An Example of a "Learning Process" Journal (using the 2 colored box format)

Book: The Nature of Mathematics 10th ed by Karl Smith.

Source: Toronto Thomson Brooks/Cole, 2004.

It is 2:00 PM (Tuesday). I have the house to myself this afternoon: a good time to try some math.

1.3 Scientific Notation and Estimation (p. 36 - 39)

Level 2 & Level 3 present no problems (assuming that numbers like a billion and a trillion are well understood - they differ in Europe and USA).

Problem Solving

55. If it takes one second to write down each digit, how long will it take to write down all the numbers from 1 to 1,000,000?

This is equivalent to asking how many digits are there in the numbers from 1 to 1, 000, 000.

Let's try by beginning with the smallest.

• there are 10 1-digit numbers (normally there are 10 unique digits, but we ignore 0 in this situation)
• there are 100 2-digit numbers but we then ignore the 10 2-digit numbers beginning with a 0)
• there are 1000 3-digit numbers minus 100 3-digit numbers with a leading 0, minus 10 numbers with 2 leading 0'setc.
• the pattern for k digits is 10*k - 10*(k-1)
• now let's consider the sums
• the sum of the digits for k is 10*k - 1 = 999,999 digits
• divide by 60 to obtain the number of minutes = 16, 666.65
• Yikes! the answer in the text is only 48 minutes and 13 seconds.
• Let's try working backwards and see if I can understand this.
  48x60 = 2880 +13 = 2893 (not an obvious number)

Sheesh! I just realized that the question is for 1 to 1,000 not 1 to 1,000,000.

1(10-1) + 2(100-10) + 3(1000-100) = 9 + 180 + 2700 = 2889

I differ by 4 seconds!

[3(900) + 2(90) +9]

Frustration. I have been on this for 30 minutes and still get 2889.

Time to move on.

1.3 Scientific Notation and Estimation (p. 36 - 39)

Problem Solving

Questions 55 - 58 are all variations on the same theme of finding the pattern of digits or zeros in all of the numbers from 1 to either 1,000 or 1,000,000.

Let's start with the numbers from 1 to 9. Easy. 9 digits, 0 zeros.

Now from 1 to 100. There are 9 1-digit numbers, 90 2-digit numbers plus 1 3-digit number

= 9 + 180 + 3

Now from 1 to 1,000.

= 9 + 2*90 + 3*900 + 4

Now I see it! I was forgetting to add the final number of digits of the last number. Finally!!

3:10 PM

I now want to move on to the next section, 1.4 Finite and Infinite. Next time.

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