example can be used to refute a conjecture, it cannot be used to prove one is always true.] propositions or conjectures

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1 Task Model 1 Task Expectations: The student is asked to give an example that refutes a proposition or conjecture; or DOK Level 2 The student is asked to give an example that supports a proposition or conjecture. [Note: Use appropriate mathematical language in asking students for a single example. While a single Test example can be used to refute a conjecture, it cannot be used to prove one is always true.] propositions or conjectures Example Item 1: with specific Primary Target 3A (Content Domain F-IF), Secondary Target 1X (CCSS F-IF.2), Tertiary Target 3F examples Target A A baseball player hits a ball toward a 20-foot wall 400 feet away. The path of the ball is modeled by the function, ( ), where x is the distance of the ball, in feet. Claire claims the domain of the function is the set of all real numbers. Steve claims the domain of the function is the set of all positive numbers. Part A: Plot a point that demonstrates that Claire is incorrect. Select None if there is not point that demonstrates that Claire is incorrect. Part B: Plot a point that demonstrates that Steve is incorrect. Select None if there is not a point that demonstrates that Steve is incorrect. Part C: Select the person with the correct domain or choose Neither. Rubric: (2 points) The student is able to identify a point on both graphs that demonstrates Claire and Steve are both incorrect. (1 point) The student is able to interpret one of them as incorrect, but not both, and can place a point that demonstrates where the student s thinking is flawed. Response Type: Graphing and Hot Spot 11 Version 2.0

2 Task Model 1 Example Item 2: Primary Target 3A (Content Domain G-SRT), Secondary Target 1X (CCSS G-SRT.2), Tertiary Target 3F DOK Level 2 Test propositions or conjectures with specific examples Target A The radius of sphere Y is twice the radius of sphere X. A student claims that the volume of sphere Y must be exactly twice the volume of sphere X. Part A: Drag numbers into the boxes to create one example to evaluate the student s claim. Part B: Decide whether the student s claim is True, False, or whether it Cannot be determined. Select the correct option. Rubric: (1 point) The student supplies an example of two radii and missing numbers for the volumes that makes the conjecture false (e.g., radii are 1 and 2, missing numbers are 1 and 8; radii are 2 and 4, missing numbers are 8 and 64; radii are 1 and 3, missing numbers are 1 and 27; etc.) and responds with False in part B. Response Type: Drag and Drop 12 Version 2.0

3 Task Model 2 DOK Levels 3, 4 Construct, autonomously, chains of reasoning that will justify or refute conjectures Target B Task Expectations: The student is presented with a mathematical phenomenon and a conjecture. Student is asked to identify or construct reasoning that justifies or refutes a conjecture. Example Item 1: Primary Target 3B (Content Domain G-CO), Secondary Target 1X (CCSS G-CO.6), Tertiary Target 3C Jose and Tina are writing a program for a computer game. They need to move Triangle A to Triangle A. To move Triangle A to Triangle A, Jose thinks: a sequence of three transformations must be performed, and there is only one possible way to do it. Tina thinks: there are other sequences of transformations that will work, and it can be done using fewer than three transformations. Part A: Describe a sequence of three transformations that maps Triangle A onto Triangle A to support Jose s thinking. Part B: If possible, support Tina s thinking by describing a sequence of fewer than three transformations that maps Triangle A onto Triangle A'. Enter your response to Part A and Part B into the response box. Be sure to number the transformations in the order they should be performed (e.g., 1, 2, 3). Label each part of your response with Part A and Part B. Rubric: (2 points) The student is able to generate three correct translations to support Jose s thinking and one or two transformations to support Tina s thinking. (1 point) The student is able to generate correct transformation(s) to support either Jose s thinking or Tina s, but not both. 13 Version 2.0

4 Task Model 2 DOK Levels 3, 4 Construct, autonomously, chains of reasoning that will justify or refute conjectures Exemplar 5 : Part A 1. reflection across the x-axis 2. reflection across the y-axis 3. translation 2 units to the right Part B 1. rotation 180 about the origin OR 1. rotation 180 about the point (1, 0) 2. translation 2 units to the right OR Any response for parts A and B that results in a correct mapping of Triangle A to Triangle A that uses the given constraints. Response Type: Short Text (hand scored) Target B 5 Exemplars only represent possible solutions. Typically, many other solutions/responses may receive full credit. The full range of acceptable responses is determined during rangefinding and/or scoring validation. 14 Version 2.0

