The kids worked in partners—most kids paired with their sibling. (It was 6 sets of siblings in the class of 12 kids—11 boys and 1 girl!) They got a kit with the pieces they’d need, and a booklet outlining step-by-step how to put the piece together. The kids were thus able to help one another if one of the pair got stuck. There was also an instructor to assist, and a big bucket of cool Lego pieces for free building once the project of the week was complete.

The first week they built the Eiffel Tower:

The second week was a windmill, including hooking it up to a motor to make it spin. Daegan got so excited when he finally got the connections the right way so it spun—so did he in the pic.

The third week they made a see-saw (teeter-totter), and the fourth a turnstile with lots of gears. Again, both projects hooked up to motors to move:

Thanks again to Lisa for setting up this class and hosting.

I began by simply playing a few rounds of tic-tac-toe with the boys, quickly moving on to having them try drawing the game board. This is not so easy—Gareth knew you played on a 3×3 grid, so he began by drawing 3 lines by 3 lines:

Gareth figured out his error and drew some boards that he and I played, while Daegan preferred to try out various placements, patterns and logic on his own:

Gareth was VERY excited when he finally beat me at a game, fair and square:

We then moved on to investigate a question I posed for the boys: which is the most important square on the tic-tac-toe board? They both sensed intuitively that it was the middle square (correct), but couldn’t explain why. Daegan came close, telling me there were 6 different ways to win using the middle square. So we investigated all the ways to win at tic-tac-toe. There are 8 of them, so I drew 8 blank boards (in groups of 3, 3, and 2) and the boys helped me fill them in. As they called out a way to win (e.g., “you could have a row of three across the bottom”), I filled in the squares so that we’d see some patterns once it was completed. Gareth filled in the final two ways to win that the boys came up with:

After a brief discussion of terminology (vertical, diagonal, horizontal…I asked the boys if they knew what a horizon was), we set to counting how many times each square is used in one winning game or another. We began by counting how many of the 8 ways to win involved filling the top left corner square—there were 3 (horizontal across the top, vertical down the left side, diagonal from top left to bottom right). We then counted how many ways used the middle square, and so on for all the squares:

We then looked at our filled in board for patterns, and implications for strategy in tic-tac-toe:

The boys now understood how and why the middle square is most important, the corners the next most important, and the middle outside edges the least. We then played a few more rounds of tic-tac-toe and they did better, but still needed reminders that theory and strategy aside, sometimes a middle outside edge is the most important square to play in a game—like when it blocks your opponent from winning!

One final thing we did was take a suggestion from the book to play on a larger grid (4×4) with 3 players, trying to make 4 in a row. This proved nearly impossible and not very interesting. But we changed the game to 3 players, 4×4 grid but needing only 3 in a row to win and the game was quite interesting. For one thing, opponents sometimes need to work together to block effectively, as in this game below. Gareth got two G’s in vertical middle squares, giving him two ways to win: top or bottom. I blocked the bottom on my turn, but Daegan didn’t block the top (he was too caught up in trying to win himself), allowing Gareth to win the match. (Daegs filled in an extra D on his next turn to get 3 in a row himself.)

Do check out some others in the Hello Math Reader series (sometimes called Hello Reader: Math). I have found several at the library, and they seem to be available used at amazon for a penny quite often.

Thank goodness the boys have had their new Nintendo DS and the Fossil Fighters game to keep them busy! My aunt sent them the game for Christmas, and the boys saved up their money to buy the DS system, which was an interesting math exercise in itself.  

