Balanced Assessment Test 2009 Core Idea Task Score Properties and Soup and Beans Representations This task asks students write equation to describe objects on a scale. Students need to use proportional reasoning or equivalent expressions to find the ratio of beans to soup. Functions and Quadratic Representations This task asks students to work with quadratics in various forms, including substitution, graphical representations, and using algebra to solve the equation. Properties and Circles and Spheres Representations This task asks students to reason about equations and their graphs. Students should be able to distinguish functions on a graph by reasoning about the difference between multiplying by a number, the square of the number, or the number cubed. Successful students were able to solve an equation for one variable in terms of another. Functions Words and Equations This task asks students to read a simple scenario and match it to an equation representing the situation described. Successful students could work with equations involving subtraction and division as well as addition and multiplication. Functions and Coffee Representations This task asks students to use a chart to solve situations involving simultaneous equations. Successful students are moving from guess and check to using algebra to express relationships and solve equations. Copyright Noyce Foundation 1
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Soup and Beans This problem gives you the chance to: make an equation and solve a problem The weight of one can of beans is x ounces. The weight of one can of soup is y ounces. Beans Beans Beans Beans Beans 1. Write an expression for the weight of the cans on the left hand side of the weighing scales. Soup Soup Soup Soup 2. Write an expression for the weight of the cans on the right hand side of the weighing scales. Beans Beans Beans Soup Soup Soup 3. Write an equation that shows the relationship between x and y. Beans Beans Beans 4. Use your equation to find the Beans number Beans of cans of beans that balance 9 cans of soup. Soup Soup Soup Soup Show your work. Soup Soup Soup Soup Soup Soup Beans Beans Beans Soup Soup Soup Soup Soup Soup? 6 Copyright Mathematics Assessment Resource Service. Soup and Beans 4
2009 Rubrics Soup and Beans Rubric The core elements of performance required by this task are: make an equation and solve a problem Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Gives correct answer: 5x + 2y 1 2. Gives correct answer: 3x + 5y 1 3. Gives correct answers: 5x + 2y = 3x + 5y or 2x = 3y 1 4. Gives correct answers: 6 Shows correct work such as: 2x = 3y 6x = 9y 1 3 Total Points 6 2 1 1 1 Copyright Mathematics Assessment Resource Service. 5
Soup and Beans Work the task and look at the rubric. What are the key mathematical ideas being assessed? Look at student work for part 1 and 2, writing expressions. How many of your students: Wrote correct expressions for the pictures? Wrote expressions with exponents? Wrote numbers sentences (no variables)? Gave values for x and y? Other? What is confusing students about algebraic notation and the use of variables? Now look at student work for part 3, writing an equation. How many of your students: Wrote a correct equation? Combined like terms (8x +7y)? Gave values for x and y? Used exponents? Other? How do their misunderstandings about variable and notation effect their equations? What other misconceptions do you see as you examine their equations? Now look at their work in part 4. How many of your students put: 6 13.5 9 10 18 7 Other Now examine their strategies. What were some of the misconceptions that you noticed in their work? What strategies did successful students use? Copyright Noyce Foundation 6
Looking at Student Work on Soup and Beans Student A is able to write expressions and equations to represent the figures. The student is able to combine like terms in the equation using symbolic manipulation to find the equation 2x =3y. The student then realizes that he is not solving for x or y but looking for an equivalent ratio. Student A Copyright Noyce Foundation 7
Student B gives a complete description of how to write the expressions and equation. The student then finds the unit ratio, y =2/3 x. The student doesn t realize that the 9 is not the value of y, but the number of y units. So the 9 cannot be plugged into the equation to solve for x. This is a very subtle distinction. What question can we pose in the classroom to help students explore this idea? Student B Copyright Noyce Foundation 8
Student C is able to write the expressions and equations. However the student seems to assign values to the soup and beans to reach a numeric value for x and y. The beans appear to be given a value 0.5. See the work in the top diagram. Can you figure out what the student is doing? Student C Copyright Noyce Foundation 9
Student D tries to combine like terms to form the equation in part three. Notice that the correct equation is under the diagram on the right. What misconceptions does the student have about solving equations? The student then uses the incorrect equation to try to solve for x. Student D Copyright Noyce Foundation 10
Student E has also tried to combine like terms and then generalize about the values. Can you follow the reasoning that led to the equation x = y +1? What would be your next steps with this student? Student E Copyright Noyce Foundation 11
Student F writes the meaning of the variable instead of the expression in parts 1 and 2, but is able to build a correct equation in part 3. The student tries to use a table to solve for the cans of beans in part 5. Students often have trouble setting up tables for themselves, because too often the logic of designing the table is done by the textbook. Students pick up only pieces of the logic of making a table. How could the table be modified to help the student find the correct number of cans? Student F Copyright Noyce Foundation 12
