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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.

Extended Constructed Response (ECR) Item - Released in 2002

A baker is making heart-shaped cakes for a wedding. To make a cake look like a heart, he cuts a circular cake in half and attaches the semicircles to a square cake, as shown below.

The baker has to make enough cakes to serve 100 people. He knows that each person will eat approximately 9 square inches of cake.

Complete the following in the answer box below:
  • How many heart-shaped cakes should he make?
  • Use mathematics to explain how you determined your answer. Use words, symbols, or both in your explanation.

The following 9 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 demonstrates a minimal understanding and analysis of the problem. The student provides an incorrect answer of 11 cakes. The student claims to calculate area but actually calculates the perimeter of the square and the circumference of the circle. Using the incorrect area values of 48 for the square and 37 for the circle, the student then calculates the number of cakes needed. This response indicates a serious flaw in reasoning regarding area and little application of a reasonable strategy to solve the problem.


Anchor Paper #2

image of student response

Score for Anchor Paper #2: Rubric Score 1

Annotation: This response demonstrates a minimal understanding and analysis of the problem. The incorrect answer of 18 cakes is given. The student correctly finds the total number of square inches of cake required to feed 100 people, but has a serious misunderstanding of the area of the cake, thinking each cake yields 48 square inches. The student attempts to calculate, by dividing 900 by 48, the number of cakes needed. This response contains a serious flaw in reasoning regarding area and indicates little application of a reasonable strategy to solve the problem.


Anchor Paper #3

image of student response

Score for Anchor Paper #3: Rubric Score 2

Annotation: This response demonstrates a conceptual understanding and analysis of the problem. The incorrect answer of 6.25 cakes is given. The explanation reveals that the student finds the number of people the square part of the cake will feed, then calculates how many cakes the baker needed to make. The entire area is not considered since no attention is given to the circular area. This response indicates an incomplete application of a reasonable strategy to solve the problem.


Anchor Paper #4

image of student response

Score for Anchor Paper #4: Rubric Score 2

Annotation: This response demonstrates a conceptual understanding and analysis of the problem. The incorrect answer of 11.11 cakes needed is given. Work reveals that the student correctly calculates the total area of one cake. The student calculates 28.566 which is the number of people one cake will serve — not the number of square inches of cake per person. The incorrect answer is the result of using the 28.566 incorrectly. This response indicates a conceptual understanding of area, but an incomplete application of a reasonable strategy to solve the problem.


Anchor Paper #5

image of student response

Score for Anchor Paper #5: Rubric Score 3

Annotation: This response demonstrates a clear understanding and analysis of the problem. The student gives an incorrect answer of 5 cakes; however, the provided work reveals that a reasonable strategy is applied. The student attempts to calculate the area of the cake, considering both the square and circular areas, but makes an error, missing one of the semicircular areas. Using the incorrect area to complete the problem results in an incorrect answer.


Anchor Paper #6

image of student response

Score for Anchor Paper #6: Rubric Score 3

Annotation: This response demonstrates a clear understanding and analysis of the problem. The student gives an incorrect answer of 5 cakes, but the explanation reveals that a reasonable strategy is applied. The student considers both the square and circular areas of the cake. However, in calculating the area of one cake the student makes an error, mistakenly using the circumference formula instead of the circle area formula. Using the incorrect area to complete the problem results in an incorrect answer.


Anchor Paper #7

image of student response

Score for Anchor Paper #7: Rubric Score 3

Annotation: This response demonstrates a clear understanding and analysis of the problem. An incorrect answer of 2 cakes is given, but the explanation reveals that a reasonable strategy is applied. The student considers both the square and circular areas of the cake. However, the student makes an error in the circular area, using diameter (d) instead of radius (r) in the correct circle area formula. Using the incorrect area to complete the problem results in an incorrect answer.


Anchor Paper #8

image of student response

Score for Anchor Paper #8: Rubric Score 4

Annotation: This response demonstrates a complete understanding and analysis of the problem. The student correctly answers that the baker needs 4 cakes, and the answer is supported by a fully developed explanation of a correct strategy. In the explanation, the student mistakenly states, "finding the volume of the square then adding the volume of the circle," but the values and math formulas clearly show that area (not volume) is found; therefore, this is a minor error.


Anchor Paper #9

image of student response
image of student response

Score for Anchor Paper #9: Rubric Score 4

Annotation: This response demonstrates a complete understanding and analysis of the problem. The student correctly answers that the baker needs 4 cakes, and that answer is supported by a fully developed explanation of a correct strategy.


Extended Constructed Response (ECR) Rubric
Print: Scoring Rubric (pdf)
Score 4

The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are 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 3

The response indicates application of a reasonable strategy that may or may not lead to a correct solution. The representations are essentially correct. The explanation and/or justification is generally well developed, feasible, and supports the solution. The response demonstrates a clear understanding and analysis of the problem.

Score 2

The response indicates an incomplete application of a reasonable strategy that 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 application of a reasonable 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 |