Balanced Assessment Test Algebra 2008


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1 Balanced Assessment Test Algebra 2008 Core Idea Task Score Representations Expressions This task asks students find algebraic expressions for area and perimeter of parallelograms and trapezoids. Successful students could show how the formula for area of a trapezoid is derived from the area of the two triangles made by decomposing the shape. Algebra Buying Chips and Candy This task asks students to form and solve a pair of linear equations in a practical situation. Successful students could use substitution or systems of equations to find their solutions. Representations Sorting Functions This task asks students to find relationships between graphs, equations, tables and rules. Successful students could describe how to look at an equation and predict the shape of the graph. Functions Sidewalk Patterns This task asks students to work with patterns and find the n th term of a sequence. Successful students could write an equation to finding the n th term. Functions and Functions Representations This task asks students to work with graphs and equations of linear and nonlinear functions. Students need to identify points on a graph, write a linear equation. Successful students knew the difference between quadratic and exponential equations and could give the equation of a parabola. Algebra
2 Algebra
3 Algebra
4 Expressions This problem gives you the chance to: work with algebraic expressions for areas and perimeters of parallelograms and trapezoids 1. Here is a parallelogram. a h b The area of a parallelogram is the product of its base times the perpendicular height. a. Which of these are correct expressions for the area of this parallelogram? Draw a circle around any that are correct. ab 1 2 ab ah 1 2 ah 2a + 2b 2(a + b) abh! b. Which of these are correct expressions for the perimeter of the parallelogram?!!!!! Draw! a circle around any! that are correct.! ab 1 2 ab!! ah 1 2 ah! 2a + 2b! 2(a + b) 2. Here is a trapezoid. It is made up of two triangles, each with height h. a! abh h b Find the area of each of the two triangles and use your results to show that the area of the 1 trapezoid is ( 2 a + b )h.! 6 Algebra Copyright 2008 by Mathematics Assessment Resource Service
5 Expressions Rubric The core elements of performance required by this task are: work with algebraic expressions for areas and perimeters of parallelograms and trapezoids Based on these, credit for specific aspects of performance should be assigned as follows points section points 1.a b. Gives correct answer: ah circled and no others circled Gives correct answers: 2a + 2b and 2(a + b) Deduct 1 point for 1 extra and 2 points for more than 1 extra. 1 2 x Provides a convincing development of the required expression such as: Shows the areas of the two triangles are Adds these two expressions to get!! 1 ah and a + b ( )h! 1 2 bh 2 x Total Points 6 Algebra Copyright 2008 by Mathematics Assessment Resource Service
6 Expressions Work the task and look at the rubric. What are the algebra tools a student needs to know to do this task? What would you like to see if for complete explanation or proof in part 3? Look at student work for part 1a, finding the area of a parallelogram. Remember the formula is given in words. Now look at student work. How many of your students put: ah Omitted ah ab abh 1/2ab 1/2ah 2a +2b 2(a+b) What surprises you about these results? How often do students get to work with algebra in the context of geometry? How might this context help students to see algebra as a sensemaking tool? Now look at student work for part 1b, finding the perimeter of a parallelogram. How many of your students put: Both formulas Omit 2a+2b Omit 2(a+b) ab 1/2ab ah 1/2ah abh Aside from the geometry, how many students didn t see the equivalency between 2a+2b and 2(a+b)? Why do you think this was difficult for students? Why do you think students had difficulty expressing perimeter in a geometric setting? Now look at work for part 2. How many students could complete the entire argument? In their work, how might you encourage them to improve their answers, make them clearer and highlight the mathematics and logic of each step? Now look at types of errors. How many students: Made no attempt on this part of the task Put either 1/2 a or 1/2 b as the area of a small triangle Tried to use numbers instead of variables to solve the problem Found the area of the small triangles but didn t combine them to try and complete the argument or proof Other errors How often do students in your class get the opportunity to make and test conjectures about geometric or other contexts using algebra? What are some of your favorite problems? Algebra
7 Looking at Student Work on Expressions Student A is able to identify the expressions for finding area and perimeter of a parallelogram. The student is able to see that 2a+2b is equivalent to 2(a+b). In part 2, Student A labels the two triangles in order to define which areas are being found by each area expression. Then the student uses words and symbols to discuss combining the two separate areas into a single expression. Student A Algebra
8 Student B uses diagrams to think about the expressions for area of each triangle. Then the student talks about combining the two expressions. Student B factors out first the height and then the 1/2 to make the combined expression equivalent to the original formula. Understanding how to close an argument and show the steps back to the original statement is an important piece of logical reasoning. Student B Student C has all the information needed to make the conclusion, but either doesn t understand how to factor out the expression or recognize a need to make the final statement the same as the original. Student C Algebra
9 Student D finds the area of each separate triangle, but does not combine the terms or make any attempt to show how that relates to the original formula. Student D Student E finds the area of the separate triangles and then seems to try to work backward from the original formula to get the two expressions. The student appears to be attempting distributive property on the right, but does not carry it through correctly. Student E Algebra
