question. As a follow up, many teachers give back the original tasks with red pens a few days later to allow students the opportunity to edit their thinking.


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Re-engagement – Confronting Misconceptions, Providing feedback on thinking, Going deeper into the mathematics. Example from 2007, 5th grade – Candies task. 1. Start with a simple problem to bring all the students along. This allows students to

clarify and articulate the mathematical ideas.

2. Make sense of another person’s strategy. Try on a strategy. Compare strategies.

Cement connection of diagram to ratio.

3. Have students analyze misconceptions and discuss why they don’t make sense. In the

process students can let go of misconceptions and clarify their misconceptions.

4. Find out how a strategy could be modified to get the right answer.

Find the seeds of mathematical thinking in student thinking.

Third Grade Goldfish Bowl Copyright © 2009 by Mathematics Assessment Resource Service. All rights reserved.

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Goldfish Bowls This problem gives you the chance to: • use numbers in a real situation 1. Dan has 3 goldfish bowls. He keeps 12 goldfish in each bowl. How many goldfish does Dan have? __________ Show how you figured it out. Dan always has the same number of fish in each of his goldfish bowls. 2. He breaks one bowl, so now he has just 2 bowls. How many fish will he need to put in each bowl? _________ 3. He buys two more bowls, so now he has 4 bowls. How many fish will he put in each bowl? _________ 4. If he has 6 bowls how many fish will he put in each bowl? ________ Show how you figured it out. 5. Dan discovers that if he has 5 fish bowls he can’t have the same number of fish in each bowl. Explain how you know he is correct. _____________________________________________________________________ _____________________________________________________________________

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Third Grade Copyright © 2009 by Mathematics Assessment Resource Service. All rights reserved.

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Goldfish Bowls Rubric The core elements of performance required by this task are: • recognize and use equivalent fractions Based on these, credit for specific aspects of performance should be assigned as follows

points

section points

1. Gives correct answer: 36 Shows work such as 12 x 3 = 36 . Accept repeated addition or diagrams.

1 1

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2. Gives correct answer: 18 (Accept 6) 1ft 1

3 Gives correct answer: 9 1ft 1

4. Gives correct answer: 6 Shows work such as 36 ÷ 6 = 6 . Accept repeated addition/subtraction or diagrams

1ft

1ft

2

5 Gives a correct explanation such as 5 does not divide into 36 equally or draws diagrams to show that: 36 ÷ 5 = 7 with 1 left over

1ft

1

Total Points 7


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Goldfish Bowls Work the task and look at the rubric. What are the big mathematical ideas in this task? What do you want students to understand about making equal groups or counting equal groups? What strategies do you think students might use to think about the equal size groups? How comfortable are students with multiplication facts? When teaching multiplication does your textbook treat multiplication and division as separate activities or are they taught together, helping students to connect the two processes? Look at student work as a whole. What strategies did students use to make sense of the equal groups? How many of your students: Multiplication/

division number

sentences

Repeated addition

Skip counting

Doubles Drew pictures

Other

In part 3, students needed to think about putting the fish into 4 bowls. How many of your students put:

9 12 6 8 48 3 Other

What type of misconception would lead to an answer of 48? What type of misconception would lead to an answer of 3? Although the magnitude of the answers is very different, what is similar in the misunderstanding? Now look at work in part 4, finding the number of fish in 6 bowls. How many of your students put:

6 2 72 12 3 Other

What are some is the logic behind some of the errors? Why might a student come up with these errors?


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In part 5 students are asked to reason about why Dan can’t use 5 bowls. How many of your students put:

• An explanation like 36 can’t be divided by 5, or gave multiples of 5 showing that it doesn’t hit 36? ____________

• Were thinking about dividing by 5 but lost track of the “whole” and talked about 30 or 12 instead of 36?______________

• Made an incorrect generalization that evens can’t be divided by odds? How could you plan a re-engagement lesson to help students explore this idea? __________

• Need more fish because they were still trying to put 12 into every bowl?_______ • Tried to divide the 5, e.g. 5 can’t be divided by 2 there will be a left over.

_________ How often do students in your class get opportunities to reason about number properties, like divisibility and why things work? How often do students get to attempt to make and prove generalizations about number?


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Looking at Student Work on Goldfish Bowl’s: Look at the work of Student A. Notice how the student shows the calculations and also explains why the operation is appropriate to solving the problem. Do you think the student needed to draw the pictures to do the calculation or is the picture just there to please the teacher? Notice the complete explanation in part 5. The student explains why 36 is not divisible by 5, but that 4 and 6 will go in evenly. Student A


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Student B also puts good detail into why he is doing each calculation. The student understands to test for divisibility in part 5, but doesn’t have the final piece about sizes of remainders. Student B


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Student C does some interesting thinking in part 2. Notice how the student draws the two bowls, reasons about the beginning amount in each and then splits the extra fish between the two bowls. Notice how the student keeps the total of 36 in mind when reasoning about the 5 bowls in the explanation. Student C


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Student D is able to reason about the total number of fish. Notice that in part 2 the student clearly knows to add more fish to the fish already in the bowl. However in part 3 the student seems to continue thinking from part 2 instead of noticing that there are now 4 bowls. Then in part 4 the student is still continuing the reasoning chain to explain part 2. The student again misses the number of bowls. In part 5 the student has completely lost the idea of total number of fish. (Part 3 should not have been marked correct) Student D


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Student E also uses pictures, repeated addition, multiplication, doubling, to verify the division sentences. The student has made an incorrect generalization in part 5. Could you think of a counter example to help the student confront the misconception? Student E


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Student F uses pictures to think about all the parts of the task. In part 5 the student loses sense of the whole and thinks about dividing up the 5 instead of the 36 goldfish. Student F

Student G makes a nice mathematical argument to show why Dan is incorrect in part 5. Notice how the student is making connections with various pieces of knowledge to complete the argument. What kinds of experiences can we provide in the classroom to help students develop this type of thinking? How do you share interesting mathematical explanations with your students so that they can learn from each other? Student G


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Student H has some difficulty with facts, but understands the idea of equal groups and using multiplication. In part 4 the student was able to make the calculation, but did not interpret the process to find a solution. However the student loses the idea of distributing the total of 36 fish into all the available bowls in part 5. Student H


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Student I is on the way to make a convincing argument in part 5. What other information would you like to see to make this complete? Student I

Student J does brainstorming while solving the problems in part 2 and 3. Because the student understands the process or purpose of using equal size groups in division, he has a strategy that could have led to a correct answer in part 3. How could the student have used the information or strategy in part 3 to find the correct amount? In part 5 the student has a strong explanation that was missed by the scorer. Student J


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Student K uses pictures and arithmetic to find solutions to the task. While the student has a strong understanding about the operation of division, the student is not clear about formal notation. Notice that the student uses multiples and number properties to make a convincing argument in part 5. Student K


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Student L also makes a good quantitative argument in part 5. Student L

