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The task
Students were given the following instructions:

Determine the cost of redecorating your room. You must carpet the room, paint two coats, and use wallpaper in some way. Draw to scale, on graph paper, each wall, including windows and doors.

This planning project requires students to be skilled with one- and two-dimensional measurement and to compute quantities appropriately and accurately. Some formulation of the problem is necessary because students decide what information is needed when buying paint and when connecting that information to bedroom measurements. Students must also use proportional reasoning to make scale drawings of the redecorated space.

This project calls for skill with and understanding of two dimensional measurement as well as numbers and operations. It can also lend itself to consideration of volume or use of optimization, e.g., minimizing cost.

Work with three dimensional measurement could easily be included in or appended to this project. For example, because bedrooms normally include at least one bed, a dresser, and other items, volume and space considerations would arise.

Circumstances of performance
This sample of student work was produced under the following conditions:
- alone in a group
in class - as homework
with teacher feedback with peer feedback
timed opportunity for revision

This was an individual project completed primarily at home. Measuring instruments and calculators were allowed.

This work sample illustrates a standard-setting
performance for the following parts of the standards:
a Number and Operation Concepts: Consistently and accurately multiply rational numbers.
c Number and Operation Concepts: Consistently and accurately apply and convert rational numbers.
e Number and Operation Concepts: Interpret percent as part of 100.
f Number and Operation Concepts: Reason proportionally to solve problems involving equivalent fractions.
d Geometry and Measurement Concepts: Determine and understand length and area.
h Geometry and Measurement Concepts: Choose appropriate units of measure and convert with ease between like units.
j Geometry and Measurement Concepts: Reason proportionally in situations with similar figures.
a Problem Solving and Mathematical Reasoning: Formulation.
b Problem Solving and Mathematical Reasoning: Implementation.
a Mathematical Skills and Tools: Compute accurately with arithmetic operations on rational numbers.
d Mathematical Skills and Tools: Measure accurately.
h Mathematical Skills and Tools: Use pencil and paper and measuring devices to achieve solutions.
a Mathematical Communication: Use mathematical language and representations with appropriate accuracy.
b Mathematical Communication: Organize work, explain a solution orally and in writing, and use other techniques to make meaning clear to the audience.
d Putting Mathematics to Work: Management and planning.
d Writing: Produce a narrative procedure.
a Conventions: Demonstrate an understanding of the rules of the English language.

 

 


What the work shows

a Number and Operation Concepts: The student consistently and accurately adds, subtracts, multiplies, and divides rational numbers using appropriate methods…, i.e., rationals written as decimals…or mixed fractions….
The student realized that some computations would be redundant. That recognition spared her some calculation. The student also recognized that the opposite walls have equal sizes, thus avoiding some redundant computations. (This paragraph is an example of how work is often more wordy than it needs to be. A table would be a more effective means of communicating the areas of wall and of woodwork.)
When rewriting areas as decimals instead of mixed fractions, some of the numbers (e.g., “241.86 square feet”) are more precise than is necessary for this kind of project.

c Number and Operation Concepts: The student consistently and accurately applies and converts the different kinds and forms of rational numbers.

e Number and Operation Concepts: The student interprets percent as part of 100 and as a means of comparing quantities of…changing sizes.

f Number and Operation Concepts: The student…reasons proportionally to solve problems involving equivalent fractions [or] equal ratios….
The computations are not shown. However, it is apparent that, even though each unit of the graph paper represents one half-foot, the student made conversions (e.g., 4 in. = foot = of one half-foot) so that she could mark off the inches (fractions of feet) with accuracy and according to scale.

d Geometry and Measurement Concepts: The student determines and understands length, area…; uses units [and] square units…of measure correctly; computes areas of rectangles….
h Geometry and Measurement Concepts: The student chooses appropriate units of measure and converts with ease between like units, e.g., inches and miles, within a customary or metric system.
The conversions between square inches, square feet, and square yards are correct and significant.

j Geometry and Number Concepts: The student reasons proportionally with measurements…to make…scale drawings.

a Problem Solving and Mathematical Reasoning: Formulation. The student:
• formulates and solves…meaningful problems….
The student planned to redecorate her room and created a realistic scenario with a constraint of $700 and the added “twist” of being able to keep 50% of the remainder, which impacts the decisions to be made in ways that the $700 upper limit could not. Formulating the problem of redecoration, imposing constraints not required by the task, and determining the information needed in order to proceed provide the evidence of .
• figures out what additional information is needed.

b Problem Solving and Mathematical Reasoning: Implementation. The student….
• invokes problem solving strategies, such as…organizing information….


• solves for unknown or undecided quantities using…sound reasoning….
The decision to purchase one gallon and one quart of paint is conceived and explained well.

a Mathematical Skills and Tools: The student computes accurately with arithmetic operations on rational numbers.
d Mathematical Skills and Tools: The student measures length [and] area,…accurately.
h Mathematical Skills and Tools: The student uses…pencil and paper, measuring devices,…[and] calculators,…to achieve solutions.

a Mathematical Communication: The student uses mathematical language and representations with appropriate accuracy, including…diagrams.
b Mathematical Communication: The student organizes work, explains facets of a solution orally and in writing,…[and] labels drawings…to make meaning clear to the audience.
This diagram is very accurately drawn to scale and labeled well for clarity. (This is one of several scale drawings the student prepared.)
This summary of costs in an organized array is clearer than additional prose would have been. Such a display would also have been appropriate at other points in this report. The summary explains the total cost of the renovation, and the subsequent diagrams show the configuration of the redecorated room.

d Putting Mathematics to Work: The student conducts a management and planning project, in which the student:
• determines the needs of the event to be managed or planned, e.g., cost, supply, scheduling.


• notes any constraints that will affect the plan.

• determines a plan.

• uses concepts from any of Standards 1 to 4, depending on the nature of the project.

• considers the possibility of a more efficient solution.

• prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the plan.

d Writing: The student produces a narrative procedure that:
engages the reader by establishing a context, creating a persona, and otherwise developing reader interest;
provides a guide to action for a relatively complicated procedure in order to anticipate a reader’s needs; creates expectations through predictable structures, e.g., headings; and provides smooth transitions between steps;
makes use of appropriate writing strategies such as creating a visual hierarchy and using white space and graphics as appropriate;
includes relevant information;
excludes extraneous information;
anticipates problems, mistakes, and misunderstandings that might arise for the reader;
provides a sense of closure to the writing.

The work engages the reader by establishing a context: redecorating a room on a budget.

The established persona is maintained throughout the work.

The student anticipated the reader’s needs and used predictable structures to fulfill those needs, such as headings (“Problem,” “Solution”), a list of supplies needed, and a series of scale drawings of the room to be redecorated.

The logical transitions for the procedure give the writing a narrative quality. However, the reference to design features are optional.

The proper use of several graphics helps to summarize the narrative.

The examples and explanations are clearly presented.

By writing clearly and concisely, the student ensured that the reader would be able to follow even the somewhat complicated explanations.

The student closed the work appropriately, summarizing the narrative in a few brief sentences and filling in the last few pieces of information the reader might require. Appropriate sales tax, although not explicitly cited in the task, is correctly applied.

a Conventions, Grammar, and Usage of the English Language: The student demonstrates an understanding of the rules of the English language in written and oral work, and selects the structures and features of language appropriate to the purpose, audience, and context of the work. The student demonstrates control of:
• grammar;
• paragraph structure;
• punctuation;
• sentence construction;
• spelling;
• usage.

The student managed the conventions, grammar, and usage of English so that they aid rather than interfere with reading. In this case, management of conventions includes consistency in the use of numbers.