Tom wants to find two real numbers, a and b, that have a sum of 10 and have a product of 10. He makes this table.


 Aubrie Newman
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1 Sum and Product This problem gives you the chance to: use arithmetic and algebra to represent and analyze a mathematical situation solve a quadratic equation by trial and improvement Tom wants to find two real numbers, a and b, that have a sum of 0 and have a product of 0. He makes this table. Sum of a and b a b Product of a and b a. How can Tom find b if he knows a? b. Draw a ring around any of these algebraic statements that express the relationships between a and b correctly. a + b =0 b = a +0 b =0 " a a =0 " b a =0b b =0a ab =0 b = 0 a! 2. Explain what happens to the product of a and b (in the last column of the table) as a increases from 0 to0.! Copyright 2007 by Mathematics Assessment Page 36 Sum and Product Test 0
2 3. Tom looks at the table and says, If the product of a and b is 0, a must be somewhere between and 2 or between 8 and 9. a. Explain how the table shows this. b. Tom tries to find the value between and 2. He decides to try a =.5 Complete this table to show his calculations. Sum of a and b a b Product of a and b 0.5 c. Which of these two statements is correct? Explain how you decided. a must be between and.5 a must be between.5 and 2 d. Find the values of a and b correct to two decimal places. Show your calculations. 8 Copyright 2007 by Mathematics Assessment Page 37 Sum and Product Test 0
3 Sum and Product Rubric The core elements of performance required by this task are: use arithmetic and algebra to represent and analyze a mathematical situation solve a quadratic equation by trial and improvement Based on these, credit for specific aspects of performance should be assigned as follows. a. Gives correct answer: Subtract a from 0 or divide 0 by a. points section points b. Draws rings around! a + b =0 b = a +0 b =0 " a a =0 " b a =0b b =0a ab =0 b = 0 a 2 2. Gives correct! explanation such as: The product starts at zero, increases up to a maximum of 25, then decreases back to zero. 3.a b c d Gives correct explanation such as: The product is less than 0 when a is and more than 0 when a is 2, so a must be between and 2. Similarly for 8 and 9. Gives correct answer: 8.5, 2.75 Gives correct explanation such as: The product is less than 0 when a is and more than 0 when a is.5, so a must be between and.5 Gives correct answers:.3 and 8.87 Shows correct work such as:.3 x 8.87 = x 8.88 = Total Points 8 5 Copyright 2007 by Mathematics Assessment Page 38 Sum and Product Test 0
4 Sum and Product Work the task and look at the rubric. What are the key mathematical ideas being assessed in this task? What do you think students might struggle with? Look at student work for b. How many of your students recognized and circled the multiplication and division relationships for a and b? Why do you think so many students struggled with this? Now look at the explanations in 2, 3a, and 3b. What was missing from students explanations? Did they focus on a and b instead of the product? Did they forget to quantify and describe how the pattern changes? Did their explanations make sense or did they describe something irrelevant? Did they describe a pattern that didn t fit all the data? What types of opportunities do students have in your class to make sense of a table of data and describe trends? How do you help students understand the qualities and expectations about mathematical explanations or descriptions? What kinds of class discussions do students engage in to help them see differences in qualities of explanations and learn the logic of a mathematically convincing argument? Look at work for part 3d. How many of your students did not attempt this part of the task? How many of your students picked values for a and b that did not add to 0? How many of your students picked negative values for a or b? How many of your students chose numbers with only decimal place? How many of your students only tried or showed evidence of one set of values, rather than testing for closer combination or to check that their solution was optimal? How many of your students used their response from the table in 3b? How many of your students tried to use algebra to solve for a and b, but got stuck because they couldn t factor the expression or didn t know what to do after they made an equation? What are the implications for instruction that you see from looking at this work? What are the big ideas that students are missing? Geometry
5 Looking at Student Work on Sum and Product Student A is able to write very specific explanations. The student is also able to use algebra to solve for a and b in part 3d. Student A Geometry
6 Many students attempted to use algebra to solve for a and b, but struggled because they did not remember the quadratic formula. See the work of Student B. Student B Geometry
7 Student C had difficulty working with decimals. The student subtracted incorrectly in 3b. The student incorrectly uses a negative sign in the quadratic formula. Student C does not check to see if the answer makes sense. A brief glance should show that the values do not at to 0 in his solution to 3d. Student C Geometry
8 Student D notices a pattern in the differences between the products, rather than trying to describe the products themselves. Do student sin your class get enough opportunities to make descriptions about trends? Student D Many students struggled with the demands in part 3d. Student E only tried one set of numbers and doesn t test to see if it can be improved. Student F finds the 2 closest sets of values, but does not understand decimal place value enough to pick the best choice. Student E Student F Geometry
9 Geometry Task 3 Sum and Product Student Task Core Idea 3 Algebraic Properties and Representations Use arithmetic and algebra to represent and analyze a mathematical situation and solve a quadratic equation by trial and improvement. Represent and analyze mathematical situations and structures using algebraic symbols. Solve equations involving radicals and exponents in contextualized problems. Mathematics of the task: Ability to read and interpret constraints and make sense of a table of values Ability to make mathematical descriptions of information in a table and quantify relationships Understand and use guess and check and/or use the quadratic formula to find an optimal solution to match the constraints of the task Recognize multiple representations of expressions for two sets of constraints to describe the relationship between two variables Based on teacher observations, this is what geometry students knew and were able to do: Solving for b and finding the product of a and b in part 3b of the task. Describing how to find b in part a of the task Describing why a must be located in a particular place on the number line based on values in the table in 3a Areas of difficulty for geometry students: Calculating and comparing with decimals Identifying algebraic expressions to show the relationship between a and b if their product is 0 Describing the trends in the product column of the table Explaining why a must be between and.5 Finding the values for a and b to two decimal points Solving a quadratic expression that doesn t factor Geometry
10 The maximum score available for this task is 8 points. The minimum score for a level 3 response, meeting standards, is 4 points. Most students 94% could fill out the guess and test table for a and the product of a and b in part 3. More than half the students, 62%, could find use division or subtraction to find a in part, describe how the product changes as the value of a increases from0 to 0, explain why the value of b falls between two values on the table, and fill in a guess and test table. 7% of the students met all the demands of the task including identifying 5 ways to express the relationship of a and b using both constraints of the problem and calculate values of a and b to two decimal places to come closest to a product of % of the students scored no points on this task. 75% of these students attempted the task. Geometry
11 Sum and Product Points Understandings Misunderstandings 0 75% of the students with this score attempted the task. Students had difficulty using the guess and test table in 3b. They couldn t compute Students could subtract.5 from 0 to solve for b and multiply with decimals to complete part 3b of the task. 4 Students with this score could usually describe the product trend in 2, explain why a is between certain values in 3a, fill in the table in 3b, and describe why the value of a is located between and.5. 5 Students could find the value of b using algebra, describe the trend in products, fill in the table, and describe where a was located. accurately with decimals for find b. Students struggled with describing the pattern in the products in the table. Students did not quantify information, giving examples such as a and b cycles, so the product also cycles. Some students gave inaccurate trends, such as the product is always 0 or the product always increases. 9% of the students did not attempt this part of the task. Many students struggled with describing how to find a in part. 9% said to use a formula. Some said it s a pattern. Others could not express the mathematics using descriptions reversing the operation, such as subtract 0 from a, or using the wrong variable, subtract b from 0. Some students used both variables in their rules, like subtract b from a or multiply a and b. Students struggled with finding all the equivalent expressions in b and solving for a and b to two decimal places. 37% did not attempt to solve for a and b in 3d. 0% attempted to use algebra, but could not complete the process using the quadratic formula. 7 Students could not find the multiplication and division expression that met the constraints for a and b in part b of the task. 70% of all students omitted these options. 8 Students could use a guess and test table to think about the relationships between two variables with different constraints. They could describe trends and reason about the value of a that would meet all the constraints. They could find the value of a and b to two decimal places. Geometry
12 Implications for Instruction Students at this grade level should be comfortable doing calculations with decimals. Many students struggled with subtracting decimal values from 0 to find b. Are there structures in place in your school geometry program to allow students to fill in holes or misconceptions in their arithmetic skills? Can students drop in and out of remedial or ramp up: classes to work on specific topics like decimal calculations, percents, or expressions and equations? Students at this grade level should understand operations, knowing the inverse relationship between addition and subtraction and the inverse relationship between multiplication and division. Students should be able to express these relations in words and symbols. Students should routinely have opportunities to solve rich tasks which incorporate a variety of current and past knowledge to solve nonroutine problems. Part of the problemsolving repertoire should involve identifying all the constraints needed in the solution. Another part of the problemsolving repertoire should include asking themselves questions, such as is this the optimal solution?, can I get closer?, how do I know? Helping students gain this skill of selftalk is important. Good students seem to have this idea, but making the ideas of selftalk explicit, helps all students to pick up this strategy. A great deal of time is devoted in traditional American algebra textbooks to factoring quadratic expressions to find solutions to the problem. In many European texts, more time is devoted to going straight to using the quadratic formula, which will work for all cases. This would be a good topic for discussion in planning for an Algebra Course or to consider when evaluating Algebra texts. How would this allow more time for other important algebraic topics? What topics would you like to see developed in more detail? Geometry
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