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!

Here is Daegan just getting started:

And here he the puzzle partway through. If you look at the top left corner, you’ll see how kakuro puzzles work. When he started the puzzle, just the 1 was filled in in the corner square. Daegan then was able to solve the square below the 1, as that column adds to 3. So he plunked down a 2. He was also able to solve across from the 1, as the sum is 6, so in went the 5. Working across from the 2, he was able to figure out that a 9 was needed (2+9=11), and now he had enough information to solve the column of three squares that sum to 20. (5+9=14, 20-14=6).

Now look again at the picture above and notice the top right-hand corner. I asked Daegan which square he wanted to solve next, and at first he wanted to continue on under the 6-7-1 and solve the three digits that sum to 14. He quickly realized this would be too difficult, and looked for another area of the puzzle to work on. He pointed to a square in the top right, and then solved it:

He went on to solve the entire puzzle without too much difficulty—start to finish he took about 10 minutes—and I asked him what he thought of  kakuro puzzles. Daegan said, “This was funner because it was more challenging and took longer to solve (than Math-U-See).” When asked how it compared with magic squares, he said, “I liked it about the same as magic squares.” We’ll be doing more of both!

I began giving each boy 1 stick. “What can you make?” I asked. We quickly found out the answer: not much with only 1 stick! You could make 1 (one), I (capital ‘aye’), or l (lower case ‘ell’). That was about it. I then gave the boys a second stick. “What can you make now?” Here’s some of the possibilities, both letters and numbers:

Note that the final two shapes could also be symbols: the equal sign, or the plus (adding) or times (multiplying) sign.

With three sticks there were even more letters and numbers, and our first shape. Here are some of them:

And with four sticks we could make a couple of shapes, and more:

I threw out words like ‘quadrilateral’ and ‘rhombus’, and pointed out that all squares are rectangles, but not all rectangles are squares. Gareth decided he’s played enough and wanted to do some drawing, but Daegan was keen add more sticks. We focussed on shapes from here on in.

With five sticks, a pentagon:

At this point we abandoned the paper underneath, as it was sliding around and making it harder than it needed to be to place the sticks in position. I told Daegan that another shape you could make with 5 sticks was a star, and showed him how to do it, using the interior angles of the pentagon as a guide:

With six sticks, Daegan got very excited about hexagons. “I know where we see those in nature! In a beehive!” And he started flapping his arms and buzzing like a bee:

He then opened the nearby cupboard and got out some Crispex cereal, another hexagon. It got gobbled up nanoseconds after I got this shot:

We then played around making shapes inside the hexagon (as we had done with the star in the pentagon), and found this:

Note that as these are all equilateral triangles, we can use this image to prove that the interior angles of a hexagon add up to 720 degrees. (Each angle in the triangle is 60 degrees, as 60 x 3 = 180 degrees, which is the number of degrees in  a triangle. Since two angles from each triangle are used to make the interior angles of the hexagon, and there are 6 triangles, you get 6 x 120 = 720 degrees). I did NOT do this with my seven year old in case you are wondering!

We quickly made a heptagon (7 sides), talking about how you don’t often see that shape anywhere, and moved on to the octagon. Daegan had an idea right away to complete the image:

Have fun with this activity; it’s quite flexible and can be adapted to different ages and abilities.

Daegan and I started with a game he’s never played before:

We adapted the rules to make it a more fair match: we counted only number of words found, not their length; I had to find words at least 5 letters long, and Daegan got twice as long (2 turns of the hourglass timer):

I was also quite generous in giving him tips on strategy, such as using “s” to make plurals, looking for rhymes, etc. Here’s the words he found on our final game:

And here’s our lists from a round earlier. I don’t think we found any words in common (and he found a 5-letter word I completely missed!):

This turned out to be a great way to have Daegs practice his spelling and penmanship. We’ll definitely give this a go again.

We then moved on to Connect 4, which gave Daegan a chance to practice being a good sport about losing (such a challenge for this 7-year old!), Make 10 Go Fish, and Guess Who?, all of which gave Daegan some practice in math and logic / reasoning. One of my goals for this year is to bring more games and play into our homeschooling: mission accomplished today!

When Gareth got home he wanted to “do homeschool” with me, and specifically wanted to do more colour-by-number puzzles. These are great for the stage Gareth is at right now: he knows his numbers, he’s beginning to read (and can read several colour words—and is SOOO proud of this!), and he needs more fine motor practice with a writing implement. He and I coloured these two puzzles together:

The boys then watched a DVD Jim had seen at the library and thought would interest them: Darwin’s secret Notebooks. Daegan decided he needed some exercise and ran on the treadmill for a bit too:

Gareth also ran for a bit while watching the show, “just like you do Mommy!” (I watch recorded soccer or Daily Show / Colbert while getting in some exercise in the mornings). Yay! I am helping them establish at least ONE good habit!

We finished off the day with a delish pasta for dinner (adults had pasta with sauteed mushrooms, sweet onion, red pepper and cherry tomato halves; I added a bit of feta to mine; Jim added olives to his. Sooooo nummy!) and an hour of reading with the boys before bed. We’re just a couple chapters from the end of Trumpet of the Swan which the boys have really enjoyed. Any suggestions for novels / chapter books to try next? 

It was a huge hit with Daegan, who seems naturally drawn to riddles and puzzles—just like his parents. He was a bit concerned at first, and quite insistent that he didn’t know what an analogy was. I kept insisting that he did know, he just didn’t know the word and symbolism, and proved it to him by working through the teaching examples they provided. Daegan was able to do them right away; he could “see” the answer:

He then worked through the first page on his own, thoroughly enjoying this type of puzzle. I am very pleased with this resource so far: these questions and flexible reasoning skills employed are timeless, the pages are perforated for easy removal, and the answers are in the back.  Highly recommended if you are looking for something a bit out of the ordinary for math / critical thinking / reasoning skills, or if you have a child who enjoys visual puzzles. Here’s Daegan’s work from today: