Exemplar for Internal Achievement Standard. Mathematics and Statistics Level 3

Transcription

1 Exemplar for Internal Achievement Standard Mathematics and Statistics Level 3 This exemplar supports assessment against: Achievement Standard Apply systems of simultaneous equations in solving problems An annotated exemplar is an extract of student evidence, with a commentary, to explain key aspects of the standard. It assists teachers to make assessment judgements at the grade boundaries. New Zealand Qualifications Authority To support internal assessment

2 Grade Boundary: Low Excellence 1. For Excellence, the student needs to apply systems of simultaneous equations, using extended abstract thinking, in solving problems. This involves one or more of: devising a strategy to investigate or solve a problem, identifying relevant concepts in context, developing a chain of logical reasoning, or proof, or forming a generalisation, and also using correct mathematical statements or communicating mathematical insight. This evidence is a student s response to the TKI task Elaine s Equations. This student has correctly solved the system of simultaneous equations for all three methods which create the third equation. This student has also interpreted these solutions geometrically (1) (2) (3). This student has identified relevant concepts in context by linking the way the third equation is created to parallel and identical planes, and explaining the general case for the third situation (4). Correct mathematical statements are used throughout the response. For a more secure Excellence, the student could develop the response to the general case for the second situation by forming a general solution for x, y and x in terms of a parameter k.

3 so Method 1 The three equations are 3x 2y z 7 These equations represent planes in 3D. Using my calculator the answers are x=5, y=2 and z =-1.5.This means that there is a unique solution and the three planes that these equations represent intersect in a unique point (5,2,-1.5). Method 2 The three equations are 6x Because I get the third equation by multiplying the first one by three they are really the same equation. This means that, in 3D we are looking for points that lie on planes one and two only. Because these two are not parallel they will intersect in a line and any points on this line will satisfy all three equations. The equations are consistent and there are multiple solutions. Method 3 The three equations are 6 In this case the first and third equations represent parallel planes because the x,y and z numbers are all equal but the constants (1 and 6) are not. In 3D we are looking for the points that lie on two parallel planes and a third one that is not parallel. They will look like this if you look at them sideways on. There are obviously no points that lie on all three planes. The planes are inconsistent and there are no solutions. This is a specific example of the general case. If two of the equations have the x,y and z numbers in proportion but the constants are not, then the two planes will be parallel, so the equations are inconsistent and there are no solutions

4 Grade Boundary: High Merit 2. For Merit, the student needs to apply systems of simultaneous equations, using relational thinking, in solving problems. This involves one or more of: selecting and carrying out a logical sequence of steps, connecting different concepts or representations, demonstrating understanding of concepts, or forming and using a model, and also relating findings to a context or communicating thinking using appropriate mathematical statements. This evidence is a student s response to the TKI task Elaine s Equations. This student has connected different concepts and representations by solving and geometrically interpreting the solutions for the methods 1 (1) and 3 (2). The student has correctly recognised the dependent equations in method 2, but incorrectly interpreted these geometrically as a triangular shape (3). The student has used appropriate mathematical statements in the response. To reach Excellence, the student could interpret the solutions for method 2 correctly, and make a general statement about the solution set of each system of equations.

5 1. original x3 x13 3x 2y z 7 x2 6x 6x 4y 6z 4 (-) 3y 0z 1 13y 52z 104 3y 0z 1 62z 93 z.5 y 4(.5) y 6 y 2 (2) 2(.5) x 5 ( 5,2, 1.5) 2. x3 6x 6x 6x lines of solutions dependent (-) 0 5 Inconsistent Unique solution consistent For the first set of equations, it is possible to solve resulting in a unique solution that is consistent. The planes will intersect at the point (5,2,-1.5) For the second set of equations, attempting to solve will result in two 6x equations (0=0). This will resemble planes always being cut by two planes making a triangular shape. This is dependent. For the third set of equations, attempting to solve will result in two equations with the same coefficients but different constants (0=-5). This is inconsistent and will resemble two parallel planes both being cut by a third plane.