Student G uses number sentences instead of writing expressions in part 1 and 2. This means that the equation will have a numerical value or be in a familiar form. What is the conceptual understanding of equality that a student needs to make an equation from the diagram? Student G Copyright Noyce Foundation 13
Student H does not understand algebraic notation and uses exponents in the expressions and equation. Notice that in part 4 the student assigns a value to the soup and beans rather than solving for x and y. Student H Copyright Noyce Foundation 14
Student I makes the assumption that x and y are equal. What question might you pose to get the student to re-examine his thinking? What evidence might the student notice that would contradict this idea? Student I Copyright Noyce Foundation 15
Task 1 Soup and Beans Student Task Core Idea 3 ic Properties and Representations Make an equation and solve a problem. 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. Use symbolic algebra to represent and explain mathematical relationships. Mathematics of this task: Writing an expression from a diagram Understanding the relationship of equality in the context of a balanced scale Using equivalent ratios in context Distinguishing between a numerical coefficient and a value for the variable Understanding when there is sufficient information to solve for a variable Understanding meaning of variable or unknown Based on teacher observations, this is what algebra students know and are able to do: Write expressions for diagrams Write equations for the diagram Areas of difficulties for algebra students: Trying to assign values to x and y Trying to combine like terms from different sides of an equality Understanding the concept of equality illustrated in the diagram Using relational thinking to find equivalent ratios Copyright Noyce Foundation 16
The maximum score available for this task is 6 points. The minimum score needed for a level 3 response, meeting standards, is 3 points. Most students, 87%, could write expressions for each side of the diagram. More than half the students, 69%, could also combine the expressions to make an equality statement about the diagram. 31% could meet all the demands of the task including using the new diagram and equivalent ratios to find the number of cans of beans to equal 9 cans of soup. Almost 11% of the students scored no points on this task. All of the students in the sample with this score attempted the task. Copyright Noyce Foundation 17
Soup and Beans Points Understandings Misunderstandings 0 All of the students in the sample with this score attempted the task. Students struggled with writing expressions from the diagram. Some students wrote number sentences, such as 5 + 2 = 7. Some students used exponents, such as x 5 + y 2. Some students tried to solve, giving answers 2 Students could write expressions to represent the relationships in the diagram 3 Students could write expressions and equations for the situation depicted by the balance scale. 6 Students could write expressions and equations for the situation depicted by the balance scale. Students realized that 9 was the number of cans, not a value for y and could use equivalent ratios to find the number of cans of beans. like x=5 or y = 2. Students struggled with writing the equation represented by the balance scale. 7% of the students wrote the expression 8x + 7y. Some students assumed that x = y or that x=y +1. Some students had equations with exponents. Others gave values for x and y. Students did not know how to combine the information in the diagram and the equation to find the number of cans of beans. 10% of the students thought the answer was 10. 10% thought the answer was 9. 8% thought the answer was 13.5. Many students got 2x = 3y and did no further work. 7% tried to substitute in 9 for y to get 2x = 27. Copyright Noyce Foundation 18
Implications for Instruction Students need more experiences using algebra to describe relationships. They should be able to understand and use equal value to write equations. Some students struggle with the basic concept of variable as a symbol describing a relationship between objects versus an answer. Students need more opportunity to work with algebraic ideas in context. Students should be able to recognize equalities in situations, such as using a balance scale. Many students tried to combine the x s to equal the y s. They didn t think about the left side of the balance being equal to the right side of the balance. Students need to understand the difference between a numerical coefficient and the value of the variable. Students saw the total of 8x and 7y and tried to define relationships with expressions like x = y +1. Students were not thinking about multiplicative relationships. They also missed basic principals of symbolic manipulation, such as using inverse operations. We want students to develop instincts about when there is enough information to solve for x or y and when there isn t enough information. This will help them when confronted with unusual problems like this one, where they needed instead to find equivalent ratios. Context is so important to developing a deeper conceptual understanding of algebraic principles because so many other ideas are brought into play that don t arise when just following procedures. Ideas for Action Research The Berkeley Lesson Study team at Willard Middle School developed a lesson on looking at numeric and symbolic strings. The Lesson, Things that Mingle, was taught by Jacob Disston. The idea was to help students begin looking at the expressions and equations as mathematical objects with varying attributes. This lesson is designed to push students to make connections