Pre-Algebra PoW Packet Anh s Code September 13, 2010 http://mathforum.org/pows/ Welcome! Standards The Problem This packet contains a copy of the problem, the answer check, our solutions, teaching suggestions and some samples of the student work we received in October, 2002, when Anh s Code first appeared. It is Library Problem #2756. The text of the problem is included below. A print-friendly version is available from the Print this Problem link on the current PreAlgPoW problem page. We invite you to visit the PoW discussion groups to explore these topics with colleagues. To access the discussions [log in using your PoW username/password], choose one of these methods: from your My PoW Work as a Teacher area use the link to PoW Member Discussions. go to prealgpow-teachers directly: http://mathforum.org/kb/forum.jspa?forumid=527 from the blue-shaded box, use the Tips/Ideas from Teachers link. Are you making the most of your PoW Membership? If you have an Individual Teacher Membership consider registering for one of our (free) Orientation Sessions to learn more about the features of your membership. Teachers with Class or School or District Memberships are welcome to take the free Orientation Session but also are encouraged to register for one of our online courses. View information, dates, and links to register here: http://mathforum.org/pd/ In Anh s Code, students are asked to decode the five mathematical operations symbolized in three equations and evaluate a given expression. The key concepts are operations, evaluating expressions, order of operations and exponentiation, although, students may have not yet learned that vocabulary word to identify that operation. If your state has adopted the Common Core State Standards, this alignment may be helpful: Grade 6: Expressions and Equations Apply and extend previous understandings of arithmetic to algebraic expressions.! 1. Write and evaluate numerical expressions involving whole-number exponents. Grade 7: Expressions and Equations Use properties of operations to generate equivalent expressions. Grades 6, 7, and 8: Mathematical Practices Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Additional alignment information can be found through the Write Math with the Math Forum service, where teachers can browse by NCTM and individual state standards, as well as popular textbook chapters, to find related problems. Anh s Code Each group in Anh s math class is making up a secret math code, where unfamiliar symbols stand for five different mathematical operations. Anh s group is going to challenge Aurora s group to crack the code used in the following equations: Aurora s group s results will be verified by asking them to evaluate this expression: 2010 Drexel University 1
What should they get? As you write your explanation of how you figured it out, use these capital letters to substitute for the symbols:! the heart is H! the star is S! the raindrop is R! the lightening bolt is L! the bull s-eye is B Answer Check The answer is 13. Your explanation should include how you determined what each symbol meant. If your answer doesn t match ours, did you remember that exponentiation is an operation? you might find this Dr. Math page on Order of Operations helpful information http://mathforum.org/library/drmath/view/58770.html did you check your arithmetic? If any of those ideas help you, you might revise your answer, and then leave a comment that tells us what you did. If you re still stuck, leave a comment that tells us where you think you need help. If your answer does match ours, have you clearly shown and explained the work you did? are you confident that you could solve another problem like this successfully? did you make any mistakes along the way? If so, how did you find them? are there any hints that you would give another student? Revise your work if you have any ideas to add. Otherwise leave us a comment that tells us how you think you did - you might answer one or more of the questions above. Our Solutions Method 1: Guess and Check I knew four numerical operations - addition, subtraction, multiplication and division. I wasn t sure what a fifth operation might be. When I looked in my math book I found out it had to do with exponents. I started the problem by using addition because just by looking at the problem, I could tell that if I used addition (H = heart) and then division (S = star), I would come up with an answer of 15 for the first equation. (5 + 40) 3 = 15 45 3 = 15 15 = 15 If my first two guesses were correct, I still had three symbols to find. I had to match subtraction, multiplication and raise to the power of to the remaining three symbols (R, L, and B). I tried the second equation and used division (S) again and tried subtraction (R): (14 S 2 R 5) L 2 = 4 (14 2-5) L 2 = 4 2 L 2 = 4 L could either be multiplication or raise to the power of because 2 * 2 = 4 but 2^2 also equals 4. At this point I am thinking that: H = + S = R = - L = * or L = ^ I still have to think about B. I think it s going to either be * or ^ and then that will help me decide for L. Now, I thought about this equation and used what I had guessed so far: 4 B (14-6) B 2 = 64 If I use * for B, it works but ^ does not work. So now I know: H = + S = R = - L = ^ B = * I substitute the operations for the symbols in the expression and I use the rules I know to evaluate the expression: 2^3 (4 * 3-10) + 6^2 * (4 + 1) (1^4 * 20) = 8 (12-10) + 36 * 5 (1 * 20) = 8 2 + 36 * 5 20 2010 Drexel University 2
= 4 + 36 * 5 20 = 4 + 180 20 = 4 + 9 = 13 The correct operations were placed in Aurora s equation and the answer is 13. Method 2: Process of Elimination I started by trying to figure out the first equation: (5 H 40) S 3 = 15 I used the operations of addition, subtraction, multiplication and division as H but I knew there could be a fifth operation. I wasn t sure how to think about that yet and so I just used the four most familiar operations. (5 + 40) S 3 = 15 (5 40) S 3 = 15 (5 40) S 3 = 15 (5 x 40) S 3 = 15 I did the same thing with the S. (5 + 40) 3 = 15 (5 40) + 3 = 15 (5 40) x 3 = 15 (5 x 40) - 3 = 15 (5 + 40) 3 = 15 was the one that worked. At this point I was thinking that H is addition and S is division. I decided to use those ideas going forward with the next two equations to see if things continued to work out. I followed the same process for the next equation. (14 S 2 R 5) L 2 = 4 I used for the S using what I thought about with the first equation. So, I have: (14 2 R 5) L 2 = 4 or if I actually do the division (14 2 = 7), I have: (7 R 5) L 2 = 4 The operations I have left to use are -, * and I know think the fifth one is ^. (14 2-5)^2 = 4 works but so would (14 2-5) * 2 = 4 So, R is subtraction and L is either multiplication or exponentiation. I continued on with the next one thinking it might help me decide which symbol should be * and which should be ^. 4 B (14 R 6) B 2 = 64 Should it be 4 * (14 6) * 2 = 64 or 4^(14 6)^ 2 = 64? Only the first equation works. So, I ve now determined that: H = + S = R = - L = ^ B = * I substitute the operations in this expression: 2 L 3 S (4 B 3 R 10) H 6 L 2 B (4 H 1) S (1 L 4 B 20) and get: 2^3/(4 * 3-10) + 6^2 * (4 + 1)/ (1^4 * 20) I simplify the expression using the rules for the order of operations and get: = 8/(12-10)+36*5/(1*20) = 8/2+36*5/20 = 4 +36*5/20 = 4 + 180/20 = 4 + 9 = 13 2010 Drexel University 3
Method 3: Use a Chart/Table to Keep Track of Your Thinking We made a chart to keep track of what we learned + - * / ^ H S R L B The first equation we consider is: ( 5 H 40 ) S 3 = 15 H cannot be - because 5-40 < 0; * because 5*40=200 and nothing you can do with this and 3 will get it down to 15; it can t be ^ because 5^40 is huge; it can't be / because there is no way to get 5/40=1/8 back to 15. So H must be +. We fill that in our chart using T for true and F for false: + - * / ^ H T F F F F S R L B Next we consider this equation and we use our decision that H is +: (40 + 5) S 3 = 15 This tells us that S is division because 45 3 = 15 and we add that to our chart: + - * / ^ H T F F F F S F F F T F R L B Using our decision that S = (14 2 R 5) L 2 = 4 ( 7 R 5) L 2 = 4 R cannot be ^ because 7^5 is huge; it can t be * because nothing you can do with 35 and 2 would get you back to 4. So it must be a minus sign. (7-5) L 2 = 4 2 L 2 = 4 So L can be * or ^ since both work. + - * / ^ H T F F F F S F F F T F R F T F F F L F F? F? B F F? F? Now we consider the third equation using what we ve decided about R = - 4 B (14-6) B 2 = 64 4 B 8 B 2 = 64 2010 Drexel University 4
If B is ^ then the result is much too big so B = *. This makes L=^. Check: Equation 1: (5+40) 3 = 15 Equation 2: (14 2-5) ^2 = 4 Equation 3: 4 *( 14-6) * 2 = 64 true true true + - * / ^ H T F F F F S F F F T F R F T F F F L F F F F T B F F T F F Now we use that information to evaluate the expression. 