10 Student F might be debating between two different area formulas, using one strategy for triangle a and a different strategy for triangle b. The students seems to settle on ah + bh. The student then factors this expression but does not know how to get the 1/2 into the problem. If you could interview this student, what question might you want to ask? If you could pose some other problems, what might help you understand where the thinking or skills break down? Notice that student F puts multiple choices for area in part 1a and doesn t recognize equivalent expressions for perimeter in part 1b. Student F Algebra
11 Many students struggled with using the distributive property on the original formula. Student G distributes the 1/2 on variables inside and outside the parentheses. Student H distributes the 1/2 over both variables and then separately distributes the h. Student G Student H Student I struggles with the concept of like and unlike terms. The student also tries to replace the variables with numbers to check the solution. Student I Algebra
12 Student J tries to use distributive property inappropriately and also teases apart the expression in an attempt to set up an equation. On the right, the student also seems to use parentheses more in a linguistic sense to separate things rather than as a mathematical notation. Student J Student K can t think with variables. To identify the expressions in 1a and b, the student needs to put in numbers to think through the process and then substitute back in the variables. In part 2 the student checks out one case of using numbers to check that the formula is true, but doesn t have the sense of the importance of variables to build a generalizable proof for all cases. How do we help students learn to think with variables? How do we help develop in students the idea of algebra as a tool for generalization instead of set of manipulations? Student K Algebra
13 Algebra Task 1 Expressions Student Task Core Idea 3 Algebraic Properties and Representations Work with algebraic expressions for areas and perimeters of parallelograms and trapezoids. Represent and analyze mathematical situations and structures using algebraic symbols. Use symbolic algebra to represent and explain mathematical relationships. Mathematics of this task: Using variables to find area and perimeter of a parallelogram Recognizing equivalent expressions by factoring or using distributive property Using algebra to make and prove a generalization Explaining the steps of factoring or using distributive property to make two expressions equivalent Based on teacher observations, this is what algebra students know and are able to do: Students were able to recognize expressions for finding perimeter of a parallelogram Many students could recognize equivalent expressions for perimeter Areas of difficulty for algebra students: Finding the area of a parallelogram/ translating from words to variables (the formula for area was given in a verbal form) Thinking with variables instead of numbers Decomposing the trapezoid into two triangles Using factoring and/or distributive.0 property to make equivalent expressions Using algebra to make a generalization Algebra
14 The maximum score available for this task is 6 points. The minimum score for a level 3 response, meeting standards, is 4 points. Many students, 80%, could find one expression for the perimeter of a parallelogram. More than half the students could either find one expression for area and perimeter of a parallelogram or find two expressions for a parallelogram. Some students, about 31%, could find two expressions for perimeter of a parallelogram, and find the area of the two small triangles. 13% of the students could meet all the demands of the task including finding an expression for the area of a parallelogram and using algebra to show how to combine and factor the expressions for the area of two triangles into the formula for the area of a trapezoid. Algebra
15 Expressions Points Understandings Misunderstandings 0 82% of the students with this score attempted the task. Students chose too many expressions for perimeter and didn t see equivalent expressions. 26% of the students omitted 2a+2b for perimeter. 31% omitted 2(a+b) for perimeter. About 5% of the students chose each of the 2 Students with this score could either find both expressions for perimeter for a parallelogram or find one expression for area and one expression for perimeter. 3 Students could identify expressions for area and perimeter. 4 Students with this score could find the expressions for perimeter and the area of the small triangles in part 2 or find one perimeter expression and explain all of part 2. following options:ah, 1/ah, ahb, ab, and 1/2 ab. Students had difficulty identifying the formula for area, even though the verbal rule was given. While many students picked ah, they often picked other choices as well. 35% of the students did not pick ah as one of the choices. Almost 14% of the students picked ab for area, confusing side length with height. 14% picked 1/2ah and 12% picked 1/2ab for the area formula, confusing area of a triangle with area for a parallelogram. About 5% picked each of the perimeter formulas for area. Students had difficulty trying to make an algebraic generalization about area of a trapezoid. Some students did not decompose the shape into two triangles. 10% of the students substituted numbers for variables. 8% left the height out of the formula for the small triangles (1/2 a or 1/2 b). 10% found the area of the small triangles, but did no further work to show how to combine the areas and create an equivalent expression to the given formula. Students applied algebra inappropriately: combining unlike terms, not being able to factor expressions, using distributive property incorrectly. 5 Students couldn t or didn t combine the areas of the small triangles to make a complete argument in part 2. 6 Students could use a geometric context to write expressions with variables for area and perimeter of a parallelogram and a trapezoid. Algebra