between ideas about equations, inequalities, and expressions. The lesson gives students opportunities to use mathematical vocabulary for a purpose: to describe, discuss, and work with these number strings. In the process it is hoped that the students begin developing instincts about these symbolic, numeric strings, what might be used for. If this were on a test what might be the question? What is the purpose of looking at numeric strings as objects and gathering global information about the whole number string rather than thinking only about individual procedures or steps? Hopefully students will begin to see the number strings as mathematical objects with their own unique set of attributes. The lesson is designed to go across several standards in the algebra strand and to help students make connections across standards. The lesson also explores various pedagogical strategies aimed at increasing student involvement, particularly from the African American and Latino students to help close the achievement gap. Students first need to walk around the room and interact with each to find people with number strings. Students are then asked to huddle at the board to discuss groupings. Cards are used for recording so that they can be moved easily if different students make different connections. Students are then placed in groups to have deeper conversations. They are also given opportunities to write and reflect as the lesson progresses. Copyright Noyce Foundation 19
Video, worksheets, discussion, student work, and a complete lesson plan are available on the Noyce website, Inside Mathematics: a professional learning community for educators. Sit down with colleagues and look at key parts of the lessons and lesson goals. Then discuss how the lesson changed the thinking of students. What changed about the attributes that they discussed? How were opportunities for conversation built into the lesson? How did features of the lesson increase the level of student participation? How might some of the ideas in this lesson help some of the struggles and misunderstandings that you saw in student work on this task. www.insidemathematics.org Copyright Noyce Foundation 20
The Math Things Mingle 7th grade mathematics lesson Willard Middle School to be taught by Jake Disston Thursday January 22, 2009 Mathematical Topic: ic Symbol Strings Making sense of the differences between equations, inequalities and expressions. Lesson Objectives: To have students sort and classify various types of equations, inequalities and expressions. To help students develop instincts and vocabulary for describing, discussing and working with equations, inequalities and expressions. To give students an opportunity to discuss and express what they know about various mathematical objects without providing a script or a procedure for them to follow. Lesson Background: Our lesson study group began the year by creating a list of big ideas that we feel students need to be comfortable with in order to succeed in 8 th grade. We deliberately focused on ideas that were big and conceptual and which from our perspective reached across specific standards and through the middle school grades, rather than focusing on specific skills or procedures described in one particular standard and belonging to one grade level. We distilled our list down to four ideas that we felt were essential for students to have significant experience with and of which they should have a strong understanding by the end of 7 th grade: variables equality operations different types of numbers This year our district adopted a new text book published by Holt. In our lesson study group, our discussions this year have largely focused on the need to balance the procedurally based material presented in the text book with opportunities for students to make sense of the big ideas in a way that enables them to both ground their mathematical understanding in the context of their own lives, and to make connections between ideas so that the knowledge they are building is coherent. Our discussions and work have also been influenced by the pedagogical practices associated with the Center For Culturally Responsive Teaching and Learning, an organization that is providing professional development at our school aimed at helping teachers incorporate routines and activities that enhance the engagement of African American and Latino students and ultimately reduce the achievement gap that is so apparent at our school. Our lesson is adapted from an activity that Ann Lawrence presented at the 2007 California Math Council Conference at Asilomar. The activity is called See what your kids know and presents students with a set of 12 cards, each containing a mathematical symbol string equations, including single and double variable equations, formulas, identities, and also expressions. Students are asked to sort the cards into sets that make sense to them, to describe the attributes that the items in the set share, and to create new examples that fit within each of the sets. We like the activity in that it creates an opportunity for students to apply their own understanding and language to work with a variety of mathematical objects simultaneously, rather than presenting a single idea and dictating what students do and how they do it, as typical text book activities do. Our plan tries to address two main concerns that we have: 1) that we create a set of cards that are appropriate for 7 th graders and which highlight the mathematical concepts of variable and equality, and 2) that we create a structure in which all students can become engaged, and that the teacher fosters a discussion in which students articulate their understanding of the ideas and connections between them that go deeper than just surface level, trivial observations at which students often stop.