Student M has lost track of the “whole”, 36 total fish. The student is now trying to divide up the 5. How does the use of labels help students to organize their thinking? Student M

One of the interesting things to notice in this set of student work is how students organized their work to help them find facts that they didn’t necessarily know by rote. Many students have used pictures or multiplication facts to find the answer to division. Student C uses tallies and writes down 6’s and 9’s as she adds to the correct total of 36. Student J uses a dealing-out strategy to put all the fish in the bowl, first putting 6 in every bowl, then 2 in every bowl. This strategy would have been successful, if the student had added the fish together and realized there were still 4 more fish to be distributed. Student K skip counts or uses multiples. Student N also uses skip counting and has a good organizational strategy to keep track of his thinking. Student N


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Student O uses an understanding of place value to solve the addition in part 1. The student uses repeated addition to solve the division problem in part 4. In part 5 the student is obviously thinking about “fiveness” when talking about the 7’s and 35. But then in talking about 8’s the student forgets that we are trying to get a number that will go 5 times. How might this student have organized his information to get a better, more complete explanation? Student O


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Student P is able to use known multiplication facts to solve the division in part 3. Then in part 4 the student uses division of the amount in 3 bowls to find the amount in 6 bowls by halving. Student P

Student Q understands the idea of dividing the total fish into equal parts in part 2, but then seems to lose the idea of distributing the fish equally into the bowls in parts 3 and 4. The student instead tries to keep the same number of fish in every bowl. Student Q


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Some students, like Student R, have trouble with recognizing or choosing operations in context. The student has learned some test taking techniques, like underlining numbers. But this does not help if the student doesn’t have a fundamental understanding of the operations and when they are used or why. How do we help students to learn about the meaning of the operation as an idea, separate from the procedure? What kinds of experiences help students to develop these ideas? Student R


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3rd Grade Task 1 Goldfish Bowls Student Task Use a real-life context for multiplication. Understand the idea of equal-

sized groups and its relationship to multiplication. Core Idea 2 Number Operations

Understand the meanings of operations and how they relate to each other, make reasonable estimates, and compute fluently.

• Understand multiplication as repeated addition, an area model, an array, and an operation on scale.

• Understand division as the inverse of multiplication, the operation of sharing.

• Develop fluency with basic number combinations for multiplication and division.

The mathematics of this task:

• Making fair shares or equal-size groups. • Using multiplication and division in context. • Developing a mathematical justification. • Keeping track of the whole

Based on teacher observation, this is what third graders knew and were able to do:

• Understand multiplication as the inverse of division and use repeated addition to find solutions

• Understand division as forming a number of equal size groups • Find the total number of fish • Use diagrams to illustrate their thinking and as a problem solving tool

Areas of difficulty for third graders:

• Keeping track of the total number of fish • Making a justification about why 36 fish can’t be put in 5 bowls equally • Having a misconception that even numbers can’t be divided by odd numbers • Distinguishing between dividing 36 fish evenly into bowls and putting the same

number of fish (12) into every bowl • Being comfortable with basic facts

Strategies used by successful students:

• Drawing diagrams and showing the changes as the story changed • Repeated addition to think about the equal size groups • Ease with multiplication facts • Underlining key parts of the question • Labeling their calculations to be clear about what the numbers represent • Writing answers in complete sentences to make sure the answers fit the context


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The maximum score available on this task is 7 points. The minimum score for a level 3 response, meeting standards, is 4 points. Most students, 90%, could find the total number of fish and show how they figured it out. Many students, 75%, could also figure out how to distribute the fish into 2 bowls. More than half the students, 62%, could find the total fish, divide them into 2 equal groups, and make sense of dividing the total fish into 4 or 6 bowls. 43% could divide the fish evenly into 4 and 6 bowls and show their thinking. Almost 22% could meet all the demands of the task including explaining why the fish couldn’t be put into 5 bowls. 4% of the students scored no points on the task. All of the students in the sample with this score attempted the task.


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Goldfish Bowls Points Understandings Misunderstandings

0 All the students in the sample with this score attempted the task.

Students had trouble identifying the correct operation to use. Their work did not show reasoning about equal-size groups.

2 Students could use pictures, addition, skip counting, or multiplication to find the total number of fish in the original 3 bowls.

Students had difficulty thinking about going from 3 bowls to 2 bowls. 4.6% of the students thought there would still be 12 fish in each bowl. Answers for part 2 ranged from 2 to 52.

3 Students could find the total fish and find the number of fish in 2 bowls.

Students had difficulty thinking about the number of fish in 4 bowls. 8.3% thought there would still be 12 fish. 7% thought there would 3 fish (12/4). About 6% each gave answers of 48 (4 x 12), 8, and 6.

4 Students could find the total fish, the number of fish in 2 bowls, and either the number of fish in 4 or 6 bowls.

About 9% of the students thought there would be 2 fish in each of the 6 bowls (12/6). These students lost track of the “whole” or total number of fish. About 6% thought there would still be 12 fish per bowl. About 5% of the students selected 3 and 5% picked 8.

6 Students could find the total number of fish, divide the total fish into 2, 4 or 6 bowls and show their work.

Students struggled with how to explain why Dan couldn’t use 5 fish bowls. Some students thought he would need more fish. Others thought 12 was not divisible by 5. Many lost track of the “whole” and tried to divide 5 by 2, explained about dividing 30 by 5, or some other number combination not relevant to solving this task. Many students had a misconception that all even numbers cannot be divided by odd numbers.

7 Students could find the total number of fish by reasoning about equal size groups. Students understood that the total number of fish needed to be maintained throughout the task. They could divide the total into equal groups for 2,4,or 6 bowls. They could also explain that 36 is not evenly divisible by 5.


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Implications for Instruction Students at this level should be developing an understanding of the operation of multiplication, making equal-size groups. This process might take the form of skip counting, dealing out equal groups pictorially and counting, repeated addition or simple multiplication facts. Teachers need to ask questions and prompt students to move up to more efficient strategies to further their mathematical development. If students are drawing concrete objects, like the goldfish, can they move to symbols or tallies. If students are using skip counting or repeated addition, they should be encouraged to move up to multiplication. Furthermore teachers need to think about the stages of the learner. For some students drawing pictures and counting is a necessary tool for doing the multiplication or for thinking about equal-size groups. However, at some point students should not need to draw the pictures, but rather they should just know the multiplication facts. At this stage they are more likely drawing the pictures just to please the teacher. Look carefully at a sample of student work. What evidence do you see that some students no longer need the pictures. These students are now ready for more abstract ways of showing their thinking by verbalizing or writing about using multiplication and division because there were equal size groups or labeling their calculations to define what they have calculated. Students need to be acknowledged for what they have accomplished and given a vision of what they should be striving for. Research projects, such as Cognitively Guided Instruction, suggests that students who learn addition and subtraction together or multiplication and division together learn the facts faster and develop a better understanding of both operations. So, students might start by thinking of 4 x 3 and then derive the facts by counting by 4’s: 4, 8, 12 or by thinking about 4 + 4 + 4. Then use the facts in context. There are _____ flowers in 3 rows. Students could then continue to think about equal size groups. Divide 12 marbles into 4 equal groups. How many marbles are there in each group? 12 divided by 4 is ____. There are ______ marbles in each group. Divide 12 marbles into groups of 4. How many groups are there? 12 divided by 4 is ______. There are ____ groups. Students need to start labeling what they have calculated and be encouraged to write their answers in sentences. This helps students to organize what they know and define what happened as a result of doing the arithmetic. Many students in this task had difficulty remembering what the “whole” or total number of fish was. This seems critical to understanding the whole later when developing ideas about fractions and fractional parts. So the way students learn and discuss beginning ideas about multiplication and division, is laying foundational knowledge to be built upon later.


