«MATHEMATICAL PROBLEM SOLVING SEMINAR Modesto Junior College Heidi Meyer TABLE OF CONTENTS Introduction Session 1: Introduction to Problem Solving ...»
MATHEMATICAL PROBLEM SOLVING SEMINAR
Modesto Junior College
TABLE OF CONTENTS
Session 1: Introduction to Problem Solving
Session 2: Look for a Pattern
Session 3: Number Theory (Divisibility)
Session 4: Use a Diagram
Session 5: Competition Problems
Session 6: Twenty-One Plus Game and Competition Reﬂection
Session 7: Puzzles and Pseudoproofs
Session 8: Use an Organized List Session 9: Pigeon Hole Principle and Ramsey Theory Session 10: State and Solve a Simpler Problem Session 11: Competition Problems Session 12: Dots and Boxes Game and Competition Reﬂection Session 13: Graph Theory Session 14: Recognitions, Party and Fourth Dimension Session 15: Changing Your Point of View, Recasting Session 16: Set Theory, Combinatorics Session 17: Parity Session 18: Cryptarithms Session 19: Competition Problems Session 20: Cats and Dogs Game and Competition Reﬂection Session 21: Inﬁnity Session 22: Guess and Check Session 23: Geometry, Draw a Picture Session 24: Eliminate Possibilities Session 25: Number Theory (Primes, Factorial, etc.) Session 26: Sprouts Game and Competition Reﬂection Session 27: Fractals Session 28: Recognitions, Party, Telescoping References Session 1: Introduction to Problem Solving - Meeting Notes “You cannot teach a man anything, you can only help him ﬁnd it within himself.” “In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual.” Galileo Galilei OPTIONAL MATERIALS: chessboard and dominoes
- time for problem solving (handout)
- discussion - introduction of this semester’s game, ‘Twenty-one Plus’
- hand out problem set and make closing comments
Begin with a welcome and possibly also a team-building activity, depending on how many students attend and whether or not they know each other. Share information from the introduction to this document, regarding what participants can expect to gain from this seminar. Explain the structure - meeting times and dates - and also that students have opportunity to take part in a nation-wide math competition, for which ﬁrst place is a $3000 scholarship. As part of the introduction be sure to talk about the diﬀerence between problems and exercises, the beneﬁts of having problem solving skills, and the fact that much of the time during our meetings will be spent in actual problem solving rather than direct instruction. Students will work individually or collaborate, depending on the topic, and they will present their ﬁndings.
COMMENTS ON HANDOUT:
The handout the participants will be working with during the session contains a variety of problems in order to introduce them to problem solving and to get across the diﬀerence between exercises and problems.
Problems 1 and 2 are rich problems which can involve strategies of using manipulative materials, making an organized list and looking for a pattern. There is an elegant pattern that appears in this problem. The number of geometric squares is the sum of square numbers. The idea is that after working problem 1, problem 2 will be an exercise rather than a problem. This is to illustrate the idea of taking any problem you solve and putting it in your ‘arsenal’ to take out to attack future problems by reducing them to exercises that you know how to approach. Never do a problem and then forget it; always ﬁle it away for future use. The answer to number 1 is 55, which is 1 + 4 + 9 + 16 + 25 and the answer to number 2 is 89, 440 or 1 + 4 + 9 + 16 + · · · + 632 + 642.
Problem 3 is simpler than it seems if you use logic. A chessboard and dominoes are suggested as manipulative materials that students can use for exploring this problem. The answer to the problem is ‘no,’ because the task is impossible. The reason is it impossible is that each domino will cover a white square and a black square, but if you remove opposite corners you have removed two corners of the same color; therefore it cannot be covered with dominoes. Exploration is a good idea, but standing back and thinking logically makes it clear why the answer is ‘no.’ Problem 4 is intended to give students experience in ‘getting their hands dirty.’ One possible solution is (800), (053), (350), (323), (620), (602), (152), (143), (440). This is assuming the jugs are in the order pictured.
Problem 5 is from a previously given SML competition, so it is intended to introduce the students to such problems. Also, there is a twist to it. It seems that the answer should be B. 1 for April 30, however the answer is C. 2 because February 29, which only occurs on a leap year, gives us 58, which is not a leap year.
Problem 6 may seem impossible or unfair or as if it doesn’t belong, but it’s intended to test the participants’ resourcefulness. They know more than they realize. In order to ﬁnd the 12 letters of the Hawaiian language, like all Hawaiian words you know - aloha, Maui, ukelele, Hawaii, luau, muu muu - listing all the letters you ﬁnd until you get 12. The letters are A, E, H, I, K, L, M, N, O, P, U, W.
After discussing as many of these problems as time allows, hand out the sheet that has this semester’s game on it, and introduce the game.
COMMENTS ON PROBLEM SET:
These are the problems the participants are to be working on for the next session. Many of these look forward to the next topic, which is the “Find a Pattern” strategy.
For problems 1 it is natural to immediately think the answer is 100, but the rate is that it takes a cat 3 minutes to kill a rat, so the answer is 3 minutes.
Problem 2 is from a previous SML competition. A good approach here is to ignore the other numbers and list the middle numbers and look for a pattern. This one is reasonably diﬃcult and leads well into the tool ‘ﬁnite diﬀerences,’ which should be introduced at the next session as a nice tool.
For problem 3 the idea is to look at combinations of numbers in the rows. In each case the ﬁrst and third numbers add to the same result that the second and fourth numbers multiply to, so the answer is 11. This might be a good place to remind students not to restrict themselves in ways that aren’t in the rules. Many people see this problem and think the answer must be a single digit because all the other entries are, but this isn’t the case.
Problem 4 is a good place to mention the strategy ‘state and solve a simpler problem’ as well as ‘look for a pattern.’ Consider powers of 2 beginning with 21. List as many as necessary until a pattern emerges upon division by 7. The answer is 1.
Problem 5 has to do with thinking outside the box. The pattern is alphabetical order, so 6 should be placed after 7.
The answer to problem 6 is B. 972. It’s a challenge, and that’s the point. This one may have to become a ‘back-burner problem.’ My experience is that students look for a geometric series ﬁrst and get stumped because it turns out that the ratio is not a whole number, which makes it hard to ﬁnd. Prime factorization is a good start, and looking for an arithmetic sequence works well from there, as there is a factor of 5 and of 7 that should give it away. There are actually inﬁnitely many sequences that can give rise to the answer, but they all lead to the same answer. Another approach at solving is to write the ﬁrst
couple of terms using formulas for geometric and arithmetic sequences:
1. How many squares of any size are there in the diagram below?
2. How many squares of any size are there on a chessboard?
3. The opposite corners of a chessboard are removed. Is it possible to cover the remaining 62 squares using 2-by-1 dominos?
4. You have three jugs, which respectively hold 3 litres, 5 litres, and 8 litres of water.
The 8 litre jug is full, the other two are empty. Your task is to divide the water into two parts, each of 4-litres, by pouring water from one jug into another. You are not allowed to estimate quantities by eye, so you can only stop pouring when one of the jugs involved becomes either full or empty.
Most or all of us probably remember some letter or number games from childhood.
Perhaps you remember games from long car trips like ﬁnding all the letters of the alphabet on signs or license plates or games you might hear on the playground like “eenie-meenie-minie-moe” or “I one it, I two it... I jumped over it and you ate it!” One
such counting game has the following rules:
This is a two person game, and the winner is the person who says “21.” We start with the number 1, and each of us can count one or two or three numbers at a time.” Play this game a number of times over the next few weeks (you might want to keep a record of which numbers each player says), think about strategy as you play, and then
answer the following questions:
1. Is there a strategy that will allow you to win every time? If so, how?
2. Can you win this game in general even if you are counting to a number other than 21 and/or can count by groups of more or less than three numbers?
At our sixth meeting this semester we will discuss strategies you’ve come up with.
We will meet on the following Thursdays from 3:00pm to 4:30pm in the Integrated Learning Center on west campus. The competition will be held on the ﬁrst Saturday in November - time and location to be announced.