Our Lesson: We will start by passing a card to each student. Each card will contain a single symbol string an equation, an inequality or an expression. Students will be asked to stand up and mingle with each other, trying to find all the other students who have a symbol string that they feel is similar to their own. The teacher will circulate, encouraging students to explain what they are seeing and how they are grouping together. Once students have grouped together into sub-sets, the teacher will call all students to the board for a huddle. On the board there will be set of the same cards. The teacher will foster a discussion in which students share their initial thoughts about the cards and the subgroups into which they can be sorted. The object of this huddle will be to allow the class to come to some consensus about the sorts of attributes that are important in sorting and distinguishing the various symbol strings, and ultimately to establish at least the three top level sub groups: equations, inequalities and expressions. Next students will be asked to sit at a table with other students with the same top level type of card. The teacher will collect these cards, and will pass out a complete set of cards to each group and ask students to sort the whole deck into the top level sub-groups. Once they have completed the first sort they will be asked to answer a reflection question on their activity recording sheet: 1) how can you tell the differences between equations, inequalities and expressions? Copyright Noyce Foundation 22
4x + 2 "18 2x "16 5x "1 # 34 B 2 > 0 m 5 > 20 X 7 < 5 3x + 2y + 2x 3a 2b + 4c - a 4p + 3p 2p 5 ( x + 2) Copyright Noyce Foundation 23
3b + 4p + 17 3x + 2y + 4z Copyright Noyce Foundation 24
105 + 2a = 101 3x + 2x = 5x a + (b + c) = (a + b) + c F 1 = F X + Y = Y + X V ( W + X) = VW + VX 3 + 2n = 8 40 = 5x Copyright Noyce Foundation 25
-4m 8 3x - 4-8 2 4 m m 2 p > 7 3 m 3p + 7m 4n - 6 y = 2 (4 2 4 a ) b c x = ¼ 3 A = L W P = 2W + 2L Copyright Noyce Foundation 26
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
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 20 15 A 10 5-4 -3-2 -1 0 1 2 3 4 5 6 B -5 D E x -10 C -15 9 Copyright 2009 by Mathematics Assessment Resource Service. Quadratic 28
Quadratic Rubric The core elements of performance required by this task are: work with a quadratic function in various forms Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. a. Gives a correct answer: 5 4 16 7 b. Gives a correct answer: - 8 c. Gives a correct answer: y = (x 1) 2 9 = x 2 2x + 1 9 = x 2 2x - 8 2. a. Gives a correct answer: C b. Gives a correct answer: A and E c. Gives a correct answer: B and D 1 1 2 4 1 1 1 3 3. a. Gives a correct answer: -2, 4 b. Gives a correct answer such as: (x+2)(x-4) = 0, so x = -2 or 4 or uses completing the square to find 1 or 2 correct answers or uses quadratic formula. 1 2 Total Points 9 1 Copyright 2009 by Mathematics Assessment Resource Service. 29
Quadratic Work the task and look at the rubric. What are the key mathematical ideas being assessed? Look at student work on part 1b, using the using the number machine. How many of your students put: -8-10 0 7 10 Other What caused some of the student errors? What did they struggle with? Now look at work on part 1c, using algebra to show that the number machine is equivalent to the equation y = x 2 2x 8. How many of your students: Wrote an expression for the number machine and then carried out the algebraic manipulations to show that the result was equal to the equation? How many tried to use substitution of values to show that both would give the same answer? How many students were unwilling to attempt this part of the task? How often do students in your class get opportunities to use algebra to make a proof or justification? Do students get opportunities to rectify different equations that come up when identifying patterns? What do students in your class seem to understand about justification? Look at work on part 2b. How many of your students put: A & E E B & D B C All No Other answer What are some of the misconceptions behind these responses? Now look at work on part 2c. How many of your students put: B &D D C B A &E All No answer Other Now look at student work on part 3a. How many of your students could use the graph to find the correct values for x? How many students made errors with the signs for x? How many of the students were not able to give a response? What other errors did you see? 30
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
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
Student A, continued 33
Student B tries to use substitution to prove that the number machine is equal to the equation y = x 2 2 8. 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
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 2 2 8. 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
Student C 36
Student D tries to use examples to prove the equality between the equation y = x 2 2 8 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
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
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
Student F 40
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
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
Student H, continued 43
Student I struggles with calculations. The student should not be given credit for the solution in 1a because the 4 is never squared. 4-9 7. 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
