What is the sum of the digits of (googol + 1)(googol - 1) when expanded?
Comments:
(1) Google 'googol' if you need some background!
(2) Does the strategy of 'make it simpler' work well here?
(3) Can you invent a similar problem or, better, have your students devise their own!
(4) Oh, BTW, NO CALCULATORS!!
(5) I felt I needed a change of pace from the heavy math ed stuff from the past few days. You too?
Wednesday, April 23, 2008
A Very Big Number Question...
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12 comments:
According to google (which derived its name from googol)
googol = 10^100
We can treat googol like a variable so
the following simplifies the product:
(googol +1)*(googol -1) =
googol*googol - googol + googol - 1 =
googol^2 -1 = 10^200 -1
Dave there can never be enough math
to think about. So keep the challenges
comming!
Btw. can we send in suggestions? Just
wondering.
Absolutely, Florian! I get some great ideas from readers and even from the search phrases used to find my blog!
BTW, you need to go one more step to finish the googol problem - I asked for the sum of the digits!
That would make it zero
Or rather 1800
Ah true!
10^200 -1 has 199 digits each being 9
therefore the sum of the digits = 9*199
=1791
Anon--
you nailed it...
Now, how do you think most students would deal with this? Algebra 1? Algebra 2? Precalculus? More interestingly, I used to give these to my 'skills' classes after developing the necessary prerequisite knowledge. I never told them the question was challenging - it was just a puzzle for them to try. Afterwards, if some got it, I would celebrate them and tell them they just solved a math contest problem!
Darn ... we cs guys always
fight with index shifts by
1 :)
Florian,
Everyone is prone to off-by-1 errors.10^200 - 1 has 200 digits, just as 10^2 - 1 has 2 digits. The answer is 1800.
I did what Eric did to double-check my answer for the dreaded off-by-one error. I'm a CS type too. :)
Cute problem. I think my middle schoolers could solve this by using the "solve a simpler problem" method. Once kids have the distributive law, they should be able to succeed with problems like this. Mine don't know the rules of combining exponents (i.e. that (10^100)^2 = 10^200 but they could get there using patterns and lower numbers, I think. Maybe I will give it a try with them if I get a chance. I have so much queued up to try with them. :) We're doing a batch of mental math at the moment, because they could use the practice.
Ran this in three classes today. I front-ended it with two easier questions:
Find the sum of the digits of
1. x + y
2. x - y
Neat side discussions about what a googol is (of course) but also, how many digits in a googol. Nice refreshers (in geometry they were necessary!) on handling exponents and multiplying binomials...
One of the algebra classes really rocked it. I ended up running the discussion, but slowly, letting kids jump ahead, letting them explain wrong answers to generate feedback, and, on hearing a good piece of reasoning, getting other kids to explain it.
Highlight though was something simple: in an algebra class a kid heard a wrong answer, and quickly deciphered the error and clarified for her mistaken classmate. Nice.
I used 10 - 15 minutes per class (in geometry we had plenty of time to explore dilations; in algebra we graphed linear inequalities - looks good, but kind of anticlimactic after all the hard work we've been doing).
Jonathan
Thanks jd!
It would be rare for any of the investigations I write to be implemented without adjustments.
What I appreciate most is that you are willing to try these activities and chronicle students' reactions. This is particularly valuable for other teachers to read. Unless one actually experiments with these kinds of challenges, it is difficult to predict what will happen.
This particular question requires KNOWLEDGE (definition of googol, knowing how to distribute, knowing a rule of exponents, knowledge of powers of ten), INSIGHT (recognizing relationship between power of ten and the number of digits), SKILLS (algebraic, arithmetic), STRATEGIES ('make it simpler', careful counting methods).
This is why I apply the 'KISS' acronym to problem-solving:
K:Knowledge
I:Insight
S:Skills
S:Strategies
Thank you again!
In my last class (doesn't meet Monday), it ran fairly smoothly.
One boy thought, this will work a lot like (100+1)(100-1), answered, generalized.
That was neat.
But I let them plug at it, and we got through florian's approach. Nice suggestions about notation (a kid explicitly made reference to the need for easy to use notation) and yet again a refresher on exponents!
Jonathan
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