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Ideas for Action Research – Re-engagement Re-engagement – Confronting misconceptions, providing feedback on thinking, going deeper into the mathematics. (See overview at beginning of toolkit). 1. Start with a simple problem to bring all the students along. This allows students to

clarify and articulate the mathematical ideas. 2. Make sense of another person’s strategy. Try on a strategy. Compare strategies. 3. Have students analyze misconceptions and discuss why they don’t make sense. In

the process students can let go of misconceptions and clarify their thinking about the big ideas.

4. Find out how a strategy could be modified to get the right answer. Find the seeds of mathematical thinking in student work.

In this task the core idea is the total number of fish. So the lesson might start by showing a piece of student work, such as: In part one, a student put: 12 x 3 = 36. What do you think the student was doing? What does the 12 represent? What does the 3 represent? How could we write the answer in a sentence? What is the 36? Why is it important? Another person, put 12 + 12 + 12, but you have just told me that 12 x 3 is the correct method? What is the person thinking? (Try to get students to verbalize that 3 equal groups of 12 is the same as 12 x 3. We want students to see the connections and make sense of the equal groups. Hopefully notice that multiplication is quicker.) Next look at the section on 2 bowls. Sometimes cognitive dissonance, creating disequilibrium, is a good learning strategy. You might say something like, “When teachers were grading this question, some students put 6 fish for an answer and some students put 18 fish for an answer. Could they both be right? How do you think the students might have gotten the answer 6? The answer 18?” Hopefully during the discussion students will talk about the total number of fish. If not, then ask them about part 3. Say to them that someone thought the answer to part 3 was 18 fish. Ask, “How might a student get this answer?” This question is trying to prompt students to verbalize that the total number of fish is always 36. Then see if they can work with a partner to find a correct solution. Now its time to push the mathematics of this task a bit. In part 5, a student said that an even number can’t be divided by an odd number. “Can anyone give me an example that would show that this is true? Do you think this is always true? Can you find an example of when it is not true? “ “How could you convince someone that the fish can’t fit in 5 bowls? Work with a partner.”

Grade Three Birthday Decoration Copyright © 2009 by Mathematics Assessment Resource Service. All rights reserved.

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Birthday Decoration This problem gives you the chance to: • find and extend a pattern Cameron is decorating the house to celebrate his mom’s birthday. He makes a pattern with silver stars and red balloons. The first piece looks like this. When he adds piece #2 it looks like this. 1. Draw piece #3 onto the pattern above. Cameron needs to know how many stars and balloons he will be using. He makes this table to help.

number of pieces 1 2 3 4 5

number of stars 2 4

number of balloons 3 6 2. Fill in the numbers in the table for three pieces. 3. How many stars and balloons will he need to make five pieces? Write your answer in the table above. 4. When he started decorating, Cameron had 26 balloons and 19 stars. He said that he would be able to make nine pieces. Is he correct? ______________ Explain your work. ___________________________________________________________________ ___________________________________________________________________ 8

Grade Three Copyright © 2009 by Mathematics Assessment Resource Service. All rights reserved.

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Birthday Decoration Rubric • • The core elements of performance required by this task are: • • find and extend a pattern • Based on these, credit for specific aspects of performance should be assigned as follows

points

section points

1. Draws the 2 stars and 3 balloons to extend the pattern 2 x 1 2

2. Fills in the table correctly: 6 stars and 9 balloons 2 x 1 2

3 Fills in the table correctly: 10 stars and 15 balloons

2 x 1

2 4 Gives correct answer: No and explanation such as:

He has 19 stars which would make 9 pieces of the pattern with one star left over. There are 26 balloons: 26 divided by 3 is 8 with a remainder of 2. So Cameron can only make 8 pieces of the pattern Partial credit For a partially correct answer.

2

(1)

2

Total Points 8

Grade Three Copyright © 2009 by Noyce Foundation All rights reserved.

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Birthday Decoration Work the task. What are the big mathematical ideas? ____________________________ How often do students in your class work problems in context? What types of opportunities do they have to interpret the answers to their calculations? Look at student work in part 1, extending the pattern. How many of your students: Made a correct

drawing Made no drawing

Forgot to draw the balloons

Made only 2 balloons

Other

How do we help students notice all the attributes of a pattern? Do you ask students questions, like what are the elements in the pattern? How many of each kind? What does the first pattern piece look like? Now look at student work for part 4. How many of your students:

• Gave a complete explanation?_____________ • Thought there were enough pieces? ____________ • Thought about individual pieces rather than the whole pattern of 5 shapes?

_________ • Added the 19 + 26 = 45 and 9 x 5 = 45, so there are enough pieces? (Didn’t

distinguish between the types of pieces?) ______________ • Thought 19 is not an even number, so it won’t work?______________ • Extended the table but couldn’t interpret the answer?______________ • Other?_____________________

Why do you think students struggled with the explanations? How do you craft lessons to help students develop the logic of making convincing arguments or explanations? How might you use some of their thinking to plan a short re-engagement lesson on part 4?


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Looking at Student Work on Birthday Decorations Student A is able to draw the pattern and extend the pattern using a table. What are the qualities in the argument that make it convincing? Student A


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Student B also makes a convincing explanation for part 4. What are the qualities in this argument that make it convincing? Student B

Student C does a lot of correct calculations in part 4. What is missing to make this a convincing argument? Student C


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Student D also does calculations, but does not know how to interpret the solutions in terms of the context. What is important about comparing the 18 and the 19? If 3 x 9 is not 26, why is this important or not important in the context of decorations? What questions might you pose to the class using this piece of student work? Student D

Look at the work of Student E. Why might he think there are enough balloons and stars? Student E

Now look at the work of Student F. He seems to have lost track of the pattern and is thinking about individual pieces. Student F


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Student G has done some thinking and has a detailed explanation. What is missing to make the explanation complete? Student G

Like Student G, Student H is sidetracked by the pattern, forgetting about the relationship of the pattern to the context. Why do you think students are so concerned with even and odd rather than the total amount of decorations? Student H


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Student I has a nice understanding of the unit of 5. The pattern grows in groups of 5. However it is also important to the pattern to have the right number of stars and balloons. Student I

Student J does not know how to interpret the chart. He tries to add all the numbers together to get a huge number of decorations. He does not see that the elements in later steps contain the elements from the previous steps. Student J


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Student K is able to do the mathematics of extending the table, but has no idea how to interpret the information. The student is just practicing skip counting. In part 4 the student tries to examine the differences rather than think about the number of total pieces needed to make the design. Student K


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3rd Grade Task 2 Birthday Decoration Student Task Find and extend a geometric pattern in drawings, tables, and numbers.

