The idea of the Club is to provide enrichment to the standard school math curriculum. The main focus is on valid reasoning and on scientific approach to solving problems, for example by extending them through asking additional questions. It is not our goal to prepare a team to participate in any specific math competition. The club is open for all third graders from Einstein Elem. though we are limited to 15 children maximum.
Einstein Elementary, Thursday 3:40 – 5:00 pm (in the library)
| Coaches: | Irina and Andrey Yatsenko |
| Parent coordinator: | Tali Guday |
We’d also like to thank parents of all our students for their constant support and feedback.
Note: Materials published here are the handouts we give to the children at the end of each class. However, discussions during the class might follow different paths or we might touch additional aspects of the problems. Sometimes our estimate of the difficulty level was way off, so we had to repeat some of the problems at the next class. We usually try to follow the same format: a few problems about the main topic (title of the class below), one or two fun problems and a game or some other interactive activity at the end.
| Class 1 (MHT)(PDF) |
Introduction Our goal was to introduce math problems from different areas, to give children the taste of how varied math can be, that it's more than just "counting". |
| Class 2 (MHT)(PDF) |
Ancient Egyptian number system The first problem about the cave men is extremely important because it teaches one-to-one correspondence. The Egyptian system itself is actually quite simple for children to grasp, they like the funny pictures, and though it's very close to Roman, it looks more intriguing. We didn't try to analyze Nim at all, just let the kids play it. |
| Class 3 (MHT)(PDF) |
Egyptian number system continued The previous class was overloaded so we backed up a little bit. Gave first three problems as a test and worked through the home problems. |
| Class 4 (MHT)(PDF) |
Working backwards When solving a problem it's very important to understand where the "start point" of the solution is. Analysis of Nim game and other problems in this class demonstrate this idea. We've also introduced the simplified variant of Randzu game. |
| Class 5 (MHT)(PDF) |
Ancient Greek number system This number system also isn't positional. We tried to show that same concepts (in this case numbers) can have different notations but notation well chosen might greatly simplify usage and speed up the progress of science. Good chunk of the class was spent discussing winning strategies in Randzu. |
| Class 6 (MHT)(PDF) |
Greek number system continued Children get so used to addition table in our decimal system that they stop noticing it. Part of the exercise of doing addition in Greek was to remind them that we do just memorize the table - there is nothing in notation to tell you that "5" + "3" equal to "8" (as opposed to the Egyptian method). Instead of a game we looked at syllogisms. |
| Class 7 (MHT)(PDF) |
Comparing Greek and Egyptian systems Finished problems from the previous class and summarized the differences between the Greek and Egyptian systems. We've tried to lead the children to "invent" a mixed system that would have the benefits of both but it proved to be too hard for them. |
| Class 8 (MHT)(PDF) |
Chinese number system This number system is the real world example of a mixed system we've tried to re-invent during the previous class. We didn't attempt to solve any problems about it however. Most of the time was devoted to a mini-tournament - we split the kids in three teams and had them play Randzu and Tangram. |
| Class 9 (MHT)(PDF) |
Big numbers Estimation problems were given during the class in a very "free form" and intentionally lacked some data. It was part of the investigation to figure out which data is missing. We've tried to demonstrate that big numbers can arise from different situations, either from the real world around us, or as a result of a process (in this case sequences). Actually, it was too much material for a single class, no time left for the Rows-and-Columns game... |
| Class 10 (MHT)(PDF) |
More of big numbers and sequences Operations on sequences is one of the doors into high algebra. Some people don't agree it fits 3rd grade level but we think the earlier kids realize that operations can apply to something else than just numbers the better, of course there is no need to introduce any special terminology. |
| Class 11 (MHT)(PDF) |
Sequences and formulas One more experiment with notations. Realizing that adding/multiplying sequences member by member can be expressed in formulas is a very important step. Properties of addition/multiplication such as commutability and distributiveness come out very clear. |