16 Implications for Instruction Students at this grade level have been working with area and perimeter of parallelograms since 4 th grade and area of triangles since 5 th grade. At this grade level students need to be able to move from the specific solution to generalizing about why the formula works and where it comes from. Students in algebra have learned a variety of tools, such as using a variable to stand for a side of any length, combining like terms, or factoring polynomials. The purpose of these procedures or tools is to be able to prove numerical relationships or why something works or discover under what circumstances it won t work. Students need to be exposed to a variety of contexts for applying their skills to make and test conjectures, to prove mathematical relationships. Procedures that can t be transferred to new situations are not useful and will probably have a short retention rate. Some of the most interesting data from algebra shows that students on cumulative tests do best on the most recently taught topic, rather than building from foundational knowledge at the beginning of a course. Students need more opportunities to connect their thinking to its use in context. Ideas for Action Research Building Logical Arguments Students, even at very young ages, are capable of learning the reasoning chains to make logical proofs. They just need to be pushed with good questioning strategies. In the book, Thinking Mathematically, Integrating Arithmetic and Algebra in Elementary School, the authors explore classroom experiences with children in 2 nd, 3 rd, and 4 th grades using variables and learning the logic of proof. In the chapter on justification and proof, secondgrader Susie is able to use number properties to justify why 5 (5) + 5= 5. She is first able to write that a + b b = a. She justifies it by saying that any number minus itself equals 0, so bb = 0. Then any number + 0 equals itself. Finally she is able to say that if both of these statements are true the original will also be true. It is productive to ask children whether their conjectures are always true and how they know that they are always true for all numbers..we consistently have been surprised at what children are capable of when given the opportunity. This book comes with a video of students making and testing conjectures. Some of the videos might be useful in the classroom to give students models of how to use and discuss mathematics. Fostering Algebraic Thinking offers some grade level appropriate activities to help students build their capacity for generalization and making proofs. Consider the problem of finding combinations of consecutive numbers to make the different answers of 1 to 35. After students have explored this problem they might be asked to describe patterns that they found with the consecutive numbers. Students might provide a range of solutions from all the numbers made of 3 consecutive numbers can be divided by three to a number N is a consecutive sum of m numbers if m divides evenly into N and m is an odd number. This resource offers students many intriguing problems to work on their abilities to make logical arguments and develop proofs using variables. It also offers suggestions on types of questions that help students build habits of mind that lead them to make better justifications. Algebra
17 A related MARS task is the Sum of Two Squares, Course , available on the Noyce website. It starts with the premise from Lewis Carroll, that 2(x 2 + y 2 ) is always the sum of two squares where x and y are a pair of nonzero integers. Students are given opportunities to investigate the conjecture with numbers, then describe the relationship in words and finally challenged to prove why this is always true using algebra. The discussion on this task would be a good place for practicing the types of questioning strategies suggested in the two references above. Algebra
18 Buying Chips and Candy This problem gives you the chance to: form and solve a pair of linear equations in a practical situation Ralph and Jody go to the shop to buy potato chips and candy bars. Ralph buys 3 bags of potato chips and 4 candy bars. He spends $3.75. Jody buys 4 bags of potato chips and 2 candy bars. She spends $3.00. Later Clancy joins Ralph and Jody and asks to buy one bag of potato chips and one candy bar from them. They need to work out how much he should pay. Ralph writes 3p + 4b = If p stands for the cost, in cents, of a bag of potato chips and b stands for the cost, in cents, of a candy bar, what does the 375 in Ralph s equation mean? 2. Write a similar equation, using p and b, for the items Jody bought. Copyright 2008 by Mathematics Assessment Resource Service 18
19 3. Use the two equations to figure out the price of a bag of potato chips and the price of a candy bar. Potato chips Show your work. Candy bar 4. Clancy has just $1. Does he have enough money to buy a bag of potato chips and a candy bar? Explain your answer by showing your calculation. 7 Copyright 2008 by Mathematics Assessment Resource Service 19
20 Buying Chips and Candy Rubric The core elements of performance required by this task are: form and solve a pair of linear equations in a practical situation Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Gives a correct explanation such as: It stands for the 375 cents that Ralph spent. (Must have correct units) 1 2. Writes a correct equation such as: 4p + 2b = 300 Partial credit For an almost correct equation. ( Left hand side of equation must be correct) 3. Gives correct answers: 45 cents or $0.45 and 60 cents or $0.60 Shows correct work such as: 8p + 4b = 6 subtract 3p + 4b = 375 5p = 225 P = 45 4 x b = 300 2b =120 b = 60 Partial credit For some correct work. 4. Gives a correct answer: no and Shows a correct calculation such as: = (1) 1 ft 2 ft (1 ft) 1 ft Total Points Copyright 2008 by Mathematics Assessment Resource Service 20
