E B F C 17 A D A D 16 B A G C 11

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1 Mystery Letters This problem gives you the chance to: form and solve equations A A A A 8 E B F C 17 A D A D 16 B A G C In this table, each letter of the alphabet represents a different number. The sum of the numbers in each row is written on the right hand side of the table. The sum of the numbers in each column is written below the table. Find the number represented by each letter. A = B = C = D = E = F = G = Show how you figured it out. 7 Copyright 2007 by Mathematics Assessment Page 80 Mystery Letters Test 7

2 Task 5: Mystery Letters Rubric The core elements of performance required by this task are: form and solve equations Based on these, credit for specific aspects of performance should be assigned as follows Gives correct answers: A = 2, B = 1, C = 5, D = 6, E = 4, F = 7, G = 3 Partial credit 6 or 5 correct values 4 points 4 or 3 correct values 3 points 2 correct values 2 points 1 correct value 1 point points 5 (4) (3) (2) (1) section points Shows some correct work. 7 Total Points 7 2 Copyright 2007 by Mathematics Assessment Page 81 Mystery Letters Test 7

3 Mystery Letters Work the task and look at the rubric. What are some of the big mathematical ideas students need to understand to work this task? Look at student work. What strategies did students use to solve the problem? Algebra All numbers on chart and calculations for each letter- Calculations for each letter- All numbers on chart, calculations for A only Calculations for A only All numbers on chart, no calculations Calculations for some letters No work What are your class norms for showing work? How does showing work help students to selfcorrect? What qualities did you like in good work? How do you help students to develop better, more efficient strategies? Look at the student work for students with scores of zeroes. Can you identify what they did not understand about the prompt? Did they do an improper operation? Did they not understand that all the A s or all the B s should represent the same number? Did they put the value from the margin that was closest to that A? Did they total all the A s? all the B s? Other? These errors represent some key basics for understanding a variable and being ready for algebra. How do you help students build this foundational piece? What might be some good next steps with these students? 7th grade

4 Looking at Student Work on Mystery Letters Students showed a variety of strategies for solving for the unknowns in Mystery Letters. Student A was the only student in the sample to use algebra to solve the problem. Student B shows the careful and logical calculations for the solving of each letter. Student C explains the first three logical choices for calculating for the unknowns, then moves to a guess and check strategy using the chart. Student D uses all guess and check. Student A 7th grade

5 Student B 7th grade

6 Student C 7th grade

7 Student D Student E is able to calculate A and substitute the value for A into all the locations in the chart. Then the student seems to guess a value for C and D that will meet the constraints of column four (A+C+C+D). However, the student does not notice that the value for D will not work for the third row (A+D+A+D) even though the values are plugged in. How do we help students look for feedback within the problem to check for the accuracy of their work? Can you find other places that should have clued the student about errors in his thinking? 7th grade

8 Student E 7th grade

9 Student F does not fill in the top row. This leads to confused thinking in other parts of the task. The student can solve for D using the third row, but forgets to add in the value of A when trying to solve for C in the 4 th column. Again, how do students build habits of mind that will help them check for errors in their thinking? Student F Student G has trouble thinking through the logic of what to solve for second. Being able to look at a list of clues and decide on order for using them is an important skill. Students at this grade level should not always be just following through a set of clues in the order given. They should be developing logic skills to look for which clues would be easiest to use. Student G solves for A. Then looking at the first column the student mistakenly tries to make E and B have the same value. This error then throws off the values in the rest of the task. What are the best clues to use after solving for A? Embedded in this error is another big mathematical idea, that variables or unknowns represent different values or different relationships. 7th grade

10 Student G 7th grade

11 Student H does not understand the constraints of the task. The student does not see the numbers outside the chart as totals, but as the possible values for the variables. Student H While Student I appears to be thinking about the constraints of the problem by showing an addition problem, the students answers are actually showing a strategy similar to H. Notice that even the tally marks have a different total than the equation. Student I s problem might be related to not being able to sort through which clue to use first. Would the student have been helped to look at row 1 first instead of column 1? Student I 7th grade

12 Student J seems to use a similar strategy for picking the value of the variable and then calculates a total for the number of times that variable appears in the table. Student J 7th grade

13 Student K does not understand that each letter needs to represent the same value each time, a fundamental principle for making sense of unknowns and solving equations. Like Student J, Student K totals all the values for each letter. What might be a next step to help this student think about unknowns? What task might you pose? Student K 7th grade

14 Student L may have done alphabet letter codes in the past. It is interesting that the values for A, B, and E actually work for the first column. How do you help students like this identify what the task is asking for and what the constraints are in the task? Student L 7th grade