| Class 12 (MHT)(PDF) |
Odd and even numbers Proving some fact for all possible numbers (e.g. that odd + odd = even) requires more than just trying a few samples. However, an example is a good place to start as it helps to form a hypothesis. |
| Class 13 (MHT)(PDF) |
Odd and even continued Mostly clean up of remaining questions from the previous class. The problem about crossing the river got enhanced by the question of how long it would actually take for all 20 soldiers to cross, which proved to be a rather hard problem of its own. |
| Class 14 (MHT)(PDF) |
More of odd and even Formal prove of the divisibility by 2 rule. Problems when odd and even numbers appear without being explicitly named. Introduced Master Mind game (on numbers). |
| Class 15 (MHT)(PDF) |
Start on divisibility, Master Mind Divisibility by 3 - introductory problems. Worked through a game of Master Mind explaining meaning of each move and what new information is added every time. |
| Class 16 (MHT)(PDF) |
Arithmetics of remainders Remainders are the very first equivalence classes that children can deal with. However operations on classes rather than particular representatives is a difficult concept, so we spent a lot of time getting used to the idea while playing with adding numbers from different columns in a table. The toothpicks (a.k.a. "safety matches") puzzles proved to be harder than we expected. |
| Class 17 (MHT)(PDF) |
Remainders continued Discussed how one could approach the toothpicks-like puzzles and worked through two of them. Built addition tables mod3 and mod5. It's important for kids to understand that remainders cannot be added "across" different addition tables (hence the problem #2) |
| Class 18 (MHT)(PDF) |
Divisibility rules Divisibility rules help to recognize that a number can be divided by another number evenly without doing the actual division. Those rules are based on how our number system works and are usually formulated in terms of digits the number we'd like to divide consists of. To prepare the children to proving the rules we began by discussing the notation and properties of positional number system with base 10. |
| Class 19 (MHT)(PDF) |
Divisibility rules continued Repeated proofs from the previous class and discussed how to come up with a divisibility rule for 4. Finished the class with Tangram puzzles, the idea was to find in each puzzle the "big" pieces we already know how to make. Proofs for divisibility rules by 2, 5 and 9 (MHT) |
| Class 20 (MHT)(PDF) |
Problems about beam scales Those problems mostly require simple logical thinking but the tricky part is to prove that the number of operations in a solution is indeed optimal. We've glossed over this, but accurately ensured that all possible outcomes for each operations are considered. |
| Class 21 (MHT)(PDF) |
The Lady or the Tiger? Problems for this class are taken from the excellent book by Raymond Smullyan. These are purely logical problems. We also brough figures of a small tiger and a princess and two tin boxes to make the class more fun. |
| Class 22 (MHT)(PDF) |
Starting graphs Graph theory is a very deep subject but there are enough entry level problems that kids are capable of solving and appreciating. Our focus during this class was to figure out which graph pictures actually depict the same structures. |
| Class 23 (MHT)(PDF) |
Graphs continued We spent this class learning about the notion of a vertex degree in a graph and how to use degrees in problems about Eulerian graph traversals. We also introduced "Sprouts" game (see M. Gardner's books). |
| Class 24 (MHT)(PDF) |
Long multiplication? Most of the children in our club knew about long multiplication, so we've decided to give them a few "trick" problems which take a lot of effort if done using the brute force long multiplication methods but have other, much easier, solutions. We also did some basic analysis of the Sprouts game. |
| Class 25 (MHT)(PDF) |
Preparing for Greatest Common Divider The plan was to run the club through June so we hoped to cover one more topic, namely, the Greatest Common Divider. To prepare for the topic we used a set of problems that don't mention GCD explicitly but actully use the concept. However, we had to finish our classes in May, so the topic didn't evolved into anything more serious. We'll work on this next year! |
| Class 26 (MHT)(PDF) |
Competition! This class was one hour written individual competition with problems on the recent topics. Problem #3 proved to be the hardest... |