21 Buying Chips and Candy Work the task and look at the rubric. What are the big mathematical ideas in this task? What does a student need to understand about algebra to be successful on this task? In part 1, the prompt is hoping to get students to explain that the decimal number has been multiplied by 100 or changed to a whole, converting it from dollars to cents. However students had difficulty with the prompt and gave an explanation about the meaning of the money in the context of the problem. In part 3, students are asked to use the 2 equations to find the cost of chips and candy. Looking at student work there was not much commonality in the incorrect answers, so instead look at the strategies students used. How many of your students used: Strategy Guess and Check Substitution Multiplication/Addition Method Add both equations without multiplication (7p+6b=675) Put equations but showed no work Made false equivalencies No work Added in extra equations Led to correct solution Led to incorrect solution How did this help you think about what students understand about algebra? What are some of the things you need to think about to be successful at substitution or to make substitution more convenient? What do you have to understand to use the multiplication/addition method? Why do you think so many students received incorrect solutions using guess and check? What ideas did they not understand? What algebraic concepts need to be understood to apply guess and check successfully? Part 4 required students to use their solutions in #3 to see if they added to more than a dollar. How many of your students: Did not attempt this part of the task? Gave an answer that wasn t supported with a quantitative reason? (They didn t total the 2 numbers.) Made arithmetic errors? Explained something else? What do you students understand about justifying an answer? What types of activities help them to internalize what is needed for a good response? 19
22 Looking at Student Work on Buying Chips and Candy Student A is able to identify the change in units in part 1 and give the correct equation for part 2. The student uses the multiplication/addition strategy for solving 2 equations with 2 unknowns. Notice that the student uses substitution into both equations to check the solutions. In part 4 Student A uses definition of givens and mathematical notation to support the conclusion that there isn t enough money for the chips. Student A 20
23 Student B solves the 4p +2b = 300 for b, then uses substitution to solve for p in the second equation. Student B 21
24 In general students who attempted substitution were less likely to get a correct solution to part 3, solving 2 equations for 2 unknowns. Student C seems to first try adding the two equations together, which doesn t help solve the problem. Then at the top right the student makes a first attempt at using substitution, but chooses the equation and variable that comes out to a fractional quantity. Then the student chooses the equation to solve for b, which would have allowed her to use substitution. At this point the student seems to switch to the multiplication/addition strategy and solve the problem successfully. This is a nice glimpse into the thought process of someone just starting to learn new material and the problemsolving necessary to see which algebraic tool to use. Success is possible because of the habit of mind of perseverance and belief in one s own ability to figure things out. How do we foster this diligence in other students? Student C 22
25 Student D is an example of a successful student using substitution with a fractional quantity. While this strategy is completed successfully, a good problem solving strategy is to think about which equation is easiest to solve for. Which variable might yield whole number solutions? How do we help students develop these types of internal questions to guide them when confronted with problem situations? Student D Student E attempts to use substitution to solve for 2 equations with 2 unknowns. However, there is no evidence of using that information to complete the process. The values for potato chips and candy bars could just as easily have come from guess and check. There is major information missing from the solution. Student E 23
26 Student F also attempts to use substitution, picking the more complicated equation to solve for first. The student tries to avoid fractions and in the process loses the exact solution due to rounding issues. How might you pose this as a question to the class to let them see and understand the potential errors caused by using decimal values? What do you want them to understand about this solution set? Student F 24
27 Student G adds the two equations together in their original format. This doesn t eliminate either variable and so the student struggles with how to solve the resulting equation. The student then seems to revert to the more comfortable process of guess and check to get the solution set. Student G Student H also adds the equations and then uses an incorrect procedure to arrive at a solution. What are some of the basic principles of solving equations that this student is missing? What habits of mind might have helped the student see her error in thinking? Student H 25
28 Student I is struggling with many of the basic ideas of algebra. In part one the student misunderstands the given equation, thinking it is finding number of items purchased rather than cost. The student doesn t seem to use the given information to write an equation, but makes up his own. In part 3 the student tries to move items across the equal sign to make the equation easier to solve. The student misses the negative sign and makes other errors in trying to use equality or balance principles. The student then uses guess and check to find a solution to the equation. However there is no attempt to relate it to the other information given in the problem. The student doesn t understand that the equation is a function with multiple solutions and that more information is needed to narrow it down to one unique solution. Student I 26