Be able to justify why something does or does not fit the pattern. Core Idea 3 Patterns, Functions and Algebra

Understand patterns and use mathematical models to represent and to understand qualitative and quantitative relationships.

• Describe and extend geometric and numeric patterns. • Represent and analyze patterns using words and/or tables • Model problem situations with objects and use representations

such as graphs and tables to draw conclusions. The mathematics of this task:

• Recognizing a pattern with different elements and extending the pattern with a drawing

• Extending the pattern using a table • Reasoning about the number of individual decorations needed to replicate the

pattern 9 times and making a justification about whether there are enough pieces • Interpreting computations in context • Interpreting remainders

Based on teacher observations, this is what third graders know and are able to do:

• Extend a pattern by drawing • Skip count • Multiply • Use a table


• Understanding that 9 pieces meant 9 parts of the pattern, not separate decoration elements

• Understanding how to interpret calculations, e.g. counting by 3’s there is no 26 does not explain that 1 more balloon is need to make 9 sets of 3

• Understanding that having enough decorations is not about odd or even, but on the total amount

• Understanding that its ok to have extras Strategies used by successful students:

• Extending the table • Using multiplication to find the number of stars and balloons needed • Making a comparison of amount needed to amount on hand • Drawing and counting • Quantifying how each part of the pattern is growing


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The maximum score available for this task is 8 points. The minimum score for a level 3 response, meeting standards, is 5 points. Most students, 83%, could extend the pattern to 3 and 5 pieces using the table. 77% could also extend the pattern by making a drawing. Some students, 27%, could also explain why there weren’t enough decorations to make nine pieces. Almost 2% of the students scored no points on the task. No students in the sample scored zero.


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Birthday Decorations Points Understandings Misunderstandings

0 No students in the sample had this score.

No information on scores lower than 4 in the sample set.

4 Students could use the table to extend the pattern to 3 and 5 pieces.

5% of the students only drew the stars and left out the balloons. 4% of the students did not make a drawing.

6 Students could draw the pattern for 3 pieces and extend the pattern using the table.

Students did not know how to make a justification for part 4. Many students put answers about even and odd, not interpreting the pattern to the context. Others thought about the total number of decorations, 45, but ignored the fact that they weren’t the right shapes. Many did calculations that could have led to the correct answer, but could not interpret the meaning.

8 Students could recognize a pattern and extend it using a drawing and a table. Students could make a convincing mathematical argument about why there weren’t enough decorations to make 9 pieces.


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Implications for Instruction Students at this grade level need to have opportunities to draw and extend patterns. Students should start to think about multiplication of equal size groups when working with patterns. Students need more experience working with problems in context, so they have to puzzle about the meaning of their calculations. Students need to think about comparisons, what does it mean to have enough? Too much? What is possible to fit this situation? Students need to start developing the ability to make convincing arguments. They should know to quantify their answers. Ideas for Action Research Students need opportunities to work with a variety of problems that require giving justifications. You might have students work a 4th grade 2003 Task, Flower Arranging, which requires logical thinking and explanations. This is a good task for thinking about constraints and how ideas match constraints. 2006 3rd grade task, Houses in a Row, is a opportunity to work with growing patterns and think about making justifications of why something is or is not possible. 2005 3rd grade task, Number Cards, gives students a chance to work with place value and making justifications. In working these tasks, think about showing different explanations and having students evaluate which explanations are more convincing. See if students can start to develop their own internal standards for making good explanations.

Third Grade Making a Doll House Copyright © 2009 by Mathematics Assessment Resource Service. All rights reserved.

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Making a Doll House This problem gives you the chance to: • recognize, name and describe shapes Jack’s dad makes doll houses. This is one of his designs. He numbers the shapes he uses.

1. Name the shapes he has used. 1__________ 2 __________ 3 ____________ 4 __________ 5__________ 6 __________ 7 ____________ 8 __________

2. Which one of these shapes does not have a line of symmetry? ______________ 3. Look at the shapes numbered 2 and 4. Write one thing that is the same about these shapes. _____________________________________________________________________ Write one thing that is different about them. _____________________________________________________________________ 4. Jack’s dad likes his doll house designs to have at least one line of symmetry. Draw the line of symmetry on this house. 8

Third Grade Copyright © 2009 by Mathematics Assessment Resource Service. All rights reserved.

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Making a Doll House Rubric The core elements of performance required by this task are: • Recognize, name and describe shapes Based on these, credit for specific aspects of performance should be assigned as follows

points

section points

1. Gives correct answers: 1. square, 2. rectangle, 3. triangle, 4. trapezoid, 5. hexagon, 6. parallelogram, 7. rhombus, 8. circle. All 8 correct Partial credit 7-6 correct 3 points 5-4 correct 2 points

3-2 correct 1 point

4

(3)

(2)

(1)

4

2. Gives correct answer: Number 6 / the parallelogram 1 1

3. Gives correct answer such as: Both have 4 sides. Gives correct answer such as: Shape 4 has no right angles and shape 2 has 4.

1 1

2

4. Draws the correct line of symmetry on the house. 1 1

Total Points 8


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Making a Doll House Work the task. Look at the rubric. What are the big mathematical ideas in this task? What types of opportunities do students have to discuss properties of geometric shapes? What attributes would you expect them to pay attention to? Look at student work on naming shapes. On shape 4, the trapezoid, how many of your students put:

Trapezoid Rectangle Parallelogram Triangle No answer Other

Look at work on shape 6, parallelogram. How many of your students put: Parallelogram Rectangle Trapezoid Triangle No answer Other

What attributes were students paying attention to when they named the shapes? What attributes were they not paying attention to? Now look at the work on part two, finding a shape with no symmetry. How many of your students put:

#6 or parallelogram

Circle Trapezoid Several shapes Other

Why do you think students might give an answer of circle? What might they be thinking about? Why might the trapezoid seem confusing to them? In part 3 students are asked to compare shapes. When naming things that are the same about the rectangle and the trapezoid, how many gave correct answers, such as:

• 4 sides?_________ • 4 angles?________ • Parallel sides?_____ • 2 sides are equal?_______

What other good attributes were mentioned?