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
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
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 -10. 8% 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
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
Now students have some background to help them re-engage with the mathematics of the task. Students did not understand what it meant to use algebra to prove that the number machine is the same as y=x 2 2x 8. I might start by showing them the work of Student B: I saw this work on a student s paper. What do you think the student was doing? I would give students first some individual think time and then let them share their ideas with a neighbor. The idea of re-engagement is to maximize the amount of talking students get to do, so that they can talk their way into understanding the concept. After everyone has had a chance to form and verbalize an opinion, I would then open up the question to a group discussion. If it doesn t come out naturally, I would also add a prompt to get students to question whether or not this is a proof. We want students at this age to understand that it is impossible to test for every possible case and that proving it for one or even many cases is not a proof. A good reading on this topic is found in Thinking Mathematically by Carpenter, Franke, and Levi, chapter 7 Justification and Proof along with videos of young students making justifications. It has some good questions to push students thinking about proof. Is that always true? How do you know that is true for all numbers? Okay, so we have seen that it works for a lot of numbers, but how do we know that there is not some number maybe a very, very big number that it will not work for? How would you get students to move from this idea to actually setting up an equality? Work with colleagues to think about the types of questions that would help students know what to set equal and how to rectify the two sides. Are there any snippets of student work that you could use from your own student work or the toolkit that would help you? Students also had difficulty with the idea of using algebra to find the solution to the equation y=x 2 2x 8. Many students used substitution, which only works if you already know the answer or can read the answer off the graph. How can we push students to think beyond this. I might start with a prompt: Dawn says that she can use factoring to find the solution to the equation. What do you think she did? This prompt gives a strategy but allows all students now the opportunity to try and apply what they know. A follow up question might be: How does this help her find a solution? 49
Next I might want students to confront a common misconception. So I might use the work of Student E: Why do you think Earl s paper is marked incorrect? What mistake has Earl made? Finally, I might push students by saying that Anita said that it was just easier to use the quadratic formula. What do you think she did? How did this help her? After students have had a chance to think through this strategy, I think it is important for them to compare strategies. I might ask them to say which strategy they think is easier and why? Then I might give them a different equation that is not factorable so that they can see that sometimes the quadratic is a better choice. I want them to start to see the purpose for knowing more than one strategy and for thinking about making decisions up front about picking a strategy that is convenient and easy for the particular situation. Look through your student work and work in the toolkits. Are there any other misconceptions that you want students to discuss explicitly? Are there some more important mathematical ideas that you want to address? How could you use student work to pose questions to address those issues? What kind of question do you want to pose at the end of lesson to get students to reflect on what they have learned? Is there a follow up problem that you want to use in a few days to see if students can transfer what they have discussed to a new situation? Would you consider giving students back their original work with a red pen and allow them to edit their papers? How would this help further their understanding? 50
Circles and Spheres This problem gives you the chance to: compare functions for length, area and volume rearrange formulas 1.The diagram shows the graphs of three functions: Area of circle against radius Volume of sphere against radius Circumference of circle against radius The formula to calculate the volume of a sphere from its radius is V = 4 / 3 πr 3 where r is the radius and V is the Volume In each case, the radius varies from 0 to 5 units, A B C 0 1 2 3 4 5 Radius For each letter, choose the correct description of the graph from the list above. A B C Say how you figured it out. Copyright 2009 by Mathematics Assessment Resource Service. Circles and Spheres 51
2. The formula to calculate the area of a circle from its radius is A = "r 2, where r units is the radius and A square units is the area. Which of the formulas below works out the radius of a circle from its area? Say yes or no for each one. r = " A r = A 2" r = 2" A r = A " 3. The formula to calculate the circumference of a circle from its radius is C = 2"r, where r units is the radius and C units is the circumference. Find a formula to work out the radius of a circle from its circumference. 8 Copyright 2009 by Mathematics Assessment Resource Service. Circles and Spheres 52