29 While Student J is able to explain that the 375 represents the total, the work in the equation shows that the student has noticed the change from dollars to cents. In fact the student also uses incorrect monetary notation in the solution to part 3. The student attempts to solve the equation by manipulating the format of the equation, moving the 2b to the right and then dividing both sides by 4. So the student has memorized an algorithm for solving equations. However the student doesn t know how to divide the 2b by 4 or the 4p by 2. The student just ignores the complicating extra variable in a need to reach a solution. What are some of the fundamental principles of variables and equations that this student is missing? How do we make these big algebraic concepts more explicit and visible in the classroom? Student J 27
30 Student K doesn t understand equality and its relationship to formulating equations. This student may still be thinking about the equal sign as the answer follows. The student sets up 4 things as equal to each other, including 300 = 375. What other errors or misunderstandings do you see in the solution process? Student K 28
31 Algebra Task 2 Buying Chips and Candy Student Task Core Idea 3 Algebraic Properties and Representations Form and solve a pair of linear equations in a practical situation. Represent and analyze mathematical situations and structures using algebraic symbols. Use symbolic algebra to represent and explain mathematical relationships. Judge the meaning, utility, and reasonableness of results of symbolic manipulation. Mathematics of the task: Understanding equalities and maintaining equalities to set up and solve equations Defining variables Combining like terms, not combining unlike terms Understanding monetary units Understanding that two linear equations with two unknowns has a unique solution, but that a linear function with two variables has an infinite number of solutions Strategies or procedures for solving sets of equations: substitution, multiplication/addition Based on teacher observations, this is what algebra students know and are able to do: Write an equation from context and understand what the variables represent Use guess and check successfully as a strategy to find a solution for two equations with two unknowns. Use substitution or multiplication/ addition to solve for the unknowns Areas of difficulty for algebra students: Quantifying answers to justify a conjecture Using distributive property correctly in solving a problem Identifying which equation and which unknown would be easiest to use when applying the substitution method Using partially remembered strategies, but could not carry them through the entire solution process or unsuccessfully combined strategies Checking work with both equations to see if the solution is true for both Understanding that functions have multiple solutions 29
32 The maximum score available for this task is 7 points. The minimum score for a level 3 response, meeting standards, is 4 points. Most students, 86%, could determine how to write an equation in part 2. Many students, 68%, could write an equation and explain that the units had changed from dollars to cents or determine whether there was enough money to buy the chips and candy. A little less than half the students could write the equation, determine the units, and write a justification about having or not having enough money to purchase the snacks. 14% of the students could use algebra to solve two equations with two unknowns, using substitution or multiplication/addition. 8% of the students scored no points on this task. 80% of the students with this score attempted the task. 30
33 Buying Chips and Candy Points Understandings Misunderstandings 0 80% of the student with this score attempted the task. Students had difficulty writing the equation. About 4% did not attempt the equation. 4% put a decimal point in the 2 Students could write an equation from a verbal description. 3 Students could write an equation and either explain the units as cents or write a convincing statement about having enough money. 4 Students could explain the units in the equations, write an equation, and determine if there was sufficient money to buy the snacks. 5 Students could solve all of the task, but used guess and check for the solution strategy in part 3. 7 Students could solve two equations with two unknowns, using algebra. Students could talk about the context of the problem in terms of the variables and numbers in the equation. Students could make a justification by quantifying supporting information. equation. Students had difficulty explaining the units in part 1. 20% explained the meaning of the number as total cost. Another 10% explained it as how much he pays. Almost 20% of the students did not attempt part 4 of the task, even though it just involved arithmetic. 11% gave correct answers, but didn t quantify their answers. Students did not know how to solve equations for two unknowns. 17% attempted substitution. 15% attempted multiplication/addition. 7% just added together the two equations, which did not eliminate one of the variables. 6% wrote the two equations but showed no work. 15% of the students did not answer part 3. 14% had answers with 75 cents as one of the two costs. 3% had incorrect monetary notation. Many students used distributive property incorrectly. 23% of all students attempted guess and check. Many were not successful, because they didn t check their answers against all the constraints. 31