6 Grade Boundary: Low Merit 3. For Merit, the student needs to apply systems of simultaneous equations, using relational thinking, in solving problems. This involves one or more of: selecting and carrying out a logical sequence of steps, connecting different concepts or representations, demonstrating understanding of concepts, or forming and using a model, and also relating findings to a context, or communicating thinking using appropriate mathematical statements. This evidence is a student s response to the TKI task Elaine s Equations. This student has connected different concepts and representations by solving and geometrically interpreting the solutions for the methods 2 (1) and 3 (2), but not completely solved the system of equations for method 1 (3). The student has used appropriate mathematical statements in the response, however there is an error in that the student finds y = -2 rather than the correct y = 2. For a more secure Merit, the student could correctly solve and geometrically interpret the solution of the system of equations for method 1.

7 a a. 3x 2y z 7 3b. 6x 6x 3c x2 x x 2y 8z x 6y 4z y 8z x 2y z x 5y x 4y x 4y 3 x = 5 y = -2 inconsistent b. 1. x x 0 0 Two planes are the same, 1 and 3, which means that there are infinite solutions along the line of intersection with plane 2. The planes are dependent. c planes are parallel to each other 1 and 3. This causes the equations to be inconsistent so there is no solution.

8 Grade Boundary: High Achieved 4. For Achieved, the student needs to apply systems of simultaneous equations in solving problems. This involves selecting and using methods, demonstrating knowledge of concepts and terms, and communicating using appropriate representations. This evidence is a student s response to the TKI task Elaine s Equations. This student has selected and solved the equations for methods 1 and 2 (1) and interpreted the nature of the solutions for method 1 geometrically (2). The student has communicated using appropriate representations. To reach Merit, the student could solve and provide the geometric nature of the solutions for both methods 2 and 3.

9 Method 1 Equation 1 Equation 2 y 4x Equation 3 3x 2y z 7 I used my graphical calculator to get x = 5, y = 2, and z= In method 1 as there is one set answer for x, y, z graphically the three planes have one point where they all cross. point of intersection Method 2 Equation 1 Equation 2 Equation 3 6x 6x Graphical calculator gives MA error x3 6x 6x (-) 6x 0 = 0 Infinite solutions

10 Grade Boundary: Low Achieved 5. For Achieved, the student needs to apply systems of simultaneous equations in solving problems. This involves selecting and using methods, demonstrating knowledge of concepts and terms and communicating using appropriate representations. This evidence is a student s response to the TKI task Elaine s Equations. This student has selected and solved the equations for method 1 (1) and interpreted the nature of the solution to method 1 geometrically (2). This student has communicated using appropriate representations. For a more secure Achieved, the student could solve or provide a geometric interpretation for another system of equations.

11 Original equation 1) 2) Rearranged equations 1) 2) Method 1: 3x 2y z 7 x = 5 y = 2 z = -5 graphical calculator Method 1: One unique solution is given at the point where all three equations meet.

12 Grade Boundary: High Not Achieved 6. For Achieved, the student needs to apply systems of simultaneous equations in solving problems. This involves selecting and using methods, demonstrating knowledge of concepts and terms and communicating using appropriate representations. This evidence is a student s response to the TKI task Elaine s Equations. This student has solved the system of equations for method 1 (1). To reach Achieved, the student could provide an appropriate representation - a geometrical interpretation of this solution.

13 3x 2y z 7 0 (1) 8 0 (2) 3x 2y z 7 0 (3) (1)x6 1 8y 2z 6 0 (4) (3)x4 1 8y 2z 28 0 (5) (4) (5) 26y 20z 22 0 (6) (2)x5 5y 20z 40 0 (7) (6) (7) 1y 62 0 y 2 Sub y = 2 in (2) (2) 4z 8 = 0 (2) 4z = 8-4z = 6 z = -1.5 sub y = 2, z = -1.5 to (5) 1 + 8(-2) + 32(-1.5) 28 = = = 0 x = 5 therefore y = 2 z = x = 5