2 L 3 S (4 B 3 R 10) H 6 L 2 B (4 H 1) S (1 L 4 B 20) = 2 ^ 3 (4 * 3-10) + 6 ^ 2 * (4 + 1) (1 ^ 4 * 20) = 8 ( 12-10) + 36 * 5 ( 1 * 20) = 8 2 + 36 * 5 20 = 4 + 36 * 1 4 = 4 + 36 4 = 4 + 9 = 13 Teaching Suggestions This problem presents an opportunity for students to practice explaining their thinking. When we first offered this problem many submitters were successful in finding a possible solution, but they were challenged to present their solutions in complete and clear ways. One key to solving the problem is to realize that the fifth operation is exponentiation. Many students find a way to describe that operation without actually naming it. Some students use the word squaring but because they are simplifying 2^3 as they work with the expression, it isn t correct to use that term. If your students keep a vocabulary list or you have a bulletin board listing math vocabulary, this problem provides an opportunity to add exponentiation to the list! This problem gives a chance to practice the Understanding the Problem strategy and activities as outlined in our Activity Series area. A link to that page is always available in the left menu when you re logged in. One less threatening way to introduce this problem is to use the Scenario Only [pdf] version. You ll find it linked from the blue-shaded rectangle where we link to resources for teachers. You ll notice that only the three equations are visible on that version. The expression is not included. Using the Scenario Only might provide an opportunity for students to notice that there are five operations. It might help them move beyond the four operations they immediately consider and talk about what that fifth operation could be. The Online Resources Page for this problem contains links to related problems in the Problem Library and to other web-based resources. If you would like one page to find all of the Current Problems as we add them throughout the 2010-2011 season, including a calendar, consider bookmarking this page (a link to the page is always available in the left menu when you re logged in): http://mathforum.org/pow/support/ Sample Student Solutions Focus on In the solutions below, we ve focused on students of the problem, meaning that they explain all the steps taken to solve the problem. Students first drafts may not be complete but as they receive feedback from you or their peers or through class discussions, perhaps, you ll have them revise their work. Problem solving is a process and revision is an important step in the process. Our hope is that these student solutions help provide insight into conversations you might have with your students as they work to improve their problem solving and communication. 2010 Drexel University 5
Ely Age: 11 Novice B=+ L=- R=/ S= * because i just gussed it I notice that Ely has admitted to guessing. I wonder if he would be surprised when I explain to him that guessing is a very good strategy for this problem. I wonder if he would have ideas about how to check his guesses. Boobyer Age: 13 Novice My final solution is 143. To solve this problem I first had to find out what each symbol meant by solving the individual problems. Then I used those math operations to solve the expression. Boobyer has written more than Ely but in a way I feel there is more to go on with Ely s short response than Boobyer s longer explanation. I wonder how he found what the symbols meant. I wonder what he means by solving the individual problems. I wonder how he got 143! There are a lot of possible questions to ask him. I would probably start by asking just one question What did you notice when you looked at the first equation? Sarah Age: 13 Apprentice Aurora's group should have gotten 4.5 as their final answer. Aurora should have gotten the following answers: Problem 1: (5H40)S3=15 (5+40)/3=15 Problem 2: (14S2R5)L2=4 (14/2-5)x2=4 Problem 3: 4B(14R6)B2=64 4x(14-6)x2=64 Letters: S=/(divide) H=+(add) R=-(subtract) L=x(multiply) B=x(multiply) Final Problem/Work: Problem: 2L3S(4B3R10)H6L2B(4H1)S(1L4B20)= Work: 2x3/(4x3-10)+6x2x(4+1)/(1x4x20)= 2x3/2+6x2x5/80= 6/2+12x2x.0625= 3+24x.0625= 3+1.5= =4.5 Final Answer: 4.5 I notice Sarah has found only four of the five operations. I would score her as an apprentice in Interpretation as well. I also notice that she has given examples of the equations and which operations work but she hasn t included any of her thinking. I also notice that when she writes each equation a second time with the operations filled in, she doesn t continue to show that the left side of the equation can be simplified to the same number written on the right side. I might ask her if this is also true: (14 2 5) 2 = 4. 2010 Drexel University 6
