# High School Algebra Reasoning with Equations and Inequalities Solve systems of equations.

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2 results of the national assessment, including the total points possible for the task, the number of core points, and the percent of students that scored at standard on the task. Related materials, including the scoring rubric, student work, and discussions of student understandings and misconceptions on the task, are included in the task packet. Grade Level Year Total Points Core Points % At Standard % 2012 Noyce Foundation

3

4 Rubric

5 Graphs The core elements of performance required by this task are: relate given line graphs to their equations Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Gives 7 correct labels (see below) Partial credit 6 correct labels 5 correct labels 4 or 3 correct labels 2 or 1 correct label(s) 2. Gives correct answer: (6,6) 1 3. Gives any correct line, for example, y = 6, x = 3, y = x + 3, y = 9 x, etc. 1 1 Total Points 7 B 5 (4) (3) (2) (1) 5 1 G C J F E A

6 Graphs Work the task and examine the rubric. What are the key mathematics the task is trying to assess? Look at student work on labeling the graph. How many of your students labeled these parts incorrectly: B G C E F J A What do you think your students had difficulty understanding? Now look at your equations for a straight line going through the point (3, 6). Make list of student equations, make tallies for those that are repeated: Correct Incorrect What do these equations suggest that your students don t understand about graphing and equations? Are there common threads to the errors? 83

7 Looking at Student Thinking on Graphs For the blank on the graph with the expected answer of B, the line x=0: 27% put A, the line y=0. Students mixed up the two axes. 5% put G, the line x + y =9, even though the bubble is pointing at 10 on the y- axis. For the blank on the graph with the expected answer of G, the line x + y = 9: 6% put I, the line y = x - 6 5% put D, the line y=6 5% put K, the solution of the simultaneous equations x + y = 9 and y = 2 x 7% put H, the line y = x + 6 5% also picked C, J, and F For the blank on the graph with the expected answer of C, the line x = 6: 13% put D, the line y = 6 4% put the answer J, the solution of the simultaneous equations x + y = 9 and y = 1/2x For the blank on the graph with the expected answer of E, the origin: 10% put A, y= 0 4% put K, the solution of the simultaneous equations x + y = 9 and y = 2x 3% put J, 3% put K, 3% put B, 3% put I For the blank on the graph with the expected answer of F, the line y = 1/2 x: 6% put I, the line y = x 6 4%% put K, the solution of the simultaneous equations x + y = 9 and y + 2x 4% put J, the solution of the simultaneous equations x + y = 9 and y = 1/2 x 3% each picked H, E, C, A, B, and H For the blank on the graph with the expected answer of J, the solution of the simultaneous equations x + y = 9 and y = 1/2x: 12% put C, the line x % put G, the line x + y = 9 9% put K, the solution of simultaneous equations s + y = 9 and y = 2x 6% picked B, the line x = 0 6% picked H, the line y = x + 6 5% picked F, the line y = 1/2 x 84

8 For the blank on the graph with the expected answer of A, the line y = 0: 30% put B, the line x 0 5% put K, the simultaneous equations x + y = 9 and y = 2x For part two, which point is on the line y = 6 and on the line x = 6, almost 30% of the students tried to give one of the letter options from part one rather than giving the coordinates (6,6). Students giving coordinates picked 6 as only one of the two coordinates. When asked to give an equation for a straight line going through the point (3,6), many students tried to use the idea of y = mx +b: Some examples of these equations: y=mx + b y=3m+b y=bx +b y=3x+6 Other choices show a confusion about equations; such as trying to find a specific solution, not understanding exponents, having solutions that would not be true for the values given if substituted into the equation: x=3+6 x 3 +y 6 =xy 18 y = x+9 9x +3y y=x+6 3x + 9y 85

9 Algebra Course One/Algebra Task 5 Graphs Student Task Core Idea 3 Alg. Properties & Representations Core Idea 1 Functions and Relations Relate line graphs to their equations. Represent and analyze mathematical situations and structures using algebraic symbols. Understand the meaning of equivalent forms of expressions, equations, inequalities, or relations Write equivalent forms of equations, inequalities, and systems of equations and solve them Analyze functions of one variable by investigating local and global behavior, including slopes as rates of change, intercepts and zeros. Based on teacher observation, this is what algebra students knew and were able to do: Students could identify the origin Students could identify the equation x + y = 9 Students could identify the equation y = 1/2 x Students could give the coordinates for the intersection of y=6 and x = 6 Areas of difficulty for algebra students: Confusing the order of x an y in a coordinate pair Recognizing the solutions for simultaneous equations Writing an equation for a line through a given point 86

10 The maximum score available on this task is 7 points. The minimum score for a level 3 response, meeting standards, is 3 points. Most students, 85%, could identify the origin and the line x + y = 9. About half the students could also identify the line y = 1/2 x. Some students, about 35%, could also identify the lines x= 0 and y = 0 and x % of the students could meet all the demands of the task including identifying the simultaneous solution for x + y =9, giving the coordinates for the intersection of y=6 and x=6, and writing the equation for a line going through (3,6). Almost 15 % of the students scored no points on this task. 57% of the students with this score attempted the task. 87

11 Graphs Points Understandings Misunderstandings 0 Only 57% of the students with this score attempted this task. Students confused the order of x and y in a coordinated pair or the x and y axis. 1 Students could identify the origin and x + y = 9. Students struggled with the solution of a simultaneous equation and the line y = 3 Students could identify the origin and x + y = 9, and either identify the solution of a simultaneous equation or the line y = 1/2x. 6 Students could also give the equation for a line going through the points (3,6). 7 Students could meet all the demands of the task by matching equations or points with their representation on a graph, finding the intersection of two lines, and writing an equation for a line passing through a given point. 1/2x. Students had difficulty finding the intersection of the lines y=6 and x =6. Many students tried to pick from the options in part 1 of the task, rather than trying to graph and find the point for themselves. Students struggled with the solution of simultaneous equations and the intersection of two lines. Implications for Instruction Students need more work with coordinate graphs and plotting points. Students at this level are still confusing x an y; for example choosing the line y= 0 foe x = 0. They may need more experience with making a table of values and plotting points in order to verify the graphs of lines. They didn t seem to use substitution of values to verify if the ordered pair would match their equation. They had difficulty finding the intersection of the equations on a graph. Is there some piece of conceptual understanding that gets missed if students don t get enough hands on experience before jumping to graphing calculators? How can you interview students or set up an experiment to find out what they understand about the connections between equations and graphs? How does understanding the intersection of two equations relate back to Printing Tickets? When teaching finding the solution to simultaneous equations is enough emphasis put on the meaning of the intersection or the connection between the solution and the graphs of the equations? How does context help to illuminate this significance? How does understanding the graphs of lines help students to develop a better understanding of the idea of a variable, something that can change or vary? What contexts help to support students development of this big mathematical idea? 88

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