15 7 th Grade Task 5 Mystery Letters Student Task Core Idea 3 Algebra and Function Form and solve equations in the context of a number puzzle. Understand relations and functions, analyze mathematical situations, and use models to solve problems involving quantity and change. Mathematics of this task: Understand an unknown as a set quantity or unit that may be repeated Use logic to sort clues into order of usefulness Set up equations or equivalencies to find values of unknowns Check values to see that they match a set of given constraints Based on teacher observation, this is what seventh graders know and are able to do: Find values for A,C,D and F Show some calculations to support correct answers Areas of difficulty for eighth graders: Checking values against all constraints Understanding the constraints of the problem, (each letter represented one numerical value and that value needed to be the same every time the letter occurred and that numbers outside the table represented totals Using algebra as a strategy for solving the problem Using logic to find the easiest clues first (many students tried to work clues with 2 unknowns first 7th grade

16 The maximum score available for this task is 7 points. The minimum score needed for a level 3 response, meeting standards, is 4 points. Many students, 80%, could find the value of one letter, usually A. More than half the students, 68%, could find two correct values and show work or find 5 or 6 values with no work. 48% of the students met all the demands of the task, including solving for 7 unknowns and showing some correct work. 19% of the students scored no points on this task. 68% of the students with this score attempted the task. 7th grade

17 Mystery Letters Points Understandings Misunderstandings 0 68% of the students attempted the task. 1 Students could find the value of one of the letters, usually A (A+A+A+A = 8). They could look through the clues to find the easiest one, 3 Students could solve for 4 values, but did not check work for against all the evidence. Students did not show work on papers to support their answers. 4 Students could either find 2 values with work or find 5 values with no supporting work. 5 Students found 4 values and showed some work to support their answers. 7 Students met all the demands of the task, including solving for 7 unknowns and showing some work to support their values. Students did not understand the constraints of the task. Many students just matched letters to one of the numbers outside the table. Half the students who missed A, picked a value from the sums. 18% of the incorrect answers in A were larger than any of the sums. Students did not know which clue to start from. Many tried to solve for the left column first instead of the top row. Students did not test their guesses against all possible information. Some students did not realize that all the B s or all the C s needed to stand for the same numerical value, so they would change the value in different locations to make it fit their original guesses. A Look at Most Common Errors A B C D E F G 8 and 1 4, 9, and 3 3, 7, and and 4 na, 5 and 1 na, 5,6,17 na,0,14, 5 na = no answer 7th grade

18 Strategies for Showing Work Numbers on Chart and calculations for each letter shown 13% Numbers on Chart, no calculations 15% Numbers on chart, calculations for A only 12% Calculations for A only 8% Calculations for every letter, no use of chart 10% Calculations for most letters, no use of chart 7% Use of algebra 1% No work shown 23% Implications for Instruction Students at this grade level should be developing skills in logical reasoning. They should be thinking about what do they know? What is easy to find? Will one set of clues be easier to use than another? Along with the reasoning skills, students should be developing habits of mind for checking their thinking. Have I checked my answers for all the evidence in the problem? Just because an answer for a letter works for row 3, will it also work for column 3? They should also have many opportunities to work rich tasks, requiring them to develop strategies or showing and organizing their work. Working with rich tasks, also helps students to develop perseverance, the idea that hard work or effort can eventually lead them to a solution to the task. Students should be building up the basic understandings about variables. They should understand that within a given context the variable should always stand for the same quantity. They should be thinking about variables as describing relationships or standing for different quantities. Some students were still struggling with mathematical literacy. They didn t understand the difference between the numbers outside of the box and their relationship to the numbers within the box. How do we help students to sort out meanings in context? Do students get ample opportunity to struggle with the meaning of word problems or do they work only with computational problems that are already set up for them? How do they learn to choose operations for themselves or develop an understanding of the meaning of operations? Reflecting on the Results for Sixth Grade as a Whole: Think about student work through the collection of tasks and the implications for instruction. What are some of the big misconceptions or difficulties that really hit home for you? If you were to describe one or two big ideas to take away and use for planning for next year, what would they be? 7th grade

19 What are some of the qualities that you saw in good work or strategies used by good students that you would like to help other students develop? Four areas that stand out for the Collaborative as a whole are: 1. Cognitive Demands of Making Information for Yourself Students had trouble generating information for themselves. In Suzi s Company, students couldn t distinguish data points from categories or frequencies. They couldn t unpack the information in the table. These issues do not even arise when working with a string of numbers without context. In Journey, students didn t understand how to plot distance on a graph. The thinking for making a graph is very different from the thinking to read information from a given graph. In Parallelogram, students had difficulty making measurements or measuring the correct dimensions for finding areas of parallelograms and triangles. While many students might have been able to label a right triangle with the correct dimensions, they were not able to draw a right triangle. 2. Logic Students had difficulty with the logic of a convincing argument for part 3 of Parallelogram. They left out information about size of known sides. They made false assumptions about the relationships between area and perimeter. They tried to use descriptions about looks, long or pointy, to make their arguments. In Mystery Letters, students weren t able to pick a logical order for using clues. Students didn t use all available answers for checking their answers (understand the logic of proof). 3. Matching Calculations to Descriptions In Work students had difficulty matching calculations to their descriptions. 4. Calculation Errors Students had difficulty finding percents and calculating with decimals. 7th grade