# AoPS Intro Counting & Probability: solve numerically?

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We are in Chapter 4 of Intro to C&P, and I'm wondering if y'all have the child calculate the answers fully by hand?

9x8x7x6, then do you require the child to calculate "3024"?

My son is totally unhappy about these calculations.  I tend to think he should just crank them out, but don't want to be unnecessarily tough on him.

ETA: the above isn't too bad, but some problems have more terms and not all of them have elegant cancellations to simplify things, either.

Edited by serendipitous journey
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How are the answers given in the solution manual? I'd use whatever form they are there, and I would allow the use of a calculator if things like that need to be multiplied out.

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How are the answers given in the solution manual? I'd use whatever form they are there, and I would allow the use of a calculator if things like that need to be multiplied out.

They are given multiplied out ... it is good to know you'd allow a calculator.  He came through okay, but it took a while, and I'm not sure how I'll weigh the advantages of arithmetic practice (which, honestly, he could use) against the advantages of simplicity and speed.

thank you so much!

Edited by serendipitous journey
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A calculator is a great tool that frees the student's mind to focus on reasoning through the problem. I allow a calculator for any middle school or high school problem where the arithmetic is complicated enough to distract from what the student is supposed to be learning.

Sometimes there is a quick mental-math solution. But when that's not true, or when you don't see the quick route right away, then the calculator is fair game. Though I did have one child who preferred to crank through the hand calculations, perhaps because it gave her mind a break from thinking hard...

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One trick is to rewrite the numbers in terms of some easy prime factors.  (I'm calculator-averse, so I like doing it this way.)

9 x 8 x 7 x 6

=9 x 7 x 6 x 8

=63 x 3 x 2 x 2 x 2 x 2

=189 x 2 x 2 x 2 x 2  <now you double the result 4 times>

=378 x 2 x 2 x 2

=756 x 2 x 2

=1512 x 2

=3024

...and I didn't use a calculator.  This is a pretty weird way to solve in real life, but I don't keep a calculator handy when I'm using AoPS, so I look for tricks to get to the solution.  You can also just crank through 72x42 as practice for 2 digit multiplication.

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We're fairly new to AoPS (working through Intro to Algebra and Intro to C & P concurrently), but Ds has found so far that if the arithmetic looks complicated or takes him more than 20-30 seconds, he's probably not on the right track. I don't make Ds write out the steps for arithmetic, so he usually does it mentally and hasn't complained. If his math problem solving ability far exceeded his skill in calculation, I think I would allow him to use a calculator while continuing to work on calculation separately.

I just asked Ds the example you gave. He explained that he would multiply from left to right:

9 x 8 = 72

72 x 7 = 490 + 14 = 504

504 x 6 = 3000 + 24  = 3024

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Can he do these calculations easily?

If so, I'd let him use a calculator.

If not, I'd still let him use a calculator, but add some arithmetic practice/drill into our day so he could calculate more rapidly without one. By the time you reach the algebra level, you shouldn't need to prove you can calculate by hand anymore.

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AoPS booksâ€™ questions that need the use of calculators do specify that in the question. I made my kids calculate by hand so that my calculator dependent DS11 wonâ€™t lose his number sense. It is also good practice for AMC and AIME which donâ€™t allow calculators.

My kids do use their calculators for physics and chemistry so they have more than enough practice on use of their calculators.

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Thanks, everybody!

The example I gave was simplified.  It is true that, for better or worse, there is often a trick to an AoPS problem that will save you some work; but as you move through C&P thare are also simply some biggish factorials to compute.  His mental math isn't good enough to handle all of them quickly, and he's a bit sloppy.

So, my reasoning being similar Arcadia's, we're keeping at it by hand.  Especially for this child, the precision he learns by doing the calculations tends to carry over into other areas: if he can't keep the arithmetic clean, he also can't keep his algebraic manipulations clean.

I'm finding that if I let him use a calculator once his precision is good he quickly loses the precision!!!

Edited by serendipitous journey
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I can't remember if it's in the Intro or Intermediate C&P , but there are some really nasty combination problems with large numbers.  In that case, I skipped the calculator because *that* was too much work and just typed in 1387 choose 97 into google to get the answer.  (I can't remember the exact numbers but it was similarly large.)

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I can't remember if it's in the Intro or Intermediate C&P , but there are some really nasty combination problems with large numbers.  In that case, I skipped the calculator because *that* was too much work and just typed in 1387 choose 97 into google to get the answer.  (I can't remember the exact numbers but it was similarly large.)

!!!  :lol:   thanks for the heads-up!  I'll let my poor child know to give me a holler if he hits numbers like that!

also: thanks for the easy Google sol'n method.

Edited by serendipitous journey
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!!!  :lol:   thanks for the heads-up!  I'll let my poor child know to give me a holler if he hits numbers like that!

also: thanks for the easy Google sol'n method.

If you want to prove he's still thinking in arithmetic, you can require him to estimate, and write down his estimation, before he reaches for the calculator.

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If you want to prove he's still thinking in arithmetic, you can require him to estimate, and write down his estimation, before he reaches for the calculator.

That is a great idea!  For this fellow, we really need the practice in detailed arithmetic ... I hadn't processed that when I posted my OP, but thinking about the responses and what my goals are for him, I realized that he really needs to stay practiced at hand calculations so that when he does need to them he can count on them.  It totally bites him in his other AoPS work if he makes careless errors early in a complex problem, so it is no kindness to let him hold on to sloppy habits.

Sigh.  Clearly, not all children require this much continued practice of careful calculation!!!

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Sigh. Clearly, not all children require this much continued practice of careful calculation!!!

:grouphug: it does get better for my DS11, at least until he gets the same teenage fog that DS12 is now having :P

All this hand calculation practice does translate well to lab report writing though for DS11. So at least there is a silver lining to all the time spent.

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