5 Task Model 2 DOK Levels 3, 4 Construct, autonomously, chains of reasoning that will justify or refute conjectures Example Item 2: Primary Target 3B (Content Domain G-CO), Secondary Target 1X (CCSS G-CO.C), Tertiary Target 3F (Source: Adapted from The Envelope, Smarter Mathematics Content Specifications, p. 70) A folded envelope is shown in the diagram as rectangle PQRS. Segments PR and SQ are diagonals of the envelope. Prove that when the rectangular envelope (PQRS) is unfolded, the shape obtained (ABCD) is a rhombus. Target B Rubric: (4 points) The student is able to justify each step in the proof, making sure to provide adequate reasoning to support the claim that all four sides of ABCD are congruent and opposite sides are parallel. Exemplar: A rhombus is a parallelogram with four congruent sides. As shown in the diagram, points A, B, C, and D all represent the same point where the diagonals of PQRS bisect each other. I ll name this point T to make the connection to points A, B, C, and D in the unfolded envelope. I know angles QTR and RTS are supplementary, because they form a straight line. Angles RTS and STP are supplementary as well, because they also form a straight line. Since point T is the same point as C and D, it follows that angles QCR and RDS supplementary, which means side BC is parallel to side AD. Also, since point T is the same point as A, it follows that and RDS and SAP are supplementary, which means side AB is parallel to side DC. This makes quadrilateral ABCD a parallelogram, because opposite sides BC AD and AB DC. Now we need to prove one pair of consecutive sides is congruent. Using the property that diagonals of a rectangle are congruent and bisect each other, PT is congruent to ST. PA and PB are both equal to PT. SA and SD are both equal to ST. PT = ST, then PA + PB = AB and SA + SD = AD or 2PT = AB and 2ST = AD. By substitution, AB = AD and the sides are also congruent. Given two adjacent sides of the parallelogram ABCD are congruent, and their opposite sides must also be congruent, proves that ABCD is a rhombus. This exemplar response represents only one possible solution. Other correct responses are possible. Response Type: Short Text (hand scored) 15 Version 2.0

6 Task Model 2 DOK Levels 3, 4 Construct, autonomously, chains of reasoning that will justify or refute conjectures Example Item 3: Primary Target 3B (Content Domain G-CO), Secondary Target 1X (CCSS G-CO.11), Tertiary Target 3F Given: and Prove: If a quadrilateral has a pair of sides that are both congruent and parallel, then the quadrilateral is a parallelogram. Statements Given Given Reasons Target B Complete the proof by providing reasons that justify each statement. ABCD is a parallelogram Rubric: (3 points) The student correctly completes the proof with reasons to support each statement. (2 points) The student is able to correctly complete four of the reasons for each statement. (1 point) The student is able to correctly complete three of the reasons for each statement. Exemplar: Alternative interior angles are congruent Reflexive property Side-Angle-Side (SAS) Corresponding parts of congruent triangles are congruent Both pairs of opposite sides are congruent; definition of a parallelogram Note: This exemplar response represents only one possible solution. Other correct responses are possible. Response Type: Fill-in Table (hand scored) 16 Version 2.0

7 Task Model 3 DOK Levels 2, 3 Task Expectations: The student is asked to connect a logical basis to its resulting conjecture. The student is presented with a mathematical or real-world scenario where an assumption is made in order to find the solution. Example Item: Primary Target 3D (Content Domain N-RN), Secondary Target 1B (CCSS N-RN.3), Tertiary Target 3F State logical assumptions being used Drag values for and into the boxes to make the paired statements true for both and Statements Example Target C If none of the values make both statements true, leave the boxes empty for that pair of statements. is a rational number is a rational number Rubric: (1 point) The student drags correct combinations of values that result in the indicated rational and irrational perimeters and areas of the rectangle. (e.g., see one possible solution at right). Other correct responses are possible. Response Type: Drag and Drop Statements is a rational number is a rational number Example 17 Version 2.0

8 Task Model 4 DOK Levels 2, 3 Use the technique of breaking an argument into cases Target D Task Expectations: The student is given different cases to consider and the student uses or shows reasoning to support the cases. Example Item: Primary Target 3D (Content Domain F-IF), Secondary Target 1L (CCSS F-IF.B), Tertiary Target 3F A student examines two graphs representing functions ( ) and ( ). The student notices that the graphs of the functions ( ) and ( ) both have a -intercept at the point (0, 5). The student makes the following claim: For any constant, the location of the -intercepts for the graphs of ( ) and ( ) is the same point. Show that this claim is true by giving the -value, in terms of, of the ordered pair (0,?) that represents the -intercept for the graphs of ( ) and ( ). Rubric: (1 point) The student is able to identify the correct -intercept value (e.g., 5 ). Response Type: Equation/Numeric 18 Version 2.0

9 Task Model 5 DOK Levels 2, 3 Distinguish correct reasoning from flawed reasoning Target E Task Expectations: The student is presented with valid or invalid reasoning. If the reasoning is flawed, the student will explain or correct the flaw. The student is asked to select the condition(s) for which an argument does or does not always apply. Two or more approaches or chains of reasoning are given and the student is asked to identify the correct method and justification OR identify the incorrect method/reasoning and the justification. Example Item: Primary Target 3E (Content Domain N-Q), Secondary Target 1C (CCSS N-Q.A), Tertiary Target Grade 7 1A Sherry wants to enlarge a photograph to 300% of its original size. The machine she is using can only make enlargements for the following percentages: 100%, 125%, 150%, and 200%. % Sequence of Enlargements 1: % 2: % 3: % Sherry thinks that she should enlarge the photograph by 100% and then by 200% to get a total enlargement of 300%. 4: % 5: % Decide if Sherry is correct. If Sherry is correct, drag the 100 and 200 from the palette into the first two answer spaces. If Sherry is incorrect, drag a sequence of enlargements she can use to get a total enlargement of 300%. Each enlargement percentage may be used more than once. If a sequence is not needed, leave the box blank. Rubric: The student drags the percentages into the appropriate boxes to show Sherry is incorrect (e.g., 200, 150). Other correct responses are possible. Response Type: Drag and Drop 19 Version 2.0