Circles and Spheres Rubric The core elements of performance required by this task are: compare functions for length, area and volume rearrange formulas Based on these, credit for specific aspects of performance should be assigned as follows 1. Gives correct answers: A: Volume of sphere against radius B: Area of circle against radius C: Circumference of circle against radius Gives correct explanations such as: C shows a linear relationship so represents length B and A must represent the quadratic relationship for area and the cubic relationship for volume comparing them shows B is the quadratic and C is the cubic graph. May calculate some values such as: R points 3x1 2 or section points 2 Partial credit 1 error 2. Gives correct answers: No, no, no, yes Partial credit One error 3. Gives correct answer: r = C 2" (1) 2 (1) 1 Total Points 8 1 5 2 Copyright 2009 by Mathematics Assessment Resource Service. 53
Circles and Spheres Work the task and look at the rubric. What are the key mathematical ideas that a student needs to understand to be successful on this task? Look at student work on part 1, matching the descriptions to the graphs. How many of your students put: Graph Volume Circumference Area Other A B C Now look at student explanations. How many of your students: Used an argument based on the power of the radius? Used an argument based on calculations for all 3 measures? Calculated only some of the measures? Used arguments based on shape of the graph or shape of the object? Didn t attempt an explanation? Now look at student work on part 2 and 3. Did students show making the conversions? Did student responses indicate a confusion about the meaning of exponents? How many of your students put r = (2π)/C? What misconception do these students have? What was the faulty logic in their solution process? How many of your students solved for C instead of 4 in part 3? How many of your students did not attempt part 3? Why do you think students had difficulty with this section? List some incorrect responses and see if you can identify the misconceptions associated with the response. 54
Looking at Student Work on Circles and Squares Student A is able to match the graphs to the descriptions by computing values for the same radius. The student is also able to manipulate equations to solve for different variables. Student A 55
Student A, part 2 Student B is able to match the descriptions to graph by reasoning about the power of the radius. The student struggles with symbol manipulation in part 2 and 3. While the student shows a good understanding of the effect of exponents on size, the errors in solving equations may indicate some other misunderstanding of exponents. Where would you go next with this student? 56
Student B 57
Student C has managed to match the descriptions with the graphs. However the explanations are unclear and don t give much indication of understanding. What is the misunderstanding shown by the answer for part 3? What doesn t the student understand about solving equations? Student C 58
Many students tried to use reasoning about the shape of the objects or graphs to help match the descriptions to the graphs. Student D uses language about rolling up. The student is also concerned that the volume isn t a function. The student looks at the formulas, but doesn t choose to quantify values. What should students know about the shape of a graph from looking at an equation? Student D 59
Student E is also concerned about the shape of objects and their graphs. In part 2 and 3 the student, while not required, shows the steps for solving the equations for the new variables. Student E 60
Student E, part 2 61
Student F also tries to use reasons about shape. Student F Student G is able to reason about volume being largest, but can t continue that train of reasoning to the other measures. Student G 62
Student H cannot find even the largest of the measures in part 1. In part 3 the student finds 2 new formulas for C instead of solving for r. Student H Student H, part 2 63
Student I thinks about volume as the space under the graph. The student misses the context of the problem in part 1. The student scores no points in part 2 and doesn t attempt part 3. Student I 64
Student J scores no points on the task. However the student has the idea of substituting values into the formula. Why do you think the student didn t finish the calculations or use the other formulas? Student J 65
Task 3 Circles and Spheres Student Task Core Idea 3 ic Properties and Representations Core Idea 2 Mathematical Reasoning Core Idea 1 Functions and Relations Find numbers that obey given rules or constraints. Find rules for sets of numbers. Use understanding of place value to solve problems in context. Represent and analyze mathematical situations and structures using algebraic symbols. Use symbolic expressions to represent relationships arising from various contexts. Compare and contrast the properties of numbers and number systems including real numbers Employ forms of mathematical reasoning and proof appropriate to the solution of the problem, including deductive and inductive reasoning, making and testing conjectures and using counterexamples and indirect proof. Show mathematical reasoning in a variety of ways, including words, numbers, symbols, pictures, charts, graphs tables, diagram and models. Draw reasonable conclusions about a situation being modeled. Understand relations and functions and select, convert flexibly among, and use various representations for them. Mathematics of the task: Know and be able to calculate volume, area, and circumference Reason about shapes of graphs, particularly relative to linear functions and quadratic or cubic functions Solve equations for different variables Based on teacher observations, this is what algebra students knew and were able to do: Recognize that volume would show the steepest growth curve Match the graphs to the descriptions Solve area formula for the radius Areas of difficulty for algebra students: Explaining why the descriptions matched the graphs Giving quantities for some measures, but not for all to make the comparison Solving a formula for a different variable 66
