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Goal 2 Geometry, Measurement, And Reasoning

Expectation 2.3 The student will apply concepts of measurement using tools and technology when appropriate.

Indicator 2.3.1 The student will use algebraic and/or geometric properties to measure indirectly.

Assessment Limits:

  • “Measure indirectly” means to use mathematical concepts such as congruence, similarity, and ratio and proportion to calculate measurements.
  • Similarity and congruence will be directly stated or implied (scale drawings, enlargements).
  • Items may require the student to make comparisons.
  • This indicator may incorporate measuring.
  • This indicator does not include right-triangle trigonometry.

Brief Constructed Response (BCR) Item - Released in 2000

The dimensions of the rectangular pool shown below are 40 yards by 20 yards. Fencing was ordered to enclose the deck. The ratio of the dimensions of the region that is fenced in to the dimensions of the pool is .

Complete the following in the answer box below:
  • What are the dimensions, in yards, of the region that is fenced in? Use mathematics to explain how you determined your answer. Use words, symbols, or both in your explanation.
     
  • How many yards of fence were purchased to enclose the deck?
     
  • Fencing costs $7.50 per yard. How much did it cost to purchase fencing to enclose the deck?
     

The following 6 Anchor Papers represent a range of score points and are used in conjunction with the rubrics to assess student responses.

Anchor Paper #1

image of student response

Score for Anchor Paper #1: Rubric Score 1

Annotation: This response indicates little to no application of a reasonable strategy to calculate the dimensions. The dimensions of the fenced-in region are incorrect. The student's explanation provides the correct proportion ( = ) that could be used to calculate the width of the fenced-in region. However, this strategy does not appear to have been employed, or was employed incorrectly, because the values presented as the dimensions (180 x 120) appear to be the perimeters of the deck and pool, respectively. The perimeter of 180 feet is correct, but by an inappropriate strategy for perimeter: "3x is the length of the fencing. X = 60, so the fence length = 180 feet." The explanation reveals serious flaws in reasoning. There is no answer for the cost of fencing of the fenced-in region. This response demonstrates a minimal understanding and analysis of the problem due to the serious flaws in reasoning.


Anchor Paper #2

image of student response

Score for Anchor Paper #2: Rubric Score 1

Annotation: The calculation of the dimensions of the fenced-in region indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The correct dimensions of the fenced-in region are clearly identified, although the student says "height," not "width." The student's explanation, while not fully developed, clearly presents a reasonable strategy ("…multiplying the dimensions of the pool by 3/2"). However, there is no evidence of an understanding of perimeter or of calculating cost. This response demonstrates a minimal understanding and analysis of the problem.


Anchor Paper #3

image of student response

Score for Anchor Paper #3: Rubric Score 2

Annotation: This student clearly presents and fully develops a strategy using proportion to calculate the dimensions of the fenced-in region. The correct dimensions of the fenced-in region are clearly identified ("…the dimensions are 60 yd by 30 yd"). The computations showing all the mathematical calculations provide an explanation to support the solution, although there is an error present. The student shows an incorrect width for the fenced-in region in the work, despite indicating correct dimensions earlier. The student mistakenly calculates area, rather than perimeter, to determine how many yards of fence were purchased. The cost to purchase fencing for the fenced-in region ($9000) is incorrect, but was correctly calculated using the value for area instead of perimeter (1200 x $7.50=$9000). The student has demonstrated a good understanding of applying a ratio to dimensions and of calculating cost, two of the main components of the problem. This response demonstrates a conceptual understanding and analysis of the problem.


Anchor Paper #4

image of student response

Score for Anchor Paper #4: Rubric Score 2

Annotation: This student clearly presents a reasonable strategy to calculate the dimensions of the fenced-in region, but an error in computation has led to the incorrect width of 20. (The area is calculated using the correct width, but this step is extraneous and does not add to or detract from the score.) The correct length but incorrect width are used to calculate the perimeter. The computations showing all the mathematical calculations provide an explanation to support the solution. The perimeter and cost of purchasing fencing for the fenced-in region are incorrect, but correctly calculated based on the student's correct and incorrect dimensions. One computation error in calculating the width was carried consistently throughout the response. The response demonstrates a conceptual understanding and analysis of the problem.


Anchor Paper #5

image of student response

Score for Anchor Paper #5: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The correct answers for dimensions, perimeter, and cost to purchase fencing for the fenced-in region are clearly identified. The student clearly presents and fully develops a strategy using proportion to calculate the dimensions of the fenced-in region. The computations showing all the mathematical calculations provide a complete explanation to support the solution. The student correctly calculates perimeter and correctly calculates the cost to purchase fencing to enclose the deck. The explanations for perimeter and cost neither detract nor add to the response since they were not required. This response demonstrates a complete understanding and analysis of the problem.


Anchor Paper #6

image of student response

Score for Anchor Paper #6: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The correct answers for dimensions, perimeter, and cost to purchase fencing for the fenced-in region are clearly identified. The student clearly presents a strategy using proportion to calculate the dimensions of the fenced-in region. The two proportions are set up to provide a complete explanation to support the solution. The student correctly calculates the perimeter and the cost to purchase fencing to enclose the deck. This response demonstrates a complete understanding and analysis of the problem.


Brief Constructed Response (BCR) Rubric
Print: Scoring Rubric (pdf)
Score 3

The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are essentially correct. The explanation and/or justification is logically sound, clearly presented, fully developed, supports the solution, and does not contain significant mathematical errors. The response demonstrates a complete understanding and analysis of the problem.

Score 2

The response indicates application of a reasonable strategy that may be incomplete or undeveloped. It may or may not lead to a correct solution. The representations are fundamentally correct. The explanation and/or justification supports the solution and is plausible, although it may not be well developed or complete. The response demonstrates a conceptual understanding and analysis of the problem.

Score 1

The response indicates little or no attempt to apply a reasonable strategy or applies an inappropriate strategy. It may or may not have the correct answer. The representations are incomplete or missing. The explanation and/or justification reveals serious flaws in reasoning. The explanation and/or justification may be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem.

Score 0

The response is completely incorrect or irrelevant. There may be no response, or the response may state, “I don't know.”

Explanation refers to the student using the language of mathematics to communicate how the student arrived at the solution.

Justification refers to the student using mathematical principles to support the reasoning used to solve the problem or to demonstrate that the solution is correct. This could include the appropriate definitions, postulates and theorems.

Essentially correct representations may contain a few minor errors such as missing labels, reversed axes, or scales that are not uniform.

Fundamentally correct representations may contain several minor errors such as missing labels, reversed axes, or scales that are not uniform.

Last Revised 8/16/00
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Resources for 2.3.1:
Skill Statements | PUBLIC RELEASE ITEMS | Lesson Plans |