Toolkit98
This problem is one of several developed by the New Standards Project at the University of Pittsburgh. Attached are the complete instructions for the problem, the scoring guide, and five student responses representing different levels of performance.
Imagine that your school principal asks you to do a special job and gives you these written directions:
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Your class will be getting a 30 gallon aquarium. The class will have $25.00 to spend on fish. You will plan which fish to buy. Use the Choosing Fish for Your Aquarium brochure to help you choose the fish. The brochure tells you things you must know about the size of the fish, how much they cost and their special needs.
Choose as many different kinds of fish as you can. Then write a letter to me explaining which fish you choose. In your letter,
1. tell me how many of each kind of fish to buy
2. give the reasons you chose those fish
3. show that you are not overspending and that the fish will not be too crowded in the aquarium. |
Choosing Fish for Your Aquarium
Planning Ahead
Use the information in this brochure to help you choose fish that will be happy and healthy in your aquarium. To choose your fish, you must know about the size of the fish, their cost, and their special needs.
Size of Fish
To be healthy, fish need enough room to swim and move around. A good rule is to have one inch of fish for each gallon of water in your aquarium. This means that in a ten gallon aquarium, the lengths of all your fish added up can be ten inches at the most.
EXAMPLE:
With a ten gallon aquarium,
here are a few of your choices:
_________________
10 INCHES
one ten-inch long fish, or
_________________
10 INCHES
a seven-inch long fish and a three-inch long fish or
__________________
10 INCHES
five fish if each is only two inches long
Cost of the Fish
Some fish cost as little as one dollar, others cost much more. The prices of each kind of fish are listed in the chart.
Special Needs
Use the chart to learn about the special needs of each kind of fish. Some fish need to live together in schools—a group of four or more of the same kind of fish—while others live in pairs or alone. A few kinds of fish have other special needs, which are listed in the chart.
| Picture* | Name | Cost | Length in Inches | Color | Special Needs, Facts |
|---|---|---|---|---|---|
 |
Zebra Danlo | $1 | 1 ½ inches | blue with gold lines | lives in schools; gets along with other kinds of fish |
 |
Marbled Hatchetfish | $1 | 2 inches | yellow | lives in schools; can leap 3-5 yards |
 |
Guppy | 2 for $3 | 2 inches | red, blue, and green | lives in schools |
 |
Red-tailed Black Shark | $5 | 4 ½ inches | black with red tail | fights with other sharks, but gets along with other kinds of fish |
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Cardinal Tetra | $5 | 1 ½ inches | red and green | lives in schools |
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Blind Cave Fish | $2 | 3 inches | silvery rose | lives in schools; uses its sense of smell and vibration to find food |
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Ramirez' Dwarf Cichlid | $5 | 2 inches | rainbow | lives in pairs; rarely lives longer than 2½ years; gets along with other fish |
 |
Velvet Cichlid | $5 | 12 ½ inches | olive with stripes | can be trained to take food from the hand and can be petted; must be kept only with other ciclids |
Student Reflections, Ideas
You can help us make these learning activities even better. Think about each of the following questions, and write to us what you honestly think. Be as clear as you can (you might want to give us examples of what you mean).
What did you enjoy about the task?
What did you not like about the task?
How is this task like other activities you do in your class? How is it different?
Purpose
In the Aquarium task, students use their knowledge of mathematics to solve a real world problem. Students use logical and numerical reasoning about money, measurement and realistic conditions to decide how best to stock an aquarium within the constraints of the situation.
This task calls for logical and numerical reasoning and justification of that reasoning. These are contained in Standard 1 (Mathematics as Problem Solving), Standard 3 (Mathematics as Reasoning), and Standard 4 (Mathematical Connections) of the NCTM Curriculum Standards for Grades K-4. Students apply understanding of measurement as described in Standard 10 (Measurement).
This task is designed to assess students' mathematical thinking and their use of information contained in a brochure which features a chart, not their ability to read the brochure and chart independently.