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Now look at some of the incorrect answers. How many students gave answers like: • Straight sides?________ • Both have vertices or sides, but with no quantity attached?________ Why is the quantity

important? What other things did students notice? Now look at the differences. How many of your students gave correct answers like:

• One has right angles the other doesn’t?_______ What other attributes did they notice? What were some of the incorrect responses? Did they give answers, such as:

• Number 4 is longer, skinnier, or more stretchy?___________ • Number 4 has pointy sides, slanted sides, diagonal sides?________ • Different shapes?__________ • Only the rectangle has parallel sides?_______ • Different number of sides?____________

What kinds of activities help students develop the ability to see and discuss attributes? How can you design lessons to give students talk time to use academic language for a purpose?


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Looking at Student Work on Making a Doll House Students had difficulty naming trapezoid, parallelogram, and rhombus. Here is a look at the data: For the trapezoid, 47% of the students named the shape incorrectly.

• 10% called it a parallelogram • 8% called it a rectangle • 5% did not attempt to name the shape • 4% thought it was a rhombus • Other names included: circumference, sphere, triangle, and pentagon

For the parallelogram, 40% of the students named the shape incorrectly. • 10% did not attempt to name the shape • 7% called it a rectangle • 6% called it a rhombus • 4% called it a trapezoid • 3% called it a triangle • Other names included: triangular prism, diamond, pentagon, triangle, hexagon, prism, and

square For the rhombus, 35% of the students named the shape incorrectly.

• 29% called it a diamond • Other names included: square, pentagon, triangle

In part 2 students were asked to name a shape with no lines of symmetry. • 28% of the students missed this part of the task. • 14% thought the circle had no line of symmetry. • 6% thought the trapezoid had no line of symmetry.

In part 3, students are asked to compare and contrast the rectangle and the trapezoid. 31% of the students thought the shapes were different because the trapezoid is longer. Here are some samples of student work: Student A and B able to compare and contrast both shapes, paying attention to important, defining attributes of the shape. Student A


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Student B

Student C mentions important characteristics, but it is unclear if the student really understands parallel lines or has memorized something about rectangles. What question might you ask to probe the student’s thinking? Student C

Student D has found another important property the shapes share. The student notices something important about the shapes, but doesn’t have the vocabulary to describe it. The student doesn’t have a word for corner or angle, but can only talk about the general impression of the shape. How do we foster development of academic language? What kinds of activities give students the opportunity to use vocabulary for a purpose? Student D

Student E is able to quantify the number of sides and angles, but again likes vocabulary. The student is not able to describe why they are different. Student E


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Student F has a misconception about corners, or angles. What do you think the student sees as angles in the trapezoid? What experiences might help this student? Student F

Student G has trouble with finding differences. The student is looking at specific attributes of the given shapes, rather than what makes them different as a category. How do we help students learn to distinguish relevant, mathematical attributes from irrelevant or less important? Student G

Student H, I and J also struggle with vocabulary and relative traits rather than traits that define the shape. This problem of vocabulary limits what students can think about. At later years, many middle school students look at a trapezoid and think triangle because they focus on the “slanty” sides rather than number of sides or angles. Student H

Student I


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Student J


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3rd Grade Task 3 Making a Doll House Student Task Recognize, name and describe geometric shapes in a diagram. Core Idea 4 Geometry and Measurement

Recognize and use characteristics, properties, and relationships of two-dimensional geometric shapes and apply appropriate techniques to determine measurements.

• Identify and compare attributes of two-dimensional shapes and develop vocabulary to describe the attributes.

• Recognize geometric ideas and relationships and apply them to problems.

Mathematics of the task:

• Identifying and naming geometric shapes • Understanding symmetry and identifying non-symmetrical shapes • Drawing lines of symmetry through a complex shape • Comparing and contrasting shapes based on geometric properties, such as number and size of

angles, number of sides, lines of symmetry, parallel or nonparallel sides Based on teacher observation, this is what third graders know and are able to do:

• Identify basic shapes • Understand symmetry • Identify common attributes

Areas of difficulty for third graders;

• Naming trapezoid, parallelogram, and rhombus (many used diamond) • Academic vocabulary, such as angle • Identifying differences that are about the shape as a category rather than the specific shapes,

i.e. longer or thinner or they look different instead of different size angles or less pairs of parallel sides


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The maximum score available for this task is 8 points. The minimum score for a level 3 response, meeting standards, is 5 points. Most students, 96%, could identify at least 5 of the eight shapes. Many students, 88%, could also find a shape that didn’t have a line of symmetry. More than half the students, 56%, could name at least 5 shapes, find the shape without a line of symmetry, and draw in a line of symmetry for the dollhouse. Some students, 37%, could also name an attribute shared by the trapezoid and rectangle. 6% of the students could meet all the demands of the task, including finding an attribute that distinguishes between a rectangle and a trapezoid. 1.6% of the students scored no points on the task. All of the students in the sample with this score attempted the task.


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Making a Dollhouse Points Understandings Misunderstandings

0 All the students with this score in the sample attempted the task.

Students did not know the names of common shapes. (See chart in Looking at Student Work)

2 Students with this score could name 4 or 5 shapes; usually the square, rectangle, triangle, circle and hexagon.

Students could not draw in a line of symmetry. About 13% of the students did not attempt to draw a line of symmetry. Some students drew in lines of symmetry for individual shapes, rather than thinking of the dollhouse as a whole shape.

5 Students could usually identify 6 shapes, draw a line of symmetry, and identify the shape that did not have line symmetry.

Many students thought the circle did not have a line of symmetry (14%). About 6% thought the trapezoid did not have a line of symmetry. Students also struggled with finding similarities in the rectangle and trapezoid. 6% thought that they were the same because they both had straight sides.

7 Students could identify most of the shapes, but may still have confused rhombus and diamond. They understood symmetry and could identify a similarity between to shapes.

Students struggled with finding differences between the shapes. They are different shapes was a common answer. 31% said the trapezoid was longer and thinner. Some students noticed properties about the shapes, but lacked language to describe them.

8 Students could identify shapes, use symmetry to find non-symmetrical shapes and draw lines of symmetry, and compare and contrast geometric shapes by important geometric properties.


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Implications for Instruction Students need to be able to name common geometric shapes and begin to talk about important attributes, e.g. number of sides, number of angles, right angles, parallel sides. Lessons need to be designed to give students talk time to practice geometric vocabulary with a purpose. For example, classifying and sorting geometric shapes helps students notice attributes and have a need for vocabulary. Students need opportunities to build designs with shapes or put together puzzles. These experiences help them to notice important attributes like side length and size of angles. Students also need to see a variety of shapes for a category, so they don’t think about triangles are only triangles if they are equilateral or have a base parallel to the bottom of the page. The richer the variety of shapes the deeper the understanding students are able to build. Students should be able to find a line of symmetry in a geometric shape. Students need to be confronted with a variety of shapes to think about symmetry, so they don’t build a misconception that lines of symmetry are always horizontal or always vertical. Ideas for Action Research – The Logic of Sorting Sorting activities help students to focus on important attributes of shapes. Sorting also gives students reasons for using academic language. Consider two activities from the Virginia Depart of Education website, What’s My Rule. (www.doe.virginia.gov/VDOE/Instruction/Elem_M/geo_elem.html) What’s My Rule

1. Choose one player to be the sorter. The sorter writes down a “secret rule” to classify the set of quadrilaterals into two or more piles and uses that rule to slowly sort the pieces as the other players observe.

