Just as good instruction accommodates differences in the ways learners construct knowledge, good assessments accommodate differences in the ways that students think about and display mathematical understanding.

Some ways of accommodating differences among learners include permitting multiple entry and exit points in an assessment and allowing students to respond in ways that reflect different levels of mathematical knowledge or sophistication. These design characteristics are critical to assessment equity just as they are critical to the content and learning principles.

Consider the two apparently parallel problems illustrated below and on the following page.5 Close inspection indicates that the first problem may be less accessible to many students. In the first mosaic there is no indication of how the colored tiles are arranged to form the picture. Therefore, to find how many of the different colored tiles would be needed for the enlarged picture, one must already known the general proposition that doubling the length of a figure quadruples its area. On the other hand, the figure in the second problem, with its explicit specification of the color of each part, opens alternative avenues for approaching the task. For example, the student might draw a sketch

Pythagoras—A Mosaic Problem

Your school's math club has designed a fancy mosaic in honor of Pythagoras, whose work united algebra and geometry. The mosaic includes a picture of the Pythagorean Theorem and its algebraic statement, as indicated below. The figure appears inside a square.

You have constructed a 2' × 2' scale model of the mosaic, which used 120 red mosaic pieces, 200 blue mosaic pieces, and 150 yellow pieces

  1. How many red, blue, and yellow pieces would be required for a 4' × 4' mosaic? Explain.

  2. How many red, blue, and yellow pieces would be required for a 6' × 6' mosaic? Explain.

  3. The school's principal likes the model of the mosaic so much that she's willing to have you build a big version that will go on permanent display in the auditorium. The only thing—she hasn't yet decided how big the mosaic should be. She wants the math club to write her a letter explaining how many tiles of each color she'll need, as soon as she makes up her mind how big the side of the square should be. Write the letter, providing an explanation of how the number of tiles can be calculated, and also why your answer is correct.

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