In Fossil Fighters, you need to find fossils, with the help of radar:

then dig up and clean the bones. There are many kinds of dinosaurs to be found (Daegan thinks around 100), but you must find the heads first, as that brings the dinosaurs back to life. (They are called vivosaurs in the game for that reason, vivo + saur = life + lizard). Here’s Gareth engrossed in cleaning some bones, including ‘blowing’ on the screen, as he is doing here:

Once you have cleaned bones, you have a vivosaur ready for battle. As you find more bones of the viviosaur, it acquires more moves to be used in battle, and as you win battles, you increase the strength of the vivosaur along various dimensions. Here’s Gareth checking out what bones he still needs to find of one of his vivosaurs, Shanshan: (note the dust! Renos will do that, ya know! )

Each vivosaur has a power type—earth, fire, air, water or neutral—and are stronger or weaker against various competitors. I am quite impressed that Gareth, who is not yet reading fluently, can cope just fine with the strategy in this game (which includes text, graphics and numbers), saying things to me like, “Vivosaur X is weak against earth-type attacks, but strong against air attacks.” Here’s a typical page you see when deciding which of your vivosaurs to put forth in a battle, which requires considering your opponents’ strengths and weaknesses, as well as your own, as you pick a small team of vivosaurs to compete, allowing you to ensure a weakness on your team in one player is compensated by a strength on your team in another:

The boys are loving the game, and with provisos in mind to keep Mom sane (we use a timer system to ensure the boys don’t spend all day on this one activity), it’s been a good find for us. Gareth certainly has incentive to keep on learning to read! Gareth says his favourite thing is the battles, getting to use his vivosaurs, and see them and their moves. Daegan days his favourite thing is reviving the vivosaurs, and finding new bones which mean new moves by his vivosaurs in battle. Both boys think kids who like dinos and computer games will enjoy this one.

Any other favourite DS games? This is the only one we have so far.

Any animal not blocked by another can go across the bridge (in the picture below, any of the animals in spots 1, 2, 3, or 4 down the left-hand side), but after that the animals must follow one of two rules: species can follow (same) species, or colour can follow (same) colour.  There are 4 species of animal (lion, giraffe, camel, hippo), and 4 colours (blue, green, yellow, red), yielding 16 unique wooden creatures (e.g., red camel, blue lion, and so on).  There are 60 different puzzles in all to solve; we only did the first 5, each one increasing in difficulty from the previous. Solutions are provided in the back of the booklet.

Here’s the boys working on puzzle 4, which they made a wrong start with originally. Daegan figured it out and he’s here explaining to Gareth which animal they should start with (which is to cross the bridge first):

And Gareth following the rules (species or colour), to a midpoint in the game:

Gareth thinking about what must come next, and looking ahead a few moves to see if it will lead to getting all the animals off the board:

There are several things I really like about this game. For the very young, simply setting up the board is a good exercise in one-to-one correspondence and visual discrimination. I like that this game makes logic concrete (and colourful and fun!), which is great for young kids. The rules to follow are simple—species follows species, colour follows colour—but you do need to think as during the course of play, even in the simple beginning puzzles, there were several points where you could pick more than one animal (e.g., you’ve just moved the red camel, and now have a choice between a red animal or another camel.) The boys were only sometimes able to “look ahead” a few moves and see which choice led down a blind alley, and which was the correct one to make. But when you chose wrong, and found yourself with no more animals to move that follow the rules (species or colour), it was very easy to backtrack, as you have the chain of animals in front of you, and the place they belong on the board on your set-up card. I like that the animals do not “move spaces” on the board during the course of the game—they go from where you set them up to (eventually) over the bridge, making it very easy to fix and learn from your mistakes—you just take the animals out of the chain, one by one, and put them back on the board in their starting spots until you reach the point where you think you made the wrong choice. 

If you’re looking for a logic game that can be played at a wide variety of levels, ages, and solo or with friends, give Animalogic a try. The same company puts out another logic game called Zoologic as well. Happy playing!

All three of my digits are the same. The sum of my digits is 6.

I am a 2-digit even number. I am less than 30. The product of my digits is 12.

I have 3 different digits, all of them odd. Digits are in order from least to greatest. I am evenly divisible by 5.