34 Implications for Instruction Students at this grade level need to learn mathematical ideas in context. They need to see a purpose or application for the various procedures that they are being shown. Within every lesson or set of lessons, students need to move from the concrete, to pictorial, to the abstract. It does students no good to be able to solve two equations with two unknowns, if they never think to use that procedure when confronted with a problem situation. Students need to be able to think through the design of the procedure, why does it work. Many students attempted to use a procedure, but could not remember the correct sequence of steps or forgot steps. In struggling with the justification of the procedure or algorithm, students make sense of the operations and have resources for thinking through what needs to be done when they forget. Students need to also develop habits of mind for checking solutions and sensemaking around their answers. For example, many students found solutions that would work correctly in one of the equations, but not in both of the equations. Students need opportunities to discuss solution strategies. Which equation or variable would it be easier to solve for if using a substitution strategy? Why? Which equation or variable would it be easier to solve for if using a multiplication and subtraction of equations strategy? Why? Students need to move from just learning procedures to thinking about the meaning of functions. Why does an equation with 2 unknowns have an infinite number of solutions? What is meant by variable? They need to see and discuss in context how the variation in one of the unknowns also causes variation in the other and why. Students also need to see and discuss why having 2 equations with 2 unknowns produces only one unique solution. Why did the second equation narrow the range of possible solutions? They should be able to talk about this meaning within the context of a situation, within the context of a graphical representation, as well as showing the numerical answer. With the ease of calculation afforded by calculators, students need to work with a variety of solutions not limited to just whole numbers. Some students show evidence of giving up or changing strategies when problems don t come out evenly. They have made an incorrect generalization based on the normal problems they work with than answers are always whole numbers.. Ideas for Action Research: Looking at Student Work to Plan Remediation: Often when planning remediation or helping students who are behind, teachers think about the students who are almost there. What are the few steps they need to be successful? But what is it that the students who are at the lowest end of the spectrum need? How are their issues different? Sit down with colleagues and examine the following pieces of student work. Consider the following questions: 32
35 1. What are the strengths, if any, that the student has? What are the concepts the students understand about the situation? How might these strengths be used to help build their understanding of the whole situation? 2. How did students use representations? Were the representations accurate? Why or why not? What would have helped the student to improve their representation? 3. What misunderstandings does the student have? What skills is the student missing? What does this suggest about a specific course of action to help this student? 4. How are the needs of each of these students the same or different? After your have carefully looked at each piece of student work, see if you can devise a plan of experiences/ discussions/ tools that might help these students to make more sense of these situations. While you don t have these exact students in your class, each member of the group will probably have students with similar misunderstandings. Identify students who you think are low and plan different approaches for attacking the problems outlined here. Have each person in the group try out a different course of action and report back on the how the lesson or series of lessons effected the targeted students. See if you can all use some similar starting problems and bring work of the students to share. What types of activities or experiences made the most noticeable improvement in student work? 33
36 Andy 34
37 Barbara 35
38 Cal Cal did not attempt part 4. 36
39 Debbie While Debbie did some work on page one of the task it was all up in the prompt area. In part 3 Debbie again does some calculations, but gives no answer for part 3 or 4. 37
40 Ed 38
41 Farrah 39
42 Sorting Functions This problem gives you the chance to: Find relationships between graphs, equations, tables and rules Explain your reasons On the next page are four graphs, four equations, four tables, and four rules. Your task is to match each graph with an equation, a table and a rule. 1. Write your answers in the following table. Graph Equation Table Rule A B C D 2. Explain how you matched each of the four graphs to its equation. Graph A Graph B Graph C Graph D Copyright 2008 by Mathematics Assessment Resource Service 40
43 Graph A Equation A Table A Rule A xy = 2 y is the same as x multiplied by x Graph B Equation B Table B Rule B y 2 = x x multiplied by y is equal to 2 Graph C Equation C Table C Rule C y = x 2 y is 2 less than x Graph D Equation D Table D Rule D y = x  2 x is the same as y multiplied by y 10 Copyright 2008 by Mathematics Assessment Resource Service 41
44 Sorting Functions The core elements of performance required by this task are: find relationships between graphs, equations, tables and rules explain your reasons points Based on these, credit for specific aspects of performance should be assigned as follows Rubric section points 1. Gives correct answers: 3 (2) (1) 3 (2) (1) 2. Equations: 4 correct 3 points Table: 4 correct 3 points Rule: 4 correct 2 points 3 or 2 correct 2 points 3 or 2 correct 2 points 3 or 2 correct 1 point 1 correct 1 point 1 correct 1 point Gives correct explanation such as: Graph A is a parabola/quadratic curve that passes through the origin and is symmetrical about the y axis (every value of y matches two values of x that are equal in size with opposite signs), so its equation is y = x 2. 2 (1) 8 Graph B is a straight line, so its equation is linear, y = x 2. Graph C is a parabola that is symmetrical about the x axis (every value of x matches two values of y that are equal in size with opposite signs), so its equation is x = y 2. Graph D: If we take any point on the graph and multiply its coordinates, say, (2, 1), we get 2. This is the equation xy = 2. Accept, we have matched the other three graphs to equations. Accept alternative correct explanations Partial credit (1) 2 or 3 correct explanations 2 Total Points 10 2 Copyright 2008 by Mathematics Assessment Resource Service 42
45 Sorting Functions Work the task and look at the rubric. What important algebraic ideas might students use to match a graph with an equation? What connections do you hope students are making to relate this information? Look at student work on matching the representations. In general did students have more difficulty with equations, tables or rules. Use this table to help you chart the information. Graph Equation Table Rule A C B A B D A C C B C D D A D B What surprised you as you charted the information? What seems most difficult for students to understand? What types of experiences or questions do students need to have to help them develop these big ideas? Now look at the student explanations for part 2. How many of your students: Could use correct algebraic ideas to think about the shapes of the graphs and the corresponding types of equations Only talked about matching graphs to tables Make a list of some of your best explanations. How could you use these as models or to pose questions for discussion to help other students develop the logic of justification? 43
46 Looking at Student Work on Sorting Functions Here are the results of students work on the table. Many students did not even understand the logic of sorting and put A, B, C, and D for each choice. That is harder to capture in the data. Graph Equation Table Rule Bold = correct response Italic= error choice for each response A C A 8% B 15% D 3% B A 6% C 9% D 10% A B 8% C 1% B D A B C A B C D C A B 7% 9% 3% 5% 5% 4% 3% 8% C B A C D C B C D D A B 8% 20% 4% 8% 1% 9% 10% 10% D A B C D D A B C B A C 9% 4% 11% 5% 4% 5% 6% 8% D 10% D 6% C 8% D 12% The second table just summarized the percent of students making errors for each part. Graph Equation Table Rule A C 26% B 25% A 19% B D 19% A 14% C 17% C B 32% C 18% D 28% D A 24% D 14% B 26% Students had a very difficult time giving reasons for matching graphs to equations. Between 14 to 20% of the students gave no response to each part of question 2. About 34% of all students just gave the vague explanation of matching graph to table and then find the equation. However some students brought out some very interesting and useful algebraic concepts to think about how to match the information. How do we help students make connections between algebraic concepts and move beyond procedural knowledge? An important piece of algebraic thinking is to move from a specific solution to making generalizations about types of solutions. What opportunities do we provide to help students to think in a more global perspective? Here are a few examples of what algebra students could do. 44
47 Student A recognizes that equations with x 2 will yield a parabola. The student uses several properties of linear functions to explain graph B. Student makes connections between similarities and differences in the graphs and equations of A and C. In part 4 the student explains why for this equation there will be no yintercept. Student A Student B makes a good case for why there is no yintercept for Graph D. Student B 45
48 Student C uses some interesting language to describe the differences between Graph A and B, giving more details about the parabolas. For graph D the student makes an argument about symmetry. What further questions or investigations could you pose for students to help them learn more about the parts of the equation that determine the symmetry or to explore how the symmetry of this graph is different from the symmetry of the parabola? Student C Student D describes how to determine which parabola is equal to graph A by looking at intercepts. The student also uses knowledge of intercepts to identify graph B. Student D 46
49 Student E uses information about the table and plotting graphs and goes into detail about they relate to each other. Notice that the student solves for y for graph D to help make sense of the shape. Student E 47
50 Student F gives the minimum descriptions to get the points. What further questions or investigations might you want to pose around the response to graph D? Student F Student G gives an implied elimination answer for graph D. Student G 48
51 Student H again uses the matching strategies but gives enough details to make it a valid explanation. What experiences or questions might push this student s thinking to the next level? Student H Student I is an example of a student whose responses are too vague for part C and D. Student I 49
52 Student J is able to think about parabolas and choose the correct representations, but struggles with the language to explain or make generalizations about B and D. Student J For graph A, 87% of the students who got the explanation correct talked about parabolas. 8% talked about quadratics. For graph B, half the students who got the explanation correct talked about it being linear. 18% talked about the equation being in the form of y=mx + b. 16% talked about the yintercept = 2. 9% talked about the slope. For graph C students talked about a sideways or strange parabola. For graph D, most students who got the explanation correct used an elimination argument. Some students gave an explanation about why there was no yintercept or the effects of multiplying by 0. A few students used a symmetry argument, solving for y, or a hyperbola to make their point. 50