On-Demand Performance
"On-demand" performance refers to performance "demanded" at a specific time and place under controlled circumstances. On-demand has the 'ready, set, go' character of a contest rather than a realistic workplace character of 'start as soon as you can, don't leave until you are finished, let me know if you need any help, I'll come by in a while to look at your work and answer any questions, if you get stuck check with your colleagues, etc.'. Tests are often on-demand while projects, studying, writing assignments , and homework are more realistic performances. On-demand responses are drafts, not final edited works. Even the best responses will typically need some editing. When scoring, focus on what has been accomplished in the response rather than what may be omitted, or may need editing.
Using the Rubric
The rubric draws upon the NCTM Curriculum Standards, 1989; scorers should be familiar with these standards and have access to them during scoring. The rubric also draws upon the professionalism of teachers in two specific ways: individual experience and collegial participation in a professional consensus. Scorers act for their profession based on consensual standards.
The overarching issue for scoring is always how well the response accomplishes the prompted purpose. The 'prompted purpose' is the purpose as understood by the students. Look at the prompt from the standpoint of the student. The rubric is written to allow for variety in the ways students approach and communicate the accomplishment of the purpose. Sometimes a student will respond with an unexpected but plausible interpretation of the prompted purpose; in such cases, judge the work on its own terms consistent with the rubric. Do not mark down from a checklist of expected response features; do not mark up from a list of anticipated ingredients; rather, judge the response as a whole: how effective is it at accomplishing the prompted purpose
Within the framework of accomplishing the task's purpose, the rubric asks the scorer to appraise the response according to how well it exemplifies the mathematical power described in the first four NCTM Standards: problem solving, communication, mathematical reasoning, and connections. In addition, mathematical ideas from those described in Standards 5 through 13 for grades K-4 will be explicit and implicit in the task. Successful responses are informed by these ideas. Finally, the scorer must judge how effectively the response uses mathematical and problem solving tools and techniques as described in the Standards for grades K-4.
The rubric unfolds from the description of the 5 response. To understand the rubric, start with the 5 criterion which includes all the elements and perspectives to be evaluated in the responses. The higher and lower levels differ from the 5 level in the quality of work. While the 5 criterion is constructed with "and", the others are constructed with "or". There are many ways to exceed a 5, or to fall short, but all 5 responses accomplish the prompted purpose with high quality work.
6 The student's response accomplishes everything a "5" response accomplishes, but with deeper mathematical insight or eloquence, or with exceptionally well-organized reasoning, illuminating connections, powerful strategy, or other demonstration of exceptional mathematical power pertaining to the task. There are many ways a response might excel; for example, it might:
5 The student's reasoning shows understanding of the two major task constraints: money and crowding, and how the selection of fish fits these constraints. The response effectively communicates the student's reasons for choosing specific fish and how the whole selection makes sense. These reasons include some reference to the additional prompted goals of variety and accounting for special needs, and the selection accomplishes these goals reasonably well. The response reflects an understanding of combining lengths to obtain reasonable results; a student might reasonably decide to exceed the crowding constraint by a slight amount with reasons. Information from the chart is interpreted usefully; although there may be minor misinterpretations, these do not invalidate the overall selection and rationale; for example, a 5 response could include a misunderstanding of "schools," of who red tailed sharks fight with, or of whether guppies are "2/pair or 2 each." The numerical operations are used effectively to obtain reasonable results; minor computational errors can only be accepted in a 5 response if the overall results are good.
4 The student's reasoning shows an understanding of the task constraints: money and crowding. In choosing fish, the student successfully attends to at least one of these constraints. The response may lack a clear, complete explanation or demonstration of how choices were made or why they make sense; or, there is no evidence of combining lengths to evaluate crowding; or, computational errors or misinterpretation of the chart lead to results which are unreasonable; or, the response does not show at least 3 kinds of fish selected.
3 The student's reasoning shows understanding that the task has constraints of money and crowding, but in choosing fish the student fails to attend successfully to one or both of these constraints. The response communicates choices, but may explain little about how they were made or why choices make sense; or, major computational errors yield nonsensical results; or, misconceptions about the chart or the constraints invalidates the reasoning.
2 The response shows a purposeful effort to choose fish and communicate but lacks an explanation of choices; the reasoning may show understanding of one or both of the task constraints but fails to attend to either constraint. There may be little evidence of using the chart, effective understanding or use of the idea of combining lengths, or appropriate computations.
