A Multiplication Based Logic Puzzle

Several weeks ago I helped a young girl with a place value page in her workbook. The largest number she needed to write out in words was less than 900 million.

The young girl was very confused so I explained it exactly the same way the book explained it – which was exactly the same way it was explained to me when I was a kid.

She easily understood the ones place, the tens place, the hundreds place, and the thousands place.  The trouble began when we started working with the ten thousands place and became even worse when I started talking about the hundred thousands place, the millions place, the ten millions place, and the hundred millions place.

The young girl could easily and correctly separate the digits of any multi-digit number into groups of three, but reading or writing that number using words baffled her.

Also translating number words back into digits and putting those digits into the right places was equally challenging for her.

After explaining what to do on EVERY single problem, I knew she really didn’t get it….even though the assignment was finally finished.

Then last week on twitter I saw a retweet of this:

I wondered what crazy, radical thing was meant by that statement so I clicked on the link and read a very clear and easy-to-understand explanation of how to teach place value. I was very impressed.

This last Friday I saw that young girl again, and I said, “I want to show you something.”

I took my red pen and wrote in her workbook,

place value places

and this time I actually taught her the concept of place value. This time she got it!

Triumphantly she wrote down a 14-digit number made from some “random” numbers that popped into her head and then read it to me perfectly. It was as thrilling for me as it was for her!

An hour before she couldn’t correctly read 90% of the whole numbers less than a million, but now she had MASTERED thousands, millions, billions, and even trillions.

I really like that blogger/tweeter Michael Tidd chose “units” as the last category because that is the natural place to say miles, meters, dollars or whatever the unit happens to be. The unit this young girl chose for her 14-digit number was CATS.

———————————————————————–

578 is the hypotenuse of two Pythagorean triples: 322-480-578 and 272-510-578. Each triple has a greatest common factor. Which factors of 578 could they be?

  • 578 is a composite number.
  • Prime factorization: 578 = 2 x 17 x 17, which can be written 578 = 2 x (17^2)
  • The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 x 3  = 6. Therefore 578 has exactly 6 factors.
  • Factors of 578: 1, 2, 17, 34, 289, 578
  • Factor pairs: 578 = 1 x 578, 2 x 289, or 17 x 34
  • Taking the factor pair with the largest square number factor, we get √578 = (√289)(√2) = 17√2 ≈ 24.04163

578 cake

2 x 17 x 17 = 578

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Comments on: "578 When I stopped teaching about thousands, it made a BIG difference" (6)

  1. Yes, I think that is intuitively how I think about place value.

    Liked by 1 person

    • It is how I intuitively think of place value as well, but for some reason I still needed somebody else to suggest how to teach it that way. I’ve designed great ways to teach other mathematical concepts, but in this case I’m grateful for social media collaboration.

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  2. It does seem to match how I think of place value, although I don’t remember how I was taught it in the first place. And I don’t know just how I would teach it. Hm. Provoking system, though.

    Liked by 1 person

    • Teaching it this way seems so obvious to me now, but it isn’t the traditional way to teach it. Just google “place value images” and you will see several examples that could be confusing to children learning this concept. Cuisenaire units, rods, flats, and blocks can be helpful in teaching children ones, tens, hundreds, and thousands, but it likely would be confusing to try representing ten thousands, hundred thousands, or millions with them.

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  3. This is an interesting and rather helpful idea, but it does seem to me that a lot of more fundamental understanding of how the number system works has to be in place before it becomes relevant. I don’t see that it gives new understanding, but it brings systematising to existing knowledge, and that’s got to be useful.

    Liked by 1 person

    • I think it helps students begin to understand the number system better than the way I and so many others were teaching it before. There may still be students who don’t get it, but I think there will not be as many.

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