Kyle Age: 16 Apprentice The answer is 3.75. The H is addition, the S is division, the R is minus, the L is multiplication, and the B is also multiplication. The first step is to take 2L3 which equals 6. The next step is to take 4B,multiply, 3R, minus,10. That answer is 2. You then dived the 2 bye the six which is 3. Next you have 6L2(4H1), which is 60. Then you dived that by 1L4B20 which is 80. Your equasions know is 3+60/80. You have to do the division first so know it is 3 +.75. You add them together and get 3.75. That is your answer. In contrast to Sarah s style, Kyle has written his steps in words. In both cases they ve told us what they did but not why they did it. I wonder how Kyle knew that 2L3 equaled 6. Why can t the L stand for division? or subtraction? or addition? Rachel and Maureen Ages: 13 Apprentice The answer to Anh's problem is 13. First we figured out what the each symbol was by using the top equations and guess and check. Here is a list of the symbols' operation: H=+(addititon) S=/(divide) R=-(subtract) L=exponent B=x(multiplication) Then we solved the bottom equation to get the final answer, which was 13,by filling in the symbols with the correct operation. I notice that Rachel and Maureen noted that they used a guess and check strategy. I wonder what they guessed first. I wonder how they checked their guess. What did they learn from it to guess a second time. Trisha and Kristen Ages: 13 & 11 Apprentice The answer Anh's group should get is 13. The process we used was guess and check. We started out by guessing what operation would fit into the equations. We made a chart with all of the symbols with the operation that they meant next to it. H= add S= divide R= subtract L= to the power B= multiply With that information we started to figure out the long expression. First we figured out the parts in the parentheses. We rewrote the equation with out using the parentheses(the answer to the parts in the parentheses we just put in the expression alone). Then we rewrote the expression again substituting the symbols with the operation that they equaled. Then we just figured out the math. We came up with a solution of 13. Trisha and Kristen have included a little more about their method than Rachel and Maureen and, yet, I m wondering similar things. What did they guess first? What did they learn as they checked it? What did they try next? 2010 Drexel University 7
Mary Age: 12 Practitioner The answer to the equation is 13. I used the guess and check method.on the first problem ( 5 H 40 ) S 3 = 15 I guessed that the H was multiplacation and S was division. That did not work so I changed the H to addition but kept the S the same. This one worked so from that information I knew that H was equal to + and S was equal to division. On the next problem I plugged in what I had found out in the last problem so it became ( 14 divided by 2 R 5 ) L 2 = 4. I guessed that R was equal to subtaction and L was equal to multiplacation. That did not work so I knew tried the same for R but L was to the power of. It worked so I knew that R was equal to - and L was equal to to the power of. I plugged this information into the next problem which became 4 B ( 14-6 ) B 2 = 64. I guessed that the B was multiplacation and it worked. I figured out that B was equal to x. I plugged all that information into the long problem which became 2 to the power of 3 / ( 4 x 3-10) + 6 to the power of 2 x (4 + 1 ) / (1 to the power of 4 x 20 ). I did all the math and got 13. Mary has a complete solution but I would encourage her to work on clarity. Ideas like breaking her explanation into several paragraphs or putting her equations on a separate line could make a big difference for the reader. Genia Age: 11 Practitioner Aurora's group should get 13. Start with the first equation. (5 H 40) S 3 = 15 H could not be - or / because then you would have negatives and fractions. Multiplication? too big. So, it's addition. 45 S 3 = 15 It's common knowledge that 45 / 3 = 15. H = + S = / Next. (14 / 2 R 5) L 2 = 4 14 / 2 = 7, So (7 R 5) L 2 = 4 You can't use division for R (you've already used it), neither can you use multiplication (too big, AGAIN) and addition is already taken. So, you have subtraction for R 2 L 2 = 4 Duh... L is multiplication. Wait! What if L is "exponentation", as on a calculator? 2^2 does equal 4. On we go. 4 B (14-6) B 2 = 64 You probably know 14-6 is 8. 4 B (8) B 2 = 64 OK, you've got to do something here that's not / + - or *. Or maybe not... I tried multiplication and I got 64: 4 * 8 = 32 32 * 2 = 64! So, then you substitute the correct operations for the symbols. This is 12-10 2^3 / (4 * 3-10) + 6^2 * (4 + 1) / (1^4 * 20) Simplify. 8 / (12-10) + 36 * (4 + 1) / (1 * 20) Go on. 8 / 2 + 36 * 5 / 20 So on and so forth... 4 + 180 / 20 And finally... 4 + 9 = 13! I noticed Genia has an engaging style to her explanation. I think a classmate not as confident as she is with this problem might enjoy listening to Genia read her solution aloud. 2010 Drexel University 8