1 Little evidence of communication or attempt to address the task.
0 No response.
LEP Response includes language other than English, route to LEP scoring table
Computation
Within a well-reasoned response, a minor computational error (see 730339) or a misinterpretation of some aspect of the chart might not necessarily exclude the response from a "5" or "6" score. Look at the response as a whole: give greater weight to the student's mathematical understanding of the task in its entirety than to flawless computation. If a student's response exceeds the crowding or money constraint by a slight amount and the crowding or money constraint by a slight amount and the student acknowledges this discrepancy, this need not prohibit a score of "5" or "6" (see 730331).
The Chart
Some otherwise quite powerful responses reflected a misunderstanding of some element contained in the chart. The most common misinterpretations:
If a students used the chart effectively except for misunderstanding one aspect of it, be flexible and evaluate the work as a whole. If it is powerful as a whole, it can score a 5.
"4" Response
"3" Response
| Serial Number | Score | Serial Number | Score |
|---|---|---|---|
| 730258 | 6 | 730245 | 3 |
| 730331 | 5 | 730458 | 3 |
| 730484 | 5 | 730457 | 3 |
| 730306 | 5 | 730451 | 3 |
| 730339 | 5 | 731068 | 2 |
| 731056 | 5 | 730445 | 2 |
| 730431 | 4 | 730256 | 2 |
| 730262 | q | 730466 | 2 |
| 730875 | 4 | 731001 | 2 |
| 730338 | 4 | 730942 | 1 |
| 730433 | 4 | 730244 | 1 |
| 730871 | 4 | 730940 | 1 |
| 730257 | 3 | 730993 | 0 |
730258
Dear Principal
I chose these fish because they get along with other fish. Also because when I imagine these fish I see buitiful ones. Thats why I chose these fish. But now I will tell you the fish I have chosen. I have chosen the guppy, the Ramirz Dwarf cichlid. And the Blind Cave fish. I also chose two more fishes and they are the Zebra Danio and also the marbled Hatch fish. Thats the fish I chose with the 30.00. Now that I've shown you the kind of fish I bought, Now then I'm going to tell you how I no these fish won't be crowded. See I've got two fish apiece of a kind and I put all the fishes I bought and put all there lenghs to gether. And it added up too 21. And the tank is 30. So thats how I know that they will not be so crowded to gether. So now that I showed you everything I did I will end this letter by saying buy.
Sincerly
Your Student
730306
Dear Mrs. Burk,
For the fish tank in my class, we have decided on getting 12 fish. We will get 5 Blind Cave fish because they don't take up to much space. They will only take up 15 gallons of water and ten dollars. Next we will get a pair of Ramirez Dwarf Cichlid because they get along with other fish. They will only take up four gallons of water and 10 dollars. Then last we will get five Zebra Danios because they too get along with other fish. They will take up to 6½ gallons and five dollars. So that leaves zero dollars left and 5½ gallons of water for the fish to swim around in. I think this is a good plan because the tank won't be crowed.
May 15, 1992
Fish Cost Gallons 1. Blind Cave Fish get 5 $10 15 gallons 2. Ramirez' Dwarf Cichlid get 2 $10 4 gallons 3. Zebra Danio get 5 $5 6½ gallons $0 left 5½ gallons left
730262
Dear Principal,
I have chose the fish for the 30 gallon aquarium. The first fish I bougt is a Red-Tailed Black Shark it will cost $5.00. I chose this fish because I thought this fish was very neat and I the class would enjoy watching it. This fish is 4½ inches. Next I bought 5 Zebra Denio. This will cost $5.00. I chose this kind of fish because it is pretty and gets along with other fish. It is 1½ inches. Last I will buy 2 Ramirez/ Dwarf Cichlid. This will cost $10.00. I chose this kind of fish because it is rainbow colored and gets along with other fish. It is 2 inches. It should not be to crowded in the aquarium.
730457
Kingman, AZ
Dear Principal,
I would like 8 zebra danio, 4 marbled hatchetfish, and 4 guppies. I chose the zebra danio because it gets along with other fish. I chose the marbled hatchetfish because they can leap 3-5 yards. I chose guppies because their very pretty. I added up how much money it would be it was $18.00 and I added up how many inches all the fish would be together they would be 24½ inches together.
Your friend,
5/14/92
730244
2 of them
2 of them
Dear Principal, I brought this and I think th |
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