Daegan also did some hundred board “What Number Am I?” logic problems, in which he needed to eliminate possibilities from the board until he was left with the only (and correct) answer. Here’s two of them, that we got off this site:

So why do both Daegan and I enjoy these sorts of questions? Aside from the obvious (we LOVE riddles and logic here), I think it is because these riddles are a fun way to learn and practice “the language of math.” In  order to solve the above riddles, one needs a solid understanding of ‘math language words’ like sum, digit, even, less than, product, etc. as well as familiarity with math concepts that form part of our ‘common knowledge’ in our culture: How many pennies in a dollar? Days in a month? etc.

There is an element of creativity to this as well when Daegan creates a problem for me to solve. I begin by having him write a number on a small piece of paper on the table where I cannot see it (so he can refer to it easily), and to think of some ‘clues’ about that number. For example, he wrote this riddle on the board for me after solving the second riddle (“I am a 2-digit even number…”) above:

I am a 2-digit even number. I am less than 100. My digits add to 12.

“How interesting!” I replied. “There’s actually several answers. 48, 66, and 84. You’ll have to make another clue for me so I can solve this riddle.” Daegan was a bit stumped at first, as in the model question 3 clue statements were sufficient information to solve the riddle. So we started talking about other math concepts and terms: greater than and less than, fractions and ratios, prime numbers, etc. He then created this additional clue for me:

My units digit [i.e., ones digit] is half my tens digit.

And I was now able to give him his answer: 84. I offered a few other possible clues he could have used: my digits are not identical; my tens digit is larger than (or double) my units digit; if you add my units digit to itself, you get my tens digit, etc. Daegan thought is was interesting how many ways you could give information in math—even identical information. (e.g., A is half of B means the same as B is double A, or B+B = A).

Have fun with “What Number Am I?” problems, and if you or your kids create some fun ones, please share! I should add that these were also among my class favourites when I taught grade 5/6, especially creating their own riddle for a classmate to solve. We made them in little lift-the-flap books (think typical birthday card shape turned sideways), with the clues on the front, and the solution and the working out of the clues on the inside (lift the flap to see the answer). Some kids got very into decorating them with math symbols, shapes, terminology to boot! We put them up on the bulletin board (wall) for parent-teacher conference night, and it was quite fun watching “math anxiety’” stricken parents learn that their child created this riddle, AND could teach them (the parent) how to solve it!

I’m found inside pillows,
I help keep birds dry;
I’m soft to the touch,
Birds need me to fly.

What am I?____________________________ (answer: feathers)

We’ve used these playful riddle books to practice penmanship (printing, upper and now lower cases), spelling, and careful reading (and reading comprehension), along with the obvious critical / creative thinking (riddle-solving). By having Daegan date his work, even he can clearly see progress. Here’s his printing from about a year and a half ago, in September 2008:

You can see that he was printing all in capitals, unevenly—letters squished together, above and below the line, lighter and darker lettering—and that he had to correct his spelling of ‘face’ (from the phonetic ‘fase’). Here is his work from a few days ago:

Quite the difference, eh?

There are several books in the Dr. Dooriddles series, of varying complexity—and I believe you can buy a computer version of the series instead if you prefer. We’ve finished books A1 and A2, and will be moving on to the “Spelling Dooriddles” next, which are somewhat more difficult, as each solution involves two words. Here’s a typical example:

I am a fluffy grain,
We eat it at our house;
With M replacing R,
I mean more than one mouse.

What am I? __________________________________ (answer: rice / mice)

Here are Daegan’s (age 7, soon to be 8 ) comments on Dr. Dooriddles:

I like it [the Dr. Dooriddles series] because it doesn’t tell you the answers. It makes you think. Some are fairly easy, but some are hard. And some answers are funny or surprising. I think most kids would like this series, because riddles are fun. 

After listing the numbers in which the digit 2 occurred, from smallest to largest, he then counted them:

And found their were 19 such numbers, but in one of them, 22, you write the digit 2 twice. So the answer to the problem of the day is 20.

Now it was time for Daegan’s favourite part, in which he creates a similar question in an attempt to stump Mom. He was giggling and quite proud of his “tricky question.” He said, “I’ve stumped you…and I’ve even stumped myself!”