2. At any time, the players can call “stop” and guess the rule. The correct identification is worth 5 points. A correct answer, but not the written one, is worth one point. Each incorrect guess results in a two-point penalty.

3. After the correct rule identification, the player who figured out the rule becomes the sorter. 4. The winner is the first one to accumulate ten points.

A similar activity can be done with triangles.


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Ideas for Action Research – The Logic of Classifying and Sorting Look at the lesson below (from Teaching Student-Centered Mathematics, Grades K-3 by John Van de Walle). How might this lesson be modified to help students think about 3-dimensional shapes? Can you design your own set of shapes? Make sure you think carefully about the question posed by the Stop Sign.

Grade Three The Math Test Copyright © 2009 by Mathematics Assessment Resource Service. All rights reserved.

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The Math Test This problem gives you the chance to: • use a bar chart This morning the Grade 3 students took a math test. The test was scored out of 40 points. The students did well and they all scored more than 30 points. Here is a graph of their results. 1. How many students scored 33 points? ______________ 2. Which score was earned by the largest number of students? ___________ 3. How many students earned more than 34 points? ____________ Show how you figured this out. 4. How many students took this test? ___________ Show how you figured this out. 5. Two children were absent this morning so they had to take the test this afternoon. They also did very well. They scored 31 and 34. Put these scores onto the graph. 8

31 32 33 34 35 36 37 38 39 40

6

5

4

3

2

1

0

7

Scores

Number of students


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The Math Test Rubric The core elements of performance required by this task are: • use a bar chart Based on these, credit for specific aspects of performance should be assigned as follows

points

section points

1. Gives correct answers: 3 1 1

2. Gives correct answer: 35 1 1

3. Gives correct answer: 9 Shows work such as: 6 + 2 + 1= 9 Accept: “I counted” with correct answer.

1 1

2

4. Gives correct answer: 22 Shows work such as: 4+5+3+1+6+2+1=22 Accept: I counted with correct answer.

1 1

2

5. Records the scores 31 and 34 accurately on the graph. 2 x 1 2

Total Points 8


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The Math Test Work the task and look at the rubric. What are the big mathematical ideas students need to know to be successful in this task?__________________________________________ This task on data is interesting because it has numbers as a category, score on test, and number as a frequency, number of people who got a certain test score. Look at student work on part 2, which score was earned by the largest number of students. How many of your students put:

35 36 37 6 32 Other

What might a student who gave an answer of 37 been confused about? What might a student who gave an answer of 6 be confused about? What does this answer show about understanding scale and bar graphs? Now look at student work on part 3, how many students earned more than 34 points. How many of your students put:

9 10 6 1 37 21 35 Over 200 Other

How do you think students might have gotten 10? 6? 37? 21? How are their misconceptions different? Now look at work on part 4, finding the total number of students who took the test. How many students put:

22 23 21 7 Over 200 Other

What types of questions do you ask students about when looking at graphs? Do you ever ask students to write their own questions about what can be answered from the graph? Have students had experiences working with graphs where both frequency and category are numbers?


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Looking at Student Work on The Math Test Student A is able to think about the mathematics of the task. In part 3 the student explains which scores are larger than 34. Student A


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Student B also has completed all the elements of the task. Notice this student shows the numbers being added. The student also seems to have a system for simplifying the addition of a string of numbers. Notice that the student makes lines on the graph to make the bar heights easier to read. Students need to see the graph as a tool that can be written on and used to help think about the task. Student B

Student C is confused about the meaning of the categorical numbers on the bottom of the graph and uses them as numerical values. What question could you pose to the class to help them discuss the logic of this issue? Student C


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Student D mistakes the frequency for a category when answering question number 2. The student gives the most students who got a particular score, rather than being able to find out the score earned by most students. In part 3 the student does not identify what the question is asking. The student finds a score larger than 34, but not the number of students who scored better than 34. The student omits the “34” score when adding up the total number of students taking the test. Student D

Student E also misuses the scale when answering question 2. The student is also thinking about 35 being more than 34, but forgets that other scores are also larger than 34. In part 4 the student uses the lowest score and adds an 8, possibly misreading the 8 from 38 on the horizontal axis, to incorrectly find the total number of students. It is likely that the student is only looking at test scores rather than number of students. Student E


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Student F recognizes that there are 3 scores higher than 34, but does not find the total number of students. In trying to count all the students, the student’s numbers take up more than one square causing a miscount. How do we help students transition to more efficient strategies like addition? Student F


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Student G had trouble with the graphing. The student added two students for each new score, rather than thinking that the scores were for two people. Student G


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3rd Grade Task 4 The Math Test Student Task Use a bar chart to answer questions and derive information. Core Idea 5 Data Analysis

Collect, organize, display and interpret data about themselves and their surroundings.

• Describe important features of a set of data (maximum, minimum, increasing, decreasing, most, least, and comparison).

• Represent data using tables, line plots, bar graphs, and pictographs.

Mathematics of this task:

• Understanding the difference between number as categorical information and number as frequency

• Reading and interpreting a graph • Understanding “more than” in the context of a graph • Finding the total represented by a bar graph • Adding data to a bar graph

Based on teacher observation, this is what third graders know and are able to do:

• Add data to a graph • Read the number of people with a score of 33 • Find the total number of students taking the test


• Showing or explaining their work • Interpreting “more than” on a graph • Knowing which numbers to use: frequency or number of students versus categorical or test

score • Calculating without errors


• Counting squares on the chart • Writing number sentences


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The maximum score available for this task is 8 points. The minimum score needed for a level 3 response, meeting standards, is 5 points. Most students, 89%, could read a number from the graph and add data to the graph. Many students, 73%, could also find the total number of students taking the test. More than half the students, 63%, could also find the score that was earned by the most students, reading from the vertical scale to the horizontal categories. 32% of the students could meet all the demands of the task, including interpreting “more than” in the context of the graph. 1.8% of the students scored no points on this task. No students in the sample had this score.


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The Math Test Points Understandings Misunderstandings

0 No students in the sample had this score.

Some students used categorical numbers for how many students, writing 33 or 35. Some students did not attempt to add data to the graph.

3 Students could read the number of students who scored 33 points from the graph and add data to the graph.

Students struggled with finding the total number of students taking the test. 6% had an answer of 21, because they didn’t count the score of 34. 9% had calculation errors with a score of 23. 4% had correct answers but did not show any work. 3% added test scores rather than students.