Jackie Age: 13 Expert They should get the answer of 13. I got the answer by first finding out what the symbols meant. The H stood for addition, S stood for division, R stood for subtraction, L stood for exponent or y to the x power, and B stood for multiplication. I figured this out by the given problems. (5 H 40)S 3=15 : If you add 5 and 40, you'll get 45. Then, divide 45 by 3 will get you 15 so H is add and S is divide. (14 S 2 R 5)L 2= 4 : 14 divided by 2 R 5 is going to come out 7 R 5. If you subtract 5 from 7,you'll get 2, and since 2 to the second power is 4, R is subtract and L is the symbol for y to the x power. ( note, if you multiply the (2) to 2, the answer will still be 4 because 2 to the second power is the same as 2x2) 4B(14R6)B2=64 : You already know R, so 14-6=8. If you multiply 4 by 8, you'll get 32 and that times 2 will equal 64. Therefore, B means to multiply. So the symbols are: H- add S- divide R- subtract L- y to the x power (exponent) B- multiply While not as clear as Andrei s solution (below) I think that Jackie s solution is very complete. One thing that stands out is that he made a point to refer to a scientific calculator which is important when discussing the order of operations. I would find it interesting to hear more about why that model of calculator is important. The last problem is 2 to the 3rd power / (4x3-10) plus 6 to the 2nd power times(4+1) / (1to the fourth power times 20) = ( the "/" is a sign for division) First you figure out the parenthesis, then exponents, multiplication or division next, and then addition or subtraction next. Then the answer would be right since it would be from order of operations. However, I used a scientific calculator, but the calculator would of done the same. Andrei Age: 13 Expert I found the following correspondence for the symbols: H - addition, S - division, R - subtraction, L exponentiation, B - multiplication. The result of the evaluation of the given expression is 13. First, I wrote the three coded equations using the capital letters in place of the symbols: (5 H 40) S 3 = 15 (1) (14 S 2 R 5) L 2 = 4 (2) 4 B (14 R 6) B 2 = 64 (3) From the beginning, I know that I must identify 5 different types of mathematical operations, which could be, most probable: addition, subtraction, multiplication, division, and exponentiation. Exponential and logarithm are less probable, being written before the number. So, I'll look first for these 5. Now, I look at the first equation. As I observe that I can write: 15 * 3 = 45 = (40 + 5) so that, comparing with equation (1) I draw the conclusion that: and: H is the symbol of addition S is the symbol of division. Now, I substitute all the symbols found into the second equation: or (14/2 R 5) L 2 = 4 (7 R 5) L 2 = 4 I notice that Andrei considers more than five possibilities for the different operations but explains why he will first consider addition, subtraction, multiplication, division, and exponentiation. He has provided a very complete and clear explanation of the thinking he did on this problem. 2010 Drexel University 9
It is easy to see that R is associated with subtraction and L could be either multiplication, or exponentiation. I use all my results in equation (3): 4 B (14-6) B 2 = 64 Looking at the numbers, I observe that B must absolutely be multiplication, so that L is exponentiation. The symbols being deduced step by step, in principle it makes not sense to verify them, but I shall do this: (5 + 40)/3 = 15 (14/2-5)^2 = 4 4 * (14-6) * 2 = 64 All are OK!. I rewrite all my results: H is addition S is division R is subtraction L is exponentiation B is multiplication Now, I substitute the symbols in the big expression: 2^3/(4 * 3-10) + 6^2 * (4 + 1)/ (1^4 * 20) = 2^3/2 + 6^2 * 5 / 20 = 4 + 9 = 13 Scoring Rubric The problem-specific scoring rubric, to help in assessing student solutions, is a separate standalone document available from a link on the problem page. We consider each category separately when evaluating the students work, thereby providing more focused information regarding the strengths and weaknesses in the work. A generic student-friendly rubric can be downloaded from the Teaching with PoWs link in the left menu (when you are logged in). We encourage you to share it with your students to help them understand our criteria for good problem solving and communication. We hope these packets are useful in helping you make the most of Pre-Algebra PoWs. Please let me know if you have ideas for making them more useful. ~ Suzanne <suzanne@mathforum.org> 2010 Drexel University 10