I began by talking about the differences between his question and the one I gave him. Counting and saying are different than writing—you don’t say “2” when you say “12 / twelve”, and I noted his representation of “1 to 1000” using the dash—I didn’t realize he knew that convention! I started my solution as Daegan did, working from smallest to largest, and then looking for patterns. Here’s my entire, very messy-looking solution—I’ll explain it bit by bit below:

I started by listing the numbers in which we’d say “2” where the digit 2 was in the ones place: 2, 22, 32, 42, 52, 62, 72, 82, 92. I then started a new column for the hundreds. At the 200s, I realized I’d need to do them separately, as you say “two” at the beginning of each number in the 200s. I continued on with the 300s, and through to the 900s, making a grid. I then counted 9 sayings of “two” in each column, and 9 columns, giving me 81 times I say “two”. (We haven’t done multiplication yet, but Daegan easily followed this when I explained it as fast skip counting by 9s).

I then went back to look at the 200s. From 200-299 there are 100 digits. I realized that for some of these numbers, you say “two” twice—like 202, 222, 232, 242…. So I found 100 times when you say “two” at the beginning of a 200s-number, and 9 times you say “two” again in the ones place.

It was then a simple matter of adding 81, 100, and 9 together to yield the final answer. Again, we’re just getting to addition with regrouping in our curriculum, but no matter—with base ten blocks Daegan was able to follow the addition on the board:

So the answer was 190. Great question, kiddo!

One thing I am finding is that the “problem of the day” is an ideal time to introduce new math terminology or symbolism. Instead of a lesson and explanation and several practice problems, the focus on ONE problem really allows Daegan to grasp the symbols / vocabulary involved. This was the case when we did the question about mean, median and mode, and yesterday I gave him an algebra question involving the “greater than” sign, something again that he has not seen a whole lot.  Once he got clear that “the alligator eats the most fish”, Daegan was able to do the question below using a “guess and check” strategy. (“The alligator eats the most fish” is the way I was taught this sign in grade school. Imagine the greater than sign, >, as an open mouth. The alligator wants to eat the larger number).

F > P
P > S
F + P + S = 6

What the the values of F, P and S? 

So if you try this approach at home, don’t restrict yourself to “grade level” or “age level” questions. Go ahead an introduce some materials that stretch your student, and lead to lots of discussion. You don’t want these questions to be too hard and induce frustration, but you certainly don’t want them to be too easy either. This is a nice, safe place to get mathematical language, symbols and other conventions clarified—something that underlies many of the difficulties of students I have taught or tutored in math.

Two resources Daegan and I are currently working with are Logic Links and Analogy Challenges, both from MindWare. Logic Links uses punch-out paper chips that are to be arranged according to various clues. It’s probably easiest to explain in pictures:

There are several things I like about this resource. I like that it uses paper chips—no writing (and erasing, and writing, and erasing…). I like that it is simple enough for Daegan to use on his own, as it lists the chips to be used in each puzzle first, and then the clues. We don’t do these every day by any means, but from time to time they are a fun way to break up other work, or to occupy Daegan while I work one-on-one with Gareth for a few minutes. The boys (ages 7 and 5) have also done the puzzles together, with Daegan reading the clues aloud to Gareth. These puzzles are challenging enough for my older child, yet because they involve moving concrete physical objects, simple enough for my younger to participate and determine if an answer is right or wrong.

The second resource we’ve been using is Analogy Challenges. We’re working through the Beginner Level, which includes a mix of pictorial and word puzzles:

The first half of the book is pictorial, the second uses words. We got to the halfway point today, which had one page of each type:

Here’s a close-up of some picture analogies:

And I liked how the word analogies involve a bonus analogy to “unscramble”. Daegan thought that was really cool!:

Again, we don’t work from this book every day, but as something playful and different it’s been great. If you know of similar resources—books, websites, computer games, whatever—please leave me a comment. Thanks!