5 Students could read information from the graph, add data, and find the total number of students represented on the graph.

Students struggled with finding the score with the most number of students. 3% picked the largest test score: 37 or 40. 13% read the number of students instead of test score. They couldn’t interpret what the question was asking.

6 Students could read and interpret a bar graph, add data, and find the total number of students.

Students had difficulty with the idea of “more than” on a graph. 12% of the students had the correct answer, but didn’t show any work. 4% had answers of ten, either because of calculation errors or because they changed their answer after adding data to the graph. 4% only considered the number of students with scores of 35, which was 6. 6% gave a test score number higher than 34 rather than number of students.

8 Students could read and interpret a bar graph, add data to the graph, and find the total number of students represented. Students could also distinguish between categorical numbers and frequency numbers. Students could reason about “more than” in the context of a graph.


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Implications for Instruction Students should be able to read and interpret bar graphs. Students need to be exposed to a variety of question types regarding graphs. They should be asked questions that require reading from the horizontal axis to the vertical axis and also from the vertical axis to the horizontal. Students should be able to make simple calculations from the data, such as how many more or how many people are represented on the bar graph. Using graphs to derive information to solve problems is a critical skill. One way of developing these skills is to have students write questions that they think can be answered by the graph. Students should be able to add additional data to the graph. Students need to be exposed to a variety of scales, that don’t always have a value of one. Question types should include finding totals, comparing differences, identifying most and least. Looking at graphs is also a good way to use the idea of more than and less than in context. Ideas for Action Research – Making Graphs to Answer Questions Students need opportunities to gather data and make their own graphs. Many of the skills or understandings of graphs don’t come from just reading information on the graph. Consider having students develop a question that they want information about, such as how do the likes of 3rd graders compare to the likes of 5th graders, what would be the best type of person to bring in for a school assembly, making a suggestion for a new lunch option in the cafeteria, what kind of party would be best for the end of year activity. Then let students devise ways of gathering the data. This is an important part of the process of using and understanding data, but is often not experienced by students. This provides a need for them to think about categorizing and sorting the data, before they record it. For example: on party ideas there might be 20 different answers in the original words of the students polled which isn’t useful for making a graph or answering the question. However some ideas might be grouped together as activities at the park or classroom games. This sorting and categorizing allows data to be combined and provide more useful information. Once students have the data organized they can then think about what type of graph to use to display the data and what would be a useful scale. Once students have finished their graphs they can show them to the class and talk about how the graph helps them answer the original question. Students can then be asked to think about what other questions can be answered from the graph. Students need experience seeing data as relevant to solving problems and answering questions. Making comparisons, such as how much more, is an important part of using derived information. Students should start to value general impressions from the graph, such as most least, and shape of the data. Are all the data bars close to the same height or wildly different? What does this mean in the context of the data? Students need to think of the graph, not as an end point, but as a starting place for getting information. Students need to be able to find a variety of derived calculations from the data, including totals, comparison subtraction, more than, less than, etc. Asking students to write their own questions that would require calculations is important because it starts them wondering about the data and makes them eager to explore what they can find out. Looking at the graph should be an adventure into what can I do with the information that might be useful.

Grade Three Valerie’s Puzzles Copyright © 2009 by Mathematics Assessment Resource Service. All rights reserved.

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Valerie’s Puzzles This problem gives you the chance to: • solve whole number problems Valerie likes to do number puzzles. Here is one of her puzzles. Each square of nine numbers must contain all the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9. Each number should be used only once in each square of nine numbers. Each row must add up to the number at the end. Three numbers are always filled in at the start of each puzzle. Complete these puzzles for Valerie.

1 = 12

2 = 15

3 = 18

3 = 12

6 = 15

9 = 18

1 = 12

4 = 15

7 = 18 9


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Valerie’s Puzzles Rubric The core elements of performance required by this task are: • solve whole number problems Based on these, credit for specific aspects of performance should be assigned as follows

points

section points

Gives correct answers: Puzzle 1 Solutions such as: 147 or 165 528 924 693 783 Puzzle 2 Solutions such as: 543 or 813 762 267 981 945 Puzzle 3 Solutions such as: 831 or 651 924 834 657 927 Allow 1 point for each correct addition. Do not allow repeats.

1x3 1x3 1x3

9

Total Points 9


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Valerie’s Puzzles Work the task and look at the rubric. What are the big mathematical ideas being assessed? _____ Look at the overall work of your students. Do you see evidence of strategies, such as:

• Marking off the numbers as they are used?_____________ • Writing number sentences?____________ • Making erasures or evidence of changing answers in an attempt to meet the

constraints?_________ • Underlining key information to keep track of the constraints or identify what is being asked?

Sometimes it is useful to get an overall picture of student performance. Make a line plot of student performance below. O 1 2 3 4 5 6 7 8 9 Now look at the work of some of the lower performing students. How many of their errors can you attribute to:

• Repeating numbers, not following the constraints?_________________ • Using numbers outside the range, such as 0 or 10?_________________ • Calculation errors, having the wrong total?_______________________ • Not attempting part of the task or leaving blanks?__________________

What types of experiences do each of these mistakes suggest that students might need? What type of classroom discussion might you pose to get students to think about some of these issues? How could you have students share useful strategies for attacking this problem? Do you think students could identify a shortcut to make puzzle 2 and puzzle 3 easier to solve?


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Looking at Student Work on Valerie’s Puzzles Success on this problem was being able to understand and keep track of the constraints: not using a number more than once, staying within the range of 1 to 9, and having the proper totals. Student A uses number sentences to help check that the totals are correct and keep track of the running totals going across. Student A


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Student B makes a tool by listing the numbers in the range, so that numbers can be marked off as used. The erasures on the numbers suggest that the same list was used for all the puzzles. Also the erasures within the puzzle show a checking for errors and persistence in finding a totally correct solution. Student B

Student C uses the list from the top of the page and also has the habit of mind to persist and make multiple efforts to find a solution. Student C


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Student D shows an understanding of the constraints with perfect scores for puzzle one and puzzle 3. In puzzle 2 the student makes calculation errors. Student D

Student E does not understand the constraint about using a number only once. The student also uses numbers outside the range. The student makes a calculation error in the last row of puzzle 3. Student E


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Student F uses numbers several times within a puzzle. In puzzle 2 and 3 the student has noticed a pattern that all the numbers need to add to 9 or add to 11. Without that constraint of using a number once these puzzles become much easier. It might be interesting to see if students in your class could find this same generalization. How might you pose that as a question? A good follow up question would be to see if students could articulate why that is true. Student F

Student G does not understand the constraints, using numbers more than once and using numbers outside the range. Also except for the first row of puzzle one, the student ignores the given number and just searches for a combination of two numbers to make the total. Student G


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3rd Grade Task 5 Valerie’s Puzzles Student Task Solve whole number puzzles while meeting a set of multiple constraints. Core Idea 2 Number Operations

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. Mathematics of the task:

• Addition and subtraction facts • Finding missing addends • Identifying and using constraints, such as range and using a number only once

Based on teacher observation, this is what third graders know and are able to do:

• Finding numbers that added to 9 or 11 in puzzle 2 and 3 • Noticing the pattern of adding to 9 or adding to 11 • Addition and subtraction facts


• Using numbers more than once • Using numbers outside the range (0, 10, 11 … ) • Calculation errors • Persevering to find the totals to all the puzzle parts


• Using a list of possible numbers and marking them off when used • Writing number sentences to find the totals • Finding number patterns to help solve puzzle 2 and 3 • Being willing to make mistakes and correct them to meet the constraints of the task

(perseverance)


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The maximum score available for this task is 9 points. The minimum score for a level 3 response, meeting standards, is 5 points. Most students, 90%, were able to get some part correct for each puzzle. Many students, 82%, could get two or three lines from a puzzle with partial success on others. More than half the students, 55%, could meet all the demands of the task, including using numbers within the range only one time per puzzle and calculating correctly. 4% of the students scored no points on the task. All the students in the sample with this score attempted the task.


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Valerie’s Puzzles Points Understandings Misunderstandings

0 All the students in the sample with this score attempted the task.

Students didn’t know what total to aim for or used only the blank numbers to get the total.

3 Students could generally solve one line per puzzle.

In puzzle one, 52% of the errors were for repeating numbers. 29% of the errors were in calculations. 8% of the errors were for using numbers outside the range. In puzzle 2, 58% of the errors were for repeating numbers. 33% were for calculation errors. 12% were for leaving blanks.

5 Students could find two or more rows for one puzzle and something from the other puzzles.

In puzzle three, 50% of the errors were for repeating numbers. 17% were for calculation errors. 11% were for leaving blanks. 15% were for using numbers outside the range.

8 Students could persist in finding a nonroutine solution, using multiple constraints. Students were fluent in addition and subtraction facts.


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Implications for Instruction Students need to be comfortable with basic addition facts. They should be able to use addition and subtraction to find missing addends. Students need more opportunities to work with problems that do not present an immediate solution strategy, that require persistence in trying a variety of numbers to get the desired answer. Students need to find ways to identify constraints and test solutions against all the constraints. Ideas for Action Research – Probing into the mathematics of the task While on the surface this task is about using simple addition and subtraction facts, there is some very interesting mathematics in the solutions to puzzle 2 and 3. In puzzle 2, all the pairs of missing addends add to 9. In puzzle 3, all the pairs of missing addends add to 11. How could you pose a question to the class that would have students try to identify patterns to help them solve puzzle 2 and 3? A follow up question might be to ask them how this information is useful. How could they use the pattern to solve the puzzle? Another probing question would be to ask them why this is true? How is the puzzle set up to make this pattern possible? Is there a way to set up the first problem to make it follow a pattern? While these are challenging, provocative questions, students need to think about mathematics as looking for patterns and generalizations and trying to make sense of why things happen. These are important mathematical ideas. All students should be exposed to interesting mathematics, even if it’s just to get a sense of something to strive to understand. Reflecting on the Results for Third Grade as a Whole: Think about student work through the collection of tasks and the implications for instruction. What are some of the big misconceptions or difficulties that really hit home for you? If you were to describe one or two big ideas to take away and use for planning for next year, what would they be? What were some of the qualities that you saw in good work or strategies used by good students that you would like to help other students develop?


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Four areas stood out for the Collaborative. These included: 1. Keeping track of the whole – In Goldfish Bowls, students in general could find the total

number of fish. But they struggled to maintain that information in later parts of the task, changing the whole to 18, 12, or 6. This idea that the whole can be a group, is a very important foundational piece to making sense of fractions and fractional parts at later grade levels. In Birthday Decoration, the reverse is true. Some students identified the whole as groups of 5, which is true. But they didn’t understand the importance that all the elements in the group were not the same.

2. Using information in context – In gold fish bowls students who were successful often labeled their calculations to keep track of what they knew and what they found out. In Birthday Decorations students did correct calculations for proving why there weren’t enough decorations to make 9 pieces, but they couldn’t or didn’t interpret that information in the context of the problem. So knowing that 19 is not an even number or 26 is not a multiple of 3, is useful mathematical information. But being able to relate it to the context that for 9 pieces there needs to 9 groups of 2 stars and 9 groups of 3 balloons is the critical piece; being able to interpret the results in the real-world situation is the crucial step. In Math Test students could read information from the graph, but had trouble with meanings or labels. Is the question asking for information about frequency, number of students, or about categories, test scores. In a string of numbers students understand the idea of more than, but couldn’t relate concept in the context or representation of a bar graph.

3. Compare and contrast – In Making a Dollhouse, students had trouble identifying important attributes of shape and being able to make comparisons based on the geometric attributes. In trying to write justifications for having enough materials in Birthday Decorations, students didn’t compare their calculations back to what was needed.

4. Identifying and working with constraints – In Valerie’s Puzzle, students did not use or understand all the constraints of the task. They used numbers more than once and used numbers outside the given range. Students need to have frequent exposure to problems with longer reasoning chains and checking their results against the constraints.

How are Third Graders Succeeding on the Ramps? Examining the Ramp: Looking at Responses of the Early 4’s (35, 36) Goldfish Bowls:

• Part 5 o Identifying the total amount of fish o Showing that the total is not divisible by 5 o Developing a justification

Birthday Decorations • Part 4

o Understanding the makeup of a unit or piece (2 stars and 3 balloons) o Finding the total needed of each type of decoration o Comparing the total to the amount of each decoration on hand o Developing a justification

Making a Dollhouse • Part 3

o Compare and contrast a rectangle and a trapezoid o Identify important geometric attributes


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The Math Test • Part 3

o Combining categorical information in the form of numbers and frequency o Understanding and interpreting more than in a graphical representation

With a group of colleagues look at student work around 33 – 36 points. Use the papers provided or pick some from your own students.

How are students performing on the ramp? What things impressed you about their performance?

What are skills or ideas they still need to work on? Are students relying on previous arithmetic skills rather than moving up to more grade level strategies? What was missing that you would hope to see from students working at this level? How do you help students at this level step up their performance or see a standard to aim for in explaining their thinking? Are our expectations high enough to these students? How do we provide models to help these students see how their work can be improved or what they are striving for? Do you think errors were caused by lack of exposure to ideas or misconceptions? What would a student need to fix or correct their errors? What is missing to make it a top-notch response? What concerns you about their work? What strategies did you see that might be useful to show to the whole class?

Amber- Total Score 33


81

Bob – Total Score 33


82

Carla – Total Score 34


83

Dan – Total Score 35


84

Erin – Total Score 36

Documents

Balanced Assessment Test – Third Grade 2009 Core … Assessment Test – Third Grade 2009 Core Idea Task Number Operations Goldfish Bowls This task asks students to use addition,