Name Date

  1. a.What is the decimal expansion of the number ? Is the number rational or irrational? Explain.
  1. What is the decimal expansion of the number ? Is the number rational or irrational? Explain.
  1. a.Write as a fraction.
  1. Write as a fraction.
  1. Brandon stated thatand are equivalent. Do you agree? Explain why or why not.
  1. Between which two positive integers doeslie?
  1. For what integer is closest to ? Explain.
  1. Identify each of the following numbers as rational or irrational. If the number is irrational, explain how you know.
  1. Order the numbers in parts (a)–(g) from least to greatest, and place on a number line.
  1. Circle the greater number in each of the pairs (a)–(e) below.
  1. Which is greater? or
  1. Which is greater? or
  1. Which is greater? or
  1. Which is greater? or
  1. Which is greater? or
  1. Put the numbers ,, and in order from least to greatest. Explain how you know which order to put them in.

  1. Between which two labeled points on the number line would be located?
  1. Explain how you know where to place on the number line.
  1. How could you improve the accuracy of your estimate?
  1. Determine the position solution for each of the following equations.
  1. The cube shown has a volume of cm3.
  1. Write an equation that could be used to determine the length, of one side.
  1. Solve the equation, and explain how you solved it.

A Progression Toward Mastery
Assessment
Task Item / STEP 1
Missing or incorrect answer and little evidence of reasoning or application of mathematics to solve the problem. / STEP 2
Missing or incorrect answer but evidence of some reasoning or application of mathematics to solve the problem. / STEP 3
A correct answer with some evidence of reasoning or application of mathematics to solve the problem, ORan incorrect answer with substantial evidence of solid reasoning or application of mathematics to solve the problem. / STEP 4
A correct answer supported by substantial evidence of solid reasoning or application of mathematics to solve the problem.
1 / a–b
8.NS.A.1 / Student makes little or no attempt to respond to either part of the problem.
OR
Student answers both parts incorrectly. / Student identifies one or both numbers as rational. Student may not write the decimal expansion for the numbers and does not reference the decimal expansions of the numbers in his or her explanation. / Student identifies both numbers as rational. Student correctly writes the decimal expansion for each number. Student may not reference the decimal expansion in his or her explanation but uses another explanation (e.g., the numbers are quotients of integers). / Student identifies both numbers as rational. Student correctly writes the decimal expansion of as , or , and as …, or Student explains that the numbers are rational by stating that every rational number has a decimal expansion that repeats eventually. Student references the decimal expansion of with the repeating decimal of zero and the decimal expansion of with the repeating decimal of.
2 / a–b
8.NS.A.1 / Student does not attempt problem or writes answers that are incorrect for both parts. / Student isable to write one of the parts (a)–(b) correctly as a fraction.
OR
Student answers both parts incorrectly but shows some evidence of understanding how to convert an infinite, repeating decimal to a fraction. / Student is able to write both parts (a)–(b) correctly as a fraction.
OR
Student writes one part correctly but makes computational errors leading to an incorrect answer for the other part. / Student correctly writesbothparts (a)–(b) as fractions. Part (a) is written as (or equivalent), and part (b) is written as (or equivalent).
c
8.NS.A.1 / Student agrees with Brandon or writes an explanation unrelated to the problem. / Student does not agree with Brandon. Student writes a weak explanation defending his position. / Student does not agree with Brandon. Student writes an explanation that shows why the equivalence was incorrect reasoning that does not equal orthat does not equal but fails to include both explanations. / Student does not agree with Brandon. Student writes an explanation that shows does not equal and that does not equal
d–e
8.NS.A.2 / Student does not attempt problem or writes answers that areincorrect for both parts (d)–(e). / Student isable to answer at least one of the parts (d)–(e) correctly. Student may or may not provide a weak justification for answer selection. / Student is able to answer both parts (d)–(e) correctly. Student may or may not provide a justification for answer selection. Explanation includes some evidence of mathematical reasoning. / Student correctly answersboth parts (d)–(e); for part (d) is between positive integers and , for part
(e) .
Student provides an explanation that included solid reasoning related to rational approximation.
3 / a–f
8.NS.A.1
