# High School Algebra Reasoning with Equations and Inequalities Solve equations and inequalities in one variable.

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3 Quadratic This problem gives you the chance to: work with a quadratic function in various forms This is a quadratic number machine. x Subtract 1 Square the result Subtract 9 y 1. a. Show that, if x is 5, y is 7. b. What is y if x is 0? c. Use algebra to show that, for this machine, y = x 2 2x 8. The diagram on the next page shows the graph of the machine s quadratic function y = x 2 2x 8 and the graphs of y = 3 and y = x. 2. a. Which point on the diagram shows the minimum value of y? b. Which point(s) on the diagram show(s) the solution(s) to the equation 3 = x 2 2x 8? c. Which point(s) on the diagram show(s) the solution(s) to the equation x = x 2 2x 8? Copyright 2009 by Mathematics Assessment Resource Service. Quadratic 27

4 3. a. Use the graph to solve the equation x 2 2x 8 = 0. Mark the solutions on the graph. x = or x = b. Use algebra to solve the same equation. y A B -5 D E x -10 C Copyright 2009 by Mathematics Assessment Resource Service. Quadratic 28

7 Now look at student work for solving the quadratic for x. How many of your students: Used factoring? Used substitution? Attempted the quadratic formula? Had no strategies for using the equation? How much work have your students had with factoring? Do you see evidence that they understand or see the purpose for factoring to solve an equation? What are the big ideas or underlying conceptual knowledge that we want students to have about quadratics and their solutions? 31

8 Looking at Student Work on Quadratic Student A is able to use the number machine to calculate values for y. The student understands how to encode the number machine into an algebraic expression. In trying to prove that the machine is equal to the equation y = x 2 2 8, the student sets it equal to the expression representing the machine. The student doesn t understand that the steps for transforming the expression need to be shown to complete the argument. The student is able to use factoring and setting the factors equal to zero to find the solutions to the quadratic equation. Student A 32

9 Student A, continued 33

10 Student B tries to use substitution to prove that the number machine is equal to the equation y = x To find the solution to the equation x = x 2 2 8, the student uses factoring. However when needing the same information in part 3b, the student uses the quadratic equation. Do you think the student knows that both are yielding the same solution? What question would you like to pose to the student to probe for understanding? Student B 34

11 Student B, continued Student C is able to calculate y using the number machine. The student does not attempt to show why the number machine is equal to the equation y = x The student understands the minimum value on the parabola. But the student doesn t connect the equations to solutions on the graph in part 2. In part 3, the student does seem to understand that the solution of a quadratic is the value of x when y = 0. However the student has not mastered how to find these solutions using algebra. Instead the student uses guess and check to find the solution. 35

12 Student C 36

13 Student D tries to use examples to prove the equality between the equation y = x and the number machine. While the student can identify solutions on a graph in part 2, the student can t name the solutions in 3a. The student attempts to use the quadratic equation to solve in part 3b, but struggles with the computations. Student D 37

14 Student D, continued Student E struggles with detail and signs. In 1b the student either doesn t square the 1 or adds 1 and 9 incorrectly. In 1c the student factors the equation y = x 2 2 8, which yields an equality, but doesn t connect the equation to the number machine. The student loses track of what needs to be proved. In part 3 the student understands that the equation needs to be factored, but doesn t use the factors to set up equations equal to zero. Thus, the student s solutions have incorrect signs. Student E 38

15 Student E, continued Student F is able to do the calculations in 1a and 1b. The student doesn t seem to understand what is meant by proving that the calculations in the number machine are the same as the equation. The student writes about what is confusing in the equation. While the student answers part of 2 correctly, the uncertainty or tentativeness of the solution is clear. In part 3, the student attempts to manipulate the equation in a manner that would solve for x. However when solving a quadratic the equation needs to be in the form 0 = ax 2 +bx +cy 2. The scorer should not have given the process point. 39

16 Student F 40

17 Student G struggles with representing ideas symbolically. The student can do the stepby-step calculations in the number machine. When attempting to express those actions in algebra the student only squares the (-1) rather than the whole quantity (x-1). The student doesn t know how to connect solutions to the graphical representation in part 2. Student G also understands that the solution for a quadratic can be found by factoring, but doesn t know how to complete the process. Student G 41

18 Student G, continued Student H is also confused by the graphical representations. First the student doesn t seem to connect the description of the graph with the idea that 3 different equations are being displayed. The student doesn t realize that minimum is a term used with quadratics and parabolas. The student does lots of symbolic manipulation but can t make sense of the information. Student H 42

19 Student H, continued 43

20 Student I struggles with calculations. The student should not be given credit for the solution in 1a because the 4 is never squared The student also errs in solving 1b. The student does not realize that the solution given in part 1c is not equal. While the student is able to guess one of the 2 points correctly in 2b and 2c, it is unclear if this is just luck. What do you think this student needs? What would be your next steps? Student I 44