In the story, the man protects the rooster from the fox, and the rooster then tells him he is a special rooster that can grant wishes. The man thinks of things to wish for, including this teapot:

The book ends by grandfather pushing the tangram pieces to his granddaughter and asking her to decide what to wish for. I did the same with the boys, who enjoyed building their own freeform shapes. You’ll never guess what they’d wish for! Here’s Gareth’s creation:

I asked Gareth what it was, and he told me, in a rather excited tone, “A T Rex! Funny that it doesn’t use the square…” (which you can see in his hand below):

Then Gareth came up with how to incorporate the square into his picture. “I know! I’ll open his mouth, and the square can be a a lawyer!” (Yes, he’s watched Jurassic Park several times—LOL):

Then it was Daegan’s turn to create:

Almost done. Where can I put the other large triangle?:

Daegan added the final piece to the body, and informed me that he had made something called an Echinodon:

Gareth then played with the tangram pieces for a while, and came to get to me show me his “funny wish."  Here’s what I saw:

I asked him what his funny wish was, and he told me “a toilet! See, here’s the lid…” (that’s what he’s pointing to in the pic above.) The kid is quite the comedian, eh?

This turned out to be a fun way to combine math, reading and art…and a bit of social studies (tangrams are Chinese in origin) to boot! If you find another good tangram book, or website, etc.. please pass it along in the comments as the boys are keen to do more.

I also recently read a book about teaching math, and one part really resonated with me: in Asian countries, a class—even in elementary school—will spend an entire lesson discussing, puzzling over, and working cooperatively on ONE problem. Not countless worksheets and timed drill of “mad math minutes.” So I started thinking about how to build more of that into our day, and somehow stumbled across this site for DynaMath magazine. The “problem of the day” pages caught my eye. I dipped a toe in by writing a problem on a whiteboard to start us off one morning, a question easy enough that Gareth was able to join in. (Day 20 here). And the next day, with a question Daegan found harder. (Day 6 here). Things got even better a few days later, when I put up a problem I knew would be a stretch: baby algebra. I adapted it from Day 17 here).

3 shamrocks = 1 leprechaun

4 leprechauns = 2 pots of gold

How many shamrocks equal 1 pot of gold?

Daegan and I solved this two ways: using manipulatives, and then again by drawing on the board. He chose his dark green cuisinaire rods to represent shamrocks, light green for leprechauns, and yellow for pots of gold. His first step was to use the rods to show the problem using these physical objects:

Our next step was to figure out what 1 pot of gold was equal to. (Recall, we need to know how many shamrocks equals 1 pot of gold.) Daegan was able to see intuitively, and with no help from me, that if 4 leprechauns (light green) = 2 pots of gold (yellow), then 2 leprechauns = 1 pot of gold. Which told me he has grasped ideas about division, ratios and multiplication…well beyond addition/subtraction facts!

So how do we relate this fact (2 leprechauns = 1 pot of gold), to the info we have about shamrocks (3 shamrocks = 1 leprechaun)? Daegan went to work changing the shamrock equation into one involving 2 leprechauns, by doubling. If 3 shamrocks = 1 leprechaun, then 6 shamrocks = 2 leprechauns.

Now, by transitivity—and Daegan could see this intuitively, he doesn’t know this term!—since 6 shamrocks = 2 leprechauns, and 2 leprechauns = 1 pot of gold, then 6 shamrocks = 1 pot of gold.

And then I added the final step to our Problem of the Day focus: I asked Daegan if he would like to construct a similar problem. He jumped at the chance! And having him create problems, and not just solve them, will allow me another way to see any gaps or errors in his reasoning. Daegan created the following—dictating to me—and then solved it intuitively and created a drawing to show his reasoning:

Note the period (.) between the three small pea pods, as they help make sense of his drawing.

We will keep doing these Problems of the Day (though not every day) and have Daegan (and eventually Gareth) create their own problems of a similar sort. The challenge of creating something to try to stump Mom is quite appealing for my two!