10 Task Model 6 Task Expectations: The student uses concrete referents to help justify or refute an argument. DOK Levels 2, 3 Base arguments on concrete referents such as objects, drawings, diagrams, and actions Target F Example Item: Primary Target 3E (Content Domain N-RN), Secondary Target 1B (CCSS N-RN.3), Tertiary Target 3C Ashley claims that when you multiply two different square roots together, the product is always rational. For example, = = 6 and = = 9. She also claims that that when you multiply two different cube roots together, the product is always irrational. For example, = and = Which statement correctly classifies Ashley s claims and provides appropriate reasoning? A. Ashley is correct because her examples support both claims. B. Ashley is correct about the product of square roots always being rational, but the product of cube roots can sometimes be rational. C. Ashley is incorrect about the product of square roots always being rational, but she is correct that the product of cube roots is always irrational. D. Ashley is incorrect because sometimes the product of square roots can be irrational, and sometimes the product of cube roots can be rational. Rubric: (1 point) The student identities the correct statement (e.g., D). Response Type: Multiple Choice, single correct response 20 Version 2.0

11 Task Model 7 Task Expectations: The student is asked to construct the conditions for which an argument does and does not apply. DOK Level 3, 4 Determine conditions under which an argument does and does not apply Target G Example Item 1: Primary Target 3G (Content Domain G-CO), Secondary Target 1X (CCSS G-CO.9), Tertiary Target 3C, Quaternary Target 3A A geometry student made this claim: If two lines are cut by a transversal, then alternate interior angles are congruent. Part A: Draw a diagram that shows congruent alternate interior angles or select None if there is not a situation to support the student s claim. Part B: Draw a diagram that shows alternate interior angles that are not congruent or select None if there is not a situation to support the student s claim. Rubric: (1 point) The student draws a transversal through two parallel lines to create congruent alternate interior angles in Part A and through two non-parallel lines for Part B. Response Type: Graphing and Hot Spot 21 Version 2.0

12 Task Model 7 Example Item 2: Primary Target 3G (Content Domain N-RN), Secondary Target 1X (CCSS N-RN.1), Tertiary Target 3A DOK Level 3, 4 Determine conditions under which an argument does and does not apply Target G An inequality is shown below. Part A: Determine the positive values of m for which the inequality is true. Enter your response as an inequality in the first response box. Part B: Determine the positive values of m for which the inequality is false. Enter your response as an inequality in the second response box. Rubric: (2 points) The student provides the values that make the given inequality true (e.g., ) and false (e.g., ). (1 point) The student errors in the use of the inequality signs (uses > instead of or instead of <) but otherwise has the correct range of values. OR The student is able to provide an example of where the statement is true (e.g., a positive value greater than or equal to 1) and a number between 0 and 1 that makes the statement false. Response Type: Equation/Numeric 22 Version 2.0

13 Task Model 7 Task Expectations: The student must determine whether a proposition or conjecture is True for all, True for some, or Not true for any and/or provide justification to support their conclusions. DOK Level 3, 4 Example Item 1: Primary Target 3G (Content Domain A-REI), Secondary Target 1X (CCSS A-REI.A), Tertiary Target 3A Determine conditions under which an argument does and does not apply Determine whether each statement is true for all values of x, true for some values of x, or not true for any value of x Target G Statement If = 9, then = True for all True for some Not true for any If, then = If ( ) =, then = ( ) Rubric: (1 point) The student correctly selects True for all, True for some, or Not true for any for each statement correctly (e.g., see below). Statement If = 9, then = True for all True for some Not true for any If, then = If ( ) =, then = (- ) Response Type: Matching Tables 23 Version 2.0

14 Task Model 7 Example Item 2: DOK Level 3, 4 Determine conditions under which an argument does and does not apply Primary Target 3G (Content Domain G-CO), Secondary Target 1X (CCSS G-CO.10), Tertiary Target 3A Consider all triangles that satisfy the given conditions: D is a point on the side of ABC = The picture shows one triangle that satisfies these conditions. Target G Determine whether each statement is true for all triangles that satisfy the given conditions true for some triangles that satisfy the given conditions not true for any triangle that satisfies the given conditions Statement True for all True for some Not true for any is perpendicular to. m ABD is less than m BAD = ½BC m ACB = ½(m ABD) Rubric: (1 point) Student correctly selects True for all, True for some, or Not true for any for each statement of equation. 24 Version 2.0

15 Statement True for all True for some Not true for any is perpendicular to. m ABD is less than m BAD = ½BC m ACB = ½(m ABD) Response Type: Matching Tables 25 Version 2.0