Thursday, September 5 - Introduction to Problem Solving Thursday, September 19 - Look for a Pattern Thursday, October 3 - Number Theory (Divisibility) Thursday, October 17 - Use a Diagram Thursday, October 31 - SML Classics Saturday, November 2 SML Competition Thursday, November 14 - Twenty-One Plus Game and Competition Reﬂection Thursday, December 5 - Puzzles and Pseudoproofs Session 1: Introduction to Problem Solving - Problem Set
3. Below are four horizontal number sequences, one of which contains a blank. Fill it in by ﬁguring out the one rule that governs all four arrangements.
Find the remainder when 22004 is divided by 7.
6. Multiplying the corresponding terms of a geometric and an arithmetic sequence yields the sequence 96, 180, 324, 567,.... Find the next term of the new sequence. SML
Begin with a discussion of the previous problem set. Problem 2 there should lend itself to an introduction to ‘ﬁnite diﬀerences’ at some point in today’s session. Math has been deﬁned as the study of pattern, so looking for patterns is a very powerful problem solving tool. There are certain patterns students should immediately recognize - evens, odds, squares, powers of two, Fibonacci numbers, cubes, triangular numbers, primes etc. These should be pointed as appropriate throughout today’s session (perhaps even ﬂat out listed before the end).
COMMENTS ON HANDOUT:
Problems 1 consists of a variety of patterns - Fibonacci numbers (be sure to mention Fibonacci-type sequences and Lucas numbers) - numbers increasing by consecutive odds, get into relationship with square here, this is also a good time to bring up ﬁnite diﬀerences;
the pattern here is n2 + 5 - clock arithmetic (familiar, but do they see it?!) - names of numbers: one, two, three, etc. - two-digit primes written with digits reversed - the ‘see and say’ pattern.
Problem 2 is Pascal’s Triangle. Students are likely to be familiar with the construction of this already, but they’ll have to notice a couple of patterns in answering the second part that the sum of each row is a power of 2 - and that summing powers of 2 yields one less than the next power of two, so the answer is 2n+1 − 1 or 220 − 1. Question - are you counting the top of the triangle as row 0 or row 1?
Problem 3 is the classic handshake problem - an excellent place to use ‘state and solve a simpler problem,’ ‘draw a picture,’ and ‘look for a pattern.’ The answer is 1275 For problem 4, the answer is 220 The formula is 2n(n + 1) Problem 5 is also a classic. It too has to do with sums of powers of two. It takes 264 − 1 moves, which is over 18 quintillion seconds or about 5.8 billion years.
COMMENTS ON PROBLEM SET:
The ﬁrst four problems continue with patterns - many similar to the ones on the handout.
The last three problems have to do with number theory and divisibility to give the students a chance to work with that before our next session, which is on that topic.
Problems 1 has to do with names of ordinal number: ﬁrst, second, third, etc.
Problem 2 requires looking at what is not there. These are the composites or non-prime numbers.
Problem 3 is similar to problem 4 on the handout, but it is a bit harder. The answer is n(n − 1) n+1 15, 150 3n + 3 = 3n Problem 4 comes from a Google Labs Aptititude Test. It has to do with lengths of number names. Each number in the sequence is the largest number that can be written with the amount of letters it is written with. The answer is A. 96.
Problem 5 may become a back-burner problem. It turns out that powers of two cannot be written as sums of consecutive integers. The proof of this has to do with parity, which we will cover third semester.
Problem 6 involves number of divisors a number has. Three numbers that will work are 24 = 16, 34 = 81, and 54 = 625. Notice we skip over 44 because that would give us more divisors. This is a good one to pull apart and do some listing and see why and how it works.
Problem 7 has to do with factoring a binomial. For example, 4 divides evenly into 156 because 4 goes into (100 + 56). Four always divides 100, so all we have to be concerned about are the last 2 digits. For 8 we are concerned about the last 3 digits, since 8 divides one thousand. Consider 1, 793, 162, 540 = (1, 793, 162, 000 + 540).