53 Algebra Task 3 Sorting Functions Student Task Find relationships between graphs, equations, tables and rules. Explain your reasons. Core Idea 1 Understand patterns, relations, and functions. Functions Understand relations and functions and select, convert flexibly among, and use various representations for them. and Relations Core Idea 3 Algebraic Properties and Representations Represent and analyze mathematical situations and structures using algebraic symbols. Use symbolic algebra to represent and explain mathematical relationships. Judge the meaning, utility, and reasonableness of results of symbolic manipulation. The mathematics of this task: Making connections between different algebraic representations: graphs, equations, verbal rules, and tables Understanding how the equation determines the shape of the graph Developing a convincing argument using a variety of algebraic concepts Being able to move from specific solutions to thinking about generalizations Based on teacher observations, this is what algebra students know and are able to do: Understand that squaring a variable yields a parabola and that the variable that is squared effects the axis around which the parabola divided Use process of elimination as a strategy Match equations to tables and graphs Look for intercepts as a strategy Use vocabulary, such as: parabola, intercept, and linear Areas of difficulty for algebra students: Knowing the difference between linear and nonlinear equations Not knowing how to explain how they matched the graph and the equation Connecting the constant to the slope, e.g. just because it s 2 doesn t meant it s negative slope Quantifying: even though they could describe the process, but didn t quantify Not knowing how or when to use the term curve or parabola 51
54 The maximum score available for this task is 10 points. The minimum score for a level 3 response, meeting standards, is 6 points. Most students, 83%, could match two or three correct graphs to the table. Many students 76% could also match at least 1 graph to an equation. More than half the students, 56%, could match two or three graphs to equations, tables, and rules. Almost half the students, 46%, could match correctly all the representations. 14% could meet all the demands of the task including explaining in detail how to match a graph to its equation using algebraic properties about graphs and equations. 13% of the students scored no points on this task. 94% of the students with this score did not attempt the task. 52
55 Sorting Equations Points Understandings Misunderstandings 0 94% of the students with this score did not attempt the task. 2 Students could match 2 or 3 graphs with tables. See table in Looking at Student Work for specific errors. 3 Students could match 1 or 2 graphs with equations and tables. See table in Looking at Student Work for specific errors. 6 Students could match some graphs See table in Looking at Student Work for with equations, tables, and rules. 8 Students had could match all the graphs with their equivalent representations in the form of equations, tables, and verbal rules. 10 Students could match graphs to equations, tables, and verbal rules and think in general terms about how equations determine the shape of graphs. specific errors. Students had difficulty giving a complete explanation of how to match a graph with an equation. Students gave vague explanations, such as matching the graph with a table. Students were not thinking about the general shapes of the graphs and the general equations that form those shapes. 53
56 Implications for Instruction Students should be able to understand the relationship between equations, graphs, rules, and tables. Students should know a variety of ways to check these relationships. Lessons should regularly focus on relating multiple representations of the same idea. It is important that algebraic ideas not be taught in isolated skill sets. Consider this quote from Fostering Algebraic Thinking by Mark Driscoll, One defining feature of algebra is that it introduces one to a set of tools tables, graphs, formulas, equations, arrays, identities, functional relations, and so on that constitute a substantial technology that can be used to discover and invent things. To master the use of these tools, learners must first understand the associated representations and how to line them together. A fluency in linking and translating among multiple represent seems to be critical in the development of algebraic thinking. The learner who can, for a particular mathematical problem, move fluidly among different mathematical representations has access to a perspective on the mathematics in the problem that is greater than the perspective any one representation can provide. Ideas for Action Research Review of the Literature Linking Multiple Representations Sometimes in the pressure to move through the curriculum, we as teachers rely too heavily on the sequence provided by our textbooks. It is important to occasionally step back and think about the subject as a whole and what are the important concepts we want students to develop. Consider taking time to read and to discuss some professional literature with colleagues. What are the important ideas being presented? What are the implications for the classroom? How can we design some specific activities or lessons to fit into our program that will help develop some of the ideas we have just read? Why is this important for students? One interesting resource related to this task would be Chapter 7 Linking Multiple Representations from the book, Fostering Algebraic Thinking. Here are some key excerpts for consideration. Issues Regarding Understanding There are challenges in thinking algebraically that go beyond learning discrete pieces of information. Often, difficulties can arise when it is assumed that students are attaching the same meanings or making the same connections that are intended by the teacher. 1. Students may not see the links between different representations of a functional relation for example, the mutual dependence between a function s graph and equation, or between its table and equation. 2. Students may interpret graphs only point wise, not globally. 3. In the course of working on a problem, students may neglect to connect the representation back to the original problem context. The chapter then goes on to give examples of classroom lessons that help develop this relational thinking and interesting problems that can be used in the classroom. 54
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