8.EE.A.2 / Student does not attempt problem or writes correct answers for one or two parts of (a)–(g). / Student correctly identifies three or four parts of (a)–(g) as rational or irrational. Student may or may not provide a weak explanation for those numbers that are irrational. / Student correctly identifies five or six parts of (a)–(g) correctly as rational or irrational. Student may provide an explanation for those numbers that are irrational but doesnot refer to their decimal expansion or any other mathematical reason. / Student correctly identifies all seven parts of (a)–(g); (a) irrational, (b) rational, (c) rational, (d) rational, (e) irrational, (f) rational, and (g) irrational. Student explains parts (a), (e), and (g) as irrational by referring to their decimal expansion or the fact that the radicand was not a perfect square.
h
8.NS.A.2 / Student correctly placeszero to two numbers correctly on the number line. / Student correctly places three or four of the numbers on the number line. / Student correctly placesfive of the six numbers on the number line. / Student correctly places all six numbers on the number line. (Correct answers noted in red below.)
4 / a–e
8.NS.A.2
8.EE.A.2 / Student correctly identifies the larger number zero to onetimes in parts (a)–(e). / Student correctly identifies the larger number two to threetimes in parts (a)–(e). / Student correctly identifies the larger number fourtimes in parts (a)–(e). / Student correctly identifies the larger number in all of parts (a)–(e); (a) , (b) , (c) numbers are equal, (d) , and (e) .
f
8.NS.A.2
8.EE.A.2 / Student does not attempt the problem or responds incorrectly. Student does not provide an explanation. / Student may correctly order the numbers from greatest to least. Student may or may not provide a weak explanation for how he or she put the numbers in order. / Student correctly orders the numbers from least to greatest. Student provides a weak explanation for how he or she put the numbers in order. / Student correctly orders the numbers from least to greatest as Explanation includes correct mathematical vocabulary (e.g., square root, cube root, between perfect squares).
5 / a–c
8.NS.A.2 / Student makes little or no attempt to do the problem.
OR
Student may or may not place correctly on the number line for part (a). For parts (b)–(c), student does not provide an explanation. / Student may or may not have placed correctly on the number line for part (a). For parts (b)–(c),student may or may not provide a weak explanation for how the number was placed. Student may or may not provide a weak explanation for how to improve accuracy of approximation. / Student place correctly on the number line for part (a). For parts (b)–(c),student may provide a weak explanation for how the number was placed. Student may provide a weak explanation for how to improve accuracy of approximation. Student references the method of rational approximation. / Student correctly places between and on the number line for part (a). For parts (b)–(c),student explains the method of rational approximation to locate the approximate position of on the number line. Student explains how to continue the rational approximation to include increasing smaller intervals to improve the accuracy of the estimate.
6 / a–b
8.EE.A.2 / Student makes little or no attempt to solve either equation or writes incorrect answers for both. / Student writes the correct answer for one of the equations but does not write the answer in the appropriate form (i.e., or instead of or ). / Student solves at least one of the equations correctly or solves both correctly but does not write the answer in the appropriate form (i.e., and instead of and ). / Student solves both equations correctly for parts (a)–(b); (a) and (b)
c–d
8.EE.A.2 / Student makes little or no attempt to solve either equation or writes incorrect answers for both. / Student may solve one equation correctly.
OR
Student uses properties of rational numbers to transform the equations but cannot determine the correct value of or makes computational errors leading to incorrect solutions for / Student solves one of the equations correctly but makes computational errors leading to an incorrect answer for the other equation. / Student solves both equations correctly for parts (c)–(d);(c) , and (d) .
e
8.EE.A.2 / Student makes little or no attempt to solve the problem. / Student may state that the length of one side of the cube is cm but does not write an equation nor solve it. / Student states that the length of one side of the cube iscm. Student correctly writes and solves an equation but provides a weak explanation for solving it. / Student states that the length of one side of the cube iscm. Student correctly writes and solvesthe equation. Student provides a clear and complete explanation for solving the equation that includes some reference to cube roots and why