The maximum score available for this task is 8 points. The minimum score for a level 3 response, meeting standards, is 5 points. Most students, 83%, could match volume to the graph. More than half the students, 58%, could match all the descriptions to the graphs. Some students, about 32%, could match the graphs to the descriptions and do some successful work on solving formulas for new variables. 6% of the students could meet all the demands of the task including justifying the matching of the descriptions to the graphs using calculations or reasoning about the power of r. 17% of the students scored no points on this task. 77% of the students with this score attempted the task. 67
Circles and Spheres Points Understandings Misunderstandings 0 77% of the students with this score attempted the task. Students could not recognize that volume was graph A. Almost 15% thought graph A was area. Almost 10% thought graph A 1 Students could match volume to the graph. 3 Students could match descriptions to the graphs. 5 Students could match descriptions to the graphs and do some of the solving of equations for new variables. 8 Students could match descriptions to graphs and use calculations or reasoning about exponents to support their choice. Students could take a formula and solve for a different variable in parts 2 and 3. was the circumference. Students had trouble discerning between the graph for area and circumference.25% of the students thought graph B was circumference. 10% thought it was volume. 20% thought graph C was area. 14% thought graph C was volume. Students struggled with solving formulas for variables. Most students picked 2 correct responses for part 2, giving them partial credit. Most common extra choices were r= A/2π and r= 2π/A. Students did not know how to explain their reasoning for ordering the descriptions. 68
Implications for Instruction Students at this grade level come to algebra with a weak background in geometry. Students need to build up their geometric foundation by exploring geometric properties using algebra. This context provides students with opportunities to think about how the radius effects the size of three different measures. Many students gave explanations about the shape of the sphere or circumference rather than thinking about the measures of area, circumference, and volume. Some students gave explanations that showed they thought radius was only connected with one of the measures or a showed no understanding of the word radius. Students should understand that quantifying answers or values makes it easier to compare. They should also realize that something needs to remain constant to make the comparison between measures. For example if the volume is calculated with a radius of 4 and the other two measures need to be calculated with a radius of 5. Otherwise no direct comparison is possible. As students build up algebra concepts they need to connect them to multiple representations. For example, students should think about how equations with exponents are not linear, but curve. They should also be able to reason about how the cubing a number would make the curve steeper. Some students thought that volume was the space underneath the graph lines. Students should be able to notice attributes of equations and get general impressions about the look of the graph. Students do not know how to solve a formula for a variable. They have some partially learned procedures, but are not consistent in applying the algebra rules for symbol manipulation. Students still struggle with expressions with squares. Many try to equate square with multiplying by 2. Ideas for Action Research This task would lend itself to a re-engagement lesson. Re-engagement Confronting misconceptions, providing feedback on thinking, going deeper into the mathematics. (See overview at beginning of toolkit). 1. Start with a simple problem to bring all the students along. This allows students to clarify and articulate the mathematical ideas. 2. Make sense of another person s strategy. Try on a strategy. Compare strategies. 3. Have students analyze misconceptions and discuss why they don t make sense. In the process students can let go of misconceptions and clarify their thinking about the big ideas. 4. Find out how a strategy could be modified to get the right answer. Find the seeds of mathematical thinking in student work. The story of this problem seems to center on students not understanding how to justify matching the descriptions to the graph. The teacher might start by saying: David thinks that circumference is larger than area because the area is 28 and the circumference is 31. Can you figure out how David got his answers? 69