21 Task 2 Quadratic Student Task Core Idea 3 ic Properties and Representations Core Idea 1 Functions and Relations Work with a quadratic function in various forms. Represent and analyze mathematical situations and structures using algebraic symbols. Approximate and interpret rates of change, from graphic and numeric data. Understand relations and functions and select, convert flexibly among, and use various representations for them. Mathematics of the task: Calculations with exponents and negative numbers Codifying calculations into a symbolic string Rectifying two forms of an equation Interpreting graphical representations of linear and quadratic equations and identifying minimum point and solutions Using algebra to find the solution to a quadratic equation (using factoring or the quadratic equation) Based on teacher observation, this is what algebra students know and are able to do: Calculate using the number machine Identify the minimum point on a parabola using a graph Areas of difficulty for algebra students: Using algebra to show that two equations are equal Finding solutions to two equations on a graph Using algebra to find the solutions to a quadratic equation 45

22 The maximum score available on this task is 9 points. The minimum score needed for a level 3 response, meeting standards, is 5 points. Most students, 92%, could either identify the minimum or use the number machine to prove that an input of 5 equals an output of 7. 82% could do both calculations with the number machine. More than half the students, 66%, could use the number machine and find the minimum point on a parabola. Some students, 30%, could calculate with the number machine, find minimum, locate solutions to two equations on a graph, and use the graph to identify the solutions of the quadratic. About 2% of the students could meet all the demands of the task including showing that two equations are equal and using algebra to solve a quadratic. 8% of the students scored no points on this task. All the students in the sample with this score attempted the task. 46

23 Quadratic Points Understandings Misunderstandings 0 All the students in the sample with this score attempted the task. Students did not know how to square numbers or calculate with negative numbers. They could not interpret the 1 Students could either solve the first number machine or locate the minimum on the graph. 2 Students could solve both number machines or solve the first number machine and find the minimum. 3 Students could use the number machine and find the minimum. 5 Students could use the number machine, find the minimum, and locate solutions to two equations on a graph. 7 Students could use the number machine, find the minimum, and locate solutions to two equations on a graph. 9 Students could use the number machine, find the minimum, and locate solutions to two equations on a graph. Students could solve quadratics graphically and algebraically. They could also use algebra to show that two expressions were equal. directions on the number machine. Students struggled with the second number machine. 8% got an answer of % got an answer of 0. Most students who struggled with the minimum put a value of 8 rather than identifying point C. Students struggled with finding solutions to two equations on the graph. For the equation 3=x 2 2x 8 common errors were D & B, E, E &D, and no attempt. For the equation x = x 2 2x 8 common errors were no attempt, D, C, B, and A & E. Students struggled with identifying the solution to a quadratic on a graph and using algebra to find the solution to a quadratic. 16% did not attempt a graphical solution. 26% did not attempt an algebraic solution. 19% used substitution from the graphical solution as their algebraic strategy. Students struggled with using algebra to prove that the number machine was equal to the equation = x 2 2x 8. 40% tried to use substitution for their proof. 19% did not attempt this part of the task. 47

24 Implications for Instruction Students need more experience using algebra as a tool. While many students may have been able to square an algebraic expression, they didn t think to pull out this tool to help show why two expressions are equal. Students didn t seem to understand how to put together algebra to make a convincing proof. They relied on the logic of proof by discrete examples. One of the big ideas of algebra is its usefulness in making and proving generalizations. This is a huge part of algebra. Students need to be able to think about what solutions mean to quadratics or to two equations. They need experiences relating solutions graphically and algebraically. Students seemed unfamiliar with using the graph to find solutions. Students also had difficult with the idea of solving quadratics. Some students knew that part of the procedure was to factor the expression, but then did not set the factors equal to zero and solve. Many students tried to solve the equation similar to the strategy of solving a linear equation putting the variable on one side and number values on the other. Other students have this idea that using substitution is a proof. Students need more experience with quadratics. Ideas for Action Research To help students learn to make generalizations and rectify equations give them an interesting pattern problem, like finding the perimeter of any number of squares in a row. As students work the task, they will come up with a variety of solutions or generalizations for the pattern. Here is a very natural way to have students work with the idea of proving the two equations are equal. Depending on how students see the pattern, they might write different description for the generalization. The first pattern might be described as every square has two sides (top and bottom) plus two additional pieces at the end. This might be expressed algebraically as p = 2x + 2. The second pattern might be described as the end squares have 3 sides and the middle squares have two sides. This might be expressed algebraically as p = 2 (x-2) + 6. The third pattern might be described as the first square has 3 sides and all other squares have 2 sides with a final end side. This might be expressed as p = 2(x-1) + 4. Now students have organic expressions to work with and try to show the equality between the equations. 48

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